# Publications

2018

Year: 2018

## Topological bifurcations of coherent structures and dimension reduction of plasma convection models

Dam, M., Brøns, M., Naulin, V. & Rasmussen, J. J. 2018 DTU Compute. 123 p. (DTU Compute PHD-2017, Vol. 461).

Publication: ResearchPh.D. thesis – Annual report year: 2018

Research in fusion energy seeks to develop a green, safe, and sustainable energy source. Nuclear fusion can be achieved by heating a hydrogen gas to temperatures of millions of kelvin. At fusion temperatures, some or all the electrons leave the atomic nucleus of the hydrogen atom. This results in an overall neutral gaseous state of negatively charged free electrons and positively charged ions. This state of matter is called plasma. To achieve and maintain fusion temperatures, the plasma must avoid direct contact with any solid material. Since the plasma consists of charged particles, it can be conﬁned with an appropriate conﬁguration of strong magnetic ﬁelds. Toroidal magnetic conﬁnement devices, such as the tokamak, are the most promising designs for a fusion reactor. A tokamak can operate in two distinct modes of operation. These are the low conﬁnement mode (L-mode) and the high conﬁnement mode (H-mode). H-mode is the preferred operating mode for a fusion reactor. The transition from L-mode to H-mode is called the L–H transition. The conﬁnement properties of a plasma are largely determined by the physics near the edge of the conﬁnement region of the plasma. The edge transport of a magnetically conﬁned plasma is predominantly caused by recurring bursts of coherent plasma structures. These structures are in L-mode called blob ﬁlaments (blobs) and in H-mode categorized into edge localized mode (ELM) ﬁlaments or inter-ELM ﬁlaments. To improve the plasma conﬁnement, it is important to understand the evolution of these structures. We apply a dynamical systems approach to quantitatively describe the time evolution of these structures. Three state variables describe blobs in a plasma convection model. A critical point of a variable deﬁnes a feature point where that variable is signiﬁcant. For a range of Rayleigh and Prandtl numbers, we analyze the bifurcations of the critical points of the three variables with time as the main bifurcation parameter. Plasma simulations can be computationally demanding. We apply a Galerkin method to approximate a plasma convection model with a reduced model. The time evolution of the energies of the pressure proﬁle, the turbulent ﬂow, and the zonal ﬂow capture the dynamic behavior of the convection model. Rayleigh decomposition splits the variables of the model into averaged variables and ﬂuctuation variables. We approximate the ﬂuctuation variables by truncated Fourier series and project the equations onto the Fourier basis functions. This results in a computationally simpler model with the spatial dimension reduced by one. Bifurcation diagrams for the energies show consistency between the bifurcation structures of the full and the reduced model.
Finally, we utilize a data-driven modeling approach called SINDy to identify a reduced model from simulation data of a convection model. The reduced model reveals a predator-prey relationship between the zonal ﬂow energy and the turbulent energy. The analytically derived bifurcation diagram for the reduced model has the same structure as the data-based bifurcation diagram for the full model.
Original language English
Publisher DTU Compute 123 Published - 2018
Series DTU Compute PHD-2017 461 0909-3192

Year: 2018

## Geometric singular perturbation analysis of systems with friction

Bossolini, E., Brøns, M. & Kristiansen, K. U. 2018 DTU Compute. 146 p. (DTU Compute PHD-2017, Vol. 454).

Publication: ResearchPh.D. thesis – Annual report year: 2018

This thesis is concerned with the application of geometric singular perturbation theory to mechanical systems with friction. The mathematical background on geometric singular perturbation theory, on the blow-up method, on non-smooth dynamical systems and on regularization is presented. Thereafter, two mechanical problems with two diﬀerent formulations of the friction force are introduced and analysed. The ﬁrst mechanical problem is a one-dimensional spring-block model describing earthquake faulting. The dynamics of earthquakes is naturally a multiple timescale problem: the timescale of earthquake ruptures is very short, when compared to the time interval between two consecutive ruptures. We identify a small parameter ε that describes the separation between the timescales, so that ε = 0 idealises the complete timescale separation. Earthquake faulting problems also have multiple spatial scales. The action of friction is generally explained as the loss and restoration of linkages between the surface asperities at the molecular scale. However, the consequences of friction are noticeable at much larger scales, like hundreds of kilometers. By using geometric singular perturbation theory and the blow-up method, we provide a detailed description of the periodicity of the earthquake episodes. In particular, we show that attracting limit cycles arise from a degenerate Hopf bifurcation, whose degeneracy is due to an underlying Hamiltonian structure that leads to large amplitude oscillations. We use a Poincaré compactiﬁcation to study the system near inﬁnity. At inﬁnity, the critical manifold loses hyperbolicity with an exponential rate. We use an adaptation of the blow-up method to recover the hyperbolicity. This enables the identiﬁcation of a new attracting manifold, that organises the dynamics at inﬁnity for ε = 0. This in turn leads to the formulation of a conjecture on the behaviour of the limit cycles as the timescale separation increases for 0 < ε 1. We illustrate our ﬁndings with numerics, and outline the proof of the conjecture. We also discuss how our results can be used to study a similar class of problems. The second mechanical problem is a friction oscillator subject to stiction. The vector ﬁeld of this discontinuous model does not follow the Filippov convention, and the concept of Filippov solutions cannot be used. Furthermore, some Carathéodory solutions are unphysical. Therefore, we introduce the concept of stiction solutions: these are the Carathéodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we ﬁnd that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we ﬁnd canard trajectories, that have the physical interpretation of delaying the slip onset. We show numerically that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family is unstable of saddle type and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.
Original language English
Publisher DTU Compute 146 Published - 2018
Series DTU Compute PHD-2017 454 0909-3192

Year: 2018

## Monge surfaces and planar geodesic foliations

Brander, D. & Gravesen, J. 2018 109, 4, 14 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2018

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and investigate the global properties of these two naturally deﬁned objects. The only compact orientable PGF surfaces are tori; these are globally Monge surfaces, and they have a simple characterization in terms of the directrix. We show how to produce many examples of Monge tori and Klein bottles, as well as tori that do not have a closed directrix
Original language English Journal of Geometry 109 4 14 0047-2468 10.1007/s00022-018-0413-7 Published - 2018

2017

Year: 2017

## Robotic system and method for manufacturing of objects

Gravesen, J., Brander, D., Bærentzen, J. A., Markvorsen, S., Bjerge Nørbjerg, T. & Hornbak Steenstrup, K. 21 Sep 2017

Publication: ResearchPatent – Annual report year: 2017

The present disclosure relates to a method and a system for manufacturing a mould (17) for creation of complex objects, such as concrete objects, by controlling and moving two end effectors (1) of a robotic system, the two end effectors (1) having a flexible cutting element (3) attached to and extending between the two end effectors (1), the method comprising the steps of: defining at least one surface (8) representing the inner surface of the mould (17); dividing the surface (8) into a number of segments represented by planar curves (9, 11, 12) on the surface (8); for each planar curve, calculating at least one elastic curve representing the planar curve; for each calculated elastic curve, calculating a set of data corresponding to placement and direction of the two end effectors (1) for configuring the flexible cutting element to a shape corresponding to the calculated elastic curve; sequentially positioning the end effectors (1) according to each set of data,

Original language English G06T 17/ 30 A I WO2017157917 21/09/2017 International Bureau of the World Intellectual Property Organization (WIPO) Published - 21 Sep 2017

Year: 2017

## Fast computation of the roots of polynomials over the ring of power series

Neiger, V., Rosenkilde, J. & Schost, É. 23 Jul 2017 ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation. The Association for Computing Machinery, Vol. Part F129312, p. 349-356 8 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field K. More precisely, given a precision d, and a polynomial Q whose coefficients are power series in x, the algorithm computes a representation of all power series f(x) such that Q(f(x)) = 0 mod xd. The algorithm works unconditionally, in particular also with multiple roots, where Newton iteration fails. Our main motivation comes from coding theory where instances of this problem arise and multiple roots must be handled. The cost bound for our algorithm matches the worst-case input and output size d deg(Q), up to logarithmic factors. This improves upon previous algorithms which were quadratic in at least one of d and deg(Q). Our algorithm is a refinement of a divide & conquer algorithm by Alekhnovich (2005), where the cost of recursive steps is better controlled via the computation of a factor of Q which has a smaller degree while preserving the roots.

Original language English ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation 8 Part F129312 The Association for Computing Machinery 23 Jul 2017 349-356 9781450350648 10.1145/3087604.3087642 Published - 23 Jul 2017 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 - Kaiserslautern, Germany

### Conference

Conference 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 42 University of Kaiserslautern Germany Kaiserslautern 25/07/2017 → 28/07/2017 Association for Computer Machinery and its Special Interest Group on Symbolic Manipulation (ACM SIGSAM)

Year: 2017

## Popov form computation for matrices of Ore polynomials

Khochtali, M., Né Nielsen, J. R. & Storjohann, A. 23 Jul 2017 ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation. The Association for Computing Machinery, Vol. Part F129312, p. 253-260 8 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

Let F[∂; s, d] be a ring of Ore polynomials over a field. We give a new deterministic algorithm for computing the Popov form P of a non-singular matrix A ∈ F[∂; s, d]n×n. Our main focus is to ensure controlled growth in the size of coefficients from F in the case F = k(z), and even k = Q. Our algorithms are based on constructing from A a linear system over F and performing a structured fraction-free Gaussian elimination. The algorithm is output sensitive, with a cost that depends on the orthogonality defect of the input matrix: the sum of the row degrees in A minus the sum of the row degrees in P. The resulting bit-complexity for the differential and shift polynomial case over ℚ(z) improves upon the previous best.

Original language English ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation 8 Part F129312 The Association for Computing Machinery 23 Jul 2017 253-260 9781450350648 10.1145/3087604.3087650 Published - 23 Jul 2017 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 - Kaiserslautern, Germany

### Conference

Conference 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 42 University of Kaiserslautern Germany Kaiserslautern 25/07/2017 → 28/07/2017 Association for Computer Machinery and its Special Interest Group on Symbolic Manipulation (ACM SIGSAM)

Year: 2017

## Matematikkommissionen - Afrapportering

Grønbæk, N., Rasmussen, A-B., Skott, C. K., Bang-Jensen, J., Jensen, K. B. S., Fajstrup, L., Schou, M. H., Christensen, M., Lumholt, M., Kjærup, R. M., Jørgensen, S., Hansen, S. L. & Markvorsen, S. 2017 47 p.

Publication: CommissionedReport – Annual report year: 2017

Matematikkommissionen er nedsat som følge af ønsket om at styrke det faglige niveau i matematik i gymnasiet. Kommissionen skal bl.a. give forslag til udvikling af fagets indhold, didaktik, prøveformer og faglige overgange dels fra grundskole til gymnasiale uddannelser og dels videre til videregående uddannelser
Original language English
Number of pages 47 Published - 2017

Year: 2017

## Decoding of interleaved Reed-Solomon codes using improved power decoding

Puchinger, S. & Rosenkilde ne Nielsen, J. 2017 Proceedings of 2017 IEEE International Symposium on Information Theory . IEEE, p. 356-60 (2017 Ieee International Symposium on Information Theory (isit)).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

We propose a new partial decoding algorithm for m-interleaved Reed-Solomon (IRS) codes that can decode, with high probability, a random error of relative weight 1 − Rm/m+1 at all code rates R, in time polynomial in the code length n. For m > 2, this is an asymptotic improvement over the previous state-of-the-art for all rates, and the first improvement for R > 1/3 in the last 20 years. The method combines collaborative decoding of IRS codes with power decoding up to the Johnson radius.
Original language English Proceedings of 2017 IEEE International Symposium on Information Theory IEEE 2017 356-60 978-1-5090-4096-4/ 10.1109/ISIT.2017.8006549 Published - 2017 2017 IEEE International Symposium on Information Theory - Aachen, Germany

### Conference

Conference 2017 IEEE International Symposium on Information Theory Germany Aachen 25/06/2017 → 30/06/2017
Series 2017 Ieee International Symposium on Information Theory (isit) 2157-8117

Year: 2017

## Further Generalisations of Twisted Gabidulin Codes

Puchinger, S., Rosenkilde, J. S. H. & Sheekey, J. 2017 Proceedings of International Workshop on Coding and Cryptography 2017. 10 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

We present a new family of maximum rank distance (MRD) codes. The new class contains codes that are neither equivalent to a generalised Gabidulin nor to a twisted Gabidulin code, the only two known general constructions of linear MRD codes.
Original language English Proceedings of International Workshop on Coding and Cryptography 2017 10 2017 Published - 2017 Tenth International Workshop on Coding and Cryptography 2017 - Saint-Petersburg, Russian Federation

### Conference

Conference Tenth International Workshop on Coding and Cryptography 2017 Russian Federation Saint-Petersburg 18/09/2017 → 22/09/2017

Year: 2017

## Improved Power Decoding of One-Point Hermitian Codes

Puchinger, S., Bouw, I. & Rosenkilde, J. S. H. 2017 Proceedings of International Workshop on Coding and Cryptography 2017. 9 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

We propose a new partial decoding algorithm for one-point Hermitian codes that can decode up to the same number of errors as the Guruswami–Sudan decoder. Simulations suggest that it has a similar failure probability as the latter one. The algorithm is based on a recent generalization of the power decoding algorithm for Reed–Solomon codes and does not require an expensive root-finding step. In addition, it promises improvements for decoding interleaved Hermitian codes.
Original language English Proceedings of International Workshop on Coding and Cryptography 2017 9 2017 Published - 2017 Tenth International Workshop on Coding and Cryptography 2017 - Saint-Petersburg, Russian Federation

### Conference

Conference Tenth International Workshop on Coding and Cryptography 2017 Russian Federation Saint-Petersburg 18/09/2017 → 22/09/2017

Year: 2017

## Operator representations of frames

Christensen, O. & Hasannasab, M. 2017 Proceedings of 2017 international conference on sampling theory and applications. IEEE, p. 207-11 (2017 International Conference on Sampling Theory and Applications (sampta)).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

The purpose of this paper is to consider representations of frames {fk}k∈I in a Hilbert space ℋ of the form {fk}k∈I = {Tkf0}k∈I for a linear operator T; here the index set I is either ℤ or ℒ0. While a representation of this form is available under weak conditions on the frame, the analysis of the properties of the operator T requires more work. For example it is a delicate issue to obtain a representation with a bounded operator, and the availability of such a representation not only depends on the frame considered as a set, but also on the chosen indexing. Using results from operator theory we show that by embedding the Hilbert space ℋ into a larger Hilbert space, we can always represent a frame via iterations of a bounded operator, composed with the orthogonal projection onto ℋ. The paper closes with a discussion of an open problem concerning representations of Gabor frames via iterations of a bounded operator.
Original language English Proceedings of 2017 international conference on sampling theory and applications IEEE 2017 207-11 978-1-5386-1565-2 10.1109/SAMPTA.2017.8024348 Published - 2017 2017 International Conference on Sampling Theory and Applications - Tallinn, Estonia

### Conference

Conference 2017 International Conference on Sampling Theory and Applications Estonia Tallinn 03/07/2017 → 07/07/2017
Series 2017 International Conference on Sampling Theory and Applications (sampta)

Year: 2017

## Surfaces foliated by planar geodesics: a model forcurved wood design

Brander, D. & Gravesen, J. 2017 Proceedings of Bridges 2017. Swart, D., Séquin, C. & Fenyvesi, K. (eds.). Tessellations Publishing, p. 487–490

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2018

Surfaces foliated by planar geodesics are a natural model for surfaces made from wood strips. We outline how to construct all solutions, and produce non-trivial examples, such as a wood-strip Klein bottle
Original language English Proceedings of Bridges 2017 David Swart, Carlo Séquin, Kristóf Fenyvesi Tessellations Publishing 2017 487–490 978-1-938664-22-9 Published - 2017 Bridges 2017 - Waterloo , Canada

### Conference

Conference Bridges 2017 University of Waterloo Canada Waterloo 27/07/2017 → 31/07/2017

Year: 2017

## On multivariate Wilson bases

Bownik, M., Jakobsen, M. S., Lemvig, J. & Okoudjou, K. A. 2017 Proceedings of 2017 International Conference on Sampling Theory and Applications. p. 192-5 4 p. (2017 International Conference on Sampling Theory and Applications (sampta)).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. In this paper we give an account of the construction of bimodular Wilson bases in higher dimensions from Gabor frames of redundancy two.
Original language English Proceedings of 2017 International Conference on Sampling Theory and Applications 4 2017 192-5 978-1-5386-1565-2 10.1109/SAMPTA.2017.8024456 Published - 2017 2017 International Conference on Sampling Theory and Applications - Tallinn, Estonia

### Conference

Conference 2017 International Conference on Sampling Theory and Applications Estonia Tallinn 03/07/2017 → 07/07/2017
Series 2017 International Conference on Sampling Theory and Applications (sampta)

Year: 2017

## Twisted Reed-Solomon Codes

Beelen, P., Puchinger, S. & Rosenkilde ne Nielsen, J. 2017 Proceedings of 2017 IEEE International Symposium on Information Theory. IEEE, p. 336-40 (2017 Ieee International Symposium on Information Theory (isit)).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

We present a new general construction of MDS codes over a finite field Fq. We describe two explicit subclasses which contain new MDS codes of length at least q/2 for all values of q ≥ 11. Moreover, we show that most of the new codes are not equivalent to a Reed-Solomon code.
Original language English Proceedings of 2017 IEEE International Symposium on Information Theory IEEE 2017 336-40 978-1-5090-4096-4 10.1109/ISIT.2017.8006545 Published - 2017 2017 IEEE International Symposium on Information Theory - Aachen, Germany

### Conference

Conference 2017 IEEE International Symposium on Information Theory Germany Aachen 25/06/2017 → 30/06/2017
Series 2017 Ieee International Symposium on Information Theory (isit) 2157-8117

Year: 2017

## On permutation polynomials over ﬁnite ﬁelds: diﬀerences and iterations

Anbar Meidl, N., Odzak, A., Patel, V., Quoos, L., Somoza, A. & Topuzoglu, A. 2017 Women in Numbers. 13 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

The Carlitz rank of a permutation polynomial f over a finite field Fq is a simple concept that was introduced in the last decade. Classifying permutations over Fq
with respect to their Carlitz ranks has some advantages, for instance f with a given Carlitz rank can be approximated by a rational linear transformation. In this note we present our recent results on the permutation behaviour of polynomials f+g, where f is a permutation over Fq of a given Carlitz rank, and g ∈ Fq[x] is of prescribed degree. We describe the relation of this problem to the well-known Chowla-Zassenhaus conjecture. We also study iterations of permutation polynomials by using the approximation property that is mentioned above.
Original language English Women in Numbers 13 2017 Published - 2017

### Bibliographical note

conference paper submitted to Women in Numbers Europe

Year: 2017

## The Exact Limit of Some Cubic Towers

Anbar Meidl, N., Beelen, P. & Nguyen, N. 2017 Proceedings of the International Conference on Arithmetic, Geometry, Cryptography and Coding theory (2015). American Mathematical Society, 17 p. (Contemporary Mathematics, Vol. 686).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2016

Recently, a new explicit tower of function fields was introduced by Bassa, Beelen, Garcia and Stichtenoth (BBGS). This resulted in currently the best known lower bound for Ihara’s constant in the case of non-prime finite fields. In particular over cubic fields, the tower’s limit is at least as good as Zink’s bound; i.e. λ(BBGS/Fq3 ) ≥ 2(q2 - 1)/(q + 2). In this paper, the exact value of λ(BBGS/Fq3 ) is computed. We also settle a question stated by Ihara.
Original language English Proceedings of the International Conference on Arithmetic, Geometry, Cryptography and Coding theory (2015) 17 American Mathematical Society 2017 Published - 2017 15th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory - Marseille, France

### Conference

Conference 15th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory 15 France Marseille 18/05/2015 → 22/05/2015 http://scientific-events.weebly.com/1193.html
Series Contemporary Mathematics 686 0271-4132

Year: 2017

## Algorithms for Zero-Dimensional Ideals Using Linear Recurrent Sequences

Neiger, V., Rahkooy, H. & Schost, É. 2017 19th International Workshop on Computer Algebra in Scientific Computing. Springer, p. 313-328 (Lecture Notes in Computer Science, Vol. 10490).

Publication: Research - peer-reviewBook chapter – Annual report year: 2017

Inspired by Faugére and Mou´s sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing of the annihilator of one or several such sequences.
Original language English 19th International Workshop on Computer Algebra in Scientific Computing Springer 2017 313-328 9783319663197 10.1007/978-3-319-66320-3_23 Published - 2017 19th International Workshop on Computer Algebra in Scientific Computing - Beijing, China

### Conference

Conference 19th International Workshop on Computer Algebra in Scientific Computing AMSS South Building China Beijing 18/09/2017 → 22/09/2017
Series Lecture Notes in Computer Science 10490 0302-9743

Year: 2017

## Rationalization in architecture with surfaces foliated by elastic curves

Nørbjerg, T. B., Gravesen, J. & Brander, D. 2017 Kgs. Lyngby: Technical University of Denmark (DTU). 101 p. (DTU Compute PHD-2016; No. 419).

Publication: ResearchPh.D. thesis – Annual report year: 2017

We develop methods for rationalization of CAD surfaces using elastic curves, aiming at a costeffective fabrication method for architectural designs of complex shapes. By moving a heated flexible metal rod though a block of expanded polystyrene, it is possible to produce shapes with both positive and negative Gaussian curvature, either for direct use or for use as moulds for concrete casting. If we can control the shape of the rod, while moving, we can produce prescribed shapes.

The flexible rod assumes at all times the shape of an Euler elastica (or elastic curve). The elastica are given in closed analytic form using elliptic functions. We use a gradient-driven optimization to approximate arbitrary planar curves by planar elastic curves. The method depends on an explicit parameterization of the space of elastic curves and on a method for finding a good initial guess for the optimization.

We approximate CAD surfaces by first extracting a collection of planar surface curves and approximating these by elastica. Providing the data for these curves to robots holding the flexible rod, we can produce an elastica-foliated surface that approximates the given CAD surface. Since not all surfaces can be closely approximated by an elastica-foliated surface, an arbitrary CAD surface must first be subdivided into segments that can be approximated. We discuss strategies for subdividing an arbitrary surface into segments that can be closely approximated, taking into account the aesthetics of the segmentation and the production constraints. If the given surface is smooth, we want the approximating surface to be smooth as well, so we must ensure smooth transition between the surface segments of the final result.

As an alternative to rationalization of arbitrary designs, we also present a method for direct generation of design surfaces using foliated Euler elastica. Here we work from a grid of blocks, so the segmentation is given, but we must still ensure smooth transition between segments.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 101 Published - 2017
Series DTU Compute PHD-2016 419 0909-3192

Year: 2017

## Optimization on Spaces of Curves

Møller-Andersen, J. & Gravesen, J. 2017 Kgs. Lyngby: Technical University of Denmark (DTU). 113 p. (DTU Compute PHD-2016; No. 432).

