Two applications of relative difference sets: Difference triangles and negaperiodic autocorrelation functions

The well-known difference sets have various connections with sequences and their correlation properties. It is the purpose of this note to give two more applications of the (not so well known) relative difference sets: we use them to construct difference triangles (based on an idea of A. Ling) and we show that a certain nonexistence result for semiregular relative difference sets implies the nonexistence of negaperiodic autocorrelation sequences (answering a question of Parker [Even length binary sequence families with low negaperiodic autocorrelation, in: Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Melbourne, 2001, Lecture Notes in Computer Science, vol. 2227, Springer, Berlin, 2001, pp. 200-209.]).     

Parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume 4

On reconstructing graphs from their sets of subgraphs

Using only the set of point-deleted subgraphs, several invariants of a graph are derived. That information is then used to show that disconnected graphs and separable graphs without endpoints are reconstructible from their sets of deleted subgraphs. Similar results are obtained in the line-deleted case.     

Generating the centrally symmetric 3-polyhedral graphs

We show that the graphs of the centrally symmetric 3-polytopes can be generated from the graphs of the cube and octahedron by applying pairs of symmetric face splittings.     

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ<Δ(X), where Δ(X) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.     

Unconditional and symmetric sets inn-dimensional normed spaces

Isoperimetric inequalities are used to obtain measure estimates on almost constancy sets of functions on product spaces. These are applied to produce almost unconditional or symmetric block sequences from given sequences. Their length, which is (logn)^1/2 in the general case, improves ton ^a where a cotype condition is imposed or when the given sequences arep-type attaining for somep<2. In thep-type attaining case, block sequences (1+ε)-equivalent to the unit vector basis ofl _p ^m can be obtained when log logm ∼ log logn. Research supported in part by NSF Grant MCS 7902489.     

An integral semiclassical method for calculating the spectra for centrally symmetric potentials

We develop a new approach to semiclassical quantization conditions that does not involve asymptotic expansions and is based on exact general properties of wave equations and their spectra. For centrally symmetric potentials, the quantization conditions depend only on a collection of integrals involving powers of the classical momentum. The energy levels calculated using these conditions are in good agreement with numerical data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 463–480, September, 2000.     

Detecting the birth and death of finite-time coherent sets

Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian, showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent sets.     

An inequality between Dirichlet and Neumann eigenvalues in a centrally symmetric domain

We prove an inequality between Dirichlet and Neumann eigenvalues of the Laplacian in a centrally symmetric Euclidean domain.

     

EXPEDIS: An exact penalty method over discrete sets

We address the problem of minimizing a quadratic function subject to linear constraints over binary variables. We introduce the exact solution method called EXPEDIS  where the constrained problem is transformed into a max-cut instance, and then the whole machinery available for max-cut can be used to solve the transformed problem. We derive the theory in order to find a transformation in the spirit of an exact penalty method; however, we are only interested in exactness over the set of binary variables. In order to compute the maximum cut we use the solver BiqMac. Numerical results show that this algorithm can be successfully applied on various classes of problems.     

Complex symmetric functions and generalized discrete Fourier transform

General systems of polynomials, satisfying prescribed symmetries and orthonormal on the unit circle with respect to weight functions belonging to a suitable symmetry class, are used in order to generalize the Discrete Fourier Transform (DFT) and the FFT algorithm.     

Resolution of the wavefront set via discrete sets

We introduce discrete wave-front sets of sup type and prove that they coincide with the Hörmander wave-front set of a distribution. To that end we recall the notion of admissible lattice pairs and wave-front sets in Fourier-Lebesgue spaces. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)