# Publications

2018

Year: 2018

## Power decoding Reed-Solomon codes up to the Johnson radius

Rosenkilde, J. S. H. 2018 12, 1, p. 81-106

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

Power decoding, or "decoding using virtual interleaving" is a technique for decoding Reed-Solomon codes up to the Sudan radius. Since the method's inception, it has been an open question if it is possible to use this approach to decode up to the Johnson radius - the decoding radius of the Guruswami-Sudan algorithm. In this paper we show that this can be done by incorporating a notion of multiplicities. As the original Power decoding, the proposed algorithm is a one-pass algorithm: decoding follows immediately from solving a shift-register type equation, which we show can be done in quasi-linear time. It is a "partial bounded-distance decoding algorithm" since it will fail to return a codeword for a few error patterns within its decoding radius; we investigate its failure behaviour theoretically as well as give simulation results.
Original language English Advances in Mathematics of Communications 12 1 81-106 1930-5346 10.3934/amc.2018005 Published - 2018

Year: 2018

## Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography

Beelen, P., Bossert, M., Puchinger, S. & Rosenkilde, J. S. H. 2018 Proceedings of 2018 IEEE International Symposium on Information Theory. IEEE, p. 946-950

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

We present a generalisation of Twisted Reed-Solomon codes containing a new large class of MDS codes. We prove that the code class contains a large subfamily that is closed under duality. Furthermore, we study the Schur squares of the new codes and show that their dimension is often large. Using these structural properties, we single out a subfamily of the new codes which could be considered for code-based cryptography: These codes resist some existing structural attacks for Reed-Solomon-like codes, i.e. methods for retrieving the code parameters from an obfuscated generator matrix.
Original language English Proceedings of 2018 IEEE International Symposium on Information Theory IEEE 2018 946-950 9781538647813 10.1109/ISIT.2018.8437923 Published - 2018 2018 IEEE International Symposium on Information Theory - Talisa Hotel, Vail, United StatesDuration: 17 Jun 2018 → 22 Jun 2018

### Conference

Conference 2018 IEEE International Symposium on Information Theory Talisa Hotel United States Vail 17/06/2018 → 22/06/2018

Year: 2018

## Explicit MDS Codes with Complementary Duals

Beelen, D. P. & Jin, L. 2018 (Accepted/In press) PP, 99, 9 p.

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

In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD codes have applications in communication system, side-channel attack (SCA) and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case.
Original language English Ieee Transactions on Information Theory PP 99 9 0018-9448 10.1109/TIT.2018.2816934 Accepted/In press - 2018

Year: 2018

## Families of spherical surfaces and harmonic maps

Brander, D. & Tari, F. 2018 23 p.

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

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely \mathcal {A}-determined map-germs from the plane to the plane can be represented by harmonic maps.
Original language English Geometriae Dedicata 23 0046-5755 10.1007/s10711-018-0389-3 Published - 2018

Year: 2018

## Frames, operator representations, and open problems

Christensen, O. & Hasannasab, M. 2018 Operator Theory: Advances and Applications. Springer, Vol. 268, p. 155-165 (Operator Theory, Vol. 268).

Research output: Research - peer-reviewBook chapter – Annual report year: 2018

A frame in a Hilbert space H is a countable collection of elements in H that allows each f Ïµ H to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.
Original language English Operator Theory: Advances and Applications 268 Springer 2018 155-165 9783319759968 10.1007/978-3-319-75996-8_8 Published - 2018
Series Operator Theory 268 0255-0156

Year: 2018

## On the Björling problem for willmore surfaces

Brander, D. & Wang, P. 2018 108, p. 411-457

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

We solve the analogue of Bj¨orling’s problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve y0 in S3, together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, lifted to
the light cone in Minkowski 5-space R5 1, we prove, using isotropic harmonic maps, that there exists a unique pair of dual Willmore surfaces y and ˆy satisfying the given values along the curve. We give explicit formulae for the generalized Weierstrass data for the surface pair. For the three dimensional target, we use the solution to explicitly describe the Weierstrass data, in terms of geometric quantities, for all equivariant Willmore surfaces. For the case that the surface has umbilic points, we apply the more general half-isotropic harmonic maps introduced by H´elein to derive a solution: in this case the map ˆy is not necessarily the dual surface, and the additional data of a derivative of ˆy must be prescribed. This solution is generalized to higher codimensions.
Original language English Journal of Differential Geometry 108 411-457 0022-040X Published - 2018