Publication: ResearchPh.D. thesis – Annual report year: 2017

This thesis is concerned with computational and theoretical aspects of Riemannian metrics on spaces of regular curves, and their applications. It was recently proved that second order constant coefficient Sobolev metrics on curves are geodesically complete. We extend this result to the case of Sobolev metrics with coefficient functions depending on the length of the curve. We show how to apply this result to analyse a wide range of metrics on the submanifold of unit and constant speed curves.

We present a numerical discretization of second order Sobolev metrics on the space of regular curves in Rd, and methods to solve the initial and boundary value problem for geodesics allowing us to compute the Karcher mean and principal components analysis of data of curves. We apply the methods to study shape variation in synthetic data in the Kimia shape database, in HeLa cell nuclei and cycles of cardiac deformations.

Finally we investigate a new application of Riemannian shape analysis in shape optimization. We setup a simple elliptic model problem, and describe how to apply shape calculus to obtain directional derivatives in the manifold of planar curves. We present an implementation based on parametrization of immersions by B-splines, which ties in naturally with Isogeometric Analysis to solve the PDE. We give numerical examples of solutions, and compare the Riemannian optimization algorithms with different choices of metrics to a naive unregularized discretize-first approach.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 113 Published - 2017
Series DTU Compute PHD-2016 432 0909-3192

Year: 2017

## Gabor frames on locally compact abelian groups and related topics

Jakobsen, M. S., Christensen, O. & Lemvig, J. 2017 Kgs. Lyngby: Technical University of Denmark (DTU). 26 p. (DTU Compute PHD-2016; No. 436).

Publication: ResearchPh.D. thesis – Annual report year: 2017

This thesis consists of four papers. The first one introduces generalized translation invariant systems and considers their frame properties, the second and third paper give new results on the theory of Gabor frames, and the fourth is a review paper with proofs and new results on the Feichtinger algebra.

The generalized translation invariant (GTI) systems provide, for the first time, a framework which can describe frame properties of both discrete and continuous systems. The results yield the well-known characterizations of dual frame pairs and Parseval frames of Gabor-, wavelet-, curvelet- and shearlet-type and for (generalized) shift-invariant systems and their continuous formulations.

This thesis advances the theory of both separable and non-separable, discrete, semicontinuous and continuous Gabor systems. In particular, the well established structure theory for separable lattice Gabor frames is extended and generalized significantly to Gabor systems with time-frequency shifts along closed subgroups in the time-frequency plane. This includes density results, the Walnut representation, the Wexler-Raz biorthogonality relations, the Bessel duality and the duality principle between Gabor frames and Gabor Riesz bases.

The theory of GTI systems and Gabor frames in this thesis is developed and presented in the setting of locally compact abelian groups, however, even in the euclidean setting the results given here improve the existing theory.

Finally, the thesis contains a review paper with proofs of all the major results on the Banach space of functions known as the Feichtinger algebra. This includes many of its different characterizations and treatment of its many equivalent norms, its minimality among all time-frequency shift invariant Banach spaces and aspects of its dual space, operators on the space and the kernel theorem for the Feichtinger algebra. The work also includes new findings such as a characterization among all Banach spaces, a forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, and new useful inequalities.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 26 Published - 2017
Series DTU Compute PHD-2016 436 0909-3192

Year: 2017

## Decoding Interleaved Gabidulin Codes using Alekhnovich's Algorithm

Puchinger, S., Müelich, S., Mödinger, D., Rosenkilde, J. S. H. & Bossert, M. 2017 57, p. 175–180

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We prove that Alekhnovich's algorithm can be used for row reduction of skew polynomial matrices. This yields an O(ℓ3n(ω+1)/2log⁡(n)) decoding algorithm for ℓ-Interleaved Gabidulin codes of length n, where ω is the matrix multiplication exponent.
Original language English Electronic Notes in Discrete Mathematics 57 175–180 1571-0653 10.1016/j.endm.2017.02.029 Published - 2017

Year: 2017

## Row Reduction Applied to Decoding of Rank Metric and Subspace Codes

Puchinger, S., Nielsen, J. S. R., Li, W. & Sidorenko, V. 2017 82, 1, p. 389–409 21 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We show that decoding of ℓ-Interleaved Gabidulin codes, as well as list-ℓ decoding of Mahdavifar–Vardy (MV) codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding ℓ-Interleaved Gabidulin codes. We obtain an algorithm with complexity O(ℓμ2) where μ measures the size of the input problem and is proportional to the code length n in the case of decoding. Further, we show how to perform the interpolation step of list-ℓ-decoding MV codes in complexity O(ℓn2), where n is the number of interpolation constraints.
Original language English Designs, Codes and Cryptography 82 1 389–409 21 0925-1022 10.1007/s10623-016-0257-9 Published - 2017

Year: 2017

## Fractional and complex pseudo-splines and the construction of Parseval frames

Massopust, P., Forster, B. & Christensen, O. 2017 314, p. 12-24

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Pseudo-splines of integer order (m, ℓ) were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies’ scaling functions. The purpose of this paper is to generalize the pseudo-splines to fractional and complex orders (z, ℓ) with α ≔ Re z ≥ 1. This allows increased flexibility in regard to smoothness: instead of working with a discrete family of functions from Cm, m∈N0, one uses a continuous family of functions belonging to the Hölder spaces Cα−1. The presence of the imaginary part of z allows for direct utilization in complex transform techniques for signal and image analyses. We also show that in analogue to the integer case, the generalized pseudo-splines lead to constructions of Parseval wavelet frames via the unitary extension principle. The regularity and approximation order of this new class of generalized splines is also discussed.
Original language English Applied Mathematics and Computation 314 12-24 1873-5649 10.1016/j.amc.2017.06.023 Published - 2017

Year: 2017

## Blowup for flat slow manifolds: Paper

Kristiansen, K. U. 2017 In : Nonlinearity. 30, 5, p. 2138-2184

Publication: Research - peer-reviewJournal article – Annual report year: 2017

In this paper, we present a way of extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimensions, they do appear naturally in certain settings; for example, in (a) the regularization of piecewise smooth systems by tanh, (b) a particular aircraft landing dynamics model, and finally (c) in a model of earthquake faulting. We demonstrate the approach using a simple model system and the examples (a) and (b).
Original language English Nonlinearity 30 5 2138-2184 0951-7715 10.1088/1361-6544/aa6449 Published - 2017

Year: 2017

## On the regularization of impact without collision: the Painlevé paradox and compliance

Hogan, S. J. & Kristiansen, K. U. 2017 473, 2202, 18 p., 20160773

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevéé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.
Original language English 20160773 Royal Society of London. Proceedings A. Mathematical, Physical and Engineering Sciences 473 2202 18 1364-5021 10.1098/rspa.2016.0773 Published - 2017

Year: 2017

## Topological fluid mechanics of the formation of the Kármán-vortex street

Heil, M., Rosso, J., Hazel, A. L. & Brøns, M. 2017 812, p. 199-221

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We explore the two-dimensional flow around a circular cylinder with the aim of elucidating the changes in the topology of the vorticity field that lead to the formation of the Kármán vortex street. Specifically, we analyse the formation and disappearance of extremal points of vorticity, which we consider to be feature points for vortices. The basic vortex creation mechanism is shown to be a topological cusp bifurcation in the vorticity field, where a saddle and an extremum of the vorticity are created simultaneously. We demonstrate that vortices are first created approximately 100 diameters downstream of the cylinder, at a Reynolds number, ReK, which is slightly larger than the critical Reynolds number, Recrit∼46, at which the flow becomes time periodic. For Re slightly above ReK, the newly created vortices disappear again a short distance further downstream. As is further increased, the points of creation and disappearance move rapidly upstream and downstream, respectively, and the Kármán vortex street persists over increasingly large streamwise distances.
Original language English Journal of Fluid Mechanics 812 199-221 0022-1120 10.1017/jfm.2016.792 Published - 2017

Year: 2017

## Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

Granados, A. 2017 Vol. 351-352, p. 14-29

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies O(1) energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a 4-dimensional Normally Hyperbolic Invariant Manifold with 5 and 4-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on.
Original language English Physica D: Nonlinear Phenomena Vol. 351-352 14-29 0167-2789 10.1016/j.physd.2017.04.003 Published - 2017

Year: 2017

## The period adding and incrementing bifurcations: from rotation theory to applications

Granados, A., Alseda, L. & Krupa, M. 2017 In : S I A M Review. 59, 2, p. 225–292

Publication: Research - peer-reviewJournal article – Annual report year: 2016

This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and \rotation” numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval.

We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasi-contractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.
Original language English S I A M Review 59 2 225–292 0036-1445 10.1137/140996598 Published - 2017

Year: 2017

## System bandwidth and the existence of generalized shift-invariant frames

Führ, H. & Lemvig, J. 2017 In : arXiv. 34 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We consider the question whether, given a countable system of lattices (Γj)j∈J in a locally compact abelian group G, there exists a sequence of functions (gj)j∈J such that the resulting generalized shift-invariant system (gj(⋅−γ))j∈J,γ∈Γj is a tight frame of L2(G). This paper develops a new approach to the study of almost periodic functions for generalized shift-invariant systems based on an \emph{unconditionally convergence property}, replacing previously used local integrability conditions. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the \emph{system bandwidth} as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calder\'on sum is uniformly bounded from below for generalized shift-invariant frames. We exhibit a condition on the lattice system for which the unconditionally convergence property is guaranteed to hold. Without the unconditionally convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system can be rather subtle, depending on analytical properties, such as the system bandwidth, as well as on algebraic properties of the lattice system.
Original language English arXiv 34 Submitted - 2017

Year: 2017

## The Fine Structure of Herman Rings

Fagella, N. & Henriksen, C. 2017 27, 3, p. 2381-2399

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3 (up to quasiconformal deformation). Shishikura’s quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we transfer McMullen’s results on the fine local geometry of Siegel disks to the Herman ring setting.
Original language English Journal of Geometric Analysis 27 3 2381-2399 1050-6926 10.1007/s12220-017-9764-9 Published - 2017

Year: 2017

## Number of solutions of systems of homogeneous polynomial equations over finite fields

Datta, M. & Ghorpade, S. R. 2017 145, p. 525-541

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the maximum value when the homogeneous polynomials have the same degree that is not too large in comparison to the size of the finite field. We show that this conjecture holds in the affirmative if the number of polynomials does not exceed the total number of variables. This extends the results of Serre (1991) and Boguslavsky (1997) for the case of one and two polynomials, respectively. Moreover, it complements our recent result that the conjecture is false, in general, if the number of polynomials exceeds the total number of variables.
Original language English American Mathematical Society. Proceedings 145 525-541 0002-9939 10.1090/proc/13239 Published - 2017

Year: 2017

## Sparse identification of a predator-prey system from simulation data of a convection model

Dam, M., Brøns, M., Rasmussen, J. J., Naulin, V. & Hesthaven, J. S. 2017 24, 2, 10 p., 022310

Publication: Research - peer-reviewJournal article – Annual report year: 2017

The use of low-dimensional dynamical systems as reduced models for plasma dynamics is useful as solving an initial value problem requires much less computational resources than fluid simulations. We utilize a data-driven modeling approach to identify a reduced model from simulation data of a convection problem. A convection model with a pressure source centered at the inner boundary models the edge dynamics of a magnetically confined plasma. The convection problem undergoes a sequence of bifurcations as the strength of the pressure source increases. The time evolution of the energies of the pressure profile, the turbulent flow, and the zonal flow capture the fundamental dynamic behavior of the full system. By applying the sparse identification of nonlinear dynamics (SINDy) method, we identify a predator-prey type dynamical system that approximates the underlying dynamics of the three energy state variables. A bifurcation analysis of the system reveals consistency between the bifurcation structures, observed for the simulation data, and the identified underlying system.
Original language English 022310 Physics of Plasmas 24 2 10 1070-664X 10.1063/1.4977057 Published - 2017

Year: 2017

## Topological bifurcations in the evolution of coherent structures in a convection model

Dam, M., Rasmussen, J. J., Naulin, V. & Brøns, M. 2017 24, 8, 7 p., 082301

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Blob filaments are coherent structures in a turbulent plasma flow. Understanding the evolution of these structures is important to improve magnetic plasma confinement. Three state variables describe blob filaments in a plasma convection model. A dynamical systems approach analyzes the evolution of these three variables. A critical point of a variable defines a feature point for a region where that variable is significant. For a range of Rayleigh and Prandtl numbers, the bifurcations of the critical points of the three variables are investigated with time as the primary bifurcation parameter. Bifurcation curves separate the parameter planes into regions with different critical point configurations for the state variables. For Prandtl number equal to 1, the number of critical points of each state variable increases with increasing Rayleigh number. For Rayleigh number equal to 104, the number of critical points is the greatest for Prandtl numbers of magnitude 100.
Original language English 082301 Physics of Plasmas 24 8 7 1070-664X 10.1063/1.4993613 Published - 2017

Year: 2017

## B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames

Christensen, O., Kim, H. O. & Kim, R. Y. 2017 p. 1-22

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters (Formula presented.) with (Formula presented.), the scaled version of (Formula presented.) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to â€œalmost perfect reconstructionâ€� within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the (Formula presented.)-spaces, as well in the time-domain as in the frequency domain.
Original language English Journal of Fourier Analysis and Applications 1-22 1069-5869 10.1007/s00041-017-9557-3 Published - 2017

Year: 2017

## Characterisations of Partition of Unities Generated by Entire Functions in Cd

Christensen, O., Kim, H. O. & Kim, R. Y. 2017 95, 2, p. 281-290 10 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Collections of functions forming a partition of unity play an important role in analysis. In this paper we characterise for any N∈N the entire functions P for which the partition of unity condition ∑n∈ZdP(x+n)χ[0,N]d(x+n)=1 holds for all x∈Rd. The general characterisation leads to various easy ways of constructing such entire functions as well. We demonstrate the flexibility of the approach by showing that additional properties like continuity or differentiability of the functions (Pχ[0,N]d)(⋅+n) can be controlled. In particular, this leads to easy ways of constructing entire functions P such that the functions in the partition of unity belong to the Feichtinger algebra.
Original language English Australian Mathematical Society. Bulletin 95 2 281-290 10 0004-9727 10.1017/S0004972716001052 Published - 2017

Year: 2017

## Gabor Frames in ℓ2(Z) and Linear Dependence

Christensen, O. & Hasannasab, M. 2017

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We prove that an overcomplete Gabor frame in (Formula presented.) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in (Formula presented.) with modulation parameter 1 / M and translation parameter N for some (Formula presented.) and generated by a finite sequence g in (Formula presented.) with K nonzero entries.
Original language English Journal of Fourier Analysis and Applications 1069-5869 10.1007/s00041-017-9572-4 Accepted/In press - 2017

Year: 2017

## Operator Representations of Frames: Boundedness, Duality, and Stability

Christensen, O. & Hasannasab, M. 2017 88, 4, p. 483-499

Publication: Research - peer-reviewJournal article – Annual report year: 2017

The purpose of the paper is to analyze frames (Formula presented.) having the form (Formula presented.) for some linear operator (Formula presented.). A key result characterizes boundedness of the operator T in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation (Formula presented.) can be achieved for an operator T that has an extension to a bounded bijective operator (Formula presented.) In this case we also characterize all the dual frames that are representable in terms of iterations of an operator V; in particular we prove that the only possible operator is (Formula presented.) Finally, we consider stability of the representation (Formula presented.) rather surprisingly, it turns out that the possibility to represent a frame on this form is sensitive towards some of the classical perturbation conditions in frame theory. Various ways of avoiding this problem will be discussed. Throughout the paper the results will be connected with the operators and function systems appearing in applied harmonic analysis, as well as with general group representations.
Original language English Integral Equations and Operator Theory 88 4 483-499 0378-620X 10.1007/s00020-017-2370-1 Published - 2017

Year: 2017

## Paley-wiener type perturbations of frames and the deviation from perfect reconstruction

Christensen, O. & Zakowicz, M. I. 2017 7, 1, p. 59-69

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Frame theory is an efficient tool to obtain expansions of elements in separable Hilbert spaces that are similar to the ones obtained via orthonormal bases, however, with considerably more exibility. In this paper we give a survey of known results about frame expansions and perturbation theory, combined with an extension to approximately dual frames. We will show, e.g., that perturbation of a pair of dual frames in the Paley-Wiener sense leads to a deviation from perfect reconstruction that can be controlled in terms of the frame bounds of the involved sequences. The paper contains an Appendix, which motivates the analysis of frames via classical results.
Original language English Azerbaijan Journal of Mathematics 7 1 59-69 2218-6816 Published - 2017

Year: 2017

## The unitary extension principle on locally compact abelian groups

Christensen, O. & Goh, S. S. 2017 33 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

The unitary extension principle (UEP) by Ron and Shen yields conditions for the construction of a multi-generated tight wavelet frame for L2(Rs) based on a given reﬁnable function. In this paper we show that the UEP can be generalized to locally compact abelian groups. In the general setting, the resulting frames are generated by modulates of a collection of functions; via the Fourier transform this corresponds to a generalized shift-invariant system. Both the stationary and the nonstationary case are covered. We provide general constructions, based on B-splines on the group itself as well as on characteristic functions on the dual group. Finally, we consider a number of concrete groups and derive explicit constructions of the resulting frames.
Original language English Applied and Computational Harmonic Analysis 33 1063-5203 10.1016/j.acha.2017.07.004 Accepted/In press - 2017

Year: 2017

## On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

Brøns, M. & Kristiansen, K. U. 2017 p. 1-23

Publication: Research - peer-reviewJournal article – Annual report year: 2017

A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.

Original language English Dynamical Systems 1-23 1468-9375 10.1080/14689367.2017.1313390 Published - 2017

Year: 2017

## Pseudospherical surfaces with singularities

Brander, D. 2017 196, 3, p. 905–928

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We study a generalization of constant Gauss curvature −1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyse the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solution is unique for most curves, but for some curves there are infinitely many solutions, and this is encoded in the curvature and torsion of the curve.
Original language English Annali di Matematica Pura ed Applicata 196 3 905–928 0373-3114 10.1007/s10231-016-0601-8 Published - 2017

Year: 2017

## Remarks on the boundary curve of a constant mean curvature topological disc

Brander, D. & Lopéz, R. 2017 62, 8, p. 1037-1043

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a condition for the surface to be totally umbilic in terms of the normal curvature.
Original language English Complex Variables and Elliptic Equations 62 8 1037-1043 1747-6933 10.1080/17476933.2016.1260558 Published - 2017

Year: 2017

## On Wilson bases in $L^2(\mathbb{R}^d)$

Bownik, M., Sielemann Jakobsen, M., Lemvig, J. & Okoudjou, K. A. 2017 49, 5, p. 3999-4023 25 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. It is well known that, starting from a tight Gabor frame for $L^{2}(\mathbb{R})$ with redundancy 2, one can construct an orthonormal Wilson basis for $L^2(\mathbb{R})$ whose generator is well localized in the time-frequency plane. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor systems. Specifically, we show that one can construct $d$-dimensional orthonormal Wilson bases starting from tight Gabor frames of redundancy $2^k$, where $k=1, 2, \hdots, d$. These results generalize most of the known results about the existence of orthonormal Wilson bases.

Original language English S I A M Journal on Mathematical Analysis 49 5 3999-4023 25 0036-1410 10.1137/17M1122190 Published - 2017

Year: 2017

## Canards in stiction: on solutions of a friction oscillator by regularization

Bossolini, E., Brøns, M. & Kristiansen, K. U. 2017 27 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We study the solutions of a friction oscillator subject to stiction. This discontinuous model is nonFilippov, and the concept of Filippov solution cannot be used. Furthermore some Carath´eodory solutions are unphysical. Therefore we introduce the concept of stiction solutions: these are the Carath´eodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we find that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories, that have the physical interpretation of delaying the slip onset. We show with numerics that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.
Original language English S I A M Journal on Applied Dynamical Systems 27 1536-0040 Accepted/In press - 2017

Year: 2017

## Singular limit analysis of a model for earthquake faulting

Bossolini, E., Brøns, M. & Kristiansen, K. U. 2017 In : Nonlinearity. 30, 7, p. 2805-34

Publication: Research - peer-reviewJournal article – Annual report year: 2016

In this paper we consider the one dimensional spring-block model describing earthquake faulting. By using geometric singular perturbation theory and the blow-up method we provide a detailed description of the periodicity of the earthquake episodes. In particular, the limit cycles arise from a degenerate Hopf bifurcation whose degeneracy is due to an underlying Hamiltonian structure that leads to large amplitude oscillations. We use a Poincar\'e compactification to study the system near infinity. At infinity the critical manifold loses hyperbolicity with an exponential rate. We use an adaptation of the blow-up method to recover the hyperbolicity. This enables the identification of a new attracting manifold that organises the dynamics at infinity. This in turn leads to the formulation of a conjecture on the behaviour of the limit cycles as the time-scale separation increases. We provide the basic foundation for the proof of this conjecture and illustrate our findings with numerics.
Original language English Nonlinearity 30 7 2805-34 0951-7715 Published - 2017

Year: 2017

## Generalized shift-invariant systems and approximately dual frames

Benavente, A., Christensen, O. & Zakowicz, M. I. 2017 8, 2, p. 177-189

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Dual pairs of frames yield a procedure for obtaining perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force us to apply sequences that do not exactly form dual frames. In this article, we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames. The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.
Original language English Annals of Functional Analysis 8 2 177-189 2008-8752 10.1215/20088752-3784315 Published - 2017

Year: 2017

## Counting generalized Reed-Solomon codes

Beelen, P., Glynn, D., Høholdt, T. & Kaipa, K. 2017 11, 4, p. 777--790

Publication: Research - peer-reviewJournal article – Annual report year: 2017

In this article we count the number of [n, k] generalized Reed–Solomon (GRS) codes, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of [n, 3] MDS codes with n = 6, 7, 8, 9.
Original language English Advances in Mathematics of Communication 11 4 777--790 1930-5346 10.3934/amc.2017057 Published - 2017

Year: 2017

## Maximum number of common zeros of homogeneous polynomials over finite fields

Beelen, P., Datta, M. & Ghorpade, S. R. 2017 13 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the
the maximum number of common zeros that r linearly independent homogeneous polynomials
of degree d in m + 1 variables with coefficients in a finite field with q elements can have in
the corresponding m-dimensional projective space over that finite field. Recently, it has been
shown by Datta and Ghorpade that this conjecture is valid if r is at most m + 1 and can
be invalid otherwise. Moreover a new conjecture was proposed for many values of r beyond
m + 1. In this paper, we prove that this new conjecture holds true for several values of r. In
particular, this settles the new conjecture completely when d = 3. Our result also includes
the positive result of Datta and Ghorpade as a special case. Further, we also determine the
maximum number of zeros in certain cases not covered by the earlier conjectures and results,
namely, the case of d = q − 1 and of d = q.
Original language English Proceedings of the American Mathematical Society 13 0002-9939 Published - 2017

Year: 2017

## A Numerical Framework for Sobolev Metrics on the Space of Curves

Bauer, M., Bruveris, M., Harms, P. & Møller-Andersen, J. 2017 10, 1, p. 47-73

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.
Original language English S I A M Journal on Imaging Sciences 10 1 47-73 1936-4954 10.1137/16M1066282 Published - 2017

Year: 2017

## Two-Point Codes for the Generalised GK curve

Barelli, É., Beelen, P., Datta, M., Neiger, V. & Rosenkilde, J. S. H. 2017 9 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We improve previously known lower bounds for the minimum distance of certain two-point AG codes constructed using a Generalized Giulietti–Korchmaros curve (GGK). Castellanos and Tizziotti recently described such bounds for two-point codes coming from the Giulietti–Korchmaros curve (GK). Our results completely cover and in many cases improve on their results, using different techniques, while also supporting any GGK curve. Our method builds on the order bound for AG codes: to enable this, we study certain Weierstrass semigroups. This allows an efficient algorithm for computing our improved bounds. We find several new improvements upon the MinT minimum distance tables.
Original language English I E E E Transactions on Information Theory 9 0018-9448 10.1109/TIT.2017.2763165 Accepted/In press - 2017

Year: 2017

## A modular interpretation of various cubic towers

Anbar Meidl, N., Bassa, A. & Beelen, P. 2017 171, p. 341-357

Publication: Research - peer-reviewJournal article – Annual report year: 2016

In this article we give a Drinfeld modular interpretation for various towers of function fields meeting Zink's bound.
Original language English Journal of Number Theory 171 341-357 0022-314X 10.1016/j.jnt.2016.07.025 Published - 2017

### Bibliographical note

Corrigendum to “A modular interpretation of various cubic towers” [J. Number Theory 171 (2017) 341–357] Published in J. Number Theory, vol. 172, (2017), page 416.

