Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints

The local dependence of static response and eigenvalues on the shape of plates and plane elastic solids is characterized. The so-called material derivative method is used. The shape sensitivity analysis includes, besides linear problems, nonlinear problems with unilateral conditions, e.g., the frictionless contact problem for an elastic body on a rigid foundation. The results on shape sensitivity analysis can be used to obtain expressions for variations of integral functionals that arise in structural optimization problems.The authors are indebted to Professor N. Olhoff and Dr. M. P. Bendsøe for stimulating discussions and valuable comments on design sensitivity analysis.     

Normal modes of oscillations of a rotating membrane with a rigid central insert (an application of Heun functions)

The eigenvalues problem eigenvalues for the equation of the transverse oscillations of a homogeneous annular membrane with a rigid insert, rotating with a constant angular velocity about its central axis, is considered. Exact analytical expressions for the eigenfunctions in terms of special functions (local Heun functions), as well as normalization integrals, are found. An explicit expression for the time-invariant shape of the membrane during regular precession of its rotation axis is obtained.     

Gaussian integral circulant digraphs

The spectrum of a digraph in general contains real and complex eigenvalues. A digraph is called a Gaussian integral digraph if it has a Gaussian integral spectrum that is all eigenvalues are Gaussian integers. In this paper, we consider Gaussian integral digraphs among circulant digraphs.     

The integral graphs with index 3 and exactly two main eigenvalues

A graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues. An eigenvalue of a graph is called main eigenvalue if it has an eigenvector such that the sum of whose entries is not equal to zero. In this paper, we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3.     

On the Laplacian integral tricyclic graphs

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.     

Algorithm for min-range multiplication of affine forms

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.     

On eigenvalues and equivalent transformation of trigonometric matrices

This paper discusses some issues related to trigonometric matrices arising from the design of finite impulse response (FIR) digital filters. A conjecture on the eigenvalues of a trigonometric matrix is posed with a partial proof given. A new result is also presented on the related equivalent transformation of this trigonometric matrix into a diagonal matrix.     

边界元近奇异积分计算的迭代Sinh-Sigmoidal组合式变换法

精确有效地消除积分的近奇异性是三维边界元法在工程应用中的首要问题.当源点与三角形积分单元间的距离无限趋近于零时,会出现近奇异积分问题,积分单元的形状和投影点的位置都是影响近奇异积分计算精度的重要因素.现有的非线性变换法大多只关注径向上积分的近奇异性,而忽略了角度方向和积分单元形状的影响,在投影点接近三角形积分单元边界的情况下,无法获得令人满意的计算精度,并且对子三角形积分单元的形状非常敏感.因此提出了一种改进的基于自适应分块技术和不同坐标变换的迭代sinh sigmoidal组合式变换法,分别消除径向和角度方向积分的近奇异性,在确保计算精度的同时,大大减小了计算规模.数值算例验证了该方法的有效性.     

Integral pentavalent Cayley graphs on abelian or dihedral groups

A graph is called integral, if all of its eigenvalues are integers. In this paper, we give some results about integral pentavalent Cayley graphs on abelian or dihedral groups.     

An initial-value method for integral operators V. Kernels depending upon a parameter

A new technique is presented for the calculation of the eigenvalues and eigenfunctions of integral operators where the kernel is an entire function of the eigenvalue parameter. The method combines a number of results from functions of a complex variable and integral equations with a new technique for converting Fredholm integral equations into an initial-value system of ordinary differential equations. Several pertinent examples are presented which illustrate the efficacy of the procedure.     

Some constructions of integral graphs

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Integral graphs are very rare and difficult to find. In this article, we introduce some general methods for constructing such graphs. As a consequence, some infinite families of integral graphs are obtained.     

Note on the Riccati method for differential eigenvalue problems showing algebraic growth

The Riccati transformation method is used to pick out eigenvalues of a system of linear ordinary differential equations on a semi-infinite interval. The eigenvalues are determined by the condition which ensures algebraic rather than exponential growth of the solutions as the independent variable tends to infinity. Numerical results which demonstrate the viability of the Riccati transformation method for this type of problem are included.     

整循环图的秩（英文）

A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulant.The rank of a graph is defined to be the rank of its adjacency matrix.This importance of the rank,due to applications in physics,chemistry and combinatorics.In this paper,using Ramanujan sums,we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.     

Integral and distance integral Cayley graphs over generalized dihedral groups

Journal of Algebraic Combinatorics - A graph is said to be integral (resp. distance integral) if all the eigenvalues of its adjacency matrix (resp. distance matrix) are integers. Let H be a finite...     

A Hilbert-Schmidt integral operator and the Weil distribution

In this paper, a positive operator is given. It is shown that the product of this positive operator and the convolution operator is a trace class Hilbert-Schmidt integral operator and has nonnegative eigenvalues. A formula is given for the trace of this product operator. It seems that this product operator is the closest trace class integral operator which has nonnegative eigenvalues and is related to the Weil distribution in the context of Connes’ program for the Riemann hypothesis. A relation is given between the trace of the product operator and the Weil distribution.

     

N-group neutron transport theory: A criticality problem in slab geometry

The steady-state equation for N-group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.     

Combined classical and generalized integral transform approaches for the analysis of the dynamic behavior of a damaged structure

The classical integral transform technique (CITT) is employed to derive an exact analytical solution for the transverse vibrations of a damaged Euler–Bernoulli beam simply supported at both ends. The exact analytical solution derived with the CITT approach requires knowledge of the eigenvalues and eigenfunctions of a related eigenvalue problem, whose hybrid solution is further obtained with the generalized integral transform technique (GITT). Expressions derived for the natural frequencies, mode shapes and frequency response function are valid for any prescribed damage field. Numerical results obtained for a damaged beam with single and multiple damages demonstrate the accuracy, convergence and robustness of the two approaches when combined.     

On a variational problem for an elastic membrane supporting a heavy ball

An elastic membrane or a soap film is investigated that is bounded by a planar curve and supports a heavy ball. The ball is allowed to move on the surface, and its position, together with the shape of the membrane will furnish a minimum to the potenial energy. This leads to an obstacle problem for Dirichlet's integral or the area functional where only the shape of the obstacle but not its position in space is given.     

On Integral Graphs with Few Cycles

A graph is called integral if the spectrum of its adjacency matrix has only integer eigenvalues. In this paper, all integral graphs with at most two cycles (trees, unicyclic and bicyclic graphs) with no eigenvalue 0 are identified. Moreover, we give some results on unicyclic integral graphs with exactly one eigenvalue 0.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.