Year: 2018

## Two-Point Codes for the Generalised GK curve

Barelli, É., Beelen, P., Datta, M., Neiger, V. & Rosenkilde, J. S. H. 2018 64, 9, p. 6268-6276 9 p.

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

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 64 9 6268-6276 9 0018-9448 10.1109/TIT.2017.2763165 Published - 2018

Year: 2018

## Monge surfaces and planar geodesic foliations

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

Research output: 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

Year: 2018

## New Basis Set for the Evaluation of Specific Rotation in Flexible Biological Molecules in Solution

Baranowska-ŁAczkowska, A., Łaczkowski, K. Z., Henriksen, C. & Fernandez, B. 2018 122, 24, p. 5477-5483 7 p.

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

A detailed theoretical investigation of specific rotation is carried out in solution for nine flexible molecules of biological importance. Systematic search for the main conformers is followed by time-dependent density functional theory (TD-DFT) calculations of specific rotation employing a wide range of basis sets. Due to conformational flexibility of the compounds under study, the possibility of basis set size reduction without deterioration of the results is investigated. The increasing size (d-)aug-cc-pVXZ (X=D, T, Q) bases of Dunning et al., and the ORP basis set, recently developed to efficiently provide molecular specific rotation, are used for this purpose. The polarizable continuum model is employed at all steps of the investigation. Comparison of the present results with the available data obtained in vacuum reveals considerable differences, being the values in solvent much closer to the experimental specific rotation data available. The ORP basis set proves to be competitive with the d-aug-cc-pVDZ set of Dunning in specific rotation calculations carried out in solvent. While having the same number of functions, the former yields in general results considerably closer to the reference triple-zeta values. We can thus recommend the ORP basis set to study the optical rotation in conformationally flexible molecules in solvent.

Original language English Journal of Physical Chemistry A 122 24 5477-5483 7 1089-5639 10.1021/acs.jpca.8b03320 Published - 2018

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).

Research output: 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

## Bézier curves that are close to elastica

Brander, D., Bærentzen, J. A., Fisker, A-S. & Gravesen, J. 2018 104, p. 36-44 9 p.

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

We study the problem of identifying those cubic B´ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are diﬃcult to work with in a digital environment. We seek a sub-class of special B´ezier curves as a proxy. We identify an easily computable quantity, which we call the λ-residual eλ, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B´ezier curve has λ-residual below 0.4, which eﬀectivelyimpliesthatthecurveiswithin1%ofitsarc-lengthtoanelasticcurveinthe L2 norm. Finally wegive two projection algorithms that take an input B´ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles ﬁxed, until it is close to an elastic curve
Original language English Computer-Aided Design 104 36-44 9 0010-4485 10.1016/j.cad.2018.05.003 Published - 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).

Research output: 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

## On the Geometry of Nanowires and the Role of Torsion

Gravesen, J. & Willatzen, M. 2018 11 p., 1800357

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

A detailed analysis of the Schrödinger equation in curved coordinates, exact to all orders in the cross sectional dimension is presented, and we discuss the implications of the frame rotation for energies of both open and closed structures. For a circular cross-section, the energy spectrum is independent of the frame orientation for an open structure. For a closed curve, the energies depend on the holonomy angle of a minimal rotating frame (MR) which is equal to the area enclosed by the tangent image on the unit sphere. In the case of a curve with a well-defined torsion at all points this is up to a multiple of 2π equal to the total torsion, a result first found in 1992 by Takagi and Tanzawa. In both cases we find that the effect on the eigenstates is a phase shift. We validate our findings by accurate numerical solution of both the exact 3D equations and the approximate 1D equations for a helix structure and find that the error is proportional to the square of the diameter of the cross section. We discuss Dirichlet versus Neumann boundary conditions and show that care has to be taken in the latter case.
Original language English 1800357 PHYSICA STATUS SOLIDI (RRL) - RAPID RESEARCH LETTERS 11 1862-6254 10.1002/pssr.201800357 Published - 2018

