an iterative algorithm for computing mean first passage times of markov chains abstract mean first passage times are an essential ingredient in both the theory and the applications of markov chains in the literature they have been expressed in elegant closed form formulas these formulas involve explicit full matrix inversions and if computed directly may incur numerical instability in this paper we present a new iterative algorithm for computing mean first passage times in a manner that does not rely on explicit full matrix inversions results regarding the convergence behavior of this algorithm are also developed

basic soules matrices and their applications abstract in this paper we study the properties of the basic soules matrices in r n n which are a special subclass of the n n soules matrices generated via the basic soules basis the basic soules basis has the sign pattern n and it corresponds to the vector e n r n of all 1 s the basic soules matrices are up to a multiple by a positive scalar symmetric and doubly stochastic we begin by investigating the permanents of basic soules matrices next for a nonsingular basic soules matrix a r n n we show that the matrix a a 1 which is known to be a nonsingular m matrix has a basic soules basis of eigenvectors furthermore we obtain explicit formulas for the eigenvalues of a a 1 in terms of the eigenvalues of a finally let a r n n be a basic soules matrix of spectral radius 1 and set q i a by investigating the sign pattern of the off diagonal entries of the group inverse q of q we determine when the perron root is a concave function in each of the off diagonal entries at a

optimizing quadratic forms of adjacency matrices of trees and related eigenvalue problems abstract let a be an adjacency matrix of a tree t with n vertices conditions are determined for the existence of a fixed permutation matrix p that maximizes the quadratic form x t p t apx over all nonnegative vectors x with entries arranged in nondecreasing order this quadratic form problem is completely solved and its answer leads to a corresponding solution for the problem of determining conditions for the existence of a fixed permutation matrix p that maximizes the largest eigenvalue of matrices of the form pdp t a over all real diagonal matrices d with nondecreasing diagonal entries it is shown that there is a tree with six vertices for which neither of the problems has a solution and all other trees with six or fewer vertices have solutions for both problems by duality the results also apply to the analogous problem of minimizing the smallest eigenvalue of matrices of the form pdp t a

graph theoretic aspects of maximizing the spectral radius of nonnegative matrices abstract let i t a te ii denote the spectral radius of the sum of an irreducible nonnegative matrix a and a matrix te ii that has a single nonzero entry namely t 0 in the i i position we consider qualitative aspects of maximizing i t especially identifying maximizing indices i and indices i and j that tie i e i t j t for all t 0 if the digraph of a is a directed cycle then all vertices tie whereas if the digraph of a is a star then the center is the unique maximizing vertex when a is the 0 1 adjacency matrix of a graph we give sufficient conditions in terms of the orbits of vertices for a tie for complete bipartite graphs and for paths vertices i are identified that maximize i t for all t 0 however even for a tree it is not in general true that some fixed vertex i maximizes i t for all t 0

a calculus for log convex interference functions the behavior of certain interference coupled multiuser systems can be modeled by means of logarithmically convex log convex interference functions in this paper we show fundamental properties of this framework a key observation is that any log convex interference function can be expressed as an optimum over elementary log convex interference functions the results also contribute to a better understanding of certain quality of service qos tradeoff regions which can be expressed as sublevel sets of log convex interference functions we analyze the structure of the qos region and provide conditions for the achievability of boundary points the proposed framework of log convex interference functions generalizes the classical linear interference model which is closely connected with the theory of irreducible nonnegative matrices perron frobenius theory we discuss some possible applications in robust communication cooperative game theory and max min fairness

markov chain sensitivity measured by mean first passage times abstract the purpose of this article is to present results concerning the sensitivity of the stationary probabilities for an n state time homogeneous irreducible markov chain in terms of the mean first passage times in the chain

