The threshold weight of a graph

The threshold weight of a graph G is introduced as a measure of the amount by which G differs from being a threshold graph. The threshold graphs are precisely the graphs whose threshold weights are 0. At the opposite extreme is the class of graphs for which the threshold weight is the maximum possible. Such graphs are defined as heavy graphs. Among the results are as following: A theorem that specifies the threshold weight of any triangle-free graph; necessary and sufficient conditions for a heavy graph in terms of the solvability of a system of linear inequalities; some sufficient conditions for a graph to be heavy and a necessary condition (conjectured to be sufficient, as well) for a heavy graph in terms of its cliques.     

Mathematical model of the pipeline with angular joint of elements

A mathematical model of the pipeline as a Koiter-Vlasov moment shell with kink lines of the surface at the junctions of the pipe segments was constructed and substantiated. The following tasks are solved: The geometric parameters of the mechanical system as a three-dimensional elastic body and as a shell are found; force factors of the shell are expressed in terms of displacements of the middle surface of the wall, taking into account the presence of a kink line; equations of pipe equilibrium are derived as Koiter-Vlasov shells with an edge along the line; forces on oblique sections are expressed as functions of shell movements; the conjugation conditions on the pipe joint line for displacements and the angle of rotation of the normal are imposed and justified; conjugation conditions for bending moments, shear forces, transverse and normal forces are imposed and justified. The presence of the solution singularity at points on the connection line of the pipe segments is theoretically established and illustrated by the numerical example.     

Mathematical analysis of investment systems

Investment systems are studied using a framework that emphasize their profiles (the cumulative probability distribution on all the possible percentage gains of trades) and their log return functions (the expected average return per trade in logarithmic scale as a function of the investment size in terms of the percentage of the available capital). The efficiency index for an investment system, defined as the maximum of the log return function, is proposed as a measure to compare investment systems for their intrinsic merit. This efficiency index can be viewed as a generalization of Shannon's information rate for a communication channel. Applications are illustrated.     

Estimation of the closeness of distributions in terms of identical moments

Estimates for Levy's distance, as well as for a uniform distance, are presented in terms of the moments of the distribution functions that are being compared. These estimates are derived on the basis of the classical problem of moments.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 64–72, 1980.     

Useful properties of the coefficients of the Slevinsky-Safouhi formula for differentiation

Properties of coefficients of the Slevinsky-Safouhi formula are derived. These properties are: matrix and orthogonality properties; the characterization of a zero array; generating functions; asymptotics; and, recurrence relations for computing a sum as efficiently as possible. These properties are useful in a numerical or computational setting.     

Qualitative results for solutions of the steady fisher-KPP equation

In this letter, the homogeneous Dirichlet problem involving the N-dimensional Fisher-KPP equation, a reaction-diffusion model which arises in study of population genetics, is investigated for a class of nonlinear polynomial growth laws. Existence and uniqueness conditions for positive (i.e., physically realistic), steady-state solutions on finite domains, or habitats, are noted and stability questions are addressed. Of particular interest are habitats that can be modeled as open balls. For these cases, two relatively recent and powerful theorems from nonlinear analysis are employed to ascertain important qualitative information. Specifically, these solutions are shown to be strictly decreasing and radially symmetric, as well as achieving a stationary maximum at the habitat's center. In addition, the function spaces containing these solutions are determined. Last, the effects of the solution parameters are investigated numerically for the physically relevant cases of N = 2 and 3, the temporal evolution of a particular solution is illustrated, and connections to nuclear reactor science, as well as other fields, are noted.     

Matrix representations of split Bezoutians

Inverses of symmetric (or skewsymmetric) Toeplitz matrices as well as of centrosymmetric (or centro-skewsymmetric) Toeplitz-plus-Hankel matrices can be represented as sums of two split Bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors. Thus it is desirable to find matrix representations for split Bezoutians B. This is the main aim of the present paper.Recursion formulas for the entries of B are presented, bases of very simple split Bezoutians or of sparse matrices are constructed, and B is represented as a corresponding linear combination. Moreover, matrix representations of Gohberg/Semencul type are established.     

Bayesian estimation of system reliability in Brownian stress-strength models

A stress-strength system fails as soon as the applied stress,X, is at least as much as the strength,Y, of the system. Stress and strength are time-varying in many real-life systems but typical statistical models for stress-strength systems are static. In this article, the stress and strength processes are dynamically modeled as Brownian motions. The resulting stress-strength system is then governed by a time-homogeneous Markov process with an absorption barrier at O. Conjugate as well as non-informative priors are developed for the model parameters and Markov chain sampling methods are used for posterior inference of the reliability of the stress-strength system. A generalization of this model is described next where the different stress-strength systems are assumed to be exchangeable. The proposed Bayesian analyses are illustrated in two examples where we obtain posterior estimates as well as perform model checking by cross-validation.     

Spreading lengths of Hermite polynomials

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments, mentioned previously, are posed.     

Lower-bounds on the connectivities of a graph

This article presents a study of the connectivities of a graph as a function of other graph parameters such as the number of vertices, the maximum degree, and the diameter. As a result, lower-bounds on the connectivities of a graph as a function of these parameters are computed. These bounds could serve as sufficient conditions for a graph to be h-edge-connected or k-connected. Consequently, the connectivity characteristics of many of the densest known graphs are determined.     