Year: 2017

## A new tower with good p-rank meeting Zink’s bound

Anbar Meidl, N., Beelen, P. & Nguyen, N. 2017 177, 4, p. 347-374 28 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

In this article we investigate the asymptotic p-rank of a new tower of function fields defined over cubic finite fields. Its limit meets Zink's bound, but the new feature of this tower is that its asymptotic
p-rank for small cubic finite fields is much smaller than that of other cubic towers for which the asymptotic p-rank is known. This is of independent interest, but also makes this new tower more interesting for theoretical applications in cryptography.
Original language English Acta Arithmetica 177 4 347-374 28 0065-1036 10.4064/aa8388-6-2016 Published - 2017

Year: 2017

## A Note on a Tower by Bassa, Garcia and Stichtenoth

Anbar Meidl, N. & Beelen, P. 2017 57, 1, p. 47-60 13 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Original language English Functiones et Approximatio Commentarii Mathematici 57 1 47-60 13 0208-6573 doi:10.7169/facm/1615 Published - 2017

Year: 2017

## Bent and bent(4) spectra of Boolean functions over finite fields

Anbar Meidl, N. & Meidl, W. 2017 46, p. 163-178

Publication: Research - peer-reviewJournal article – Annual report year: 2017

For c is an element of F(2)n, a c-bent4 function f from the finite field F(2)n to F-2 is a function with a fiat spectrum with respect to the unitary transform V-f(c), which is designed to describe the component functions of modified planar functions. For c = 0 the transform V-f(c) reduces to the conventional Walsh transform, and hence a 0-bent4 function is bent. In this article we generalize the concept of partially bent functions to the transforms V-f(c). We show that every quadratic function is partially bent, and hence it is plateaued with respect to any of the transforms V-f(c). In detail we analyse two quadratic monomials. The first has values as small as possible in its spectra with respect to all transforms V-f(c), and the second has a flat spectrum for a large number of c. Moreover, we show that every quadratic function is c-bent4 for at least three distinct c. In the last part we analyse a cubic monomial. We show that it is c-bent(4) only for c = 1, the function is then called negabent, which shows that non-quadratic functions exhibit a different behaviour. (C) 2017 Elsevier Inc. All rights reserved.
Original language English Finite Fields and Their Applications 46 163-178 1071-5797 10.1016/j.ffa.2017.03.008 Published - 2017

Year: 2017

## Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

Anbar Meidl, N., Meidl, W. M. & Topuzoglu, A. 2017 82, 1, p. 265–291 27 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2016

Original language English Designs, Codes and Cryptography 82 1 265–291 27 0925-1022 10.1007/s10623-016-0213-8 Published - 2017

### Bibliographical note

This paper is dedicated to the memory of Tosun Terzioglu

Year: 2017

## Modified planar functions and their components

Anbar Meidl, N. & Meidl, W. M. 2017 p. 1–15 15 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

Zhou ([20]) introduced modified planar functions in order to describe (2n; 2n; 2n; 1) relative difference sets R as a graph of a function on the finite field F2n, and pointed out that projections of R are difference sets that can be described by negabent or bent4 functions, which are Boolean functions given in multivariate form. One of the objectives of this paper is to contribute to the understanding of these component functions of modified planar functions. Moreover, we obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function. We finally point out that though these components behave somewhat different than the multivariate bent4 functions, they are bent or semibent functions shifted by a certain quadratic term, a property which they share with their multivariate counterpart.
Original language English Cryptography and Communications 1–15 15 1936-2447 10.1007/s12095-017-0218-9 Published - 2017

Year: 2017

## On the difference between permutation poynomials over finite fields

Anbar Meidl, N., Odzak, A., Patel, V., Quoos, L., Somoza, A. & Topuzoglu, A. 2017 In : arXiv. 12 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2017

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if p > (d 2 − 3d + 4)2 , then there is no complete mapping polynomial f in Fp[x] of degree d ≥ 2. For arbitrary finite fields Fq, a similar non-existence result is obtained recently by I¸sık, Topuzo˘glu and Winterhof in terms of the Carlitz rank of f. Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f + g are both permutation polynomials of degree d ≥ 2 over Fp, with p > (d 2−3d+4)2 , then the degree k of g satisfies k ≥ 3d/5, unless g is constant. In this article, assuming f and f + g are permutation polynomials in Fq[x], we give lower bounds for k in terms of the Carlitz rank of f and q. Our results generalize the above mentioned result of I¸sık et al. We also show for a special class of polynomials f of Carlitz rank n ≥ 1 that if f + x k is a permutation over Fq, with gcd(k + 1, q − 1) = 1, then k ≥ (q − n)/(n + 3).
Original language English arXiv 12 Published - 2017

Year: 2017

## A complete characterization of Galois subfields of the generalized Giulietti–Korchmáros function field

Anbar, N., Bassa, A. & Beelen, P. 2017 48, p. 318-330

Publication: Research - peer-reviewJournal article – Annual report year: 2017

We give a complete characterization of all Galois subfields of the generalized Giulietti–Korchmáros function fields Cn/Fq2n for n ≥5. Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields.
Original language English Finite Fields and Their Applications 48 318-330 1071-5797 10.1016/j.ffa.2017.08.006 Published - 2017

2016

Year: 2016

## Cuttable Ruled Surface Strips for Milling

Steenstrup, K. H., Nørbjerg, T. B., Søndergaard, A., Bærentzen, J. A. & Gravesen, J. 2016 Advances in Architectural Geometry 2016. Adriaenssens, S., Gramazio, F., Kohler, M., Menges, A. & Pauly, M. (eds.). vdf Hochschulverlag AG an der ETH Zürich, p. 328-342

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2016

This paper proposes a novel pre-processing method for industrial robotic CNC-milling. The method targets a hybrid machining process, in which the main bulk of material is removed through robotic hot or abrasive wire cutting, after which regular CNC-machining is employed for removal of the remaining material volume. Hereby, the roughing process is significantly sped up, reduc-ing overall machining time. We compare our method to the convex hull and re-move between 5% and 75% more material; on most models we obtain a 50% improvement. Our method ensures that no overcutting happens and that the result is cuttable by wire cutting.
Original language English Advances in Architectural Geometry 2016 Sigrid Adriaenssens, Fabio Gramazio, Matthias Kohler, Achim Menges, Mark Pauly vdf Hochschulverlag AG an der ETH Zürich 2016 328-342 978-3-7281-3778-4 10.3218/3778-4_22 Published - 2016 Advances in Architectural Geometry (AAG 2016) - Zurich, Switzerland

### Conference

Conference Advances in Architectural Geometry (AAG 2016) ETH Zurich Switzerland Zurich 09/09/2016 → 13/09/2016 http://www.aag2016.ch/

Year: 2016

## Algorithms for Simultaneous Padé Approximations

Rosenkilde, J. S. H. & Storjohann, A. 2016 Proceedings of the 41st International Symposium on Symbolic and Algebraic Computation (ISSAC '16). Association for Computing Machinery, p. 405-412

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2016

We describe how to solve simultaneous Padé approximations over a power series ring K[[x]] for a field K using O~(nω - 1 d) operations in K, where d is the sought precision and $n$ is the number of power series to approximate. We develop two algorithms using different approaches. Both algorithms return a reduced sub-bases that generates the complete set of solutions to the input approximations problem that satisfy the given degree constraints. Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite Padé approximations.
Original language English Proceedings of the 41st International Symposium on Symbolic and Algebraic Computation (ISSAC '16) Association for Computing Machinery 2016 405-412 978-1-4503-4380-0 Published - 2016 41st International Symposium on Symbolic and Algebraic Computation (ISSAC '16) - Waterloo, Ontario, Canada

### Conference

Conference 41st International Symposium on Symbolic and Algebraic Computation (ISSAC '16) 41 Canada Waterloo, Ontario 19/07/2016 → 22/07/2016 http://www.issac-symposium.org/2016/

Year: 2016

## Sculpturing Surfaces with Cartan Ribbons

Raffaelli, M., Bohr, J. & Markvorsen, S. 2016 Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture. Tessellations Publishing, p. 457-460 (Bridges (Winfield)).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2016

Using the concepts of Cartan development and rolling from differential geometry we develop a method for sculpturing any surface with the use of Cartan ribbons.
Original language English Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture Tessellations Publishing 2016 457-460 978-1-938664-19-9 Published - 2016 Bridges 2016 - Jyväskylä, Finland

### Conference

Conference Bridges 2016 University of Jyväskylä Finland Jyväskylä 09/08/2016 → 13/08/2016 http://bridgesmathart.org/bridges-2016/
Series Bridges (Winfield) 1099-6702

Year: 2016

## Designing for hot-blade cutting: Geometric Approaches for High-Speed Manufacturing of Doubly-Curved Architectural Surfaces

Brander, D., Bærentzen, J. A., Clausen, K., Fisker, A-S., Gravesen, J., Lund, . M. N., Nørbjerg, T. B., Steenstrup, K. H. & Søndergaard, A. 2016 Advances in Architectural Geometry 2016. Adriaenssens, S., Gramazio, F., Kohler, M., Menges, A. & Pauly, M. (eds.). vdf Hochschulverlag AG an der ETH Zürich, p. 306-327

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2016

In this paper we present a novel method for the generation of doubly-curved, architectural design surfaces using swept Euler elastica and cubic splines. The method enables a direct design to production workflow with robotic hot-blade cutting, a novel robotic fabrication method under development by authors of the paper, which facilitates high-speed production of doubly-curved foam moulds. Complementary to design rationalisation, in which arbitrary surfaces are translated to hot-blade-cuttable geometries, the presented method enables architects and designers to design directly with the non-trivial constraints of blade-cutting in a bottom-up fashion, enabling an exploration of the unique architectural potential of this fabrication approach. The method is implemented as prototype design
tools in MatLAB, C++, GhPython, and Python and demonstrated through cutting of expanded polystyrene foam design examples.
Original language English Advances in Architectural Geometry 2016 Sigrid Adriaenssens, Fabio Gramazio, Matthias Kohler, Achim Menges, Mark Pauly vdf Hochschulverlag AG an der ETH Zürich 2016 306-327 978-3-7281-3778-4 10.3218/3778-4_21 Published - 2016 Advances in Architectural Geometry (AAG 2016) - Zurich, Switzerland

### Conference

Conference Advances in Architectural Geometry (AAG 2016) ETH Zurich Switzerland Zurich 09/09/2016 → 13/09/2016 http://www.aag2016.ch/

Year: 2016

## Hot Blade Cuttings for the Building Industries

Brander, D., Bærentzen, J. A., Evgrafov, A., Gravesen, J., Markvorsen, S., Nørbjerg, T. B., Nørtoft, P. & Steenstrup, K. H. 2016 Proceedings of the KoMSO Challenge Workshop: Math for the Digital Factory (2014). Ghezzi, L., Hömberg, D. & Landry, C. (eds.). Springer, 19 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2016

The constructions of advanced architectural designs are presently very labour intensive, time consuming, and expensive. They are therefore only applied to a few prestige projects, and it is a major challenge for the building industry to bring the costs down and thereby offer the architects more variability in the (economically allowed) designs - i.e., to allow them to think out of the box. To address this challenge The Danish National Advanced Technology Foundation (now InnovationsFonden) is currently supporting the BladeRunner project that involves several Danish companies and public institutions. The project aims to reduce the amount of manual labour as well as production time by applying robots to cut expanded polystyrene (EPS) moulds for the concrete to form doubly curved surfaces. The scheme is based upon the so-called Hot Wire or Hot Blade technology where the surfaces are essentially swept out by driving an Euler elastica through a block of EPS. This paper will be centered around the mathematical challenges encountered in the implementation of this idea. Since the elastica themselves are well known and described in the works of Euler et al. already in eighteenth century, these new challenges are mainly concerned with the rationalization of the architects’ CAD drawings into surfaces that can be created via this particular sweeping and cutting technology.
Original language English Proceedings of the KoMSO Challenge Workshop: Math for the Digital Factory (2014) L. Ghezzi, D. Hömberg, Ch. Landry 19 Springer 2016 Published - 2016 KoMSO Challenge Workshop - Berlin, Germany

### Workshop

Workshop KoMSO Challenge Workshop Weierstrass Institut für Angewandte Analysis und Stochastik Germany Berlin 07/05/2014 → 09/05/2014 https://www.wias-berlin.de/workshops/MaDiFa/

Year: 2016

## Regularization by External Variables

Bossolini, E., Edwards, R., Glendinning, P. A., Jeffrey, M. R. & Webber, S. 2016 Extended Abstracts Spring 2016: Nonsmooth Dynamics. Colombo, A., Lázaro, J. T., Jeffrey, M. & Olm, J. M. (eds.). Springer, p. 19-24

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2017

Regularization was a big topic at the 2016 CRM Intensive Research Program on Advances in Nonsmooth Dynamics. There are many open questions concerning well known kinds of regularization (e.g., by smoothing or hysteresis). Here, we propose a framework for an alternative and important kind of regularization, by external variables that shadow either the state or the switch of the original system. The shadow systems are derived from and inspired by various applications in electronic control, predator-prey preference, time delay, and genetic regulation.
Original language English Extended Abstracts Spring 2016 : Nonsmooth Dynamics Alessandro Colombo, J. Tomàs Lázaro, Mike Jeffrey, Josep M. Olm Springer 2016 19-24 978-3-319-55641-3 10.1007/978-3-319-55642-0_4 Published - 2016 Intensive Research Program on Advances in Nonsmooth Dynamics 2016 - Barcelona, Spain

### Seminar

Seminar Intensive Research Program on Advances in Nonsmooth Dynamics 2016 Centre de Recerca Matemàtica​ ​​​ Spain Barcelona 01/02/2016 → 29/04/2016

Year: 2016

## Robotic Hot-Blade Cutting: An Industrial Approach to Cost-Effective Production of Double Curved Concrete Structures

Søndergaard, A., Feringa, J., Nørbjerg, T. B., Steenstrup, K. H., Brander, D., Gravesen, J., Markvorsen, S., Bærentzen, J. A., Petkov, K., Hattel, J. H., Clausen, K., Jensen, K., Knudsen, L. & Kortbek, J. 2016 Robotic Fabrication in Architecture, Art and Design 2016. Reinhardt, D., Saunders, R. & Burry, J. (eds.). Springer, p. 150-164

Publication: Research - peer-reviewBook chapter – Annual report year: 2016

This paper presents a novel method for cost-effective, robotic production of double curved formwork in Expanded Polystyrene (EPS) for in situ and prefabricated concrete construction. A rationalization and segmentation procedure is developed, which allows for the transliteration of double curved NURBS surfaces to Euler elastica surface segments, while respecting various constraints of production. An 18 axis, tri-robot system approximates double curved NURBS surfaces by means of an elastically deformed and heated blade, mounted on the flanges of two manipulators. Re-orienting or translating either end of the blade dynamically deforms the blade’s curvature. The blade follows the contours of the rationalized surface by continuous change in position and orientation of the end-effectors. The concept’s potential is studied by a pilot production of a full-scale demonstrator panel assembly.
Original language English Robotic Fabrication in Architecture, Art and Design 2016 Dagmar Reinhardt, Rob Saunders, Jane Burry Springer 2016 150-164 978-3-319-26376-2 978-3-319-26378-6 10.1007/978-3-319-26378-6_11 Published - 2016

Year: 2016

## Rationalization with ruled surfaces in architecture

Steenstrup, K. H., Gravesen, J. & Bærentzen, J. A. 2016 Kgs. Lyngby: Technical University of Denmark (DTU). 70 p. (DTU Compute PHD-2016; No. 413).

Publication: ResearchPh.D. thesis – Annual report year: 2016

This thesis addresses the problems of rationalizing and segmenting large scale 3D models, and how to handle difficult production constraints in this area. The design choices when constructing large scale architecture are influenced by the budget. Therefore I strive to minimize the amount of time and material needed for production. This makes advanced free form architecture viable for low cost projects, allowing the architects to realize their designs.

By pre-cutting building blocks using hot wire robots, the amount of milling necessary can be reduced drastically. I do this by rationalizing the intended shape as a piecewise ruled surface; the developed method was able to cut away up to 95% of the excess material. Methods were developed to minimize the number of blocks necessary to build advanced large scale 3D shapes. Using stochastic optimization to guide the segmentation, it was possible to remove up to 48% of the building blocks. Hot blade cutting for constructing models with positive Gauss curvature is an upcoming technology. Three segmentation algorithms were developed to solve construction constraints that arises when using this technique. One of the algorithms focusses on creating an aesthetic segmentation.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 70 Published - 2016
Series DTU Compute PHD-2016 413 0909-3192

Year: 2016

Balci, A. & Brøns, M. 2016 Kgs. Lyngby: Technical University of Denmark (DTU). 194 p. (DTU Compute PHD-2015; No. 379).

Publication: ResearchPh.D. thesis – Annual report year: 2016

In an incompressible fluid flow, streamline patterns and their bifurcations are investigated close to wall for two-dimensional system and close to free and viscous surfaces in three-dimensional system. Expanding the velocity field in a Taylor series, we conduct a local analysis at the given expansion point. Applying the boundary conditions, some relations are obtained among the coefficients of the expansions. Series of coordinate transformations, which preserves the boundary conditions, are used to reduce the number of coefficients. Finally, using the normal form and unfolding theory, the velocity field is analysed structurally and bifurcation diagrams are obtained.

First, two-dimensional viscous flow close to wall for non-simple degenerate critical point is considered depending on three-parameter space. Second, threedimensional axisymmetric, viscous and steady flow is analysed close to free and viscous surfaces into three situations: Local analysis close to center axis; away from the axis and close to a stationary wall. Next, in the absence of axisymmetric condition, three-dimensional viscous flow is consider close to a free surface.

