adjacency spectra of random and complete hypergraphs abstract we present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs there is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all ones hypermatrix several of the ingredients along a possible path to this conjecture are established and may be of independent interest in spectral hypergraph hypermatrix theory in particular we provide a bound on the spectral radius of the symmetric bernoulli hyperensemble and show that the spectrum of the complete k uniform hypergraph for k 2 3 is close to that of an appropriately scaled all ones hypermatrix

levy ornstein uhlenbeck and subordination spectral vs jump description unlike brownian motion which propagates diffusively and whose sample path trajectories are continuous non brownian levy motions propagate via jumps flights and their sample path trajectories are purely discontinuous when analyzing systems involving non brownian levy motions the common practice is to use either spectral or fractional calculus methods in this manuscript we suggest an alternative analytical approach using the poisson superposition jump structure of non brownian levy motions we demonstrate this approach in two exemplary topics i systems governed by l evy driven ornstein uhlenbeck dynamics and ii systems subject to temporal levy subordination we show that this approach yields answers and insights that are not attainable using spectral methods alone

equitable partition theorem of tensors and spectrum of generalized power hypergraphs abstract in this paper we present an equitable partition theorem of tensors which gives the relations between h eigenvalues of a tensor and its quotient equitable tensor and extends the equitable partitions of graphs to hypergraphs furthermore with the aid of it some properties and h eigenvalues of the generalized power hypergraphs are obtained which extends some known results including some results of yuan qi and shao 20

a new class of generalized set valued implicit variational inclusions in banach spaces with an application in this paper we introduce and study a new class of generalized set valued implicit variational inclusions in real banach spaces by using nadler s theorem and the resolvent operator technique for m accretive mapping in real banach spaces we construct some new iterative algorithms for solving this class of generalized set valued implicit variational inclusions we prove the existence of solution for this kind of generalized set valued implicit variational inclusions without compactness and the convergence of iterative sequences generated by the algorithms in banach spaces we also give an application to generalized set valued implicit variational inequalities in real hilbert spaces

singular value inclusion sets of rectangular tensors abstract in this paper two singular value inclusion sets for rectangular tensors via a digraph and a lifting square tensor are given respectively furthermore it is shown that these two singular value inclusion sets are both tighter than the known result given by zhao and li in theorem 2 1 of 25 and singular value inclusion sets via a digraph in special case are tighter than the inclusion sets via a lifting square tensor numerical examples are still given to show the relation among the inclusion sets via a digraph the inclusion sets via a lifting square tensor and the known result given by zhao and li 25 finally by using the lifting square tensors the result of tensors provided by bu et al 24 is extended to rectangular tensors

stationary states in langevin dynamics under asymmetric levy noises properties of systems driven by white non gaussian noises can be very different from these of systems driven by the white gaussian noise we investigate stationary probability densities for systems driven by ensuremath alpha stable l evy type noises which provide natural extension to the gaussian noise having however a new property namely a possibility of being asymmetric stationary probability densities are examined for a particle moving in parabolic quartic and in generic double well potential models subjected to the action of ensuremath alpha stable noises relevant solutions are constructed by methods of stochastic dynamics in situations where analytical results are known they are compared with numerical results furthermore the problem of estimation of the parameters of stationary densities is investigated

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

on some properties of the determinants of tensors abstract we use the hilbert s nullstellensatz hilbert s zero point theorem to give a direct proof of the formula for the determinants of the products of tensors by using this determinant formula and using tensor product to represent the transformations of the slices of tensors we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices we also study the determinants of tensors after two types of transposes we use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors and give a canonical form of the weakly reducible tensors

the drazin inverse of an even order tensor and its application to singular tensor equations abstract the notion of the moore penrose inverses of matrices was recently extended from matrix space to even order tensor space with einstein product in the literature in this paper we further study the properties of even order tensors with einstein product we define the index and characterize the invertibility of an even order square tensor we also extend the notion of the drazin inverse of a square matrix to an even order square tensor an expression for the drazin inverse through the core nilpotent decomposition for a tensor of even order is obtained as an application the drazin inverse solution of the singular linear tensor equation a x b will also be included