Year: 2018

## Preface: Seventh International Symposium on Bifurcations and Instabilities in Fluid Dynamics (BIFD2017): Preface

Bar-Yoseph, P. Z., Brøns, M., Friedman, B., Gelfgat, A., Mikishev, A. & Oron, A. 2018 50, 5, 2 p., 050001

Research output: Research - peer-reviewEditorial – Annual report year: 2018

Original language English 050001 Fluid Dynamics Research 50 5 2 0169-5983 10.1088/1873-7005/aad0c5 Published - 2018

Year: 2018

## A new family of maximal curves

Beelen, P. & Montanucci, M. 2018 (Accepted/In press) 21 p.

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

In this article we construct for any prime power q and odd n ≥ 5, a new Fq2n -maximal curve Xn. Like the Garcia–G¨uneri–Stichtenoth maximal curves, our curves generalize the Giulietti–Korchmaros maximal curve, though in a different way. We compute the full automorphism group of Xn, yielding that it has precisely q(q2 − 1)(qn + 1) automorphisms. Further, we show that unless q = 2, the curve Xn is not a Galois subcover of the Hermitian curve. Finally, up to our knowledge, we find new values of the genus spectrum of Fq2n -maximal curves, by considering some Galois subcovers of Xn.
Original language English Journal of the London Mathematical Society 21 0024-6107 10.1112/jlms.12144 Accepted/In press - 2018

Year: 2018

## Julia Sets of Orthogonal Polynomials

Christiansen, J. S., Henriksen, C., Pedersen, H. L. & Petersen, C. L. 2018 (Accepted/In press) p. 1-13

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

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials fPng to properties of the support. More precisely we relate the Julia set of Pn to the outer boundary of the support, the lled Julia set to the polynomial convex hull K of the support, and the Green's function associated with Pn to the Green's function for the complement of K.
Original language English Potential Analysis 1-13 0926-2601 10.1007/s11118-018-9687-5 Accepted/In press - 2018

Year: 2018

## Designing interactively with elastic splines

Brander, D., Bærentzen, J. A., Fisker, A-S. & Gravesen, J. 2018 62, p. 181-191 11 p.

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

We present an algorithm for designing interactively with C1 elastic splines. The idea is to design the elastic spline using a C1 cubic polynomial spline where each polynomial segment is so close to satisfying the Euler-Lagrange equation for elastic curves that the visual difference becomes negligible. Using a database of cubic Bézier curves we are able to interactively modify the cubic spline such that it remains visually close to an elastic spline.
Original language English Computer-Aided Geometric Design 62 181-191 11 0167-8396 10.1016/j.cagd.2018.03.007 Published - 2018

Year: 2018

## Weierstrass semigroups on the Giulietti–Korchmáros curve

Beelen, P. & Montanucci, M. 2018 52, p. 10-29

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

In this article we explicitly determine the structure of the Weierstrass semigroups H(P) for any point P of the Giulietti–Korchmáros curve X. We show that as the point varies, exactly three possibilities arise: one for the Fq2 -rational points (already known in the literature), one for the Fq6 ∖Fq2 -rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of H(P) in case P is an Fq6 ∖Fq2 -rational point.
Original language English Finite Fields and Their Applications 52 10-29 1071-5797 10.1016/j.ffa.2018.03.002 Published - 2018