As an application of the bifurcation diagrams for three-dimensional axisymmetric viscous flow, three different shaped container driven by a rotating top disk is considered. Using a spectral collocation method, a code is constructed to obtain the meridional and swirl velocities. In a result of this code, all structural changes on the streamline patterns are observed and the occurring bifurcations are determined. These bifurcations are compared with the bifurcations obtained from topologically.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 194 Published - 2016
Series DTU Compute PHD-2015 379 0909-3192

Year: 2016

## Sixth International Symposium on Bifurcations and Instabilities in Fluid Dynamics (BIFD2015): Foreword

Bar-Yoseph, P. Z., Brøns, M., Gelfgat, A., Oron, A., Tuckerman, L. S. & Wesfreid, J. E. 2016 48, 6, 2 p., 061001

Publication: Research - peer-reviewEditorial – Annual report year: 2016

Hydrodynamic stability is of fundamental importance in fluid dynamics. As a well-established subject of scientific investigation, it continues to attract great interest in the fluid mechanics community. Bifurcations and instabilities are observed in all areas of fundamental and applied fluid dynamics and remain a challenge for experimental, theoretical and computational studies. Examples of prototypical hydrodynamic instabilities are the Rayleigh–Bénard, Taylor–Couette, Bénard–Marangoni, Rayleigh–Taylor, and Kelvin–Helmholtz instabilities. A fundamental understanding of bifurcation patterns requires the identification of mechanisms responsible for the instability. From an applied point of view, such knowledge is also necessary in order to design reliable and efficient industrial processes, such as melting, mixing, crystal growth, coating, and welding. Modeling of instability mechanisms in biological and biomedical devices is currently a very active and rapidly developing area of research with important biotechnological and medical applications, such as biofilm engineering and wound healing. The understanding of symmetry-breaking in hemodynamics could have important consequences for vascular diseases, such as atherosclerotic and vulnerable plaques, abdominal aortic aneurisms, carotid artery disease, and pulmonary embolisms and implications for vascular interventions such as grafting and stenting. The collection of papers in this issue is a selection of the presentations given at the Sixth International Symposium on Instability and Bifurcations in Fluid Dynamics (BIFD) held at the ESPCI, Paris, 15–17 July2015. With four invited and nearly 400 contributed talks, the symposium gave an overview of the state of the art of the field including experimental, theoretical, and computational approaches to convection, effects of magnetic fields, wake flows, rotating flows, and manyother problems. The complete program can be found at the conference website http://bifd2015.sciencesconf.org/.
Original language English 061001 Fluid Dynamics Research 48 6 2 0169-5983 10.1088/0169-5983/48/6/061001 Published - 2016

Year: 2016

## On Various R-duals and the Duality Principle

Stoeva, D. T. & Christensen, O. 2016 84, 4, p. 577-590

Publication: Research - peer-reviewJournal article – Annual report year: 2016

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.
Original language English Integral Equations and Operator Theory 84 4 577-590 0378-620X 10.1007/s00020-016-2283-4 Published - 2016

Year: 2016

## Counterexamples to the B-spline Conjecture for Gabor Frames

Lemvig, J. & Nielsen, K. H. 2016 22, 6, p. 1440–1451 12 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2016

The frame set conjecture for B-splines Bn, n≥2, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form ab=r, where r is a rational number smaller than one and a and b denote the sampling and modulation rates, respectively, has infinitely many pieces, located around b=2,3,…, not belonging to the frame set of the nth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines Bn, n≥2.
Original language English Journal of Fourier Analysis and Applications 22 6 1440–1451 12 1069-5869 10.1007/s00041-016-9462-1 Published - 2016

Year: 2016

## On some Hermite series identities and their applications to Gabor analysis

Lemvig, J. 2016 182, 4, p. 899–912

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We prove some infinite series identities for the Hermite functions. From these identities we disprove the Gabor frame set conjecture for Hermite functions of order (Formula presented.) and (Formula presented.) for (Formula presented.). The results hold not only for Hermite functions, but for two large classes of eigenfunctions of the Fourier transform associated with the eigenvalues (Formula presented.) and i, and the results indicate that the Gabor frame set of all such functions must have a rather complicated structure.
Original language English Monatshefte fuer Mathematik 182 4 899–912 0026-9255 10.1007/s00605-016-0930-0 Published - 2016

Year: 2016

## Exponential estimates of symplectic slow manifolds

Kristiansen, K. U. & Wulff, C. 2016 261, 1, 46 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2016

In this paper we prove the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom for which the error field is exponentially small. We allow for infinitely many fast degrees of freedom. The method we use is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom.
Original language English Journal of Differential Equations 261 1 46 0022-0396 10.1016/j.jde.2016.03.003 Published - 2016

Year: 2016

## Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen, M. S. & Lemvig, J. 2016 22, 1, p. 36-70 35 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler–Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.
Original language English Journal of Fourier Analysis and Applications 22 1 36-70 35 1069-5869 10.1007/s00041-015-9407-0 Published - 2016

Year: 2016

## Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

Jakobsen, M. S. & Lemvig, J. 2016 368, 12, p. 8447-8480

Publication: Research - peer-reviewJournal article – Annual report year: 2016

Original language English American Mathematical Society. Transactions 368 12 8447-8480 0002-9947 10.1090/tran/6594 Published - 2016

Year: 2016

## Estimates of the first Dirichlet eigenvalue from exit time moment spectra

Hurtado, A., Markvorsen, S. & Palmer, V. 2016 365, 3, p. 1603-1632

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung–Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.
Original language English Mathematische Annalen 365 3 1603-1632 0025-5831 10.1007/s00208-015-1316-7 Published - 2016

Year: 2016

## Explicit constructions and properties of generalized shift-invariant systems in L2(R)

Christensen, O., Hasannasabjaldehbakhani, M. & Lemvig, J. 2016 43, 2, p. 443–472

Publication: Research - peer-reviewJournal article – Annual report year: 2016

Generalized shift-invariant (GSI) systems, originally introduced by Hernández et al. and Ron and Shen, provide a common frame work for analysis of Gabor systems, wavelet systems, wave packet systems, and other types of structured function systems. In this paper we analyze three important aspects of such systems. First, in contrast to the known cases of Gabor frames and wavelet frames, we show that for a GSI system forming a frame, the Calderón sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calderón sum is bounded by the lower frame bound and show that under a weak assumption the condition is equivalent with the local integrability condition introduced by Hernández et al. Second, we provide explicit and general constructions of frames and dual pairs of frames having the GSI-structure. In particular, the setup applies to wave packet systems and in contrast to the constructions in the literature, these constructions are not based on characteristic functions in the Fourier domain. Third, our results provide insight into the local integrability condition (LIC).
Original language English Advances in Computational Mathematics 43 2 443–472 1019-7168 10.1007/s10444-016-9492-x Published - 2016

Year: 2016

## On Partition of Unities Generated by Entire Functions and Gabor Frames in L2(Rd) and ℓ2(Zd)

Christensen, O., Kim, H. O. & Kim, R. Y. 2016 22, 5, p. 1121-1140

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We characterize the entire functions P of d variables, d≥2, for which the Zd-translates of Pχ[0,N]d satisfy the partition of unity for some N∈N. In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where P is a trigonometric polynomial, we characterize the maximal smoothness of Pχ[0,N]d, as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in L2(Rd), with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in ℓ2(Zd).
Original language English Journal of Fourier Analysis and Applications 22 5 1121-1140 1069-5869 10.1007/s00041-015-9450-x Published - 2016

Year: 2016

## On the Gabor frame set for compactly supported continuous functions

Christensen, O., Kim, H. O. & Kim, R. Y. 2016 2016, 17 p., 94

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines BN, N≥2, and certain ‘continuous and truncated’ versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.
Original language English 94 Journal of Inequalities and Applications 2016 17 1025-5834 10.1186/s13660-016-1021-4 Published - 2016

Year: 2016

## From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

Burke, J., Desroches, M., Granados, A., Kaper, T. J., Krupa, M. & Vo, T. 2016 26, 2, p. 405-451 47 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010).
Original language English Journal of Nonlinear Science 26 2 405-451 47 0938-8974 10.1007/s00332-015-9279-0 Published - 2016

Year: 2016

## Approximation by planar elastic curves

Brander, D., Gravesen, J. & Nørbjerg, T. B. 2016 19 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven optimization is then used to find the approximating elastic curve.
Original language English Advances in Computational Mathematics 19 1019-7168 10.1007/s10444-016-9474-z Published - 2016

Year: 2016

## Spherical Surfaces

Brander, D. 2016 25, 3, p. 257-272

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the geometric Cauchy data for the generic singularities, as well as for the cuspidal beaks and cuspidal butterfly singularities. We consider the bifurcations of generic one parameter families of spherical fronts and provide evidence that suggests that these are the cuspidal beaks, cuspidal butterfly and one other singularity. We also give the loop group potentials for spherical surfaces with finite order rotational symmetries and for surfaces with embedded isolated singularities.
Original language English Experimental Mathematics 25 3 257-272 1058-6458 10.1080/10586458.2015.1077359 Published - 2016

Year: 2016

## Wavelets for Non-expanding Dilations and the Lattice Counting Estimate

Bownik, M. & Lemvig, J. 2016 2017, 23, p. 7264-7291 28 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2016

We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the powers of a dilation matrix A∈GL(n,R). This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically. We show that the lattice counting estimate holds for all dilations A with |detA|≠1 and for almost every lattice Γ with respect to the invariant probability measure on the set of lattices. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and almost every choice of a dilation A with respect to the Haar measure on GL(n,R).
Original language English International Mathematics Research Notices 2017 23 7264-7291 28 1073-7928 10.1093/imrn/rnw222 Published - 2016

Year: 2016

## Autorotation: Invited Comment

Bohr, J. & Markvorsen, S. 2016 In : Physica Scripta. 91, 2, 9 p., 023005

Publication: Research - peer-reviewJournal article – Annual report year: 2016

A continuous autorotation vector field along a framed space curve is defined, which describes the rotational progression of the frame. We obtain an exact integral for the length of the autorotation vector. This invokes the infinitesimal rotation vector of the frame progression and the unit vector field for the corresponding autorotation vector field. For closed curves we define an autorotation number whose integer value depends on the starting point of the curve. Upon curve deformations, the autorotation number is either constant, or can make a jump of (multiples of) plus-minus two, which corresponds to a change in rotation of multiples of 4π. The autorotation number is therefore not topologically conserved under all transformations. We discuss this within the context of generalised inflection points and of frame revisit points. The results may be applicable to physical systems such as polymers, proteins, and DNA. Finally, turbulence is discussed in the light of autorotation, as is the Philippine wine dance, the Dirac belt trick, and the 4π cycle of the flying snake.
Original language English 023005 Physica Scripta 91 2 9 0281-1847 10.1088/0031-8949/91/2/023005 Published - 2016

Year: 2016

## The structure of dual Grassmann codes

Beelen, P. & Pinero, F. 2016 79, 3, p. 451-470

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classification the lines lying on the Grassmannian variety play a central role. Related codes, namely the affine Grassmann codes, were introduced more recently in Beelen et al. (IEEE Trans Inf Theory 56(7):3166–3176, 2010), while their duals were introduced and studied in Beelen et al. (IEEE Trans Inf Theory 58(6):3843–3855, 2010). In this paper we also classify and count the minimum weight codewords of the dual affine Grassmann codes. Combining the above classification results, we are able to show that the dual of a Grassmann code is generated by its minimum weight codewords. We use these properties to establish that the increase of value of successive generalized Hamming weights of a dual Grassmann code is 1 or 2.
Original language English Designs, Codes and Cryptography 79 3 451-470 0925-1022 10.1007/s10623-015-0085-3 Published - 2016

Year: 2016

## New basis set for the prediction of the specific rotation in flexible biological molecules

Baranowska-Łaczkowska, A., Z. Łaczkowski, K. Z. Ł., Henriksen, C., Fernandez, B., Kozak, M. & Zielinska, S. 2016 In : RSC Advances. 6, 24, p. 19897-19902

Publication: Research - peer-reviewJournal article – Annual report year: 2016

Using a novel method based on increasingly accurate calculations, we obtain the main conformers of a set of flexible molecules. We then employ the recently developed ORP basis set for calculating the specific rotation of the found set carried out at the TD-DFT level of theory. The results are compared to those obtained with the (d-)aug-cc-pVXZ (X = D, T and Q) basis sets of Dunning et al. The ORP values are in good overall agreement with the aug-cc-pVTZ results making the ORP a good basis set for routine TD-DFT optical rotation calculations of conformationally flexible molecules. The results presented for the investigated chiral azido alcohols are to our knowledge the first estimations of their specific rotations.
Original language English RSC Advances 6 24 19897-19902 2046-2069 10.1039/c5ra20186a Published - 2016

Year: 2016

## Bifurcations of a creeping air–water flow in a conical container

Balci, A., Brøns, M., Herrada, M. A. & Shtern, V. N. 2016 30, 5, p. 485-496

Publication: Research - peer-reviewJournal article – Annual report year: 2016

This numerical study describes the eddy emergence and transformations in a slow steady axisymmetric air–water flow, driven by a rotating top disk in a vertical conical container. As water height (Formula presented.) and cone half-angle (Formula presented.) vary, numerous flow metamorphoses occur. They are investigated for (Formula presented.), and (Formula presented.). For small (Formula presented.), the air flow is multi-cellular with clockwise meridional circulation near the disk. The air flow becomes one cellular as (Formula presented.) exceeds a threshold depending on (Formula presented.). For all (Formula presented.), the water flow has an unbounded number of eddies whose size and strength diminish as the cone apex is approached. As the water level becomes close to the disk, the outmost water eddy with clockwise meridional circulation expands, reaches the interface, and induces a thin layer with anticlockwise circulation in the air. Then this layer expands and occupies the entire air domain. The physical reasons for the flow transformations are provided. The results are of fundamental interest and can be relevant for aerial bioreactors.
Original language English Theoretical and Computational Fluid Dynamics 30 5 485-496 0935-4964 10.1007/s00162-016-0391-z Published - 2016

Year: 2016

## Patterns of a slow air-water flow in a semispherical container

Balci, A., Brøns, M., Herrada, M. A. & Shtern, V. N. 2016 58, p. 1-8

Publication: Research - peer-reviewJournal article – Annual report year: 2016

This numerical study analyzes the development of eddies in a slow steady axisymmetric air-water flow in a sealed semispherical container, driven by a rotating top disk. As the water height, Hw, increases, new flow cells emerge in both water and air. First, an eddy emerges near the axis-bottom intersection. Then this eddy expands and reaches the interface, inducing a new cell in the air flow. This cell appears as a thin near-axis layer which then expands and occupies the entire air domain. As the disk rotation intensifies at Hw = 0.8, the new air cell shrinks to the axis and disappears. The bulk water circulation becomes separated from the interface by a thin layer of water counter-circulation. These changes in the flow topology occur due to (a) competing effects of the air meridional flow and swirl, which drive meridional motions of opposite directions in water, and (b) feedback of water flow on the air flow. In contrast to flows in cylindrical and conical containers, there is no interaction with Moffatt corner vortices here.
Original language English European Journal of Mechanics B - Fluids 58 1-8 0997-7546 10.1016/j.euromechflu.2016.03.004 Published - 2016

Year: 2016

## On the Number of Rational Points on Prym Varieties over Finite Fields

Aubry, Y. & Haloui, S. 2016 58, 1, p. 55-68

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.
Original language English Glasgow Mathematical Journal 58 1 55-68 0017-0895 10.1017/S0017089515000063 Published - 2016

2015

Year: 2015

## A flow meter for ultrasonically measuring the flow velocity of fluids.

Willatzen, M., Gravesen, J. & Nørtoft, P. 7 May 2015

Publication: ResearchPatent – Annual report year: 2015

The invention regards a flow meter for ultrasonically measuring the flow velocity of fluids comprising a duct having a flow channel with an internal cross section comprising variation configured to generate at least one acoustic resonance within the flow channel for a specific ultrasonic frequency, and at least two transducers for generating and sensing ultrasonic pulses, configured to transmit ultrasonic pulses at least at said specific ultrasonic frequency into the flow channel such that the ultrasonic pulses propagate through a fluid flowing in the flow channel, wherein the flow meter is configured to determine the flow velocity of the fluid flowing in the flow channel based on a change in transit time, phase and/or pulse such as amplitude and/or form, of the ultrasonic pulses.

Original language English G01F 1/ 66 A I WO2015063079 07/05/2015 International Bureau of the World Intellectual Property Organization (WIPO) Published - 7 May 2015

### Bibliographical note

Also published as: EP3063508 (A1)

Year: 2015

## On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes

Anbar, N., Meidl, W. M. & Topuzoglu, A. 2015

Publication: Research - peer-reviewPaper – Annual report year: 2015

Original language English 2015 10 Published - 2015 9th International Workshop on Coding and Cryptography (WCC 2015) - Paris, France

### Conference

Conference 9th International Workshop on Coding and Cryptography (WCC 2015) 9 France Paris 13/04/2015 → 17/04/2015 Organized by INRIA http://wcc2015.inria.fr/

Year: 2015

## Regularization of Piecewise Smooth Two-Folds

Kristiansen, K. U. & Hogan, S. J. 2015

Publication: Research - peer-reviewPoster – Annual report year: 2015

Original language English 2015 1 Published - 2015 EquaDiff 2015 - Lyon, France

### Conference

Conference EquaDiff 2015 Université Claude Bernard Lyon France Lyon 06/07/2015 → 10/07/2015 http://equadiff2015.sciencesconf.org/

Year: 2015

## Bifurcation Analysis of Structures in a Convection Model

Dam, M., Brøns, M., Rasmussen, J. J. & Naulin, V. 2015

Publication: Research - peer-reviewPoster – Annual report year: 2015

Original language English 2015 1 Published - 2015 XXXV Dynamics Days Europe 2015 - Exeter, United Kingdom

### Conference

Conference XXXV Dynamics Days Europe 2015 35 University of Exeter United Kingdom Exeter 06/09/2015 → 10/09/2015

Year: 2015

## VirtualTable: a projection augmented reality game

Dal Corso, A., Olsen, M. D., Steenstrup, K. H., Wilm, J., Jensen, S. H. N., Paulsen, R. R., Eiríksson, E. R., Nielsen, J. B., Frisvad, J. R., Einarsson, G. & Kjer, H. M. 2015 Proceedings of SIGGRAPH Asia 2015 Posters. Association for Computing Machinery, 1 p. 40

Publication: Research - peer-reviewConference abstract in proceedings – Annual report year: 2015

VirtualTable is a projection augmented reality installation where users are engaged in an interactive tower defense game. The installation runs continuously and is designed to attract people to a table, which the game is projected onto. Any number of players can join the game for an optional period of time. The goal is to prevent the virtual stylized soot balls, spawning on one side of the table, from reaching the cheese. To stop them, the players can place any kind of object on the table, that then will become part of the game. Depending on the object, it will become either a wall, an obstacle for the soot balls, or a tower, that eliminates them within a physical range. The number of enemies is dependent on the number of objects in the field, forcing the players to use strategy and collaboration and not the sheer number of objects to win the game.
Original language English Proceedings of SIGGRAPH Asia 2015 Posters 1 Association for Computing Machinery 2015 40 978-1-4503-3926-1 10.1145/2820926.2820950 Published - 2015 8th ACM SIGGRAPH Conference and Exhibition on Computer Graphics and Interactive Techniques in Asia (2015) - Kobe, Japan

### Conference

Conference 8th ACM SIGGRAPH Conference and Exhibition on Computer Graphics and Interactive Techniques in Asia (2015) 8 Kobe Convention Center Japan Kobe 02/11/2015 → 05/11/2015 http://sa2015.siggraph.org/en/

Year: 2015

## A characterization of tight and dual generalized translation invariant frames

Jakobsen, M. S. & Lemvig, J. 2015 Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015). IEEE Press, p. 96-100

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

We present results concerning generalized translation invariant (GTI) systems on a second countable locally compact abelian group G. These are systems with a family of generators {gj, P}jεJ, pεPJ ⊂ L2(G), where J is a countable index set, and Pj, j ε J are certain measure spaces. Furthermore, for each j we let Γj, be a closed subgroup of G such that G/Γj is compact. A GTI system is then the collection of functions UjεJ{gj, p(· - γ}γεΓj, pεPj. Many well known systems, such as wavelet, shearlet and Gabor systems, both the discrete and continuous types, are GTI systems. We characterize when such systems form tight frames, and when two GTI Bessel systems form dual frames for L2(G). In particular, this offers a unified approach to the theory of discrete and continuous frames and, e.g., yields well known results for discrete and continuous Gabor and wavelet systems.
Original language English Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015) IEEE Press 2015 96-100 978-1-4673-7353-1 10.1109/SAMPTA.2015.7148858 Published - 2015 11th International Conference on Sampling Theory and Applications - Washington DC, United States

### Conference

Conference 11th International Conference on Sampling Theory and Applications 11 United States Washington DC 25/05/2015 → 29/05/2015 http://www.american.edu/cas/mathstat/sampta2015/index.cfm

Year: 2015

## Duality results for co-compact Gabor systems

Jakobsen, M. S. & Lemvig, J. 2015 Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015). IEEE Press, p. 144-147

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

In this paper we give an account of recent developments in the duality theory of Gabor frames. We prove the Wexler-Raz biorthogonality relations and the duality principle for co-compact Gabor systems on second countable, locally compact abelian groups G. Our presentation does not rely on the existence of uniform lattices in G.
Original language English Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015) IEEE Press 2015 144-147 978-1-4673-7353-1 10.1109/SAMPTA.2015.7148868 Published - 2015 11th International Conference on Sampling Theory and Applications - Washington DC, United States

### Conference

Conference 11th International Conference on Sampling Theory and Applications 11 United States Washington DC 25/05/2015 → 29/05/2015 http://www.american.edu/cas/mathstat/sampta2015/index.cfm

Year: 2015

## Linear complexity for multidimensional arrays - a numerical invariant

Gomez-Perez, D., Høholdt, T., Moreno, O. & Rubio, I. 2015 Proceedings of the IEEE International Symposium on Information Theory (ISIT 2015). IEEE, p. 2697-2701

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

Linear complexity is a measure of how complex a one dimensional sequence can be. In this paper we extend the concept of linear complexity to multiple dimensions and present a definition that is invariant under well-orderings of the arrays. As a result we find that our new definition for the process introduced in the patent titled “Digital Watermarking” produces arrays with good asymptotic properties.
Original language English Proceedings of the IEEE International Symposium on Information Theory (ISIT 2015) IEEE 2015 2697-2701 978-1-4673-7704-1 10.1109/ISIT.2015.7282946 Published - 2015 IEEE International Symposium on Information Theory (ISIT 2015) - Hong Kong, Hong Kong

### Conference

Conference IEEE International Symposium on Information Theory (ISIT 2015) Hong Kong Hong Kong 14/06/2015 → 19/06/2015 http://www.isit2015.org/

Year: 2015

## On R-duals and the duality principle

Christensen, O. & Stoeva, D. 2015 Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015). IEEE, p. 352-356

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

In 2004 Casazza, Kutyniok and Lammers introduced the R-duals of sequences in a general Hilbert space. The purpose was to obtain a general version of the duality principle in Gabor analysis. It was shown that the R-duals cover the duality principle for tight Gabor frames and Gabor Riesz bases. In this paper we discuss the relationship between the R-duals and a variant, called R-duals of type III, introduced in 2014. In contrast to the original R-duals, it is known that the R-duals of type III generalize the duality principle for all Gabor frames, but we believe that a smaller and more convenient class will work as well. The purpose of the paper is to give a focussed presentation of the R-duals of type I and III that can trigger the research on this.
Original language English Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015) IEEE 2015 352-356 978-1-4673-7353-1 10.1109/sampta.2015.7148911 Published - 2015 11th International Conference on Sampling Theory and Applications - Washington DC, United States

### Conference

Conference 11th International Conference on Sampling Theory and Applications 11 United States Washington DC 25/05/2015 → 29/05/2015 http://www.american.edu/cas/mathstat/sampta2015/index.cfm

Year: 2015

## Surfaces with Natural Ridges

Brander, D. & Markvorsen, S. 2015 Proceedings of Bridges Baltimore 2015: Mathematics, Music, Art, Architecture, Culture. p. 379-382

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

We discuss surfaces with singularities, both in mathematics and in the real world. For many types of mathematical surface, singularities are natural and can be regarded as part of the surface. The most emblematic example is that of surfaces of constant negative Gauss curvature, all of which necessarily have singularities. We describe a method for producing constant negative curvature surfaces with prescribed cusp lines. In particular, given a generic space curve, there is a unique surface of constant curvature K = -1 that contains this curve as a cuspidal edge. This is an effective means to easily generate many new and beautiful examples of surfaces with constant negative curvature.

Original language English Proceedings of Bridges Baltimore 2015 : Mathematics, Music, Art, Architecture, Culture 2015 379-382 Published - 2015 Bridges Baltimore 2015 - Baltimore, Maryland, United States

### Conference

Conference Bridges Baltimore 2015 The University of Baltimore United States Baltimore, Maryland 29/07/2015 → 01/08/2015 http://bridgesmathart.org/

Year: 2015

## Curve Matching with Applications in Medical Imaging

Bauer, M., Bruveris, M., Harms, P. & Møller-Andersen, J. 2015 Proceedings of the 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (MFCA 2015). p. 83-94

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical properties. We propose numerical solutions for the initial and boundary value problems of _nding geodesics. These two methods are combined in a Riemannian gradientbased optimization scheme to compute the Karcher mean. We apply this to a study of the shape variation in HeLa cell nuclei and cycles of cardiac deformations, by computing means and principal modes of variations.
Original language English Proceedings of the 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (MFCA 2015) 2015 83-94 Published - 2015 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (MFCA 2015) - Munich, Germany

### Workshop

Workshop 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (MFCA 2015) 5 Germany Munich 09/10/2015 → … In conjunction with MICCAI 2015. http://www-sop.inria.fr/asclepios/events/MFCA15/

### Bibliographical note

The proceedings of the workshop are available as a collection of open archive papers.

