Abstract:
We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm?en-Lindelof theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.

Abstract:
In this paper, we propose a fast algorithm for computing the DGFT (Discrete Generalized Fourier Transforms) on hexagon domains 6], based on the geometric properties of the domain. Our fast algorithm (FDGFT) reduces the computation complexity of DGFT from O(N4) to O(N2logN). In particulary, for N = 2P23P34P45P56P6, the floating point computation working amount equals to (17/2P-2 + 16p3 + 135/8P4 + 2424/25p5 + 201/2p6)3N2. Numerical examples are given to access our analysis.

Abstract:
The discrete Fourier transform defines a unitary matrix operator. The logarithm of this operator is computed, along with the projection maps onto its eigenspaces. A geometric interpretation of the discrete Fourier transform is also given.

Abstract:
In this paper, we investigate the stationarity of stochastic processes in the fractional Fourier domains. We study the stationarity of a stochastic process after performing fractional Fourier transform (FRFT), and discrete fractional Fourier transform (DFRT) on both continuous and discrete stochastic processes, respectively. Also we investigate the stationarity of the fractional Fourier series (FRFS) coefficients of a continuous time stochastic process, and the stationarity of the discrete time fractional Fourier transform (DTFRFT) of a discrete time stochastic process. Closed formulas of the input process autocorrelation function and pseudo-autocorrelation function after performing the fractional Fourier transform are derived given that the input is a stationary stochastic process. We derive a formula for the output autocorrelation as a function of the $a^{th}$ power spectral density of the input stochastic process, also we derived a formula for the input fractional power spectral density as a function of the fractional Fourier transform of the output process autocorrelation function. We proved that, the input stochastic process must be zero mean to satisfy a necessary but not a sufficient condition of stationarity in the fractional domains. Closed formulas of the resultant statistics are also shown. It is shown that, in case of real input process, the output process is stationary if and only if the input process is white. On the other hand, if the input process is a complex process, it should be proper white process to obtain a stationary output process.

Abstract:
A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^3$.

Abstract:
A recent result of G. Cz\'edli and E.\,T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding "forks". We give a construction that accomplishes the same by deleting elements, by "resections".

Abstract:
The continuum theory of domain structures in ferromagnetic/superconducting bilayers fails when the equilibrium domain size becomes comparable with effective penetration depth $\Lambda$. Instead, a lattice of discrete vortices must be considered. Here, we report our results on the discrete vortex lattices in stripe domain structures of ferromagnetic/superconducting bilayers. The vortices are assumed to be situated periodically on chains in stripe domains. We study the configurations containing up to three chains per domain, and calculate their equilibrium energies, equilibrium domain size and vortex positions through a method based on London-Maxwell equations. In equilibrium, the vortices in the neighbor domains are half-way shifted while they are next to each other in the same domain. Additionally, more vortex chains per domain appear spontaneously depending on magnetization and domain wall energy.

Abstract:
We establish a new method called Discrete Weierstrass Fourier Transform to approximate data sets, which is a generalization of Discrete Fourier Transform. The theory of this method as well as some experiments are analyzed in this paper. In some examples, this method has a faster convergent speed than Discrete Fourier Transform.

Abstract:
The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.

Abstract:
In this article, we prove that finite semidistributive lattices are dismantlable if and only if they are planar. This extends a well-known result by Kelly and Rival that states the same property for finite distributive lattices. Moreover, we show how the breadth of finite semidistributive lattices can be computed with the help of canonical join representations. We use this result to conclude that the breadth of a finite semidistributive dismantlable lattice cannot exceed $2$.