Year: 2018

## The Four-Band Spin-Less Kane Model in Curvilinear Coordinates

Gravesen, J. & Willatzen, M. 2018 (Accepted/In press) 5 p., 1800305

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

The possibility to fabricate complicated nanostructure geometries with novel topological effects so as to tailor physical properties makes it adamant to develop advanced analytical and numerical methods. The first multiband k · p model in general curvilinear coordinates based on Kane's four‐band spin‐less model for the upper conduction and valence band states is developed. The model captures the combined effects of electron and light‐hole bandstructure coupling and curvature effects. The formulation in curvilinear coordinates allows to obtain a simple set of equations, displaying directly the influence of the local curvature, and the explicit equation sets in the cases of a torus and a helix‐shaped nanowire structure with a square cross section are given. The presented derivation can be generalized to other types of k · p multiband models.
Original language English 1800305 Physica Status Solidi. Rapid Research Letters 5 1862-6254 10.1002/pssr.201800305 Accepted/In press - 2018

Year: 2018

## Ab initio study of the CO-N2 complex: a new highly accurate intermolecular potential energy surface and rovibrational spectrum

Cybulski, H., Henriksen, C., Dawes, R., Wang, X-G., Bora, N., Avila, G., Carrington, T. & Fernández, B. 2018 13 p.

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

A new, highly accurate ab initio ground-state intermolecular potential-energy surface (IPES) for the CO-N2 complex is presented. Thousands of interaction energies calculated with the CCSD(T) method and Dunning's aug-cc-pVQZ basis set extended with midbond functions were fitted to an analytical function. The global minimum of the potential is characterized by an almost T-shaped structure and has an energy of -118.2 cm-1. The symmetry-adapted Lanczos algorithm was used to compute rovibrational energies (up to J = 20) on the new IPES. The RMSE with respect to experiment was found to be on the order of 0.038 cm-1 which confirms the very high accuracy of the potential. This level of agreement is among the best reported in the literature for weakly bound systems and considerably improves on those of previously published potentials.
Original language English Physical Chemistry Chemical Physics 13 1463-9076 10.1039/c8cp01373j Published - 2018

Year: 2018

Kristiansen, K. U. & Hogan, S. J. 2018 17, 1, p. 859-908

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

We consider the problem of a slender rod slipping along a rough surface. Painleve [C. R. Seances Acad. Sci., 121 (1895), pp. 112-115; C. R. Seances Acad. Sci., 141 (1905), pp. 401-405; C. R. Seances Acad. Sci., 141 (1905), pp. 546-552] showed that the governing rigid body equations for this problem can exhibit multiple solutions (the indeterminate case) or no solutions at all (the inconsistent case), provided the coefficient of friction mu exceeds a certain critical value mu(p). Subsequently Genot and Brogliato [Eur. J. Mech. A Solids, 18 (1999), pp. 653-677] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead the rod will either stop slipping and stick or it will lift off from the surface. Between these two cases is a special solution for mu > mu(c) > mu(p), where mu(c) is a new critical value of the coefficient of friction. Physically, the special solution corresponds to the rod slipping until it reaches a singular "0/0" point P. Even though the rigid body equations cannot describe what happens to the rod beyond the singular point P, it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a canard [E. Benoit et al., Collect. Math., 31-32 (1981), pp. 37-119]. To overcome the inadequacy of the rigid body equations beyond P, the rigid body assumption is relaxed in the neighborhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point P and the special solution under this regularization, in the limit of vanishing compliance. In this paper, we prove the existence of a canard orbit in a reduced four-dimensional slow-fast phase space, connecting a two-dimensional focus-type slow manifold with the stable manifold of a two-dimensional saddle -type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup.
Original language English S I A M Journal on Applied Dynamical Systems 17 1 859-908 1536-0040 10.1137/17M1122256 Published - 2018

Year: 2018

## Generalized Hamming weights of affine Cartesian codes

Beelen, P. & Datta, M. 2018 51, p. 130-145

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

Let F be any field and A1,…,Am be finite subsets of F. We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F[x1,…,xm] can have in A1×…×Am. In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed–Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well.
Original language English Finite Fields and Their Applications 51 130-145 1071-5797 10.1016/j.ffa.2018.01.006 Published - 2018

Year: 2018

## Frame properties of systems arising via iterated actions of operators

Christensen, O. & Hasannasabjaldehbakhani, M. 2018 (Accepted/In press) 15 p.