Year: 2015

## Second order elastic metrics on the shape space of curves

Bauer, M., Bruveris, M., Harms, P. & Møller-Andersen, J. 2015 Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV) 2015. Drira, H., Kurtek, S. & Turaga, P. (eds.). BMVA Press, p. 1-11

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. The combination of these algorithms allows to compute Karcher means in a Riemannian gradient-based optimization scheme. Our framework has the advantage that the constants determining the weights of the zero, first, and second order terms of the metric can be chosen freely. Moreover, due to its generality, it could be applied to more general spaces of mapping. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing physical objects.
Original language English Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV) 2015 H. Drira, S. Kurtek, P. Turaga BMVA Press 2015 1-11 1-901725-56-1 Published - 2015 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV) 2015 - Swansea, United Kingdom

### Workshop

Workshop 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV) 2015 1 United Kingdom Swansea 10/09/2015 → … Part of the 26th British Machine Vision Conference (BMVC 2015) http://bmvc2015.swan.ac.uk/?p=2681

Year: 2015

## Twisted Polynomials and Forgery Attacks on GCM

Abdelraheem, M. A. A. M. A., Beelen, P., Bogdanov, A. & Tischhauser, E. W. 2015 Advances in Cryptology – EUROCRYPT 2015: Proceedings of the 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Part 1. v, E. & Fischlin, M. (eds.). Springer, p. 762-786 (Lecture Notes in Computer Science, Vol. 9056).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

Polynomial hashing as an instantiation of universal hashing is a widely employed method for the construction of MACs and authenticated encryption (AE) schemes, the ubiquitous GCM being a prominent example. It is also used in recent AE proposals within the CAESAR competition which aim at providing nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem.

In this paper, we comprehensively address this issue. We propose to use twisted polynomials from Ore rings as forgery polynomials. We show how to construct sparse forgery polynomials with full control over the sets of roots. We also achieve complete and explicit disjoint coverage of the key space by these polynomials. We furthermore leverage this new construction in an improved key recovery algorithm.

As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which is a CAESAR candidate.
Original language English Advances in Cryptology – EUROCRYPT 2015 : Proceedings of the 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Part 1 Elisabeth v, Marc Fischlin Springer 2015 762-786 978-3-662-46799-2 978-3-662-46800-5 10.1007/978-3-662-46800-5_29 Published - 2015 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2015) - Sofia, Bulgaria

### Conference

Conference 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2015) 34 Bulgaria Sofia 26/04/2015 → 30/04/2015 https://www.cosic.esat.kuleuven.be/eurocrypt_2015/
Series Lecture Notes in Computer Science 9056 0302-9743

Year: 2015

## Gabor Analysis for Imaging

Christensen, O., Feichtinger, H. G. & Paukner, S. 2015 Handbook of Mathematical Methods in Imaging. Scherzer, O. (ed.). 2 ed. Springer, Vol. 1, p. 1717-1757

Publication: Research - peer-reviewBook chapter – Annual report year: 2016

In contrast to classical Fourier analysis, time–frequency analysis is concerned with localized Fourier transforms. Gabor analysis is an important branch of time–frequency analysis. Although significantly different, it shares with the wavelet transform methods the ability to describe the smoothness of a given function in a location-dependent way.

The main tool is the sliding window Fourier transform or short-time Fourier transform (STFT) in the context of audio signals. It describes the correlation of a signal with the time–frequency shifted copies of a fixed function (or window or atom). Thus, it characterizes a function by its transform over phase space, which is the time–frequency plane (TF-plane) in a musical context or the location–wave-number domain in the context of image processing.

Since the transition from the signal domain to the phase space domain introduces an enormous amount of data redundancy, suitable subsampling of the continuous transform allows for complete recovery of the signal from the sampled STFT. The knowledge about appropriate choices of windows and sampling lattices has increased significantly during the last three decades. Since the suggestion goes back to the idea of D. Gabor [45], this branch of TF analysis is called Gabor analysis . Gabor expansions are not only of interest due to their very natural interpretation but also algorithmically convenient due to a good understanding of algebraic and analytic properties of Gabor families.

In this chapter, we describe some of the generalities relevant for an understanding of Gabor analysis of functions on Rd. We pay special attention to the case d = 2, which is the most important case for image processing and image analysis applications.

The chapter is organized as follows. Section 2 presents central tools from functional analysis in Hilbert spaces, e.g., the pseudo-inverse of a bounded operator and the central facts from frame theory. In Sect. 3, we introduce several operators that play important roles in Gabor analysis. Gabor frames on L2(Rd) are introduced in Sect. 4, and their discrete counterpart are treated in Sect. 5. Finally, the application of Gabor expansions to image representation is considered in Sect. 6.
Original language English Handbook of Mathematical Methods in Imaging Otmar Scherzer 1 Springer 2015 2 1717-1757 978-1-4939-0789-2 978-1-4939-0790-8 10.1007/978-1-4939-0790-8_54 Published - 2015

Year: 2015

## Mathematical modelling of membrane separation

Vinther, F., Brøns, M. & Meyer, A. S. 2015 Kgs. Lyngby: Technical University of Denmark (DTU). 130 p. (DTU Compute PHD-2013; No. 320).

Publication: ResearchPh.D. thesis – Annual report year: 2015

This thesis concerns mathematical modelling of membrane separation. The thesis consists of introductory theory on membrane separation, equations of motion, and properties of dextran, which will be the solute species throughout the thesis. Furthermore, the thesis consist of three separate mathematical models, each with a different approach to membrane separation.

The first model is a statistical model investigating the interplay between solute shape and the probability of entering the membrane. More specific the transition of solute particles from being spherical to becoming more elongated as prolate ellipsoids with the same volume. The porous membrane is assumed isotropic such that the model reduces to a two dimensional model. With this assumption ellipsoids with the same volume reduces to ellipses with the same area. The model finds the probability of entering the pore of the membrane. It is found that the probability of entering the pore is highest when the largest of the radii in the ellipse is equal to half the radius of the pore, in case of molecules with circular radius less than the pore radius. The results are directly related to the macroscopic distribution coefficient and the rejection coefficient.

The second model is a stationary model for the flux of solvent and solute in a hollow fibre membrane. In the model we solve the time independent equations for transport of solvent and solute within the hollow fibre. Furthermore, the flux of solute and solvent through the membrane is coupled through the boundary conditions. The model investigates how the true and observed rejection coefficient depends on the transmembrane pressure, the average inlet velocity, and the molecular weight. Furthermore, the effect of concentration dependent viscosity on the rejection coefficients is investigated. The results show that the true rejection coefficient is increasing as a function of increasing transmembrane pressure, increasing inlet velocity, and decreasing molecular weight. Furthermore, it is found that a concentration dependent viscosity decreases the true rejection. The observed rejection is increasing for decreasing molecular weight and increasing inlet velocities. The observed rejection can be either increasing or decreasing as a function of increasing transmembrane pressure. Moreover, the observed rejection is reduced when the viscosity depends on the concentration. The study is a time dependent model of back-shocking. During back-shocking the pressure difference across the membrane is reversed for a given time. This implies that the concentration polarization at the membrane surface is flushed away. When the pressure is reversed back to normal the membrane performs better resulting in an increased average flux. Two models models of the problem was made.

In a two dimensional model, limited to capture the dynamics close to the membrane, a positive effect was observed on both the observed rejection and the average solvent flux. Furthermore, an analytical upper estimate for the optimal back-shock time is given. In a three dimensional model, where the flow within the entire hollow fibre is modelled, the mentioned upper estimate is used to obtain a positive effect on both the observed rejection and the average solvent flux. Moreover, the effect of a concentration dependent viscosity was investigated. It was found that the average flux compared to the steady-state solution increased when the viscosity depends on the concentration.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 130 Published - 2015
Series DTU Compute PHD-2013 320 0909-3192

Year: 2015

## An algebraic approach to graph codes

Pinero, F. & Beelen, P. 2015 Kgs. Lyngby: Technical University of Denmark (DTU). 96 p. (DTU Compute PHD-2014; No. 352).

Publication: ResearchPh.D. thesis – Annual report year: 2015

This thesis consists of six chapters. The first chapter, contains a short introduction to coding theory in which we explain the coding theory concepts we use. In the second chapter, we present the required theory for evaluation codes and also give an example of some fundamental codes in coding theory as evaluation codes. Chapter three consists of the introduction to graph based codes, such as Tanner codes and graph codes. In Chapter four, we compute the dimension of some graph based codes with a result combining graph based codes and subfield subcodes. Moreover, some codes in chapter four are optimal or best known for their parameters. In chapter five we study some graph codes with Reed–Solomon component codes. The underlying graph is well known and widely used for its good characteristics. This helps us to compute the dimension of the graph codes. We also introduce a combinatorial concept related to the iterative encoding of graph codes with MDS component code. The last chapter deals with affine Grassmann codes and Grassmann codes. We begin with some previously known codes and prove that they are also Tanner codes of the incidence graph of the point–line partial geometry of the Grassmannian. We expect that the techniques exposed in chapter six are also applicable to other codes as well.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 96 Published - 2015
Series DTU Compute PHD-2014 352 0909-3192

Year: 2015

## Good towers of function Fields

Nguyen, N. & Beelen, P. 2015 Kgs. Lyngby: Technical University of Denmark (DTU). 125 p. (DTU Compute PHD-2015; No. 394).

Publication: ResearchPh.D. thesis – Annual report year: 2015

Algebraic curves are used in many different areas, including error-correcting codes. In such applications, it is important that the algebraic curve C meets some requirements. The curve must be defined over a finite field GF(q) with q elements, and then the curve also should have many points over this field. There are limits on how many points N(C) an algebraic curve C defined over a finite field can have.

An invariant of the curve which is important in this context is the curve’s genus g(C). Hasse and Weil proved that N(C)≤q+1+2g(C) √q and this bound can in general not be improved. However if the genus is large compared with q, the bound can be improved. Drinfeld and Vladut showed the asymptotic result:

A(q)≔limsup ( N(C)/ g(C)→∞ g(C)) ≤ √q-1.

The quantity A(q) is called Ihara’s constant. If q is a square, it is known that A(q)=√q-1, while the value of the A(q) is unknown for all other values of q.

In this thesis, we study a construction using Drinfeld modules that produces explicitly defined families of algebraic curves that asymptotically achieve Ihara’s constant. Such families of curves can also be described using towers of function fields. Restated in this language the aim of the project is to find good and optimal towers. Using the theory of Drinfeld modules and computer algebraic techniques, some new examples of good towers are obtained. We analyse towers of Drinfeld modular curves describing certain equivalence classes of rank 2 Drinfeld modules. Using rank 3 Drinfeld modules further examples of good towers are produced.

Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 125 Published - 2015
Series DTU Compute PHD-2015 394 0909-3192

Year: 2015

## Theoretical Study of the Pyridine-Helium van der Waals Complexes

v, H., Henriksen, C. & Fernandez, B. 2015 119, 44, p. 10999-11006

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this study we evaluate a high-level ab initio ground-state intermolecular potential-energy surface for the pyridine–He van der Waals complex, using the CCSD(T) method and Dunning’s augmented correlation consistent polarized valence double-ζ basis set extended with a set of 3s3p2d1f1g midbond functions. The potential is characterized by two symmetric global minima of −93.2 cm–1 that correspond to geometries where the distance between the helium atom and the pyridine center of mass is 3.105 Å and the angle with respect to the pyridine c rotational axis is 3.9°. Six local minima can be observed for geometries with the helium atom in the plane cotaining the pyridine molecule. To further analyze the nature of the intermolecular interactions in the complex, we use symmetry-adapted perturbation theory (SAPT). Additional consideration of the pyridine–He2 complex provides a better insight into many-body nonadditive contributions to intermolecular interactions in systems with more helium atoms.
Original language English Journal of Physical Chemistry Part A: Molecules, Spectroscopy, Kinetics, Environment and General Theory 119 44 10999-11006 1089-5639 10.1021/acs.jpca.5b08492 Published - 2015

Year: 2015

## On R-Duals and the Duality Principle in Gabor Analysis

Stoeva, D. T. & Christensen, O. 2015 21, 2, p. 383-400

Publication: Research - peer-reviewJournal article – Annual report year: 2015

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et al., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames.
Original language English Journal of Fourier Analysis and Applications 21 2 383-400 1069-5869 10.1007/s00041-014-9376-8 Published - 2015

Year: 2015

## Isogeometric analysis of sound propagation through laminar flow in 2-dimensional ducts

Nørtoft, P., Gravesen, J. & Willatzen, M. 2015 284, p. 1098-1119

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We consider the propagation of sound through a slowly moving fluid in a 2-dimensional duct. A detailed description of a flow-acoustic model of the problem using B-spline based isogeometric analysis is given. The model couples the non-linear, steady-state, incompressible Navier-Stokes equation in the laminar regime for the flow field, to a linear, time-harmonic acoustic equation in the low Mach number regime for the sound signal. B-splines are used both to represent the duct geometry and to approximate the flow and sound fields. This facilitates an exact representation of complex duct geometries, as well as high continuity approximations of state variables. Acoustic boundary conditions on artificial truncation boundaries are treated using a mode matching formulation. We validate the model against known acoustic modes for a uniform flow through a straight duct. Improved error convergence rates are found when the acoustic pressure is approximated by higher order polynomials. Based on the model, we examine how the acoustic signal varies with sound frequency, flow speed and duct geometry. A combination of duct geometry and sound frequency is identified for which the acoustic signal is particularly sensitive to the flow speed.
Original language English Computer Methods in Applied Mechanics and Engineering 284 1098-1119 0045-7825 10.1016/j.cma.2014.11.035 Published - 2015

Year: 2015

## Sub-quadratic decoding of one-point hermitian codes

Nielsen, J. S. R. & Beelen, P. 2015 61, 6, p. 3225-3240

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami-Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the matrix minimization algorithms from computer algebra, yielding similar asymptotic complexities.
Original language English IEEE Transactions on Information Theory 61 6 3225-3240 0018-9448 10.1109/TIT.2015.2424415 Published - 2015

### Bibliographical note

Copyright 2015 IEEE. IEEE. Personal use of this material is permitted. Permission
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Year: 2015

## Perturbative Semiclassical Trace Formulae for Harmonic Oscillators

Møller-Andersen, J. & Ögren, M. 2015 75, 3, p. 359-382 24 p.

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this article we extend previous semiclassical studies by including more general perturbative potentials of the harmonic oscillator in arbitrary spatial dimensions. Our starting point is a radial harmonic potential with an arbitrary even monomial perturbation, which we use to study the resulting U(D) to O(D) symmetry breaking. We derive the gross structure of the semiclassical spectrum from periodic orbit theory, in the form of a perturbative (ħ → 0) trace formula. We then show how to apply the results to even-order polynomial potentials, possibly including mean-field terms. We have drawn the conclusion that the gross structure of the quantum spectrum is determined from only classical circular and diameter orbits for this class of systems.
Original language English Reports on Mathematical Physics 75 3 359-382 24 0034-4877 10.1016/S0034-4877(15)30011-2 Published - 2015

Year: 2015

Markvorsen, S. 2015 28, p. 208-228

Publication: Research - peer-reviewJournal article – Annual report year: 2015

One of the finest and most powerful assets of Finsler geometry is its ability to model, describe, and analyze in precise geometric terms an abundance of physical phenomena that are genuinely asymmetric, see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9]. In this paper we show how wildfires can be naturally included into this family. Specifically we show how the celebrated and much applied Richards’ equations for the large scale elliptic wildfire spreads have a rather simple Finsler-geometric formulation. The general Finsler framework can be explicitly ‘integrated’ to provide detailed - and curvature sensitive - geodesic solutions to the wildfire spread problem. The methods presented here stem directly from first principles of 2-dimensional Finsler geometry, and they can be readily extracted from the seminal monographs [10] and [11], but we will take special care to introduce and exemplify the necessary framework for the implementation of the geometric machinery into this new application - not least in order to facilitate and support the dialog between geometers and the wildfire modelling community. The ‘integration’ part alluded to above is obtained via the geodesics of the ensuing Finsler metric which represents the local fire templates. The ‘paradigm’ part of the present proposal is thus concerned with the corresponding shift of attention from the actual fire-lines to consider instead the geodesic spray - the ‘fire-particles’ - which together, side by side, mold the fire-lines at each instant of time and thence eventually constitute the local and global structure of the wildfire spread.
Original language English Nonlinear Analysis: Real World Applications 28 208-228 1468-1218 10.1016/j.nonrwa.2015.09.011 Published - 2015

Year: 2015

## Computation of saddle-type slow manifolds using iterative methods

Kristiansen, K. U. 2015 14, 2, p. 1189–1227

Publication: Research - peer-reviewJournal article – Annual report year: 2015

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, which require mesh refinements to ensure uniform convergence with respect to , appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples, including a model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables, and the computation of homoclinic connections in the FitzHugh–Nagumo system.
Original language English S I A M Journal on Applied Dynamical Systems 14 2 1189–1227 1536-0040 10.1137/140961948 Published - 2015

Year: 2015

## On the use of blowup to study regularizations of singularities of piecewise smooth dynamical systems in R^3

Kristiansen, K. U. & Hogan, S. J. 2015 14, 1, p. 382–422

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this paper we use the blowup method of Dumortier and Roussarie, in the formulation due to Krupa and Szmolyan, to study the regularization of singularities of piecewise smooth dynamical systems in R3. Using the regularization method of Sotomayor and Teixeira, we first demonstrate the power of our approach by considering the case of a fold line. We quickly extend a main result of Reves and Seara in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain nonresonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.
Original language English S I A M Journal on Applied Dynamical Systems 14 1 382–422 1536-0040 10.1137/140980995 Published - 2015

Year: 2015

## Periodic orbits near a bifurcating slow manifold

Kristiansen, K. U. 2015 259, 9, p. 4561–4614

Publication: Research - peer-reviewJournal article – Annual report year: 2015

This paper studies a class of $1\frac12$-degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of $\ln^2\epsilon^{-1}$-many periodic orbits that all stay within an $\mathcal O(\epsilon^{1/3})$-distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation. Here $\epsilon\ll 1$ measures the time scale separation. These periodic orbits are predominantly unstable. The proof is based on averaging of two blowup systems, allowing one to estimate the effect of the singularity, combined with results on asymptotics of the second Painleve equation. The stable orbits of smallest amplitude that are {persistently} obtained by these methods remain slightly further away from the slow manifold being distant by an order $\mathcal O(\epsilon^{1/3}\ln^{1/2}\ln \epsilon^{-1})$.
Original language English Journal of Differential Equations 259 9 4561–4614 0022-0396 10.1016/j.jde.2015.06.006 Published - 2015

Year: 2015

## Regularizations of two-fold bifurcations in planar piecewise smooth systems using blowup

Kristiansen, K. U. & Hogan, S. J. 2015 14, 4, p. 1731-1786

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We use blowup to study the regularization of codimension one two-fold singularities in planar piecewise smooth (PWS) dynamical systems. We focus on singular canards, pseudo-equlibria and limit cycles that can occur in the PWS system. Using the regularization of Sotomayor and Teixeira [30], we show rigorously how singular canards can persist and how the bifurcation of pseudo-equilibria is related to bifurcations of equilibria in the regularized system. We also show that PWS limit cycles are connected to Hopf bifurcations of the regularization. In addition, we show how regularization can create another type of limit cycle that does not appear to be present in the original PWS system. For both types of limit cycle, we show that the criticality of the Hopf bifurcation that gives rise to periodic orbits is strongly dependent on the precise form of the regularization. Finally, we analyse the limit cycles as locally unique families of periodic orbits of the regularization and connect them, when possible, to limit cycles of the PWS system. We illustrate our analysis with numerical simulations and show how the regularized system can undergo a canard explosion phenomenon
Original language English S I A M Journal on Applied Dynamical Systems 14 4 1731-1786 1536-0040 10.1137/15M1009731 Published - 2015

Year: 2015

## Density and duality theorems for regular Gabor frames

Jakobsen, M. S. & Lemvig, J. 2015 270, 1, p. 229-263

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We investigate Gabor frames on locally compact abelian groups with time–frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the “size” of the subgroup. From these density results we are able to extend the classical Wexler–Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time–frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new.
Original language English Journal of Functional Analysis 270 1 229-263 0022-1236 10.1016/j.jfa.2015.10.007 Published - 2015

Year: 2015

## The Metric of Colour Space

Gravesen, J. 2015 82, November, p. 77-86

Publication: Research - peer-reviewJournal article – Annual report year: 2015

The space of colours is a fascinating space. It is a real vector space, but no matter what inner product you put on the space the resulting Euclidean distance does not correspond to human perception of difference between colours.

In 1942 MacAdam performed the first experiments on colour matching and found the MacAdam ellipses which are often interpreted as defining the metric tensor at their centres. An important question is whether it is possible to define colour coordinates such that the Euclidean distance in these coordinates correspond to human perception.