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

multiple nontrivial solutions for critical growth quasilinear elliptic systems abstract in the present paper we consider the following critical growth quasilinear elliptic system p u i 1 p d u i f u i u i q 2 u i in i 1 k u i 0 on i 1 k by exploiting the nehari manifold and variational methods we obtain the existence and multiplicity results of nontrivial solutions for the system

some results on eigenvalues of finite type resolvents and riesz projections abstract the present paper considers a separable hilbert space h and a bounded linear operator a h h that has an eigenvalue of finite type at 0 c using the image and the kernel of certain operators that are defined recursively starting from 0 a a construction of the local smith form and extended canonical system of root functions of a at 0 is provided this allows to express the partial multiplicities the algebraic multiplicity the jordan structure and jordan chains of a at 0 in terms of the quantities defined by the recursion similarly formulas for the rank image and kernel of the operators in the principal part of the resolvent a 1 at 0 and thus also of the riesz projection are given in terms of the quantities defined by the recursion

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

quantum mechanics on profinite groups and partial order inverse limits and profinite groups are used in a quantum mechanical context two cases are considered a quantum system with positions in the profinite group and momenta in the group and a quantum system with positions in the profinite group and momenta in the group the corresponding schwatz bruhat spaces of wavefunctions and the heisenberg weyl groups are discussed the sets of subsystems of these systems are studied from the point of view of partial order theory it is shown that they are directed complete partial orders it is also shown that they are topological spaces with t0 topologies and this is used to define continuity of various physical quantities the physical meaning of profinite groups non archimedean metrics partial orders and t0 topologies in a quantum mechanical context is discussed

quantum mechanics on p adic numbers a quantum system with positions in and momenta in is studied displacement operators in the phase space of this system and the corresponding heisenberg weyl group are studied it is shown that such a system can be constructed from a semi infinite chain of spins which are coupled in a particular way

resolvent expansions of matrices and applications abstract various explicit expansions of the resolvent of a square complex matrix in a neighborhood of the origin including the well known laurent expansion are obtained simple proofs using algebraic arguments rather than the theory of complex functions are given cesaro average expansions of the resolvent on the spectral circle are also developed several applications of these expansions are given in particular these include some new inverse nonnegativity properties of the drazin inverses of m matrices

derivatives of the perron root of a positive reciprocal matrix with application to the analytic hierarchy process the derivatives of the perron root of an essentially nonnegative matrix have been thoroughly studied in the literature this paper presents a refinement of these results to a special class of essentially nonnegative matrices namely positive reciprocal matrices this class of matrices arises in saaty s analytic hierarchy process a decision analysis method which is based upon pairwise comparisons among a set of alternatives over various criteria formulas for the first and second partial derivatives of the perron root of a positive reciprocal matrix are developed and the use of these formulas in conjunction with saaty s method is illustrated

symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents this paper deals with a class of singular biharmonic elliptic systems involving critical exponents in a bounded symmetric domain by using the variational method and the symmetric criticality principle of palais we obtain several existence and multiplicity results of g symmetric solutions for the systems

on the first and second order derivatives of the perron vector abstract in a previous work 5 the authors developed formulas for the second order partial derivatives of the perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix these formulas which involve the group generalized inverse of an associated m matrix were used to investigate the concavity and convexity of the perron root as a function of the entries the authors now combine the above results together with an approach taken in an earlier joint paper 6 of the second author with l elsner and c johnson and they develop formulas for the second order derivatives of an appropriately normalized perron vector with respect to the matrix entries which again are given in terms the group generalized inverse of an associated m matrix convexity properties of the perron vector as a function of the entries of the matrix are then examined in addition formulas for the first derivative of the perron vector resulting from different normalizations of this eigenvector are also given a by product of one of these formulas yields that the group generalized inverse of a singular and irreducible m matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant

existence and multiplicity of positive solutions to elliptic systems involving critical exponents abstract in this paper a singular elliptic system involving multiple critical exponents and the caffarelli kohn nirenberg inequality is investigated by using the extremals of the best hardy sobolev constants the existence and multiplicity of positive solutions to the system are established