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

Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences View the MathML source for some bounded operators Tj and elements φj in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators φ1,…,φJ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system View the MathML source but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.
Original language English Applied and Computational Harmonic Analysis 15 1063-5203 10.1016/j.acha.2018.04.002 Accepted/In press - 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 G06T 17/ 30 A I21 Sep 2017 14 Mar 2016EP20160160088

Research output: 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) 14/03/2016 EP20160160088 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. Association for Computing Machinery, Vol. Part F129312, p. 349-356 8 p.

Research output: 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 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 - University of Kaiserslautern, Kaiserslautern, GermanyDuration: 25 Jul 2017 → 28 Jul 2017Conference number: 42

### 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

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

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

Research output: 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

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

Christensen, O. & Hasannasab, M. 2017 (Accepted/In press)

Research output: 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

## 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.

Research output: 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

## Operator representations of frames

Christensen, O. & Hasannasab, M. 2017 Proceedings of 2017 international conference on sampling theory and applications. IEEE, p. 207-11

Research output: 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, EstoniaDuration: 3 Jul 2017 → 7 Jul 2017

### Conference

Conference 2017 International Conference on Sampling Theory and Applications Estonia Tallinn 03/07/2017 → 07/07/2017

Year: 2017

## Dynamical sampling and frame representations with bounded operators

Christensen, O., Hasannasab, M. & Rashidi, E. 2017 463, p. 634-644

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

The purpose of this paper is to study frames for a Hilbert space H, having the form {Tnφ}n=0∞ for some φ∈H and an operator T:H→H. We characterize the frames that have such a representation for a bounded operator T, and discuss the properties of this operator. In particular, we prove that the image chain of T has finite length N in the overcomplete case; furthermore {Tnφ}n=0∞ has the very particular property that {Tnφ}n=0N−1∪{Tnφ}n=N+ℓ∞ is a frame for H for all ℓ∈N0. We also prove that frames of the form {Tnφ}n=0∞ are sensitive to the ordering of the elements and to norm-perturbations of the generator φ and the operator T. On the other hand positive stability results are obtained by considering perturbations of the generator φ belonging to an invariant subspace on which T is a contraction.
Original language English Journal of Mathematical Analysis and Applications 463 634-644 0022-247X 10.1016/j.jmaa.2018.03.039 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

Research output: 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

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).

Research output: 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 - AMSS South Building, Beijing, ChinaDuration: 18 Sep 2017 → 22 Sep 2017

### 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

## The Fine Structure of Herman Rings

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

Research output: 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

## 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).

Research output: 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, FranceDuration: 18 May 2015 → 22 May 2015Conference number: 15http://scientific-events.weebly.com/1193.html

### 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

## 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).

Research output: 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

## Counting generalized Reed-Solomon codes

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

Research output: 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

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

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

Research output: 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 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 2202 18 1364-5021 10.1098/rspa.2016.0773 Published - 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 33, 1, p. 136-158

Research output: 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 33 1 136-158 1468-9367 10.1080/14689367.2017.1313390 Published - 2017

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).

Research output: 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

## 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

Research output: 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

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

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

Research output: 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

## 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.

Research output: 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

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

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

Research output: 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

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

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

Research output: 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 0096-3003 10.1016/j.amc.2017.06.023 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

Research output: 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

## 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.

Research output: 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.

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

In this note, we prove that the tower given by Bassa, Garcia and Stichtenoth in [4] is a subtower of the one given by Anbar, Beelen and Nguyen in [2]. This completes the study initiated in [16, 2] to relate all known towers over cubic finite fields meeting Zink’s bound with each other.
Original language English Functiones et Approximatio Commentarii Mathematici 57 1 47-60 13 0208-6573 10.7169/facm/1615 Published - 2017

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).

Research output: 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

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

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

Research output: 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

## 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

Research output: 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

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

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

Research output: 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

## 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

Research output: 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

## 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.

Research output: 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

## 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

Research output: 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

## Blowup for flat slow manifolds: Paper

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

Research output: 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

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

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

Research output: 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

## A modular interpretation of various cubic towers

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

Research output: 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

## 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.

Research output: 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 Bulletin of the Australian Mathematical Society 95 2 281-290 10 0004-9727 10.1017/S0004972716001052 Published - 2017

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)).

