A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate an...

Most real world engineering design problems, such as cross-country water mains, include combinations of continuous, discrete, and binary value decision variables. Very often, the binary decision variables associate with the presence and/or absence of some nominated alternatives or project’s components. This study extends an existing continuous Ant Colony Optimization (ACO) algorithm to simultan...

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions – the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical c...

this paper consists of three sections. in the first section an efficient method is used for decomposition of the canonical matrices associated with repetitive structures. to this end, cylindrical coordinate system, as well as a special numbering scheme were employed. in the second section, divide and conquer method have been used for eigensolution of these structures, where the matrices are in ...

The d-dimensional discrete Schrödinger operator whose potential is supported on the subspace Zd2 of Zd is considered: H = Ha+VM , where Ha = H0+Va, H0 is the d-dimensional discrete Laplacian, Va is a constant “surface” potential, Va(x) = aδ(x1), x = (x1, x2), x1 ∈ Zd1 , x2 ∈ Zd2 , d1 + d2 = d, and VM (x) = gδ(x1) tanπ(α · x2 + ω) with α ∈ Rd2 , ω ∈ [0, 1). It is proved that if the components of...

Saddle dynamics is a time continuous to efficiently compute the any-index saddle points and construct solution landscape. In practice, needs be discretized for numerical computations, while corresponding analyses are rarely studied in literature, especially high-index cases. this paper we propose convergence analysis of discrete dynamics. To specific, prove local linear rates schemes dynamics, ...

The linearized Vlasov equation for a plasma system in a uniform magnetic field and the corresponding linear Vlasov operator are studied. The spectrum and the corresponding eigenfunctions of the Vlasov operator are found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and complex. Interestingly, the real eigenvalues are infinitely degenerat...

The field $Q_{p}$ of $p$-adic numbers is defined as the completion of the field of the rational numbers $Q$ with respect to the  $p$-adic norm $|.|_{p}$. In this paper, we study the continuous and discrete $p-$adic shearlet systems on $L^{2}(Q_{p}^{2})$. We also suggest discrete $p-$adic shearlet frames. Several examples are provided.

Certain solutions to Harper’s equation are discrete analogues of (and approximations to) the Hermite–Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscill...

A computational scheme for solving elliptic boundary value problems with axially symmetric confining potentials using different sets of one-parameter basis functions is presented. The efficiency of the proposed symbolic-numerical algorithms implemented in Maple is shown by examples of spheroidal quantum dot models, for which energy spectra and eigenfunctions versus the spheroid aspect ratio wer...