minimization of norms and the spectral radius of a sum of nonnegative matrices under diagonal equivalence abstract we generalize in various directions a result of friedland and karlin on a lower bound for the spectral radius of a matrix that is positively diagonally equivalent to a doubly stochastic matrix the original result characterizes the equality case for two special zero patterns of the doubly stochastic matrix here we characterize the equality cases for doubly stochastic matrices of general zero pattern we further generalize the results to sums of matrices that are diagonally equivalent to doubly stochastic matrices our claims follow from inequalities we prove on norms of matrices finally we prove the corresponding inequalities and equalities for nonnegative matrices that are not sums of matrices diagonally equivalent to doubly stochastic matrices

lie superalgebras and the multiplet structure of the genetic code i codon representations it has been proposed by hornos and hornos phys rev lett 71 4401 4404 1993 that the degeneracy of the genetic code i e the phenomenon that different codons base triplets of dna are transcribed into the same amino acid may be interpreted as the result of a symmetry breaking process in their work this picture was developed in the framework of simple lie algebras here we explore the possibility of explaining the degeneracy of the genetic code using basic classical lie superalgebras whose representation theory is sufficiently well understood at least as far as typical representations are concerned in the present paper we give the complete list of all typical codon representations typical 64 dimensional irreducible representations whereas in the second part we shall present the corresponding branching rules and discuss which of them reproduce the multiplet structure of the genetic code

improved bounds for a condition number for markov chains abstract let p be the transition matrix for an n state homogeneous ergodic markov chain set q i p and let q q i j be the group generalized inverse of q a well known condition number due to funderlic and meyer which is used in the error analysis for the computation of the stationary distribution vector 1 n t of the chain is 4 max 1 i j n q i j in this paper we refine two upper estimates on 4 due to meyer in the course of proving one of our results we show that q i j j 1 i q 1 j where q j is the n 1 n 1 principal submatrix of q obtained from deleting its j th row and column and we characterize the case of equality the fact that we have a tight upper bound on the individual entries of the group inverse allows us to apply it in other contexts in which the group inverse arises for an irreducible nonnegative matrix such applications include for instance bounds on the second order partial derivative of the perron root with respect to any entry of the matrix and on the elasticity of the perron root with respect to any entry of the matrix

ultrametrics in the genetic code and the genome abstract ultrametric approach to the genetic code and the genome is considered and developed p adic degeneracy of the genetic code is pointed out ultrametric tree of the codon space is presented it is shown that codons and amino acids can be treated as p adic ultrametric networks ultrametric modification of the hamming distance is defined and noted how it can be useful ultrametric approach with p adic distance is an attractive and promising trend towards investigation of bioinformation

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

on the elliptic problems with critical weighted sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we deal with the conditions that ensure the existence of nontrivial solutions for the elliptic equation div x 2 a u u x 2 1 a u p 2 x b p u u with dirichlet boundary condition which involving the caffarelli kohn nirenberg inequalities the results depend crucially on the parameters a b and

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

a comparative study of algorithms for matrix balancing the problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics urban planning statistics demography and stochastic modeling these problems are called matrix balancing problems we describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing matrix scaling and nonlinear optimization the algorithms we study are the ras algorithm a diagonal similarity scaling algorithm and a truncated newton algorithm for network optimization we present results from computational experiments with large scale problems based on producing consistent estimates of social accounting matrices for developing countries

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

group inverses of m matrices associated with nonnegative matrices having few eigenvalues abstract we continue here earlier investigations of the structure of group generalized inverses a of singular and irreducible m matrices a ri m where m is an n n nonnegative and irreducible matrix whose spectral radius is r and where m is assumed to belong to one of several special classes of nonnegative matrices we now focus on cases where m has only a few distinct eigenvalues we are particularly interested in instances of such matrices m which cause all or almost all of the off diagonal entries of the corresponding a to be nonpositive we then apply our results to the study of the sign pattern of the off diagonal entries of the group generalized inverse of a ri m where m comes from one of the following families of nonnegative matrices 1 the magic square of order n 4k generated by matlab 2 the doubly regular tournament matrices the latter type is an example of a special type of a nonnegative matrix m for which the group inverse of the associated m matrix satisfies a a t for some 0