Using cubic splines to represent the colour coordinates and an optimisation approach we find new colour coordinates that make the MacAdam ellipses closer to uniform circles than the existing standards.
Original language English Graphical Models 82 November 77-86 1524-0703 10.1016/j.gmod.2015.06.005 Published - 2015

Year: 2015

## Ends, fundamental tones and capacity of minimal submanifolds via extrinsic comparison theory

Gimeno, V. & Markvorsen, S. 2015 42, 4, p. 749-774

Publication: Research - peer-reviewJournal article – Annual report year: 2014

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.
Original language English Potential Analysis 42 4 749-774 0926-2601 10.1007/s11118-014-9456-z Published - 2015

Year: 2015

## Directional Time-frequency Analysis via Continuous Frames

Christensen, O., Forster, B. & Massopust, P. 2015 92, 2, p. 268-281

Publication: Research - peer-reviewJournal article – Annual report year: 2015

Grafakos and Sansing [‘Gabor frames and directional time–frequency analysis’, Appl. Comput. Harmon. Anal.25 (2008), 47–67] have shown how to obtain directionally sensitive time–frequency decompositions in L2(Rn) based on Gabor systems in L2(R). The key tool is the ‘ridge idea’, which lifts a function of one variable to a function of several variables. We generalise their result in two steps: first by showing that similar results hold starting with general frames for L2(R), in the settings of both discrete frames and continuous frames, and second by extending the representations to Sobolev spaces. The first step allows us to apply the theory to several other classes of frames, for example wavelet frames and shift-invariant systems, and the second one significantly extends the class of examples and applications. We consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretise the representations using ϵ-nets.
Original language English Australian Mathematical Society. Bulletin 92 2 268-281 0004-9727 10.1017/S0004972715000337 Published - 2015

Year: 2015

## Fourier-like frames on locally compact abelian groups

Christensen, O. & Goh, S. S. 2015 192, p. 82-101

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We consider a class of functions, defined on a locally compact abelian group by letting a class of modulation operators act on a countable collection of functions. We derive sufficient conditions for such a class of functions to form a Bessel sequence or a frame and for two such systems to be dual frames. Explicit constructions are obtained via various generalizations of the classical B-splines to the setting of locally compact abelian groups. (C) 2014 Elsevier Inc. All rights reserved.
Original language English Journal of Approximation Theory 192 82-101 0021-9045 10.1016/j.jat.2014.11.002 Published - 2015

Year: 2015

## On extensions of wavelet systems to dual pairs of frames

Christensen, O., Kim, H. O. & Kim, R. Y. 2015 42, 2, p. 489-503

Publication: Research - peer-reviewJournal article – Annual report year: 2016

It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.
Original language English Advances in Computational Mathematics 42 2 489-503 1019-7168 10.1007/s10444-015-9432-1 Published - 2015

Year: 2015

## On Gabor frames generated by sign-changing windows and B-splines

Christensen, O., Kim, H. O. & Kim, R. Y. 2015 39, 3, p. 534-544

Publication: Research - peer-reviewJournal article – Annual report year: 2015

For a class of compactly supported windows we characterize the frame property for a Gabor system {EmbTnag}m,nZ, for translation parameters a belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of "curves." For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a,b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gröchenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain "hyperbolic strip."
Original language English Applied and Computational Harmonic Analysis 39 3 534-544 1063-5203 10.1016/j.acha.2015.02.006 Published - 2015

Year: 2015

## Mixed-Mode Oscillations Due to a Singular Hopf Bifurcation in a Forest Pest Model

Brøns, M., Desroches, M. & Krupa, M. 2015 22, 2, p. 71-79

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In a forest pest model, young trees are distinguished from old trees. The pest feeds on old trees. The pest grows on a fast scale, the young trees on an intermediate scale, and the old trees on a slow scale. A combination of a singular Hopf bifurcation and a “weak return” mechanism, characterized by a small change in one of the variables, determines the features of the mixed-mode oscillations. Period-doubling and saddle-node bifurcations lead to closed families (called isolas) of periodic solutions in a bifurcation corresponding to a singular Hopf bifurcation.
Original language English Mathematical Population Studies 22 2 71-79 0889-8480 10.1080/08898480.2014.925344 Published - 2015

Year: 2015

## Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential

Brander, D. & Dorfmeister, J. 2015 XIV, 2, p. 645-675

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space E3 which preserve the Hopf differential. We prove that, given a CMC H surface f, either minimal or not, and a fixed basepoint z0 on this surface, there is a naturally defined family fh, for all real h, of CMC h surfaces that are tangent to f at z0, and which have the same Hopf differential. Given the classical Weierstrass data for a minimal surface, we give an explicit formula for the generalized Weierstrass data for the non-minimal surfaces fh, and vice versa. As an application, we use this to give a well-defined dressing action on the class of minimal surfaces. In addition, we show that symmetries of certain types associated with the basepoint are preserved under the deformation, and this gives a canonical choice of basepoint for surfaces with symmetries. We use this to define new examples of non-minimal CMC surfaces naturally associated to known minimal surfaces with symmetries.
Original language English Annali della Scuola Normale Superiore di Pisa - Classe di Scienze XIV 2 645-675 0391-173X 10.2422/2036-2145.201302_012 Published - 2015

Year: 2015

## Extending the zero-derivative principle for slow–fast dynamical systems

Benoît, E., Brøns, M., Desroches, M. & Krupa, M. 2015 66, 5, p. 2255-2270

Publication: Research - peer-reviewJournal article – Annual report year: 2015

Slow–fast systems often possess slow manifolds, that is invariant or locally invariant sub-manifolds on which the dynamics evolves on the slow time scale. For systems with explicit timescale separation, the existence of slow manifolds is due to Fenichel theory, and asymptotic expansions of such manifolds are easily obtained. In this paper, we discuss methods of approximating slow manifolds using the so-called zero-derivative principle. We demonstrate several test functions that work for systems with explicit time scale separation including ones that can be generalized to systems without explicit timescale separation. We also discuss the possible spurious solutions, known as ghosts, as well as treat the Templator system as an example.
Original language English Zeitschrift fuer Angewandte Mathematik und Physik 66 5 2255-2270 0044-2275 10.1007/s00033-015-0552-8 Published - 2015

Year: 2015

## Linear codes associated to determinantal varieties

Beelen, P., Ghorpade, S. R. & Hasan, S. U. 2015 338, 8, p. 1493-1500

Publication: Research - peer-reviewJournal article – Annual report year: 2015

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2×2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number of matrices of rank 1 in a linear space of matrices of a given dimension over a finite field. In particular, we determine the structure and the maximum possible dimension of linear spaces of matrices in which every nonzero matrix has rank 1.
Original language English Discrete Mathematics 338 8 1493-1500 0012-365X 10.1016/j.disc.2015.03.009 Published - 2015

Year: 2015

## Good families of Drinfeld modular curves

Bassa, A., Beelen, P. & Nguyen, N. 2015 18, 1, p. 699-712

Publication: Research - peer-reviewJournal article – Annual report year: 2016

In this paper, we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach for obtaining explicit defining equations for some of these towers and, in particular, give a new explicit example of an optimal tower over a quadratic finite field.
Original language English London Mathematical Society. Journal of Computation and Mathematics 18 1 699-712 1461-1570 10.1112/S146115701500025X Published - 2015

Year: 2015

## Towers of Function Fields over Non-prime Finite Fields

Bassa, A., Beelen, P., Garcia, A. & Stichtenoth, H. 2015 15, 1, p. 1-29

Publication: Research - peer-reviewJournal article – Annual report year: 2015

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara’s quantity A(ℓ), for ℓ = pn with p prime and n > 3 odd. We relate the explicit equations to Drinfeld modular varieties.
Original language English Moscow Mathematical Journal 15 1 1-29 1609-3321 Published - 2015

Year: 2015

## Codimension three bifurcation of streamline patterns close to a no-slip wall: A topological description of boundary layer eruption

Balci, A., Andersen, M., Thompson, M. C. & Brøns, M. 2015 27, 5, 14 p., 053603

Publication: Research - peer-reviewJournal article – Annual report year: 2015

A vortex close to a no-slip wall gives rise to the creation of new vorticity at the wall. This vorticity may organize itself into vortices that erupt from the separated boundary layer. We study how the eruption process in terms of the streamline topology is initiated and varies in dependence of the Reynolds number Re. We show that vortex structures are created in the boundary layer for Re around 600, but that these disappear again without eruption unless Re > 1000. The eruption process is topologically unaltered for Re up to 5000. Using bifurcation theory, we obtain a topological phase space for the eruption process, which can account for all observed changes in the Reynolds number range we consider. The bifurcation diagram complements previously analyzes such that the classification of topological bifurcations of flows close to no-slip walls with up to three parameters is now complete.
Original language English 053603 Physics of Fluids 27 5 14 1070-6631 10.1063/1.4921527 Published - 2015

Year: 2015

## Vortex breakdown in a truncated conical bioreactor

Balci, A., Brøns, M., Herrada, M. A. & Shtern, V. N. 2015 47, 6, 26 p., 065503

Publication: Research - peer-reviewJournal article – Annual report year: 2015

This numerical study explains the eddy formation and disappearance in a slow steady axisymmetric air–water flow in a vertical truncated conical container, driven by the rotating top disk. Numerous topological metamorphoses occur as the water height, Hw, and the bottom-sidewall angle, α, vary. It is found that the sidewall convergence (divergence) from the top to the bottom stimulates (suppresses) the development of vortex breakdown (VB) in both water and air. At α = 60°, the flow topology changes eighteen times as Hw varies. The changes are due to (a) competing effects of AMF (the air meridional flow) and swirl, which drive meridional motions of opposite directions in water, and (b) feedback of water flow on AMF. For small Hw, the AMF effect dominates. As Hw increases, the swirl effect dominates and causes VB. The water flow feedback produces and modifies air eddies. The results are of fundamental interest and can be relevant for aerial bioreactors.
Original language English 065503 Fluid Dynamics Research 47 6 26 0169-5983 10.1088/0169-5983/47/6/065503 Published - 2015

2014

Year: 2014

## Bifurcation Analysis and Dimension Reduction of a Predator-Prey Model for the L-H Transition

Dam, M., Brøns, M., Juul Rasmussen, J., Naulin, V. & Xu, G. 2014

Publication: ResearchPoster – Annual report year: 2014

Original language English 2014 1 Published - 2014 51st Culham Plasma Physics Summer School - Abingdon, Oxfordshire, United Kingdom

### Course

Course 51st Culham Plasma Physics Summer School 51 Culham Science Center United Kingdom Abingdon, Oxfordshire 14/07/2014 → 25/07/2014

Year: 2014

## Towards fusion energy as a sustainable energy source: Activities at DTU Physics

Rasmussen, J., Christensen, A. S., Dam, M., Jacobsen, A. S., Jensen, T., Jessen, M., Korsholm, S. B., Leipold, F., Madsen, J., Magnussen, M. L., Naulin, V., Nielsen, A. H., Nielsen, S. K., Nimb, S. R., Juul Rasmussen, J., Salewski, M., Stejner Pedersen, M. & Tophøj, L. E. H. 2014 Abstract Book - DTU Sustain Conference 2014. Kgs. Lyngby: Technical University of Denmark (DTU), 1 p.

Publication: Research - peer-reviewConference abstract in proceedings – Annual report year: 2014

Nuclear fusion – the process from which the Sun derives its energy – holds the potential to become a clean,safe, highly efficient, and virtually inexhaustible energy source for the future. To mimic this process on earth, experimental fusion devices seek to heat gas to millions of degrees (creating a fusion plasma) and to confine it within magnetic fields. Learning how such plasmas behave and can be controlled is a crucial step towards realizing fusion as a sustainable energy source.At the Plasma Physics and Fusion Energy (PPFE) section at DTU Physics, we are exploring these issues,focusing on areas of high priority on the way towards a working fusion power plant. On the theoreticalfront, we are simulating plasma turbulence and transport of heat and particles in fusion plasmas (Fig. 1a). These issues play a key role in determining how the plasma behaves globally and how well it remains confined in the magnetic field of the fusion device. Understanding this is important for optimizing plasmaperformance and for controlling the heat load onto the walls of the confining vessel.Experimentally, we operate equipment to measure key plasma properties in experimental fusion devices such as ASDEX Upgrade in Germany (Fig. 1b+c). Using a technique called collective Thomson scattering(CTS), we can infer the plasma composition and the dynamics of energetic ions in the plasma. Control of these parameters is vital for achieving a high fusion yield in future power plants. We are also designing CTS equipment for the next-step fusion device ITER (Fig. 1d), in which plasma temperatures will exceed 200million C. This machine is currently being built in France in a large international effort to experimentally demonstrate fusion as a viable energy source and pave the way for the first fusion power plant.
Original language English Abstract Book - DTU Sustain Conference 2014 1 Kgs. Lyngby Technical University of Denmark (DTU) 2014 Published - 2014 DTU Sustain Conference 2014 - Lyngby, Denmark

### Conference

Conference DTU Sustain Conference 2014 Technical University of Denmark Denmark Lyngby 17/12/2014 → 17/12/2014 http://www.sustain.dtu.dk/

Year: 2014

## Rotation reversal in a 1D turbulence spreading model

Naulin, V., Juul Rasmussen, J., Dam, M. & Brøns, M. 2014 Proceedings of the 41st EPS Conference on Plasma Physics: Europhysics Conference Abstracts. Vol. 38F, 4 p. P2.067

Publication: Research - peer-reviewConference abstract in proceedings – Annual report year: 2014

Original language English Proceedings of the 41st EPS Conference on Plasma Physics : Europhysics Conference Abstracts 4 38F 2014 P2.067 Published - 2014 41st EPS Conference on Plasma Physics - Berlin, Germany

### Conference

Conference 41st EPS Conference on Plasma Physics 41 Germany Berlin 23/06/2014 → 27/06/2014

Year: 2014

## Planar Parametrization in Isogeometric Analysis

Gravesen, J., Evgrafov, A., Nguyen, D-M. & Nørtoft, P. 2014 Mathematical Methods for Curves and Surfaces: 8th International Conference, MMCS 2012, Oslo, Norway, June 28 – July 3, 2012, Revised Selected Papers. Springer, 189 p. (Lecture Notes in Computer Science, Vol. 8177).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2014

Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis-oriented methods. Their performance is illustrated through a few numerical examples.
Original language English Mathematical Methods for Curves and Surfaces : 8th International Conference, MMCS 2012, Oslo, Norway, June 28 – July 3, 2012, Revised Selected Papers 189 Springer 2014 978-3-642-54381-4 978-3-642-54382-1 10.1007/978-3-642-54382-1_11 Published - 2014 8th International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2012) - Oslo, Norway

### Conference

Conference 8th International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2012) Norway Oslo 28/06/2012 → 03/07/2012 http://heim.ifi.uio.no/~cagd/2012/
Series Lecture Notes in Computer Science 8177 0302-9743

Year: 2014

## Frames and extension problems I

Christensen, O. 2014 Fractals, Wavelets, and their Applications: Contributions from the International Conference and Workshop on Fractals and Wavelets. Springer, p. 219-234 (Springer Proceedings in Mathematics & Statistics, Vol. 92).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

In this article we present a short survey of frame theory in Hilbert spaces. We discuss Gabor frames and wavelet frames and set the stage for a discussion of various extension principles; this will be presented in the article Frames and extension problems II (joint with H.O. Kim and R.Y. Kim).
Original language English Fractals, Wavelets, and their Applications : Contributions from the International Conference and Workshop on Fractals and Wavelets Springer 2014 219-234 978-3-319-08104-5 978-3-319-08105-2 10.1007/978-3-319-08105-2__14 Published - 2014 International Conference and Workshop on Fractals and Wavelets - Kerala, India

### Conference

Conference International Conference and Workshop on Fractals and Wavelets India Kerala 08/11/2013 → 16/11/2013
Series Springer Proceedings in Mathematics & Statistics 92 2194-1009

Year: 2014

## Frames and extension problems II

Christensen, O., Kim, H. O. & Kim, R. Y. 2014 Fractals, Wavelets, and their Applications: Contributions from the International Conference and Workshop on Fractals and Wavelets. Springer, p. 235-243 (Springer Proceedings in Mathematics & Statistics, Vol. 92).

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2015

This article is a follow-up on the article Frames and Extension Problems I. Here we will go into more recent progress on the topic and also present some open problems.
Original language English Fractals, Wavelets, and their Applications : Contributions from the International Conference and Workshop on Fractals and Wavelets Springer 2014 235-243 978-3-319-08104-5 978-3-319-08105-2 10.1007/978-3-319-08105-2_15 Published - 2014 International Conference and Workshop on Fractals and Wavelets - Kerala, India

### Conference

Conference International Conference and Workshop on Fractals and Wavelets India Kerala 08/11/2013 → 16/11/2013
Series Springer Proceedings in Mathematics & Statistics 92 2194-1009

Year: 2014

## Differential Geometry Applied to Rings and Möbius Nanostructures

Lassen, B., Willatzen, M. & Gravesen, J. 2014 Physics of Quantum Rings. Fomin, V. M. (ed.). Springer, Vol. Part III, Ch. 16, p. 409-435 (NanoScience and Technology).

Publication: Research - peer-reviewBook chapter – Annual report year: 2014

Nanostructure shape effects have become a topic of increasing interest due to advancements in fabrication technology. In order to pursue novel physics and better devices by tailoring the shape and size of nanostructures, effective analytical and computational tools are indispensable. In this chapter, we present analytical and computational differential geometry methods to examine particle quantum eigenstates and eigenenergies in curved and strained nanostructures. Example studies are carried out for a set of ring structures with different radii and it is shown that eigenstate and eigenenergy changes due to curvature are most significant for the groundstate eventually leading to qualitative and quantitative changes in physical properties. In particular, the groundstate in-plane symmetry characteristics are broken by curvature effects, however, curvature contributions can be discarded at bending radii above 50 nm. In the second part of the chapter, a more complicated topological structure, the Möbius nanostructure, is analyzed and geometry effects for eigenstate properties are discussed including dependencies on the Möbius nanostructure width, length, thickness, and strain.
Original language English Physics of Quantum Rings Vladimir M. Fomin Part III Springer 2014 409-435 16 978-3-642-39196-5 978-3-642-39197-2 10.1007/978-3-642-39197-2_16 Published - 2014
Series NanoScience and Technology 1434-4904

Year: 2014

## Six problems in frame theory

Christensen, O. 2014 Professor Butzer on the occasion of his eighty-fifth anniversary. Birkhäuser Verlag GmbH, 27 p.

Publication: Research - peer-reviewBook chapter – Annual report year: 2014

We discuss various problems in frame theory that have been open for some years. A short discussion of frame theory is also provided, but it only contains the information that is necessary in order to understand the open problems and their role.
Original language English Professor Butzer on the occasion of his eighty-fifth anniversary 27 Birkhäuser Verlag GmbH 2014 Published - 2014

### Bibliographical note

Dedicated to Professor Butzer on the occasion of his eighty-fifth anniversary.

Year: 2014

## Good Towers of Function Fields

Bassa, A., Beelen, P. & Nguyen, N. 2014 Algebraic Curves and Finite Fields. Walter de Gruyter, (Radon Series on Computational and Applied Mathematics, Vol. 16).

Publication: Research - peer-reviewBook chapter – Annual report year: 2013

In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory). In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any non-prime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields.
Original language English Algebraic Curves and Finite Fields Walter de Gruyter 2014 978-3-11-031791-6 Published - 2014
Series Radon Series on Computational and Applied Mathematics 16 1865-3707

Year: 2014

## Mathematical modelling of dextran filtration through hollow fibre membranes

Vinther, F., Pinelo, M., Brøns, M., Jonsson, G. E. & Meyer, A. S. 2014 125, p. 21-36

Publication: Research - peer-reviewJournal article – Annual report year: 2014

In this paper we present a mathematical model of an ultrafiltration process. The results of the model are produced using standard numerical techniques with Comsol Multiphysics. The model describes the fluid flow and separation in hollow fibre membranes. The flow of solute and solvent within the hollow fibre is modelled by solving the Navier-Stokes equation along with the continuity equation for both the solute and the solvent. The flux of solute and solvent through the membrane are given by the solution diffusion model, since ultrafiltration occurs at high rejections. For a given set of parameters describing the characteristics of the membrane, effect on the observed and the intrinsic rejection of the membrane are investigated for the different working parameters: inlet velocity, molecular weight, and transmembrane pressure. Furthermore, the model investigates the effect of a concentration dependent viscosity. The model shows that both the observed and intrinsic rejection increase when the inlet velocity increases. Moreover, the intrinsic rejection increases as a function of transmembrane pressure, but the observed rejection has a characteristic maximum. Therefore, the observed rejection can either increase or decrease as a function of pressure. The influence of a concentration dependent viscosity is to increase the concentration on the membrane surface. This leads to a decrease in both the observed and the intrinsic rejection, when compared to a constant viscosity. For small values of the solute permeability the concentration dependent viscosity decreases the volumetric flux through the membrane at high pressures. This effect is due to a very high concentration at the membrane surface. The model is related to experimental data. There is a good qualitative and a reasonable quantitative agreement between simulations and experimental data.
Original language English Separation and Purification Technology 125 21-36 1383-5866 10.1016/j.seppur.2014.01.034 Published - 2014

Year: 2014

## Predicting optimal back-shock times in ultrafiltration hollow fibre modules through path-lines

Vinther, F., Pinelo, M., Brøns, M., Jonsson, G. E. & Meyer, A. S. 2014 470, p. 275-293

Publication: Research - peer-reviewJournal article – Annual report year: 2014

This paper presents a two dimensional mathematical model of back-shocking in ultrafiltration. The model investigates the effect of back-shocking on concentration polarization. The model shows a positive effect on both the volumetric flux and the observed rejection when back-shocking is applied as compared to the steady-state solution. Furthermore, the effect of changing different parameters such as inlet velocity, forward and backwards pressure on the back-shock time, the increase in volumetric flux and observed rejection, is presented. Moreover, two analytical estimates for the optimal back-shock time derived from calculating the path-lines during a back-shock cycle are presented. Both of these expressions are in good agreement with the results obtained from the mathematical model and data collected from the literature. Based on this, a simple expression for an optimal back-shock time in a multi-parameter problem is provided.
Original language English Journal of Membrane Science 470 275-293 0376-7388 10.1016/j.memsci.2014.07.031 Published - 2014

Year: 2014

## On the subfield subcodes of Hermitian codes

Pinero, F. & Janwa, H. 2014 70, 1-2, p. 157-173

Publication: Research - peer-reviewJournal article – Annual report year: 2013

We present a fast algorithm using Gröbner basis to compute the dimensions of subfield subcodes of Hermitian codes. With these algorithms we are able to compute the exact values of the dimension of all subfield subcodes up to q ≤ 32 and length up to 215. We show that some of the subfield subcodes of Hermitian codes are at least as good as the previously known codes, and we show the existence of good long codes.
Original language English Designs, Codes and Cryptography 70 1-2 157-173 0925-1022 10.1007/s10623-012-9736-9 Published - 2014

Year: 2014

## Contaminant ingress into multizone buildings: An analytical state-space approach

Parker, S., Coffey, C., Gravesen, J., Kirkpatrick, J., Ratcliffe, K., Lingard, B. & Nally, J. 2014 7, 1, p. 57-71

Publication: Research - peer-reviewJournal article – Annual report year: 2014

The ingress of exterior contaminants into buildings is often assessed by treating the building interior as a single well-mixed space. Multizone modelling provides an alternative way of representing buildings that can estimate concentration time series in different internal locations. A state-space approach is adopted to represent the concentration dynamics within multizone buildings. Analysis based on this approach is used to demonstrate that the exposure in every interior location is limited to the exterior exposure in the absence of removal mechanisms. Estimates are also developed for the short term maximum concentration and exposure in a multizone building in response to a step-change in concentration. These have considerable potential for practical use. The analytical development is demonstrated using a simple two-zone building with an inner zone and a range of existing multizone models of residential buildings. Quantitative measures are provided of the standard deviation of concentration and exposure within a range of residential multizone buildings. Ratios of the maximum short term concentrations and exposures to single zone building estimates are also provided for the same buildings.
Original language English Building Simulation 7 1 57-71 1996-3599 10.1007/s12273-013-0136-5 Published - 2014

Year: 2014

## Pilot-wave hydrodynamics in a rotating frame: Exotic orbits

Oza, A. U., Wind-Willassen, Ø., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2014 26, 17 p., 082101

Publication: Research - peer-reviewJournal article – Annual report year: 2014

We present the results of a numerical investigation of droplets walking on a rotating vibrating fluid bath. The drop’s trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. As the forcing acceleration is progressively increased, stable circular orbits give way to wobbling orbits, which are succeeded in turn by instabilities of the orbital center characterized by steady drifting then discrete leaping. In the limit of large vibrational forcing, the walker’s trajectory becomes chaotic, but its statistical behavior reflects the influence of the unstable orbital solutions. The study results in a complete regime diagram that summarizes the dependence of the walker’s behavior on the system parameters. Our predictions compare favorably to the experimental observations of Harris and Bush [“Droplets walking in a rotating frame: from quantized orbits to multi-modal statistics,” J. Fluid Mech. 739, 444–464 (2014)].
Original language English 082101 Physics of Fluids 26 17 1070-6631 10.1063/1.4891568 Published - 2014

