On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.     

A survey of credibility theory

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.     

On regularization of a certain class of distributions

We construct a regularization of a distribution of the form , where f is a tempered distribution and a function is infinitely differentiable outside a semialgebraic set N and has power singularities together with any its derivative.     

A characterization of certain discrete exponential families

For independent random variables X and Y, define % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabofaruWrL9MCNLwyaGqbciaa-bcacqGHHjIUcaWFGaGaa8hw% aiaa-TcacaWFzbaaaa!4551!\[{\rm{S}} \equiv X + Y\]. When the conditional expectations % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadweacaGGBbqefCuzVj3zPfgaiuGajaaqcaWFNbGccaGGOaGa% amiwaiaacMcacaGG8bGaam4uaiaac2facqGHHjIUcaWGHbGaaiikai% aadofacaGGPaaaaa!4BC4!\[E[g(X)|S] \equiv a(S)\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadweacaGGBbGaamiAaiaacIcacaWGybGaaiykaiaacYhacaWG% tbGaaiyxaiabggMi6kaadkgacaGGOaGaam4uaiaacMcaaaa!4894!\[E[h(X)|S] \equiv b(S)\]are given, then under certain assumptions, the density function of X has the form of u(x)k()e^ax, where u(x) is uniquely determined by the functions a(·) and b(·).     

Approximation of inverse moments of discrete distributions

For a wide class of discrete distributions, we derive a representation of the inverse (negative) moments through the Stirling numbers of the first kind and inverse factorial moments. We specialize the results for the Poisson, binomial, hypergeometric and negative binomial distributions.     

Generalization of some results concerning eigenvalues of a certain class of matrices and some applications

In this note, we present a generalization of some results concerning the spectral properties of a certain class of block matrices. As applications, we study some of its implications on nonnegative matrices and doubly stochastic matrices as well as on graph spectra and graph energy.     

Statistical inference for a certain class of bivariate distributions

The paper deals with statistical inference for a certain class of bivariate distributions. The class of marginal distributions is given and is shown to include distributions with only location and scale parameters. A normalizing transformation is applied to the marginal distributions and the parameters are estimated by maximum likelihood. For this class there is a great deal of simplification in the calculations for the asymptotic covariance matrix of the vector of parameter estimators. Statistics for tests of zero correlation are discussed. Also, the analysis is carried out for exponential marginal distributions.     

The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems

In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.     

The structure of the distribution of a couple of observable random variables in credibility theory

We consider Bühlmann's classical model in credibility theory and we assume that the set of possible values of the observable random variables X₁, X₂,… is finite, say with n elements. Then the distribution of a couple (X_r, X_s) (r ≠ s) amounts to a square real matrix of order n, that we call a credibility matrix. In order to estimate credibility matrices or to adjust roughly estimated credibility matrices, we study the set of all credibility matrices and some particular subsets of it.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

Maximal determinant over a certain class of matrices and its application to D-optimality of designs

It is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices. In this context we study the problem of maximizing the determinant of $\alpha {\mathbf{I}}_t-(\mathbf{P}+{\mathbf{P}}^T)$, $\alpha >2$, over the class of t-by-t permutation matrices, and the determinant of $\alpha {\mathbf{I}}_t+\mathbf{P}+{\mathbf{P}}^T$, $\alpha \ge 2.5$, over the class of t-by-t permutation matrices with zero diagonal (derangement matrices). The results are used to characterize D-optimal complete block designs under an interference model.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.     

On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices

In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity.     

A characterization of certain diffusions on the Sierpinski gasket by the exit distributions

LetK be the Sierpinski gasket with verticesa ₁,a ₂,a ₃ forming an equilateral triangle. Suppose that (X _t ,P ^x ) and (Y _t ,Q ^x ) are diffusions onK with the same hitting probabilities to verticesa ₁,a ₂,a ₃. We show that ifX is an asymptotically one-dimensional diffusion or ap-stream diffusion, thenY is a time-change ofX.Part of this work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany.     

A two-parameter of weighted exponential distributions

In this paper, a class of distributions called the two-parameter weighted exponential distributions is introduced (TWE). This new class of distributions generalizes the ones-weighted exponential distributions (WE)-proposed by Gupta and Kundu (2009). The main properties of this new class of distributions are investigated. Several statistical properties and statistical inferences are then obtained and studied. Two real data sets of which one is a right censored data set were analyzed, and it is shown that in both two cases our model fits much better than WE or some other existing models.     

A large population job search game with discrete time

A job search problem is considered, in which there is a large population of jobs initially available and a large population of searchers. The ratio of the number of searchers to the number of jobs is α. Each job has an associated value from a known distribution. At each of N moments the searchers observe a job, whose value comes from the distribution of the values of currently available jobs. If a searcher accepts a job, s/he ceases searching and the job becomes unavailable. Hence, the distribution of the values of available jobs changes over time. Also, the ratio of the number of those still searching to the number of available jobs changes. The model is presented and Nash equilibrium strategies for such problems are considered. By definition, when all the population use a Nash equilibrium strategy, the optimal response of an individual is to use the same strategy. Conditions are given that ensure the existence of a unique Nash equilibrium strategy. Examples are given to illustrate the model and present different approaches to solving such problems.     

Conditions of matrices in discrete tension spline approximations of DMBVP

Some splines can be defined as solutions of differential multi-point boundary value problems (DMBVP). In the numerical treatment of DMBVP, the differential operator is discretized by finite differences. We consider one dimensional discrete hyperbolic tension spline introduced in (Costantini et al. in Adv Comput Math 11:331–354, 1999), and the associated specially structured pentadiagonal linear system. Error in direct methods for the solution of this linear system depends on condition numbers of corresponding matrices. If the chosen mesh is uniform, the system matrix is symmetric and positive definite, and it is easy to compute both, lower and upper bound, for its condition. In the more interesting non-uniform case, matrix is not symmetric, but in some circumstances we can nevertheless find an upper bound on its condition number. This research was supported by Grant 0037114, by the Ministry of Science, Education and Sports of the Republic of Croatia.     

Three solutions to a perturbed nonlinear discrete Dirichlet problem

Using critical point theory, we study the existence of at least three solutions for a perturbed nonlinear Dirichlet boundary value problem for difference equations depending on two positive parameters.     

A characterization of the type of the Cauchy-Hua measure on real symmetric matrices

The Cauchy-Hua measure is the probability on the set ofn×n real symmetric matrices of densitys C _n/det(I + s ₂)_{(n + 1)/2} . For any probability measure on the set ofn×n real symmetric matrices, we define (if verifies an additional condition) the image of by a 2n×2n real symplectic matrix, and we introduce the type of . Then, the type of the Cauchy-Hua measure is characterized from its invariance by the symplectic group.