proximity in group inverses of m matrices and inverses of diagonally dominant m matrices in this paper we connect generalize and broaden properties of matrices related to i the triangular inequality for mean first passage times in finite homogeneous ergodic markov chains ii the triangle inequality for proximities in laplacian matrices of undirected weighted graphs and iii the metzler property of the column entrywise diagonal dominance of inverses of diagonally dominant m matrices

existence of solutions for singular elliptic systems with critical exponents abstract let 0 be an open bounded domain in r n n 3 and 2 2 n n 2 we consider the following elliptic system of two equations in h 0 1 h 0 1 u t u x 2 2 u 2 u v u v t v x 2 2 u v 2 v v where 0 and 1 satisfy 2 using moser iteration we prove the asymptotic behavior of solutions for at the origin by exploiting the mountain pass theorem we establish the existence of solutions to we also discuss sign changing solutions for in this paper

on group inverses of m matrices with uniform diagonal entries abstract in two recent works the condition of the diagonal entries of the group inverse of a singular and irreducible m matrix being uniform constant has arisen in resistive electrical circuits and in the effect upon the perron root of certain diagonal perturbation of a nonnegative matrix in this paper we first show a negative result that the group inverse of the laplacian matrix of an undirected weighted graph g on n vertices with a cutpoint cannot have uniform diagonal this includes the case when g is a tree we characterize however all weighted n cycles g whose laplacian has a group inverse with a uniform diagonal finally we consider the mean first passage matrix m of an ergodic markov chain with a doubly stochastic transition matrix t we show that if the group inverse of i t has a uniform diagonal then the group inverse of the m matrix m i m is again an m matrix

derivatives of the perron root at an essentially nonnegative matrix and the group inverse of an m matrix abstract the results of the work in this paper can be divided into two major parts in the first part we develop formulae using group inverses of irreducible m matrices for the second order partial derivatives with respect to the entries of the perron root r at an essentially nonnegative and irreducible matrix a we use these formulae to obtain explicit expressions for the matrices 2 r a ij 2 and 2 r a ii jj and examine these expressions at some special cases when for example in addition to a being essentially nonnegative and irreducible a is row stochastic or a is of rank 1 and so on in the second major part of this paper we investigate for a singular and irreducible m matrix q a certain matrix g which is a positive diagonal scaling of the group inverse q of q we show for instance that z gz 0 for all real vectors z and use this result and some of the formulae obtained in the first part of the paper to provide a matrix theoretic proof of a theorem due to j e cohen math proc cambridge philos soc 86 1979 343 350 which states that for an essentially nonnegative and irreducible matrix a the perron root is a convex function of the main diagonal of a

graph theoretic analysis of network input output processes the control of dynamical processes in networks is considered in the case where measurement and actuation capabilities are sparse and possibly remote specifically we study control of a canonical network dynamics when only one network component s state can be measured and only one in general different component can be actuated to do so we characterize the finite and infinite zeros of the resulting siso system in terms of the graph topology using these results we establish graph theoretic conditions under which there are zeros in the closed right half plane these conditions depend on the length strength and number of the paths from the component where the input is applied to the component where the measurements are made then we present the implications of these conditions on the controller design task focusing in stabilizations destabilization of network processes under static negative feedback

perron frobenius theory over real closed fields and fractional power series expansions abstract some of the main results of the perron frobenius theory of square nonnegative matrices over the reals are extended to matrices with elements in a real closed field we use the results to prove the existence of a fractional power series expansion for the perron frobenius eigenvalue and normalized eigenvector of real square nonnegative irreducible matrices which are obtained by perturbing a possibly reducible nonnegative matrix further we identify a system of equations and inequalities whose solution yields the coefficients of these expansions for irreducible matrices our analysis assures that any solution of this system yields a fractional power series with a positive radius of convergence