Year: 2014

## Prime tight frames

Lemvig, J., Miller, C. & Okoudjou, K. A. 2014 40, 2, p. 315-334

Publication: Research - peer-reviewJournal article – Annual report year: 2012

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames and use thischaracterization to suggest effective analysis and synthesis computation strategies for such frames. Finally, we describe all prime frames constructed from the spectral tetris method, and, as a byproduct, we obtain a characterization of when the spectral tetris construction works for redundancies below two.
Original language English Advances in Computational Mathematics 40 2 315-334 1019-7168 10.1007/s10444-013-9309-0 Published - 2014

Year: 2014

## Sparse Matrices in Frame Theory

Lemvig, J., Krahmer, F. & Kutyniok, G. 2014 29, 3-4, p. 547-568

Publication: Research - peer-reviewJournal article – Annual report year: 2012

Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.
Original language English Computational Statistics 29 3-4 547-568 0943-4062 10.1007/s00180-013-0446-1 Published - 2014

Year: 2014

## An Iterative Method for the Approximation of Fibers in Slow-Fast Systems

Kristiansen, K. U., Brøns, M. & Starke, J. 2014 13, 2, p. 861–900

Publication: Research - peer-reviewJournal article – Annual report year: 2014

In this paper we extend a method for iteratively improving slow manifolds so that it also can be used to approximate the fiber directions. The extended method is applied to general finite-dimensional real analytic systems where we obtain exponential estimates of the tangent spaces to the fibers. The method is demonstrated on the Michaelis--Menten--Henri model and the Lindemann mechanism. The latter example also serves to demonstrate the method on a slow-fast system in nonstandard slow-fast form. Finally, we extend the method further so that it also approximates the curvature of the fibers.
Original language English S I A M Journal on Applied Dynamical Systems 13 2 861–900 1536-0040 10.1137/120889666 Published - 2014

Year: 2014

## On the sizes of expander graphs and minimum distances of graph codes

Høholdt, T. & Justesen, J. 2014 325, p. 38-46

Publication: Research - peer-reviewJournal article – Annual report year: 2014

We give lower bounds for the minimum distances of graph codes based on expander graphs. The bounds depend only on the second eigenvalue of the graph and the parameters of the component codes. We also give an upper bound on the size of a degree regular graph with given second eigenvalue.
Original language English Discrete Mathematics 325 38-46 0012-365X 10.1016/j.disc.2014.02.005 Published - 2014

Year: 2014

## Optimal codes as Tanner codes with cyclic component codes

Høholdt, T., Pinero, F. & Zeng, P. 2014 76, 1, p. 37-47

Publication: Research - peer-reviewJournal article – Annual report year: 2014

In this article we study a class of graph codes with cyclic code component codes as affine variety codes. Within this class of Tanner codes we find some optimal binary codes. We use a particular subgraph of the point-line incidence plane of A(2,q) as the Tanner graph, and we are able to describe the codes succinctly using Gröbner bases.
Original language English Designs, Codes and Cryptography 76 1 37-47 0925-1022 10.1007/s10623-014-9962-4 Published - 2014

Year: 2014

## Moving least squares simulation of free surface flows

Felter, C. L., Walther, J. H. & Henriksen, C. 2014 91, p. 47-56

Publication: Research - peer-reviewJournal article – Annual report year: 2014

In this paper a Moving Least Squares method (MLS) for the simulation of 2D free surface flows is presented. The emphasis is on the governing equations, the boundary conditions, and the numerical implementation. The compressible viscous isothermal Navier–Stokes equations are taken as the starting point. Then a boundary condition for pressure (or density) is developed. This condition is applicable at interfaces between different media such as fluid–solid or fluid–void. The effect of surface tension is included. The equations are discretized by a moving least squares method for the spatial derivatives and a Runge–Kutta method for the time derivatives. The computational frame is Lagrangian, which means that the computational nodes are convected with the flow. The method proposed here is benchmarked using the standard lid driven cavity problem, a rotating free surface problem, and the simulation of drop oscillations. A new exact solution to the unsteady incompressible Navier–Stokes equations is introduced for the rotating free surface problem. © 2013 Elsevier Ltd. All rights reserved.
Original language English Computers & Fluids 91 47-56 0045-7930 10.1016/j.compfluid.2013.12.006 Published - 2014

Year: 2014

## Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment

Elmegård, M., Krauskopf, B., Osinga, H. M., Starke, J. & Thomsen, J. J. 2014 77, p. 951–966

Publication: Research - peer-reviewJournal article – Annual report year: 2014

A piecewise-linear model with a single degree of freedom is derived from first principles for a driven vertical cantilever beam with a localized mass and symmetric stops. The aim is to show that this model constitutes a considerable step toward developing a vibro-impact model that is able to make qualitative and quantitative predictions of the observed dynamics. The resulting piecewise-linear dynamical system is smoothed by a switching function (nonlinear homotopy). For the chosen smoothing function, it is shown that the smoothing can induce bifurcations in certain parameter regimes. These induced bifurca tions disappear when the transition of the switching is sufficiently and increasingly localized as the impact becomes harder. The bifurcation structure of the impact oscillator response is investigated via the one- and twoparameter continuation of periodic orbits in the driving frequency and/or forcing amplitude. The results are in good agreement with experimental measurements.
Original language English Nonlinear Dynamics 77 951–966 0924-090X 10.1007/s11071-014-1353-x Published - 2014

Year: 2014

## Iso-geometric shape optimization of magnetic density separators

Dang Manh, N., Evgrafov, A., Gravesen, J. & Lahaye, D. 2014 In : Compel. 33, 4, p. 1416-1433

Publication: Research - peer-reviewJournal article – Annual report year: 2014

Purpose
The waste recycling industry increasingly relies on magnetic density separators. These devices generate an upward magnetic force in ferro-fluids allowing to separate the immersed particles according to their mass density. Recently, a new separator design has been proposed that significantly reduces the required amount of permanent magnet material. The purpose of this paper is to alleviate the undesired end-effects in this design by altering the shape of the ferromagnetic covers of the individual poles.

Design/methodology/approach
The paper represents the shape of the ferromagnetic pole covers with B-splines and defines a cost functional that measures the non-uniformity of the magnetic field in an area above the poles. The authors apply an iso-geometric shape optimization procedure, which allows us to accurately represent, analyze and optimize the geometry using only a few design variables. The design problem is regularized by imposing constraints that enforce the convexity of the pole cover shapes and is solved by a non-linear optimization procedure. The paper validates the implementation of the algorithm using a simplified variant of the design problem with a known analytical solution. The algorithm is subsequently applied to the problem posed.

Findings
The shape optimization attains its target and yields pole cover shapes that give rise to a magnetic field that is uniform over a larger domain.

Research limitations/implications
This increased magnetic field uniformity is obtained at the cost of a pole cover shape that differs per pole. This limitation has negligible impact on the manufacturing of the separator. The new pole cover shapes therefore lead to improved performance of the density separation.

Practical implications
Due to the larger uniformity the generated field, these shapes should enable larger amounts of waste to be processed than the previous design.

Originality/value
This paper treats the shapes optimization of magnetic density separators systematically and presents new shapes for the ferromagnetic poles covers.
Original language English Compel 33 4 1416-1433 0332-1649 10.1108/COMPEL-07-2013-0234 Published - 2014

Year: 2014

## A short introduction to frames, Gabor systems, and wavelet systems

Christensen, O. 2014 4, 1, p. 25-39

Publication: Research - peer-reviewJournal article – Annual report year: 2014

In this article we present a short survey of frame theory in Hilbert spaces. We discuss Gabor frames and wavelet frames, and a recent transform that allows to move results from one setting into the other and vice versa.
Original language English Azerbaijan Journal of Mathematics 4 1 25-39 2218-6816 Published - 2014

Year: 2014

## From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa

Christensen, O. & Goh, S. S. 2014 36, 2, p. 198-214

Publication: Research - peer-reviewJournal article – Annual report year: 2014

We discuss an elementary procedure that allows us to construct dual pairs of wavelet frames based on certain dual pairs of Gabor frames and vice versa. The construction preserves tightness of the involved frames. Starting with Gabor frames generated by characteristic functions the construction leads to a class of tight wavelet frames that include the Shannon (orthonormal) wavelet, and applying the construction to Gabor frames generated by certain exponential B-splines yields wavelet frames generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knot sequences. On the other hand, the pendant of the Meyer wavelet turns out to be a tight Gabor frame generated by a C∞(R) function with compact support. As an application of our results we show that for each given pair of bandlimited dual wavelet frames it is possible to construct dual wavelet frames for any desired scaling and translation parameters.
Original language English Applied and Computational Harmonic Analysis 36 2 198-214 1063-5203 10.1016/j.acha.2013.03.005 Published - 2014

Year: 2014

## On entire functions restricted to intervals, partition of unities, and dual Gabor frames

Christensen, O., Kim, H. O. & Kim, R. Y. 2014 38, 1, p. 72-86

Publication: Research - peer-reviewJournal article – Annual report year: 2014

Partition of unities appears in many places in analysis. Typically it is generated by compactly supported functions with a certain regularity. In this paper we consider partition of unities obtained as integer-translates of entire functions restricted to finite intervals. We characterize the entire functions that lead to a partition of unity in this way, and we provide characterizations of the “cut-off” entire functions, considered as functions of a real variable, to have desired regularity. In particular we obtain partition of unities generated by functions with small support and desired regularity. Applied to Gabor analysis this leads to constructions of dual pairs of Gabor frames with low redundancy, generated by trigonometric polynomials with small support and desired regularity.
Original language English Applied and Computational Harmonic Analysis 38 1 72-86 1063-5203 10.1016/j.acha.2014.03.005 Published - 2014

Year: 2014

## On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

Christensen, O., Kim, H. O. & Kim, R. Y. 2014 57, 2, p. 254-263

Publication: Research - peer-reviewJournal article – Annual report year: 2014

The unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.
Original language English Canadian Mathematical Bulletin 57 2 254-263 0008-4395 10.4153/CMB-2013-015-9 Published - 2014

Year: 2014

## Experimental bifurcation analysis of an impact oscillator – Determining stability

Bureau, E., Schilder, F., Elmegård, M., Santos, I., Thomsen, J. J. & Starke, J. 2014 333, 21, p. 5464–5474

Publication: Research - peer-reviewJournal article – Annual report year: 2014

We propose and investigate three different methods for assessing stability of dynamical equilibrium states during experimental bifurcation analysis, using a control-based continuation method. The idea is to modify or turn off the control at an equilibrium state and study the resulting behavior. As a proof of concept the three methods are successfully implemented and tested for a harmonically forced impact oscillator with a hardening spring nonlinearity, and controlled by electromagnetic actuators. We show that under certain conditions it is possible to quantify the instability in terms of finite-time Lyapunov exponents. As a special case we study an isolated branch in the bifurcation diagram brought into existence by a 1:3 subharmonic resonance. On this isola it is only possible to determine stability using one of the three methods, which is due to the fact that only this method guarantees that the equilibrium state can be restored after measuring stability.
Original language English Journal of Sound and Vibration 333 21 5464–5474 0022-460X 10.1016/j.jsv.2014.05.032 Published - 2014

Year: 2014

## Vorticity generation and conservation for two-dimensional interfaces and boundaries

Brøns, M., Thompson, M. C., Leweke, T. & Hourigan, K. 2014 758, p. 63-93

Publication: Research - peer-reviewJournal article – Annual report year: 2015

The generation, redistribution and, importantly, conservation of vorticity and circulation is studied for incompressible Newtonian fluids in planar and axisymmetric geometries. A generalised formulation of the vorticity at the interface between two fluids for both no-slip and stress-free conditions is presented. Illustrative examples are provided for planar Couette flow, Poiseuille flow, the spin-up of a circular cylinder, and a cylinder below a free surface. For the last example, it is shown that, although large imbalances between positive and negative vorticity appear in the wake, the balance is found in the vortex sheet representing the stress-free surface.
Original language English Journal of Fluid Mechanics 758 63-93 0022-1120 10.1017/jfm.2014.520 Published - 2014

Year: 2014

## Constant Gaussian curvature surfaces in the 3-sphere via loop groups

Brander, D., Inoguchi, J-I. & Kobayashi, S. 2014 269, 2, p. 281-303

Publication: Research - peer-reviewJournal article – Annual report year: 2015

In this paper we study constant positive Gauss curvature K surfaces in the 3-sphere S3 with 0<K<1, as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in S3 with Gauss curvature K<1 is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if K is constant. We give a uniform loop group formulation for all such surfaces with K≠0, and use the generalized d’Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with K<0 and those with 0<K<1.
Original language English Pacific Journal of Mathematics 269 2 281-303 0030-8730 10.2140/pjm.2014.269.281 Published - 2014

Year: 2014

## Timelike Constant Mean Curvature Surfaces with Singularities

Brander, D. & Svensson, M. 2014 24, 3, p. 1641-1672

Publication: Research - peer-reviewJournal article – Annual report year: 2013

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.
Original language English Journal of Geometric Analysis 24 3 1641-1672 1050-6926 10.1007/s12220-013-9389-6 Published - 2014

Year: 2014

## An Improvement of the Gilbert–Varshamov Bound Over Nonprime Fields

Bassa, A., Beelen, P., Garcia, A. & Stichtenoth, H. 2014 60, 7, p. 3859-3861

Publication: Research - peer-reviewJournal article – Annual report year: 2014

Original language English IEEE Transactions on Information Theory 60 7 3859-3861 0018-9448 10.1109/TIT.2014.2316531 Published - 2014

Year: 2014

## Galois towers over non-prime finite fields

Bassa, A., Beelen, P., Garcia, A. & Stichtenoth, H. 2014 164, 2, p. 163-179

Publication: Research - peer-reviewJournal article – Annual report year: 2014

In this paper we construct Galois towers with good asymptotic properties over any non-prime finite field Fℓ; i.e., we construct sequences of function fields N=(N1⊂N2⊂⋯) over Fℓ of increasing genus, such that all the extensions Ni/N1 are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties are important for applications in various fields including coding theory and cryptography.
Original language English Acta Arithmetica 164 2 163-179 0065-1036 10.4064/aa164-2-6 Published - 2014

Year: 2014

## 5th International Symposium on Bifurcations and Instabilities in Fluid Dynamics (BIFD2013) Foreword

Bar-Yoseph, P. Z., Brøns, M., Gelfgat, A. & Oron, A. 2014 46, 4, 2 p., 041001

Publication: Research - peer-reviewJournal article – Annual report year: 2014

Original language English 041001 Fluid Dynamics Research 46 4 2 0169-5983 10.1088/0169-5983/46/4/041001 Published - 2014

Year: 2014

## Topology of helical fluid flow

Andersen, M. & Brøns, M. 2014 25, 03, p. 375-396

Publication: Research - peer-reviewJournal article – Annual report year: 2014

Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature obtained with other approaches. In particular, we discuss the role of the stream function for the topology of the streamline pattern in incompressible flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Hardin, J. C. 1982 Phys. Fluids 25, 1949–1952) contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Okulov, V. L. 1995 Russ. J. Eng. Thermophys. 5, 63–75) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. By including the self-induced velocity on the vortex filament by a localised induction approximation, the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of parameters.
Original language English European Journal of Applied Mathematics 25 03 375-396 0956-7925 10.1017/S0956792514000084 Published - 2014

2013

Year: 2013

## Estimates of the first Dirichlet eigenvalue from exit time moment spectra

Hurtado, A., Markvorsen, S. & Palmer, V. 2013 23 p.

Publication: Research - peer-reviewWorking paper – Annual report year: 2013

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung--Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.
Original language English 23 Published - 2013

Year: 2013

## Multi-Trial Guruswami–Sudan Decoding for Generalised Reed–Solomon Codes

Nielsen, J. S. R. & Zeh, A. 2013 WCC 2013 International Workshop on Coding and Cryptography. 11 p.

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2013

An iterated refinement procedure for the Guruswami–Sudan list decoding algorithm for Generalised Reed–Solomon codes based on Alekhnovich’s module minimisation is proposed. The method is parametrisable and allows variants of the usual list decoding approach. In particular, finding the list of closest codewords within an intermediate radius can be performed with improved average-case complexity while retaining the worst-case complexity.
Original language English WCC 2013 International Workshop on Coding and Cryptography 11 2013 Published - 2013 International Workshop on Coding and Cryptography (WCC 2013) - Bergen, Norway

### Conference

Conference International Workshop on Coding and Cryptography (WCC 2013) Norway Bergen 15/04/2013 → 19/04/2013 http://www.selmer.uib.no/WCC2013/

Year: 2013

## On transforms between Gabor frames and wavelet frames

Christensen, O. & Goh, S. S. 2013 Proceedings of the 10th International Conference on Sampling Theory and Applications (SampTA). European Association for Signal Processing (EURASIP), p. 81-84

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2013

We describe a procedure that enables us to construct dual pairs of wavelet frames from certain
dual pairs of Gabor frames. Applying the construction to Gabor frames generated by appropriate exponential Bsplines gives wavelet frames generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knot sequences. There is also a reverse transform, which yields pairs of dual Gabor frames when applied to certain wavelet frames.
Original language English Proceedings of the 10th International Conference on Sampling Theory and Applications (SampTA) European Association for Signal Processing (EURASIP) 2013 81-84 Published - 2013 10th International Conference on Sampling Theory and Applications (SampTA 2013) - Bremen, Germany

### Conference

Conference 10th International Conference on Sampling Theory and Applications (SampTA 2013) Germany Bremen 01/07/2013 → 05/07/2013 http://www.univie.ac.at/nuhag-php/event_NEW/make.php?event=SampTA2013

Year: 2013

## An iterative method for the canard explosion in general planar systems

Brøns, M. 2013 Proceedings of the 9th AIMS International Conference: Dynamical Systems and Differential Equations, DCDS Supplement 2013. American Institute of Mathematical Sciences, p. 77-83

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2013

The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls the slow/fast dynamics. However, canard explosions are also observed in systems where no such parameter can obviously be identied. Here we show how the iterative method of Roussel and Fraser, devised to construct regular slow manifolds, can be used to determine a canard point in a general planar system of nonlinear ODEs. We demonstrate the method on the van der Pol equation, showing that the asymptotics of the method is correct, and on a templator model for a self-replicating system.
Original language English Proceedings of the 9th AIMS International Conference : Dynamical Systems and Differential Equations, DCDS Supplement 2013 American Institute of Mathematical Sciences 2013 77-83 Published - 2013 The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications - Orlando, FL, United States

### Conference

Conference The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications United States Orlando, FL 01/07/2012 → 05/07/2012

Year: 2013

## On the Dimension of Graph Codes with Reed–Solomon Component Codes

Beelen, P., Høholdt, T., Pinero, F. & Justesen, J. 2013 2013 IEEE International Symposium on Information Theory Proceedings (ISIT). IEEE, p. 1227-1231

Publication: Research - peer-reviewArticle in proceedings – Annual report year: 2013

We study a class of graph based codes with Reed-Solomon component codes as affine variety codes. We give a formulation of the exact dimension of graph codes in general. We give an algebraic description of these codes which makes the exact computation of the dimension of the graph codes easier.
Original language English 2013 IEEE International Symposium on Information Theory Proceedings (ISIT) IEEE 2013 1227-1231 978-1-4799-0446-4 10.1109/ISIT.2013.6620422 Published - 2013 IEEE International Symposium on Information Theory (ISIT 2013) - Istanbul, Turkey

### Conference

Conference IEEE International Symposium on Information Theory (ISIT 2013) Turkey Istanbul 07/07/2013 → 12/07/2013 http://www.isit2013.org/

Year: 2013

## List Decoding of Algebraic Codes

Nielsen, J. S. R., Beelen, P. & Høholdt, T. 2013 Kgs. Lyngby: Technical University of Denmark (DTU). 178 p. (DTU Compute PHD-2013; No. 309).

Publication: ResearchPh.D. thesis – Annual report year: 2013

We investigate three paradigms for polynomial-time decoding of Reed–Solomon codes beyond half the minimum distance: the Guruswami–Sudan algorithm, Power decoding and the Wu algorithm. The main results concern shaping the computational core of all three methods to a problem solvable by module minimisation; by applying the fastest known algorithms for this general problem, we then obtain realisations of each paradigm which are as fast or faster than all previously known methods. An element of this is the “2D key equation”, a heavily generalised form of the classical key equation, and we show how to solve such using module minimisation, or using our new Demand–Driven algorithm which is also based on module minimisation.

The decoding paradigms are all derived and analysed in a self-contained manner, often in new ways or examined in greater depth than previously. Among a number of new results, we give: a fast maximum-likelihood list decoder based on the Guruswami–Sudan algorithm; a new variant of Power decoding, Power Gao, along with some new insights into Power decoding; and a new, module based method for performing rational interpolation for theWu algorithm. We also show how to decode Hermitian codes using Guruswami–Sudan or Power decoding faster than previously known, and we show how to Wu list decode binary Goppa codes.
Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 178 Published - 2013
Series DTU Compute PHD-2013 309 0909-3192

Year: 2013

## Topology of streamlines and vorticity contours for two - dimensional flows

Andersen, M. & Brøns, M. 2013 Kgs. Lyngby: Technical University of Denmark (DTU). 150 p. (DTU Compute PHD-2013; No. 314).

Publication: ResearchPh.D. thesis – Annual report year: 2014

Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature. For inviscid flow where a velocity field is generated by a sum of helical vortex filaments with same pitch we use the established results to prove briefly that the velocity field is helical. We discuss the role of the stream function for the topology of the streamlines in incompressible, helical flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Phys. Fluids 25, 1982) contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Russ. J. Eng. Thermophys. 5, 1995) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. Including the self-induced velocity on the vortex filament by the localised induction approximation the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of the parameters. The analysis of the closed form show promise for analysing more complex flow with helical symmetry e.g. multiple helical vortex filaments inside a cylinder which has industrial relevance.

We then change focus and study creation, destruction and interaction of vortices in two dimensional flow. A vortex is advected above a wall causing a viscous response near the wall which generates a new vortex structure. The problem is studied numerically relying on the code developed by Prof. M. Thompson and his group at Monash University, Australia. We also investigate the problem analytically using normal form theory. It is not a simple task to define a vortex in a proper way that allow the study of creation and destruction of vortices. We investigate three sound choices: the vorticity extrema, the streamline centers in a coordinate system with zero wall speed and the streamline centers in a frame moving with constant velocity as predicted by a point vortex above a wall in inviscid fluid. There is no reason to a priori expect equivalent results of the three vortex definitions. However, the study is mainly motivated by the findings of Kudela & Malecha (Fluid Dyn. Res. 41, 2009) who find good agreement between the vorticity and streamlines in the fixed wall system. For small Re no new vortices are observed. Creation of a vortex occurs for sufficiently large Re for all the applied vortex definitions. The new vortex alters the generating vortex motion by slowing its horizontal motion and lifting it further from the wall. In the fixed wall system vortex eruption happens through a characteristic ’figure 8’ bifurcation. Considering the other coordinate system there is no topological change indicating when a vortex has left the boundary layer. However, here there is remarkable good agreement between streamlines and the vorticity contours even for short-lived vortices close to the wall.