Research output: 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, GermanyDuration: 25 Jun 2017 → 30 Jun 2017

### 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

## Popov form computation for matrices of Ore polynomials

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

Research output: 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 Association for Computing Machinery 2017 253-260 9781450350648 10.1145/3087604.3087650 Published - 2017 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 - University of Kaiserslautern, Kaiserslautern, GermanyDuration: 25 Jul 2017 → 28 Jul 2017Conference number: 42

### 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

## 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

Research output: 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

## The unitary extension principle on locally compact abelian groups

Christensen, O. & Goh, S. S. 2017 (Accepted/In press) 33 p.

Research output: 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 multivariate Wilson bases

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

Research output: 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 IEEE 2017 192-5 978-1-5386-1565-2 10.1109/SAMPTA.2017.8024456 Published - 2017 2017 International Conference on Sampling Theory and Applications - Tallinn, EstoniaDuration: 3 Jul 2017 → 7 Jul 2017

### Conference

Conference 2017 International Conference on Sampling Theory and Applications Estonia Tallinn 03/07/2017 → 07/07/2017

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

Research output: 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 - University of Waterloo , Waterloo , CanadaDuration: 27 Jul 2017 → 31 Jul 2017

### Conference

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

Year: 2017

## Modified planar functions and their components

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

Research output: 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

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

Bossolini, E., Brøns, M. & Kristiansen, K. U. 2017 16, 4, p. 2233–2258

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

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 16 4 2233–2258 1536-0040 10.1137/17M1120774 Published - 2017

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.

Research output: 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 FederationDuration: 18 Sep 2017 → 22 Sep 2017

### Conference

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

Year: 2017

## Generalized shift-invariant systems and approximately dual frames

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

Research output: 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

## 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)).

Research output: 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, GermanyDuration: 25 Jun 2017 → 30 Jun 2017

### 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

## 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

Research output: 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

## 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.

Research output: 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

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

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

Research output: 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 10.1090/proc/13863 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.

Research output: 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

## 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.

Research output: 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

## Singular limit analysis of a model for earthquake faulting

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

Research output: 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

## 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.

Research output: 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 FederationDuration: 18 Sep 2017 → 22 Sep 2017

### Conference

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

Year: 2017

## Pseudospherical surfaces with singularities

Brander, D. 2017 196, 3, p. 905–928

Research output: 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

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

Research output: 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

Research output: 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

## 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.

Research output: 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 - Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, GermanyDuration: 7 May 2014 → 9 May 2014https://www.wias-berlin.de/workshops/MaDiFa/

### 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

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

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

Research output: 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

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

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

Research output: 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

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

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

Research output: 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

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

Research output: 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

## Approximation by planar elastic curves

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

Research output: 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

## 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.

Research output: 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

## 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

Research output: 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

## Exponential estimates of symplectic slow manifolds

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

Research output: 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

## 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

Research output: 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) - ETH Zurich, Zurich, SwitzerlandDuration: 9 Sep 2016 → 13 Sep 2016http://www.aag2016.ch/

### Conference

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

Year: 2016

## Autorotation: Invited Comment

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

Research output: 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

## Spherical Surfaces

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

Research output: 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

## 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

Research output: Research - peer-reviewConference abstract 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 - Centre de Recerca Matemàtica​ ​​​, Barcelona, SpainDuration: 1 Feb 2016 → 29 Apr 2016

### 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

## 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

Research output: 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, CanadaDuration: 19 Jul 2016 → 22 Jul 2016Conference number: 41http://www.issac-symposium.org/2016/

### 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

## 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

Research output: 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

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

Research output: 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

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

Lemvig, J. 2016 182, 4, p. 899–912

Research output: 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

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

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

Research output: 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

## 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

Research output: 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

## The structure of dual Grassmann codes

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

Research output: 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

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

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

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

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