Narrow Results

In this note, we apply a spectral method to the graph of alternating bilinear forms. In this way, we obtain upper bounds on the size of an alternating rank-metric code for given values of the minimum rank distance. We computationally compare our results with Delsarte’s linear programming bound, observing that they give the same value. For small values of the minimum rank distance, we are able to establish the equivalence of the two methods. The problem remains open for larger values.

For a graph $G$ with $n$ vertices and $m$ edges, the energy (or the adjacency energy) is the sum of the absolute values of the eigenvalues (or adjacency eigenvalues) of $G$, that is, ${ℰ}(G)={\sum }_{i=1}^n|{\lambda }_i(G)|$, where ${\lambda }_i(G)$ is the $i\mathrm{\text{th}}$ adjacency eigenvalue of $G$. In this paper, we discuss the characteristic polynomial and the energy of dendrimers. We first show the formula obtained for the characteristic polynomial and the energy of tree dendrimers considered in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045] are incorrect. Using a different algorithmic procedure we will derive the correct formula for the characteristic polynomial and the energy of dendrimer $d(3,k)$. Further, we obtain the energy of $d(l,k)$ for $k=4$ and $k=5$, which partially answers a problem raised in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045].

The Lanczos method has rapidly become the preferred method of solution for the generalized eigenvalue problems. The recent emergence of parallel computers has aroused much interest in the practical implementation of the Lanczos algorithm on these high performance computers. This paper describes an implementation of a generalized Lanczos algorithm on a distributed memory parallel computer, with specific application to structural dynamic analysis.

One major cost in the parallel implementation of the generalized Lanczos procedure is the factorization of the (shifted) stiffness matrix and the forward and backward solution of triangular systems. In this paper, we review a parallel sparse matrix factorization scheme and propose a strategy for inverting the principal block submatrix factors to facilitate the forward and backward solution of triangular systems on distributed memory parallel computers. We also discuss the different strategies in the implementation of mass-matrix-vector multiplication and how they are used in the implementation of the Lanczos procedure. The Lanczos procedure implemented includes partial and external selective reorthogonalizations. Spectral shifts are introduced when memory space is not sufficient for storing the Lanczos vectors. The tradeoffs between spectral shifts and Lanc-zos iterations are discussed. Numerical results on Intel’s parallel computers, the iPSC/860 hypercube and the Paragon machines will be presented to illustrate the effectiveness and scalability of the parallel generalized Lanczos procedure.

We investigate the asymptotic behavior of large eigenvalues for a class of finite difference self-adjoint operators with compact resolvent in l².

The eigenvalues and eigenfunctions of the one-dimensional quasi-relativistic Hamiltonian (-ℏ²c²d²/dx² + m²c⁴)^1/2 + V_well(x) (the Klein–Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) are studied. Eigenvalues represent energies of a "massive particle in the box" quasi-relativistic model. Approximations to eigenvalues λ_n are given, uniformly in n, ℏ, m, c and a, with error less than C₁ℏca^-1exp(-C₂ℏ^-1mca)n^-1. Here 2a is the width of the potential well. As a consequence, the spectrum is simple and the nth eigenvalue is equal to (nπ/2 - π/8)ℏc/a + O(1/n) as n → ∞. Non-relativistic, zero mass and semi-classical asymptotic expansions are included as special cases. In the final part, some L² and L^∞ properties of eigenfunctions are studied.

We consider a Schrödinger operator with a Hermitian $2\times 2$ matrix-valued potential which is lattice-periodic and can be diagonalized smoothly on the whole ${ℝ}^n.$ In the case of the potential taking its minimum only on the lattice, we prove that the well-known semiclassical asymptotics of the first band spectrum for a scalar potential remains valid for our model.

We show that if the Ricci curvature of a left-invariant metric g on S³ is greater than that of the standard metric g₀, then the eigenvalues of Δ_g are greater than the corresponding eigenvalues of Δ_go.

In this paper, we investigate universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem on a bounded domain in an n-dimensional polydisk 𝔻ⁿ. Moreover, from the domain monotonicity of the eigenvalue, we can prove that if the first eigenvalue of the Dirichlet Laplacian tends to when the domain tends to the polydisk 𝔻ⁿ, then all of the eigenvalues tend to .

On a compact surface endowed with any Spin^c structure, we give a formula involving the Energy-Momentum tensor in terms of geometric quantities. A new proof of a Bär-type inequality for the eigenvalues of the Dirac operator is given. The round sphere 𝕊² with its canonical Spin^c structure satisfies the limiting case. Finally, we give a spinorial characterization of immersed surfaces in 𝕊² × ℝ by solutions of the generalized Killing spinor equation associated with the induced Spin^c structure on 𝕊² × ℝ.

Let $x:M\to {ℝ}^N$ be an $n$-dimensional compact self-shrinker in ${ℝ}^N$ with smooth boundary $\partial M$. In this paper, we study eigenvalues of the operator ${{ℒ}}_r$ on $M$, where ${{ℒ}}_r$ is defined by

In this paper, we show an alternative and accurate solution of the radial Schrödinger equation for the exponential cosine screened Coulomb potential within the framework of the asymptotic iteration method. Unlike other methods, which require approximations for the centrifugal or exponential terms, we show that it is possible to solve the full potential without making any approximations within the framework of this method. The bound state energy eigenvalues are obtained for any n and l values and the results are compared with the findings of different methods for several screening parameters. Moreover, we study the screening parameter δ = 0 case to obtain the energy eigenvalues and corresponding eigenfunctions of this potential in a closed-form.

