Spectrum of the Kerzman-Stein Operator for the Ellipse

The skew-hermitian part of the Cauchy operator, defined with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coefficient in the asymptotic expansion of the eigenvalues.     

Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals

Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of ℝ. If k belongs to class defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C¹ case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out.     

On the Spectrum of the Weighted p-Biharmonic Operator with Weight

In this work we give a non decreasing sequence of positive eigenvalues of the weighted p-biharmonic operator with weight and with Navier boundary conditions, then we study the simplicity and the isolation of the first positive eigenvalue. Finally, we study the one dimensional case.     

On the 2-factor index of a graph

The 2-factor index of a graph G, denoted by f(G), is the smallest integer m such that the m-iterated line graph L^m(G) of G contains a 2-factor. In this paper, we provide a formula for f(G), and point out that there is a polynomial time algorithm to determine f(G).     

Eigenvalues of integral operators onL 2(I) given by analytic kernels

Let {_n} _n=0 be the eigenvalue sequence of a symmetric Hilbert-Schmidt operator onL ₂(I). WhenI is an open interval, a necessary condition for {_n} _n=0 to be in the sequence space is obtained. WhenI is a closed bounded interval, sufficient conditions for {_n} _n=0 to be in the sequence space ^– are obtained.     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.     

On an Inverse Spectral Problem for a Convolution Integro-differential Operator

The operator of double differentiation perturbed by the composition of a Volterra convolution operator and the differentiation one on a finite interval with Dirichlet boundary conditions is considered. It is proved that the standard asymptotics is necessary and sufficient for an arbitrary sequence of complex numbers to be the spectrum of such an operator, which is determined uniquely. A constructive procedure for solving the inverse problem is given. Received: March 5, 2007.     

Operator norm inequalities in semi-Hilbertian spaces

In this work we extend Cordes inequality, McIntosh inequality and CPR-inequality for the operator seminorm defined by a positive semidefinite bounded linear operator A.     

Explicit matchings in the middle levels of the Boolean lattice

New classes of explicit matchings for the bipartite graph (k) consisting of the middle two levels of the Boolean lattice on 2k+1 elements are constructed and counted. This research is part of an ongoing effort to show that (k) is Hamiltonian.Supported by Office of Naval Research contract N00014-85K-0494.Supported by National Science Foundation grant DMS-8041281.     

On the Burkard-Hammer condition for hamiltonian split graphs

A graph G=(V,E) is called a split graph if there exists a partition V=I∪K such that the subgraphs of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary but not sufficient condition for hamiltonian split graphs with |I|<|K|. In this paper, we show that the Burkard-Hammer condition is also sufficient for the existence of a Hamilton cycle in a split graph G such that 5≠|I|<|K| and the minimum degree δ(G)?|I|-3. For the case 5=|I|<|K|, all split graphs satisfying the Burkard-Hammer condition but having no Hamilton cycles are also described.     

On the discrete version of Picone's identity

For a real number p with 1<p we consider the first eigenvalues of the p-Laplacian on graphs, and estimates for the solutions of p-Laplace equations on graphs. We provide a discrete version of Picone's identity and its application. More precisely, we prove a Barta-type inequality for graphs with boundary. Finally, we provide a discrete version of the anti-maximum principle.     

Transposition generation of alternating permutations

A permutation _{1 2}... _n is alternating if ₁< ₂> ₃< ₄.... Alternating permutations are counted by the Euler numbers. Here we show that alternating permutations can be listed so that successive permutations differ by a transposition, ifn is odd. Extensions and open problems are mentioned.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

On a nonlocal diffusion model with Neumann boundary conditions

We study a nonlocal diffusion model analogous to heat equation with Neumann boundary conditions. We prove the existence and uniqueness of solutions and a comparison principle. Furthermore, we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable.     

Chromatic sums of general maps on the sphere and the projective plane

In this paper, we study the chromatic sum functions of rooted general maps on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of rooted loopless maps, bipartite maps and Eulerian maps are also derived. Moreover, some explicit expressions of enumerating functions are also derived.     

The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I

We compute the number of rhombus tilings of a hexagon with sidesN,M,N, N,M,N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of lengthM.     

The total graph and regular graph of a commutative ring

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.     

The energy change of weighted graphs

The energy of an (edge)-weighted graph is the sum of the absolute values of the eigenvalues of its (weighted) adjacency matrix. We study how the energy of a weighted graph changes when the weights change. We give some sufficient conditions so that the energy of a weighted graph increases when the positive weight increases. We also characterize some classes of weighted graphs satisfying these sufficient conditions.     

Measurable chromatic number of geometric graphs and sets without some distances in euclidean space

LetH be a set of positive real numbers. We define the geometric graphG _H as follows: the vertex set isR ⁿ (or the unit circleS ¹) andx, y are joined if their distance belongs toH. We define the measurable chromatic number of geometric graphs as the minimum number of classes in a measurable partition into independent sets. In this paper we investigate the difference between the notions of the ordinary and measurable chromatic numbers. We also prove upper and lower bounds on the Lebesgue upper density of independent sets.