p adic models of ultrametric diffusion constrained by hierarchical energy landscapes we demonstrate that the p adic analysis is a natural basis for the construction of a wide variety of models of ultrametric diffusion constrained by hierarchical energy landscapes a general analytical description in terms of the p adic analysis is given for a class of models two exactly solvable examples i e the ultrametric diffusion constrained by the linear energy landscape and the ultrametric diffusion with a reaction sink are considered we show that such models can be applied to both the relaxation in complex systems and the rate processes coupled to rearrangement of the complex surrounding

forest matrices around the laplacian matrix abstract we study the matrices q k of in forests of a weighted digraph and their connections with the laplacian matrix l of the i j entry of q k is the total weight of spanning converging forests in forests with k arcs such that i belongs to a tree rooted at j the forest matrices q k can be calculated recursively and expressed by polynomials in the laplacian matrix they provide representations for the generalized inverses the powers and some eigenvectors of l the normalized in forest matrices are row stochastic the normalized matrix of maximum in forests is the eigenprojection of the laplacian matrix which provides an immediate proof of the markov chain tree theorem a source of these results is the fact that matrices q k are the matrix coefficients in the polynomial expansion of adj i l thereby they are precisely faddeev s matrices for l

schwinger boson formulation and solution of the crow kimura and eigen models of quasispecies theory we express the crow kimura and eigen models of quasispecies theory in a functional integral representation we formulate the spin coherent state functional integrals using the schwinger boson method in this formulation we are able to deduce the long time behavior of these models for arbitrary replication and degradation functions we discuss the phase transitions that occur in these models as a function of mutation rate we derive for these models the leading order corrections to the infinite genome length limit

replica symmetry breaking related to a general ultrametric space i replica matrices and functionals abstract a family of replica matrices related to general ultrametric spaces is introduced these matrices generalize the known parisi matrices some functionals of the replica approach are computed

p adic description of characteristic relaxation in complex systems this work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p adic analysis we show that three types of relaxation fitted into the kohlrausch williams watts law the power decay law and the logarithmic decay law are similar random processes inherently these processes are ultrametric and are described by the p adic master equation the physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape we also discuss relations between the relaxation kinetics and the energy landscapes

some eigenvalue inequalities for a class of jacobi matrices abstract we consider the class of jacobi tridiagonal matrices t l d where l is the negative of the discrete laplacian and d is a diagonal matrix we prove the inequality 1 t 1 t where 1 t represents the lowest eigenvalue of the matrix t and where t l d with d being the symmetric increasing rearrangement of d the proof follows from rearrangement inequalities going back at least to hardy littlewood and polya and is the one dimensional discrete analogue of a well known result for schrodinger operators we also prove that the gap 2 1 is increased by strictly symmetric increasing perturbations in the case that d is symmetric finally we give an inequality relating the lowest eigenvalues of four jacobi matrices of the form t l d when their potentials d obey certain conditions

inequalities and equalities associated with the campbell youla generalized inverse of the indefinite admittance matrix of resistive networks abstract matrix theoretic interpretations are obtained for the campbell youla inverse and its associated terminal weight numbers and network diameter these concepts are useful for the effective modeling of resistive electrical networks in our companion paper we explore some related graph theory geometric interpretations were recently given by fiedler

more on generalizations of matrix monotonicity abstract following berman and plemmons 5 werner 17 poole and barker 13 and others we investigate some generalizations of matrix monotonicity and consider their relation to mp matrices we prove some observations on almost monotone mp and group monotone matrices however of much interest to us is the problem whether a symmetric positive semidefinite matrix is mp if and only if it is monotone on its range to consider this question we obtain several results on the structure of range monotone matrices

algebraic model for the evolution of the genetic code a search for symmetries based on the compact simple lie algebras is performed to verify to what extent the genetic code is a manifestation of some underlying symmetry an exact continuous symmetry group cannot be found to reproduce the present genetic code however a unique approximate symmetry group is compatible with codon assignment for the fundamental amino acids and the termination codon in order to obtain the actual genetic code the symmetry must be slightly broken

