The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence of the spectral formula chi(G)=sum_x (-1)^dim(x) = p(G)-n(G), where p(G) is the number of positive eigenvalues and n(G) is the number of negati...

A real n-dimensional homogeneous polynomial f (x) of degree m and a real constant c define an algebraic hypersurface S whose points satisfy f (x) = c. The polynomial f can be represented by Axm where A is a real mth order n-dimensional supersymmetric tensor. In this paper, we define rank, base index and eigenvalues for the polynomial f , the hypersurface S and the tensor A. The rank is a nonneg...

Counting lattice points inside a ball of large radius in Euclidean space is classical problem analytic number theory, dating back to Gauss. We propose variation on this problem: studying the asymptotics measure an integer affine planes ball. The first term volume ball; we study size remainder term. While equivalent counting eigenvalues Laplace operator torus, our corresponds integrated density ...

Let n be any positive integer and Fn be the friendship (or Dutch windmill) graph with 2n+1 vertices and 3n edges. Here we study graphs with the same adjacency spectrum as Fn. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let G be a graph cospectral with Fn. Here we prove that if G has no cycle of length 4 or 5, then G ∼= Fn. Moreover if G...

We study 1D discrete Schrödinger operators H with integer-valued potential and show that, (i), invertibility (in fact, even just Fredholmness) of always implies its half-line compression H+ (zero Dirichlet boundary condition, i.e. matrix truncation). In particular, the eigenvalues avoid zero – all other integers. use this result to conclude (ii), finite section method (approximate inversion via...

We consider Schrödinger operators with periodic potentials in the positive quadrant on plane Dirichlet boundary conditions. show that for any integer N and interval I there exists a potential such operator has eigenvalues counted multiplicity this is no other spectrum interval. Furthermore, to right left of it essential spectrum. Moreover, we prove similar results product an orthant Euclidean s...

This paper presents a combinatorial study of sums integer powers the cotangent. Our main tool is realization cotangent values as eigenvalues simple self-adjoint matrix with complex entries. We use trace method to draw conclusions about and series expansions generating function provide explicit evaluations; it remarkable that throughout calculations combinatorics are governed by higher tangent a...

Abstract We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $${{\,\mathrm{\mathbb {T}}\,}}^2$$ T 2 in the homotopy class any linear expanding map with integer eigenvalues. These examples exhibit b...

In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...

So far, various components of image characteristics have been used for steganalysis, including the histogram characteristic function, adjacent colors distribution, and sample pair analysis. However, some certain steganography methods have been proposed that can thwart some analysis approaches through managing the embedding patterns. In this regard, the present paper is intended to introduce a n...