The structure of the centralizer of a permutation

Relative to the symmetric groups S_n the structure of centralizers of permutations are known as direct products of certain general wreath products. A recent generalization of the cycle notation for partial one-one transformations (charts) is applied to show that relative to the symmetric inverse semigroups C_n the structure of centralizers of permutations are also direct products of certain subsemigroups of a wreath product, and this latter wreath product includes the former as a subgroup. A necessary and sufficient condition is given for two charts to commute and the approach for the C_n-case parallels and generalizes the one for the S_n-case. As a result, the C_n-case yields the standard known characterizations of commuting permutations, as well as formulas for the orders of centralizers as corollaries. It is an open problem to extend these results to the centralizers of arbitrary charts in C_n. This research was partially supported by a Mary Washington College Faculty Development Grant.     

Inverse Laplace transforms of a class of non-rational fractional functions

Vieta's famous product using factors that are nested radicals is the oldest infinite product as well as the first non-iterative method for finding π. In this paper a simple geometric construction intimately related to this product is described. The construction provides the same approximations to π as are given by partial products from Vieta's formula. Upper and lower bounds for the error are obtained in a visual way.     

The structure and construction of bi-Frobenius Hom-algebras

Bi-Frobenius Hom-algebras are introduced in this paper. They provide a common generalization of finite dimensional monoidal Hom-Hopf algebras and of bi-Frobenius algebras introduced by Doi and Takeuchi. The different conditions for finite dimensional monoidal Hom-algebras to be bi-Frobenius Hom-algebras are discussed. The substructures, quotient structures as well as morphisms of bi-Frobenius Hom-algebras are also studied. In addition, the Radford’s formula for S⁴ of a bi-Frobenius Hom-algebra is shown. The semisimplicity and separability for a special class of finite dimensional bi-Frobenius Hom-algebras are researched finally, which presents a version of Maschke’s theorem for this family.     

Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry

Rodolfo Rodríguez ^{The aim of this paper is to analyse a mixed finite-element methodfor computing the vibration modes of a Timoshenko curved rodwith arbitrary geometry. Optimal order error estimates are provedfor displacements, rotations and shear stresses of the vibrationmodes, as well as a double order of convergence for the vibrationfrequencies. These estimates are essentially independent ofthe thickness of the rod, which leads to the conclusion thatthe method is locking-free. Numerical tests are reported inorder to assess the performance of the method.}     

A simulation-based approach to the study of coefficient of variation of Gompertz distribution under progressive first-failure censoring

In applied statistics, the coefficient of variation is widely used. However, inference concerning the coefficient of variation of non-normal distributions are rarely reported. In this article, a simulation-based Bayesian approach is adopted to estimate the coefficient of variation (CV) under progressive first-failure censored data from Gompertz distribution. The sampling schemes such as, first-failure censoring, progressive type II censoring, type II censoring and complete sample can be obtained as special cases of the progressive first-failure censored scheme. The simulation-based approach will give us a point estimate as well as the empirical sampling distribution of CV. The joint prior density as a product of conditional gamma density and inverted gamma density for the unknown Gompertz parameters are considered. In addition, the results of maximum likelihood and parametric bootstrap techniques are also proposed. An analysis of a real life data set is presented for illustrative purposes. Results from simulation studies assessing the performance of our proposed method are included.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

Studying the sensitivity of pollutants’ concentrations caused by variations of chemical rates

A systematic procedure for sensitivity analysis of a case study in the area of air pollution modeling has been performed. Contemporary mathematical models should include a large set of chemical and photochemical reactions to be established as a reliable simulation tool. The Unified Danish Eulerian Model is in the focus of our investigation as one of the most advanced large-scale mathematical models that describes adequately all physical and chemical processes.Variance-based methods are one of the most often used approaches for providing sensitivity analysis. To measure the extent of influence of the variation of the chemical rate constants in the mathematical model over pollutants’ concentrations the Sobol’ global sensitivity indices are estimated using efficient techniques for small sensitivity indices to avoid a loss of accuracy. Studying relationships between input parameters and the model’s output as well as internal mechanisms is very useful for a verification and an improvement of the model and also for development of monitoring and control strategies of harmful emissions, for a reliable prediction of the final output of scenarios when the concentration levels of pollutants are exceeded. The proposed procedure can also be applied when other large-scale mathematical models are used.     

Diffraction of light by superposed parallel supersonic waves,one being then-th harmonic of the other. Solution of the system of difference-differential equations for the amplitudes by a series expansion method

In the problem of the diffraction of light by two parallel supersonic waves, consisting of a fundamental tone and itsn-th harmonic, the solution of the system of difference-differential equations for the amplitudes has been reduced to the integration of a partial differential equation. The expressions for the amplitudes of the diffracted light waves are obtained as the coefficients of the Laurent expansion of the solution of this partial differential equation. The latter has been integrated for two approximations:

1. Forρ = 0, the results of Murty’s elementary theory are reestablished.
2. Forρ ≤ 1, a power series inρ, the terms of which are calculated as far as the third one, leads to a new expression for the intensities of the diffracted light waves, verifying the general symmetry properties obtained by Mertens.

     

Estimates of Steiner subratio and Steiner-Gromov ratio

For an arbitrary metric space, exact lower bounds for n-point Steiner subratio and Steiner-Gromov ratio are obtained. The exact value of these ratios for several metric spaces are calculated, for example, for philogenetic spaces as a corollary from the main theorem. It is also proved that any number from 0.5 to 1 could appear as the Steiner subratio and as the Steiner-Gromov ratio of a certain metric space.     

A differential equations approach to the modal location for a family of bivariate gamma distributions

Analytical and numerical results are given for determining the location of the mode of a class of bivariate gamma densities as a function of the parameters. The model location for a class of bivariate gammas as considered by Kibble (1941, Sankhya A 5 137–150) is shown to satisfy a nonlinear differential equation in ϱ, the correlation coefficient for fixed shape parameter. Qualitative and asymptotic properties of the modal location are also given. Whenever the shape parameters are unequal, analytical and numerical results are used to provide a conjecture for the modal location in the general case.