minimum maximum rank of sign pattern tensors and sign nonsingular tensors abstract in this paper we define the sign pattern tenors minimum maximum rank of sign pattern tenors term rank of tensors and sign nonsingular tensors the necessity and sufficiency for the minimum rank of sign pattern tenors to be 1 is given we show that the maximum rank of a sign pattern tensor is not less than the term rank and the minimum rank of the sign pattern of a sign nonsingular tensor is not less than its dimension we get some characterizations of tensors having sign left or sign right inverses

on the inverse of a tensor abstract in this paper we consider the left right inverse of a tensor we characterize the existence of any order k left right inverse of a tensor and show the expression of left right inverse of a tensor we also present a result for the similarity of tensors

a langevin approach to the log gauss pareto composite statistical structure abstract the distribution of wealth in human populations displays a log gauss pareto composite statistical structure its density is log gauss in its central body and follows power law decay in its tails this composite statistical structure is further observed in other complex systems and on a logarithmic scale it displays a gauss exponential structure its density is gauss in its central body and follows exponential decay in its tails in this paper we establish an equilibrium langevin explanation for this statistical phenomenon and show that i the stationary distributions of langevin dynamics with sigmoidal force functions display a gauss exponential composite statistical structure ii the stationary distributions of geometric langevin dynamics with sigmoidal force functions display a log gauss pareto composite statistical structure this equilibrium langevin explanation is universal as it is invariant with respect to the specific details of the sigmoidal force functions applied and as it is invariant with respect to the specific statistics of the underlying noise

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

a general product of tensors with applications abstract we study a general product of two n dimensional tensors a and b with orders m 2 and k 1 this product satisfies the associative law and is a generalization of the usual matrix product using this product many concepts and known results of tensors can be simply expressed and or proved and a number of applications of it will be given using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables we define the similarity and congruence of tensors which are also the generalizations of the corresponding relations for matrices and prove that similar tensors have the same characteristic polynomials thus the same spectra we study two special kinds of similarity permutational similarity and diagonal similarity and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors we also define the direct product of tensors in matrix case it is also called the kronecker product and give its applications in the study of the spectra of two kinds of the products of hypergraphs we also give applications of this general product in the study of nonnegative tensors including a characterization of primitive tensors the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors

the tensor splitting with application to solve multi linear systems abstract in this paper firstly we introduce the variant tensor splittings and present some equivalent conditions for a strong m tensor based on the tensor splitting secondly the existence and uniqueness conditions of the solution for multi linear systems are given thirdly we propose some tensor splitting algorithms for solving multi linear systems with coefficient tensor being a strong m tensor as an application a tensor splitting algorithm for solving the multi linear model of higher order markov chains is proposed numerical examples are given to demonstrate the efficiency of the proposed algorithms

further results on moore penrose inverses of tensors with application to tensor nearness problems abstract the notion of the moore penrose inverse of a matrix has been extended to tensor case with einstein product recently in this paper we continue this work and propose the full rank decomposition of tensors by introducing a new kind of tensor rank which gives a novel representation of the moore penrose inverse moreover we also obtain several formulae related to the moore penrose inverse of a tensor and its i j k inverses in addition we consider the tensor nearness problem associated with a tensor equation a generalization of matrix nearness problems that arise in many areas of applied matrix computations it is shown that the unique solution to the tensor nearness problem can be represented by using the moore penrose inverses of the given tensors

on the z eigenvalues of the adjacency tensors for uniform hypergraphs abstract the adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs and some fundamental properties for the adjacency tensor and its z eigenvalues of a uniform hypergraph are obtained in particular some bounds on the smallest and the largest z eigenvalues of the adjacency tensors for uniform hypergraphs are presented