A very common problem in science is the numerical diagonalization of symmetric or hermitian 3 × 3 matrices. Since standard "black box" packages may be too inefficient if the number of matrices is large, we study several alternatives. We consider optimized implementations of the Jacobi, QL, and Cuppen algorithms and compare them with an alytical method relying on Cardano's formula for the eigenvalues and on vector cross products for the eigenvectors. Jacobi is the most accurate, but also the slowest method, while QL and Cuppen are good general purpose algorithms. The analytical algorithm outperforms the others by more than a factor of 2, but becomes inaccurate or may even fail completely if the matrix entries differ greatly in magnitude. This can mostly be circumvented by using a hybrid method, which falls back to QL if conditions are such that the analytical calculation might become too inaccurate. For all algorithms, we give an overview of the underlying mathematical ideas, and present detailed benchmark results. C and Fortran implementations of our code are available for download from .

The framework of principal component analysis (PCA) based on singular value decomposition (SVD) is applied to the monthly sea surface temperature (SST) observations in the North Atlantic Ocean for the time interval 1856–2008. Multiyear time series of SST for each month are used to investigate the statistical relationship between SST variations from the 12 months. To obtain approximate stationary conditions, the trend and a multidecadal oscillation are removed from the data. The remaining SST residuals exhibit remarkable correlation between successive months, due largely to persistence. PCA demonstrates the dimension reduction of the data sets and provides a robust way of analyzing multivariate observations describing the climate system.

A new family of superlattice structures modulated by the pseudo-Gaussian potential is proposed. The correlation between the configuration of minibands and the superlattice geometry is revealed using the transfer matrix approach.

We present a numerical technique for counting and computing the eigenvalues of a complex tridiagonal matrix, lying in a given region of the complex plane. First, we evaluate the integral of the logarithmic derivative p′(λ)/p(λ), where p(λ) is the characteristic polynomial of the tridiagonal matrix, on a simple closed contour, being the closure of that region. The problem of evaluating this integral is transformed into the equivalent problem of numerically solving a complex initial value problem defined on an ordinary first-order differential equation, integrated along this contour, and solved by the Fortran package dcrkf54.f95 (developed recently by the authors). In accordance with the "argument principle," the value of the contour integral, divided by 2πi, counts the eigenvalues lying in the region. Second, a Newton–Raphson (NR) method, fed by random guesses lying in the region, computes the roots one by one. If, however, NR signals that |p′(λ)| converges to zero, i.e. a multiple root probably exists at the current value λ, then counting the roots within an elementary square centered at λ reveals the multiplicity of this root.

We introduce a quantity of contextuality, i.e. the maximum contextuality of a two-qubit state in 4-cycle measurement scenarios, to study quantum contextuality of two-qubit states for the 4-cycle noncontextual inequalities. Subsequently, we prove several appealing features of this quantity of contextuality, including the faithfulness, the boundedness, the invariance under unitary operations and the convexity. Remarkably, for arbitrary two-qubit state, we reformulate this quantity of contextuality analytically by eigenvalues of the state. Therefore, in 4-cycle measurement scenarios, we establish an efficient way to distinguish two-qubit states that can reveal contextuality from ones that cannot.

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form (defined, without loss of generality but for definiteness, in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation; every Hamiltonian in this class of operators consists of the relativistic kinetic energy , where β > 0 allows for the possibility of more than one particles of mass m, and a spherically symmetric attractive potential V(r), r ≡ |x|. In general, accurate eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of a numerical approximation procedure. Our main emphasis, however, is on the derivation of rigorous semianalytical expressions for both upper and lower bounds to the energy levels of such operators. We compare the bounds obtained within different approaches and present relationships existing between the bounds.

The classical limit of wave quantum mechanics is analyzed. It is shown that the basic requirements of continuity and finiteness to the solution of the form ψ(x) = Ae^{i ϕ (x)} + Be^{-i ϕ (x)}, where and W(x) is the reduced classical action of the physical system, give the asymptote of the wave equation and general quantization condition for the action W(x), which yields the exact eigenvalues of the system.

In the context of relativistic quantum mechanics, we obtain a nonlinear first order differential equation for the energy as a function of the coupling constant of a central potential. This differential equation is only exact for power law and logarithmic potentials in the massless limit. For other potentials, we discuss under which conditions the differential equation yields rigorous upper and lower limits on the value of energy levels. These results are applied to the Cornell potential used in meson spectroscopy. We also show that the method applies to noncentral potentials.

The Klein–Gordon and Dirac equations are considered in a semiinfinite laboratory (x > 0) in the presence of background metrics ds² = u²(x)η_μν dx^μ dx^ν and ds² = -dt² + u²(x)η_ij dxⁱ dx^j with u(x) = e^±gx. These metrics have nonconstant scalar-curvatures. Various aspects of the solutions are studied. For the first metric with u(x) = e^gx, it is shown that the spectra are discrete, with the ground state energy for spin-0 particles. For u(x) = e^-gx, the spectrums are found to be continuous. For the second metric with u(x) = e^-gx, each particle, depends on its transverse-momentum, can have continuous or discrete spectrum. For Klein–Gordon particles, this threshold transverse-momentum is , while for Dirac particles it is g/2. There is no solution for u(x) = e^gx case. Some geometrical properties of these metrics are also discussed.