The normal form approach does not reveal simple connections between the streamline topology and the vorticity contour topology. Only for a non simple degenerate on wall critical point may a bifurcation occur in both the streamlines and the vorticity contours. The streamline bifurcations in this normal form contain the lower part of the ’figure 8’ bifurcation observed in numerics. The similarities and differences of the streamlines in the two different coordinate systems are well described by normal form theory.

We derive the criterion, u⋅∇ ω = 0, for exactly matching contours of the vorticity contours and streamlines. This is fulfilled when the Navier - Stokes equations and the heat equation have identical solutions.

Finally we focus on the superposition of two rotational invariant vortices in R2. The topology of the streamlines and the topology of the vorticity conturs are determined by the zeros of a single real function. For the canonical example of two Gaussian vortices three parameters exist. Three structurally stable topologies are observed. For the streamlines two of the topologies are well known for the corresponding situation of two point vortices when the singularities are treated as centers. The last topology is a single center which is consistent with the powerful result on the long time behaviour proved by Gallay & Wayne (Comm. in Math. Phys. 255, 2005). The case of three critical points of the streamlines is a subset of three critical points of the vorticity. This explains an observation in the simulations of vortex generation near a wall. Here, a long living erupted vortex disappears due to viscosity. This happens first considering the streamlines while being more robust when considering the vorticity formulation.

Original language English
Place of Publication Kgs. Lyngby Technical University of Denmark (DTU) 150 Published - 2013
Series DTU Compute PHD-2013 314 0909-3192

Year: 2013

## Recent progress in the relative equilibria of point vortices — In memoriam Hassan Aref

Beelen, P., Brøns, M., Krishnamurthy, V. S. & Stremler, M. A. 2013 7, p. 3-12

Publication: Research - peer-reviewConference article – Annual report year: 2013

Hassan Aref, who sadly passed away in 2011, was one of the world's leading researchers in the dynamics and equilibria of point vortices. We review two problems on the subject of point vortex relative equilibria in which he was engaged at the time of his death: bilinear relative equilibria and the geometry of the three-vortex problem as it relates to equilibria. A set of point vortices is in relative equilibrium if it is at most rotating rigidly around the center of vorticity, and the configuration is bilinear if the vortices are placed on two orthogonal lines in the co-rotating frame. A very complete characterisation of the bilinear case can be obtained when one of the lines contains only two vortices. The classic three-vortex problem can be viewed anew by considering the dy- namics of the circle circumscribing the vortex triangle and the interior angles of that triangle. This approach leads naturally to the observation that the equilateral triangle is the only equilibrium configuration for three point vortices, regardless of their strength values.
Original language English I U T A M. Procedia 7 3-12 2210-9838 10.1016/j.piutam.2013.03.002 Published - 2013 Topological Fluid Dynamics II (IUTAM Symposium) - Cambridge, United Kingdom

### Conference

Conference Topological Fluid Dynamics II (IUTAM Symposium) United Kingdom Cambridge 23/07/2012 → 27/07/2012 http://www.newton.ac.uk/programmes/TOD/todw01.html

Year: 2013

## Isogeometric shape optimization in fluid mechanics

Nørtoft, P. & Gravesen, J. 2013 48, 5, p. 909-925

Publication: Research - peer-reviewJournal article – Annual report year: 2013

The subject of this work is numerical shape optimization in fluid mechanics, based on isogeometric analysis. The generic goal is to design the shape of a 2-dimensional flow domain to minimize some prescribed objective while satisfying given geometric constraints. As part of the design problem, the steady-state, incompressible Navier-Stokes equations, governing a laminar flow in the domain, must be solved. Based on isogeometric analysis, we use B-splines as the basis for both the design optimization and the flow analysis, thereby unifying the models for geometry and analysis, and, at the same time, facilitating a compact representation of complex geometries and smooth approximations of the flow fields. To drive the shape optimization, we use a gradient-based approach, and to avoid inappropriate parametrizations during optimization, we regularize the optimization problem by adding to the objective function a measure of the quality of the boundary parametrization. A detailed description of the methodology is given, and three different numerical examples are considered, through which we investigate the effects of the regularization, of the number of geometric design variables, and of variations in the analysis resolution, initial design and Reynolds number, and thereby demonstrate the robustness of the methodology.
Original language English Structural and Multidisciplinary Optimization 48 5 909-925 1615-147x 10.1007/s00158-013-0931-8 Published - 2013

Year: 2013

## He-, Ne-, and Ar-phosgene intermolecular potential energy surfaces

Munteanu, C. R., Henriksen, C., Felker, P. M. & Fernández, B. 2013 117, 18, p. 3835-3843

Publication: Research - peer-reviewJournal article – Annual report year: 2013

Using the CCSD(T) model, we evaluated the intermolecular potential energy surfaces of the He-, Ne-, and Ar-phosgene complexes. We considered a representative number of intermolecular geometries for which we calculated the corresponding interaction energies with the augmented (He complex) and double augmented (Ne and Ar complexes) correlation-consistent polarized valence triple-ζ basis sets extended with a set of 3s3p2d1f1g midbond functions. These basis sets were selected after systematic basis set studies carried out at geometries close to those of the surface minima. The He-, Ne-, and Ar-phosgene surfaces were found to have absolute minima of -72.1, -140.4, and -326.6 cm -1 at distances between the rare-gas atom and the phosgene center of mass of 3.184, 3.254, and 3.516 Å, respectively. The potentials were further used in the evaluation of rovibrational states and the rotational constants of the complexes, providing valuable results for future experimental investigations. Comparing our results to those previously available for other phosgene complexes, we suggest that the results for Cl2-phosgene should be revised.
Original language English Journal of Physical Chemistry Part A: Molecules, Spectroscopy, Kinetics, Environment and General Theory 117 18 3835-3843 1089-5639 10.1021/jp3110024 Published - 2013

Year: 2013

## Vorticity generation and wake transition for a translating circular cylinder: Wall proximity and rotation effects

Hourigan, K., Rao, A., Brøns, M., Leweke, T., Thompson, M. C. & Yaojun Ge, S. C. 2013 122, p. 2-9

Publication: Research - peer-reviewJournal article – Annual report year: 2013

The wake transitions of generic bluff bodies, such as a circular cylinder, near a wall are important because they provide understanding of different transition paths towards turbulence, and give some insight into the effect of surface modifications on the flow past larger downstream structures. In this article, the fundamentals of vorticity generation and transport for the two-dimensional flow of incompressible Newtonian fluids are initially reviewed. Vorticity is generated only at boundaries by tangential pressure gradients or relative acceleration. After generation, it can cross-annihilate with opposite-signed vorticity, and can be stored at a free surface, thus conserving the total vorticity, or circulation. Vorticity generation, diffusion and storage are demonstrated for a cylinder translating and rotating near a wall. The wake characteristics and the wake transitions are shown to change dramatically under the influence of cylinder rotation and wall proximity. At gaps between the cylinder and the wall of less than approximately 0.25 cylinder diameter, the wake becomes three dimensional prior to becoming unsteady, while for larger gaps the initial transition is to an unsteady two-dimensional wake. At a gap of 0.3 cylinder diameter, we observe a sharp increase in the critical Reynolds number at which three-dimensionality sets in. As the gap is further increased, the critical Reynolds number initially decreases before increasing to that for an isolated cylinder. The effect of cylinder rotation on these transitions is also quantified, with forward (prograde) rotation enhancing three-dimensional instability and reverse (retrograde) rotation stabilising the wake. High retrograde rotation leads to suppression of three-dimensional flow until beyond the highest Reynolds number investigated (Re=750).
Original language English Journal of Wind Engineering & Industrial Aerodynamics 122 2-9 0167-6105 10.1016/j.jweia.2013.06.009 Published - 2013 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications - Shanghai, China

### Conference

Conference The Seventh International Colloquium on Bluff Body Aerodynamics and Applications 7 Jin Jiang Tower China Shanghai 02/09/2012 → 06/09/2012 http://bbaa7.org/index.aspx

Year: 2013

## Refined ab initio intermolecular ground-state potential energy surface for the He-C2H2 van der Waals complex

Fernández, B., Henriksen, C. & Farrelly, D. 2013 111, 9-11, p. 1173-1177

Publication: Research - peer-reviewJournal article – Annual report year: 2013

A refined CCSD(T) intermolecular potential energy surface is developed for the He-C2H2 van der Waals complex. For this, 206 points on the intermolecular potential energy surface, evaluated using the CCSD(T) method and the aug-cc-pVQZ basis set extended with a set of 3s3p2d1f1g midbond functions, are fitted to a 15-parameter analytic function. The potential is characterised by minima of-24.21 cm-1 at distances between the rare gas atom and the C2H2 centre of mass of 4.3453 Å, and with the complex in a linear configuration. At intermediate distances the surface is rather similar to that developed previously by Munteanu and Fernández (J. Chem. Phys., 123, 014309, 2005) but differs notably at short range. The improved potential energy surface should, therefore, be particularly useful for computations of collision line broadening. Dynamical calculations of a number of rovibrational bound state energies and wave functions are presented. Inspection of the nodal surfaces of several low lying excited states shows that the complex is close to the free rotor limit.
Original language English Molecular Physics 111 9-11 1173-1177 0026-8976 10.1080/00268976.2013.793837 Published - 2013

Year: 2013

## Bifurcation analysis and dimension reduction of a predator-prey model for the L-H transition

Dam, M., Brøns, M., Juul Rasmussen, J., Naulin, V. & Xu, G. 2013 20, 10, p. 102302

Publication: Research - peer-reviewJournal article – Annual report year: 2013

The L-H transition denotes a shift to an improved confinement state of a toroidal plasma in a fusion reactor. A model of the L-H transition is required to simulate the time dependence of tokamak discharges that include the L-H transition. A 3-ODE predator-prey type model of the L-H transition is investigated with bifurcation theory of dynamical systems. The analysis shows that the model contains three types of transitions: an oscillating transition, a sharp transition with hysteresis, and a smooth transition. The model is recognized as a slow-fast system. A reduced 2-ODE model consisting of the full model restricted to the flow on the critical manifold is found to contain all the same dynamics as the full model. This means that all the dynamics in the system is essentially 2-dimensional, and a minimal model of the L-H transition could be a 2-ODE model.
Original language English Physics of Plasmas 20 10 102302 1070-664X 10.1063/1.4823719 Published - 2013

Year: 2013

## Construction of smooth compactly supported windows generating dual pairs of gabor frames

Christiansen, L. H. & Christensen, O. 2013 6, 1, 13 p., 1350011

Publication: Research - peer-reviewJournal article – Annual report year: 2013

Let g be any real-valued, bounded and compactly supported function, whose integer-translates {Tkg}k∈ℤ form a partition of unity. Based on a new construction of dual windows associated with Gabor frames generated by g, we present a method to explicitly construct dual pairs of Gabor frames. This new method of construction is based on a family of polynomials which is closely related to the Daubechies polynomials, used in the construction of compactly supported wavelets. For any k ∈ ℕ ∪ {∞} we consider the Meyer scaling functions and use these to construct compactly supported windows g ∈ Ck(ℝ) associated with a family of smooth compactly supported dual windows . For any n ∈ ℕ the pair of dual windows g, hn ∈ Ck(ℝ) have compact support in the interval [-2/3, 2/3] and share the property of being constant on half the length of their support. We therefore obtain arbitrary smoothness of the dual pair of windows g, hn without increasing their support.
Original language English 1350011 Asian-European Journal of Mathematics 6 1 13 1793-5571 10.1142/S1793557113500113 Published - 2013

Year: 2013

## Extensions of Bessel sequences to dual pairs of frames

Christensen, O., Kim, H. O. & Kim, R. Y. 2013 34, 2, p. 224-233

Publication: Research - peer-reviewJournal article – Annual report year: 2013

Tight frames in Hilbert spaces have been studied intensively for the past years. In this paper we demonstrate that it often is an advantage to use pairs of dual frames rather than tight frames. We show that in any separable Hilbert space, any pairs of Bessel sequences can be extended to a pair of dual frames. If the given Bessel sequences are Gabor systems in L2(R), the extension can be chosen to have Gabor structure as well. We also show that if the generators of the given Gabor Bessel sequences are compactly supported, we can choose the generators of the added Gabor systems to be compactly supported as well. This is a significant improvement compared to the extension of a Bessel sequence to a tight frame, where the added generator only can be compactly supported in some special cases. We also analyze the wavelet case, and find sufficient conditions under which a pair of wavelet systems can be extended to a pair of dual frames. © 2012 Elsevier Inc. All rights reserved.
Original language English Applied and Computational Harmonic Analysis 34 2 224-233 1063-5203 10.1016/j.acha.2012.04.003 Published - 2013

Year: 2013

## On frame properties for Fourier-like systems

Christensen, O. & Osgooei, E. 2013 172, p. 47-57

Publication: Research - peer-reviewJournal article – Annual report year: 2013

Fourier-like systems are formed by multiplying a class of exponentials with a set of window functions. Via the Fourier transform they are equivalent to shift-invariant systems. We present sufficient and easily verifiable conditions for such systems to form a frame with a dual frame having the same structure. An attractive class of frames is formed by letting the window functions be trigonometric polynomials, restricted to compact intervals. We prove, under weak conditions, that such systems generate a frame with a dual that is also generated by a trigonometric polynomial. For polynomial windows, a result of this type does not hold. Throughout the paper the results are related to the well established theory for Gabor systems.
Original language English Journal of Approximation Theory 172 47-57 0021-9045 10.1016/j.jat.2013.04.004 Published - 2013

Year: 2013

## Regularity of Dual Gabor Windows

Christensen, O., Kim, H. O. & Kim, R. Y. 2013 2013, 8 p., 747268

Publication: Research - peer-reviewJournal article – Annual report year: 2013

Original language English 747268 Abstract and Applied Analysis 2013 8 1085-3375 10.1155/2013/747268 Published - 2013

### Bibliographical note

Copyright © 2013 Ole Christensen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Year: 2013

## The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

Brander, D. & Svensson, M. 2013 93, 1, p. 37-66

Publication: Research - peer-reviewJournal article – Annual report year: 2013

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and suffcient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.
Original language English Journal of Differential Geometry 93 1 37-66 0022-040X Published - 2013

Year: 2013

## Ribbon Crystals

Bohr, J. & Markvorsen, S. 2013 In : P L o S One. 8, 10, 7 p., e74932

Publication: Research - peer-reviewJournal article – Annual report year: 2013

A repetitive crystal-like pattern is spontaneously formed upon the twisting of straight ribbons. The pattern is akin to a tessellation with isosceles triangles, and it can easily be demonstrated with ribbons cut from an overhead transparency. We give a general description of developable ribbons using a ruled procedure where ribbons are uniquely described by two generating functions. This construction defines a differentiable frame, the ribbon frame, which does not have singular points, whereby we avoid the shortcomings of the Frenet–Serret frame. The observed spontaneous pattern is modeled using planar triangles and cylindrical arcs, and the ribbon structure is shown to arise from a maximization of the end-to-end length of the ribbon, i.e. from an optimal use of ribbon length. The phenomenon is discussed in the perspectives of incompatible intrinsic geometries and of the emergence of long-range order.
Original language English e74932 P L o S One 8 10 7 1932-6203 10.1371/journal.pone.0074932 Published - 2013

### Bibliographical note

2013 Bohr, Markvorsen. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Year: 2013

## Bounding the number of points on a curve using a generalization of Weierstrass semigroups

Beelen, P. & Ruano, D. 2013 66, 1-3, p. 221-230

Publication: Research - peer-reviewJournal article – Annual report year: 2013

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup (J Pure Appl Algebra 207(2), 243–260, 2006) for an n-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in (J Pure Appl Algebra 213(6), 1152–1156, 2009).
Original language English Designs, Codes and Cryptography 66 1-3 221-230 0925-1022 10.1007/s10623-012-9685-3 Published - 2013

Year: 2013

## On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes

Beelen, P., Høholdt, T., Nielsen, J. S. R. & Wu, Y. 2013 59, 6, p. 3269-3281

Publication: Research - peer-reviewJournal article – Annual report year: 2013

We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami–Sudan algorithm, immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.
Original language English I E E E Transactions on Information Theory 59 6 3269-3281 0018-9448 10.1109/TIT.2013.2243800 Published - 2013

### Bibliographical note

Copyright 2015 IEEE. IEEE. Personal use of this material is permitted. Permission
from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

Year: 2013

## Mathematical modeling of the hypothalamic–pituitary–adrenal gland (HPA) axis, including hippocampal mechanisms

Andersen, M., Vinther, F. & Ottesen, J. T. 2013 246, 1, p. 122-138

Publication: Research - peer-reviewJournal article – Annual report year: 2013

This paper presents a mathematical model of the HPA axis. The HPA axis consists of the hypothalamus, the pituitary and the adrenal glands in which the three hormones CRH, ACTH and cortisol interact through receptor dynamics. Furthermore, it has been suggested that receptors in the hippocampus have an influence on the axis.A model is presented with three coupled, non-linear differential equations, with the hormones CRH, ACTH and cortisol as variables. The model includes the known features of the HPA axis, and includes the effects from the hippocampus through its impact on CRH in the hypothalamus. The model is investigated both analytically and numerically for oscillating solutions, related to the ultradian rhythm seen in data, and for multiple fixed points related to hypercortisolemic and hypocortisolemic depression.The existence of an attracting trapping region guarantees that solution curves stay non-negative and bounded, which can be interpreted as a mathematical formulation of homeostasis. No oscillating solutions are present when using physiologically reasonable parameter values. This indicates that the ultradian rhythm originate from different mechanisms.Using physiologically reasonable parameters, the system has a unique fixed point, and the system is globally stable. Therefore, solutions converge to the fixed point for all initial conditions. This is in agreement with cortisol levels returning to normal, after periods of mild stress, in healthy individuals.Perturbing parameters lead to a bifurcation, where two additional fixed points emerge. Thus, the system changes from having a unique stable fixed point into having three fixed points. Of the three fixed points, two are stable and one is unstable. Further investigations show that solutions converge to one of the two stable fixed points depending on the initial conditions. This could explain why healthy people becoming depressed usually fall into one of two groups: a hypercortisolemic depressive group or a hypocortisolemic depressive group.
Original language English Mathematical Biosciences 246 1 122-138 0025-5564 10.1016/j.mbs.2013.08.010 Published - 2013

2012

Year: 2012

## Shearlets and Optimally Sparse Approximations

Kutyniok, G., Lemvig, J. & Lim, W-Q. 2012 Shearlets: Multiscale Analysis for Multivariate Data. Kutyniok, G. & Labate, D. (eds.). Birkhäuser Verlag GmbH, p. 145-198

Publication: Research - peer-reviewBook chapter – Annual report year: 2012

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.
Original language English Shearlets: Multiscale Analysis for Multivariate Data Gitta Kutyniok, D. Labate Birkhäuser Verlag GmbH 2012 145-198 978-0-8176-8316-0 Published - 2012

2005

Year: 2005

## Beneath the Wheel - Greenwood Engineering

Dias, K., Gravesen, J., Hjorth, P. G., Larsen, P., Please, C., Radulovic, N., Wang, L. & Aagaard Pedersen, L. 2005 University of Southern Denmark. 14 p.

Publication: ResearchReport – Annual report year: 2005

This is the report from the 54th European Study Group wiht Industry of the Greenwood Engineering problem. We model pavement response to both a point and distributed loads and compare with data from Greenwood's High Speed Deflectograph Measurements.
Original language English
Publisher University of Southern Denmark 14 Published - 2005

### Bibliographical note

ESGI-54 Final Report

1993

Year: 1993

## A characterization of spheres, circles and cardioids

Gæde, L., Gravesen, J. & Markvorsen, S. Jun 1993 60, 6, p. 579-590

Publication: Research - peer-reviewJournal article – Annual report year: 1993

Original language English Archiv der Mathematik 60 6 579-590 0003-889X 10.1007/BF01236085 Published - Jun 1993

1990

Year: 1990

## Loop groups and Yang-Mills theory in dimension two

Gravesen, J. Feb 1990 127, 3, p. 597-605 9 p.

Publication: Research - peer-reviewJournal article – Annual report year: 1990

Given a connection ω in a G-bundle over S2, then a process called radial trivialization from the poles gives a unique clutching function, i.e., an element γ of the loop group ΩG. Up to gauge equivalence, ω is completely determined by γ and a map f:S2 →g into the Lie algebra. Moreover, the Yang-Mills function of ω is the sum of the energy of γ and the square of a certain norm of f. In particular, the Yang-Mills functional has the same Morse theory as the energy functional on ΩG. There is a similar description of connections in a G-bundle over an arbitrary Riemann surface, but so far not of the Yang-Mills functional.

Original language English Communications in Mathematical Physics 127 3 597-605 9 0010-3616 10.1007/BF02104504 Published - Feb 1990

1989

Year: 1989

## On the topology of spaces of holomorphic maps

Gravesen, J. Jul 1989 162, 1, p. 247-286

Publication: Research - peer-reviewJournal article – Annual report year: 1989

Original language English Acta Mathematica 162 1 247-286 0001-5962 10.1007/BF02392839 Published - Jul 1989

Year: 1989

## Complex structures in the Nash-Moser category

Gravesen, J. Jan 1989 7, 2, p. 155-161

Publication: Research - peer-reviewJournal article – Annual report year: 1989

Working in the Nash-Moser category, it is shown that the harmonic and holomorphic differentials and the Weierstrass points on a closed Riemann surface depend smoothly on the complex structure. It is also shown that the space of complex structures on any compact surface forms a principal bundle over the Teichmüller space and hence that the uniformization maps of the closed disk and the sphere depend smoothly on the complex structure.

Original language English Annals of Global Analysis and Geometry 7 2 155-161 0232-704X 10.1007/BF00127865 Published - Jan 1989

1983

Year: 1983

## Catastrophe Theory and Caustics

Gravesen, J. 1983 In : S I A M Review. 25, 2, p. 239-247 9 p.

Publication: Research - peer-reviewJournal article – Annual report year: 1983

It is shown by elementary methods that in codimension two and under the assumption that light rays are straight lines, a caustic is the catastrophe set for a time function. The general case is also discussed.
Original language English S I A M Review 25 2 239-247 9 0036-1445 10.1137/1025046 Published - 1983

Year: 1983

## WHITNEY O-INFINITY-TOPOLOGIES AND THE BAIRE PROPERTY

Gravesen, J. 1983 52, 1, p. 58-60

Publication: Research - peer-reviewJournal article – Annual report year: 1983

Original language English Mathematica Scandinavica 52 1 58-60 0025-5521 Published - 1983