on some properties of three different types of triangular blocked tensors abstract we define and study three different types of upper and lower triangular blocked tensors which are all generalizations of the triangular blocked matrices and are also generalizations of reducible or weakly reducible tensors we study some common properties as well as different properties of these three types of triangular blocked tensors we obtain the formulas for the determinants characteristic polynomials and spectra of the first and second type triangular blocked tensors and give an example to show that these formulas no longer hold for the third type triangular blocked tensors we prove that the product of any two n 1 n r upper or lower triangular blocked tensors of the first or second or third type is still an n 1 n r upper or lower triangular blocked tensor of the same type we also prove that if an n 1 n r upper triangular blocked tensor of the first or second or third type has a left k inverse then its unique left k inverse is still an n 1 n r upper triangular blocked tensor of all the three types also if it has a right k inverse then all of its right k inverses are still n 1 n r upper triangular blocked tensors of all the three types by showing that the left k inverse if any of a weakly irreducible nonsingular m tensor is a positive tensor we show that the left k inverse if any of a first or second or third type normal n 1 n r upper triangular blocked nonsingular m tensor is an n 1 n r upper triangular blocked tensor of all the three types all of whose diagonal blocks are positive tensors we also show that every order m dimension n tensor is permutation similar to some third type normal upper triangular blocked tensor all of whose diagonal blocks are irreducible we give an example to show that this is not true for the first type normal upper triangular blocked tensor

from shape to randomness a classification of langevin stochasticity abstract the langevin equation perhaps the most elemental stochastic differential equation in the physical sciences describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise the langevin equation is in effect a mechanism that maps the stochastic white noise input to a stochastic output a stationary steady state distribution in the case of potential wells and a transient extremum distribution in the case of potential gradients in this paper we explore the degree of randomness of the langevin equation s stochastic output and classify it a la mandelbrot into five states of randomness ranging from infra mild to ultra wild we establish closed form and highly implementable analytic results that determine the randomness of the langevin equation s stochastic output based on the shape of the langevin equation s potential field

the inverse rank and product of tensors abstract in this paper we give some basic properties for the left right inverse rank and product of tensors the existence of order 2 left right inverses of tensors is characterized we obtain some equalities and inequalities on the tensor rank we also show that the rank of a uniform hypergraph is independent of the ordering of its vertices and the laplacian tensor and the signless laplacian tensor have the same rank for odd bipartite even uniform hypergraphs

on eigenvalue problems of real symmetric tensors abstract we use variational methods to give a positive answer to a conjecture posed by liqun qi l qi eigenvalues of a real supersymmetric tensor j symbolic comput 40 2005 1302 1324 regarding the real eigenvalues of certain higher order tensors

systems of nonempty intersection theorems with applications in this paper we study the systems of nonempty intersection theorems by the systems of nonempty intersection theorems we can study the existence theorem of systems of quasi kkm problems the existence theorems of systems of quasi variational inclusions problems the existence theorems of systems of quasi variational disclusions problems and fixed point theorem we also establish some of their equivalent relations

collective fixed points and maximal elements with applications to abstract economies abstract in this paper we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is condensing as an application of our collective fixed points theorem we derive the coincidence theorem for two families of multivalued maps defined on product spaces then we give some existence results for maximal elements for a family of ls majorized multivalued maps whose product is condensing we also prove some existence results for maximal elements for a family of multivalued maps which are not ls majorized but their product is condensing as applications of our results some existence results for equilibria of abstract economies are also derived the results of this paper are more general than those given in the literature

the misconception of mean reversion the notion of random motion in a potential well is elemental in the physical sciences and beyond quantitatively this notion is described by reverting diffusions asymptotically stationary diffusion processes which are simultaneously i driven toward a reversion level by a deterministic force and ii perturbed off the reversion level by a random white noise the archetypal example of reverting diffusions is the ornstein uhlenbeck process which is mean reverting in this paper we analyze reverting diffusions and establish that i if the magnitude of the perturbing noise is constant then the diffusion s stationary density is unimodal and the diffusion is mode reverting ii if the magnitude of the perturbing noise is non constant then in general neither is the diffusion s stationary density unimodal nor is the diffusion mode reverting in the latter case we further establish a result asserting when unimodality and mode reversion do hold in particular we demonstrate that the notion of mean reversion which is fundamental in economics and finance is a misconception as mean reversion is an exception rather than the norm

