(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family, allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms, involving the symmetric and antisymmetric multivariate sine and cosine functions.Comment: 25 pages, no figures; LaTaX; corrected typo

Six types of E−E-functions of the Lie groups O(5) and G(2)

New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently found in the literature. All the families are fully characterized, their most important properties are described, namely their continuous and discrete orthogonalities and decompositions of their products.Comment: 25 pages, 13 figure

Image Sampling with Quasicrystals

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary materials, please visit at http://www.Eyemaginary.com/Portfolio/Publications.htm

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2

Three variable exponential functions of the alternating group

New class of special functions of three real variables, based on the alternating subgroup of the permutation group $S_3$, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the $E-$functions of $C_3$ is shown. Continuous interpolation of the three dimensional data is studied and exemplified.Comment: 10 pages, 3 figure

A well-posed optimal spectral element approximation for the Stokes problem

A method is proposed for the spectral element simulation of incompressible flow. This method constitutes in a well-posed optimal approximation of the steady Stokes problem with no spurious modes in the pressure. The resulting method is analyzed, and numerical results are presented for a model problem

High granularity tracker based on a Triple-GEM optically read by a CMOS-based camera

The detection of photons produced during the avalanche development in gas chambers has been the subject of detailed studies in the past. The great progresses achieved in last years in the performance of micro-pattern gas detectors on one side and of photo-sensors on the other provide the possibility of making high granularity and very sensitive particle trackers. In this paper, the results obtained with a triple-GEM structure read-out by a CMOS based sensor are described. The use of an He/CF$_4$ (60/40) gas mixture and a detailed optimization of the electric fields made possible to obtain and very high GEM light yield. About 80 photons per primary electron were detected by the sensor resulting in a very good capability of tracking both muons from cosmic rays and electrons from natural radioactivity

On E-functions of Semisimple Lie Groups

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of $G$. We distinguish two cases of even Weyl groups -- one is the direct product of even Weyl groups of simple components of $G$, the second is the full even Weyl group of $G$. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two -- we describe in detail $E-$transforms of semisimple Lie groups of rank 3.Comment: 17 pages, 2 figure

Four dimensional Lie symmetry algebras and fourth order ordinary differential equations

Realizations of four dimensional Lie algebras as vector fields in the plane are explicitly constructed. Fourth order ordinary differential equations which admit such Lie symmetry algebras are derived. The route to their integration is described.Comment: 12 page