The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that fundamental domains all such satisfy a very special analytic property: Dirichlet eigenfunctions for Laplace eigenvalue equation are trigonometric functions. In 2008, McCartin two di...

Consider a compact metric space $$(M, d_M)$$ and $$X = M^{{\mathbb {N}}}$$ . We prove Ruelle’s Perron Frobenius Theorem for class of subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from continuous function $$A : M \times \rightarrow {\mathbb {R}}$$ that determines the set admissible sequences. In particular, this includes fi...

where p > 0, c > 0, and λ > 0 are parameters and Ω is an open bounded region with boundary ∂Ω in class C2 in Rn for n ≥ 1. Here g :Ω→ R is a Cα function while h :Ω→ R is a nonnegative Cα function with ‖h‖∞ = 1. When p = 1, (1.1) arises in population dynamics where 1/λ is the diffusion coefficient and ch(x) represents the constant yield harvesting. In this case (p = 1), when g(x) is a positive c...

We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1,∞) and that th...

This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < ǫh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as ǫ → 0, and find the twoterm asymptotics in ǫ → 0 of ea...

The problem of the time required for a diffusing molecule, within a large bounded domain, to first locate a small target is prevalent in biological modeling. Here we study this problem for a small spherical target. We develop uniform in time asymptotic expansions in the target radius of the solution to the corresponding diffusion equation. Our approach is based on combining expansions of a long...

Consider an operator dQ(f) = d dxk (Q(x)f(x)) where Q(x) is some fixed polynomial of degree k. One can easily see that dQ has exactly one polynomial eigenfunction pn(x) in each degree n ≥ 0 and its eigenvalue λn,k equals (n+k)! n! . A more intriguing fact is that all zeros of pn(x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each pn(x) enj...

We present some new error estimates for the eigenvalues and eigenfunctions obtained by the Rayleigh-Ritz method, the common variational method to solve eigenproblems. The errors are bounded in terms of the error of the best approximation of the eigenfunction under consideration by functions in the ansatz space. In contrast to the classical theory, the approximation error of eigenfunctions other...

A method for solving the descriptor discrete-time linear system is focused. For easily, it is converted to a standard discrete-time linear system by the definition of a derivative state feedback. Then partial eigenvalue assignment is used for obtaining state feedback and solving the standard system. In partial eigenvalue assignment, just a part of the open loop spectrum of the standard linear s...

We consider random Schrödinger operators of the form ∆ + ξ , where ∆ is the lattice Laplacian on Zd and ξ is an i.i.d. random field, and study the extreme order statistics of the Dirichlet eigenvalues for this operator restricted to large but finite subsets of Zd . We show that, for ξ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into ...