bifurcation bimodality and finite variance in confined levy flights we investigate the statistical behavior of levy flights confined in a symmetric quartic potential well u x proportional variant x 4 at stationarity the probability density function features a distinct bimodal shape and decays with power law tails which are steep enough to give rise to a finite variance in contrast to free levy flights from a delta initial condition a bifurcation of the unimodal state is observed at t c 0 the nonlinear oscillator with potential u x ax 2 2 bx 4 4 a b 0 shows a crossover from unimodal to bimodal behavior at stationarity depending on the ratio a b

fractal levy correlation cascades the correlation structure of a wide class of random processes driven by stable non gaussian levy noise sources is explored since these noises are of infinite variance correlations cannot be measured via auto covariance functions exploiting the underlying poissonian structure of levy noises we present a cascade of poissonian correlation functions which characterize the correlation structure and the process distribution of the processes under consideration the theory developed is applied to various examples including motions ornstein uhlenbeck and moving average processes and fractional motions and noises all driven by stable non gaussian levy noises

from gene families and genera to incomes and internet file sizes why power laws are so common in nature we present a simple explanation for the occurrence of power law tails in statistical distributions by showing that if stochastic processes with exponential growth in expectation are killed or observed randomly the distribution of the killed or observed state exhibits power law behavior in one or both tails this simple mechanism can explain power law tails in the distributions of the sizes of incomes cities internet files biological taxa and in gene family and protein family frequencies

e cospectral hypergraphs and some hypergraphs determined by their spectra two k uniform hypergraphs are said to be cospectral e cospectral if their adjacency tensors have the same characteristic polynomial e characteristic polynomial a k uniform hypergraph h is said to be determined by its spectrum if there is no non isomorphic k uniform hypergraph cospectral with h in this note we give a method for constructing e cospectral hypergraphs which is similar with godsil mckay switching some hypergraphs are shown to be determined by their spectra

some results on systems of quasi variational inclusion problems and systems of generalized quasi variational inclusion problems abstract in this paper we study systems of quasi variational inclusion problem and systems of quasi variational disclusion problem from the existence theorems of solution for these two types of problems we study various types of systems of quasi variational inclusion problems systems of quasi equilibrium problems systems of quasi kkm theorem abstract economics and system of kkm theorem we also show their equivalent relations we study further existence theorems of solution for generalized quasi variational inclusion problem our results are different from any existence result in the literature

the solution existence of general variational inclusion problems we propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous sufficient conditions for the solution existence are established as applications we derive consequences for several special cases of variational inclusion problems quasioptimization problems equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers

the bounds on spectral radius of nonnegative tensors via general product of tensors abstract in 1 j y shao proposed the general product of tensors based on this product the minc type bound on spectral radius of nonnegative tensors is given and the generalized minimax theorem of nonnegative tensors is presented moreover we give some bounds on the spectral radius of the general product of tensors

tensors product and hyperdeterminant of boundary formats product abstract in this paper a new product of tensors is given which is a generalization of the product of tensors introduced by gelfand kapranov zelevinsky this product satisfies the associative law using the associative law it is concluded that the product of tensors given by bu and others and the general product of tensors introduced by shao can be represented by this new tensors product as an application of this new tensors product and the associative law the hyperdeterminant of the product of a kind of tensors is studied i e some formulas for the hyperdeterminant of boundary formats product are shown

the studies of systems of variational inclusions problems and variational disclusions problems with applications abstract in this paper we study existence theorems of solutions for systems of variational inclusions problems and systems of variational disclusions problems from these existence results we establish existence theorems of solutions for systems of generalized vector quasiequilibrium problems and systems of quasioptimization problems