m matrix generalized inverses of m matrices abstract let p be an n n nonnegative irreducible and stochastic matrix and consider the associated singular and irreducible m matrix q i p we show that without further restrictive spectral or combinatorial assumptions on p it is reasonable to expect that the group inverse q of q will be an m matrix only when p lies in a small wedge about a rank 1 nonnegative matrix a wedge whose existence is itself dependent on p we go on to consider the question of when q has a nonzero generalized inverse not necessarily the group inverse which is an m matrix we give examples of all sorts of classes of p for which such is the case but point out that the general problem remains unsolved

on the singular elliptic systems involving multiple critical sobolev exponents in this paper a singular elliptic system is investigated which involves multiple critical sobolev exponents and hardy type terms by using variational methods and analytical techniques the existence of positive and sign changing solutions to the system is established

on convexity properties of the spectral radius of nonnegative matrices abstract elementary matrix theoretic proofs are given for the following well known results r d max re an eigenvalue of a d and s d ln e d a are convex here d is diagonal a a nonnegative n n matrix and the spectral radius

conditions for strict inequality in comparisons of spectral radii of splittings of different matrices we present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices but we also obtain new results for nonstrict inequalities of the spectral radii we emphasize two different approaches one combinatorial and the other analytic and discuss their merits in the light of the results obtained we try to get fairly general results and indicate by counter examples that some of our hypotheses cannot be relaxed in certain directions

a solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices abstract let p and e be two n n complex matrices such that for sufficiently small positive e p ee is nonnegative and irreducible it is known that the spectral radius of p ee and corresponding normalized eigenvector have fractional power series expansions the goal of the paper is to develop an algorithm for computing the coefficients of these expansions under two restrictive assumptions namely that p has a single jordan block corresponding to its spectral radius and that the unique up to scalar multiples left and right eigenvectors of p corresponding to its spectral radius say v and w satisfy v t ew 0 our approach is to consider an associated countable system of nonlinear equations and solve this system recursively at each step we consider the coefficients of the expansion of the spectral radius of p ee as parameters and solve a related linear system parametrically the next coefficient of the expansion of the spectral radius is then determined from feasibility considerations for a linear system this solution method is novel and seems useful for computing coefficients of corresponding expansions when the two restrictive assumptions are relaxed also interestingly the coefficients we compute yield a preferred basis of the generalized eigenspace corresponding to the spectral radius of the unperturbed matrix p

a note on the perturbation of positive matrices by normal and unitary matrices abstract in a recent paper neumann and sze considered for an n n nonnegative matrix a the minimization and maximization of a s the spectral radius of a s as s ranges over all the doubly stochastic matrices they showed that both extremal values are always attained at an n n permutation matrix as a permutation matrix is a particular case of a normal matrix whose spectral radius is 1 we consider here for positive matrices a such that a n is a nonnegative matrix for all normal matrices n whose spectral radius is 1 the minimization and maximization problems of a n as n ranges over all such matrices we show that the extremal values always occur at an n n real unitary matrix we compare our results with a less recent work of han neumann and tastsomeros in which the maximum value of a x over all n n real matrices x of frobenius norm n was sought

a stochastic p adic model of the capillary flow in porous random medium abstract we develop the p adic model of propagation of fluids e g oil or water in capillary networks in a porous random medium the hierarchic structure of a system of capillaries is mathematically modeled by endowing trees of capillaries with the structure of an ultrametric space considerations are restricted to the case of idealized networks represented by homogeneous p trees with p branches leaving each vertex where p 1 is a prime number such trees are realized as the fields of p adic numbers we introduce and study an inhomogeneous markov process describing the penetration of fluid into a porous random medium

acute perturbation of the group inverse abstract for any n n complex matrix a let a g be the group inverse of a when a is singular a matrix b a e is said to be an acute perturbation of a if e is small and the spectral radius b g b a g a 1 the acute perturbation coincides with the stable perturbation of the group inverse if the matrix b satisfies condition r b n a 0 n b r a 0 which was introduced by castro gonzalez et al 2008 8 furthermore several examples are provided to illustrate the acute perturbation of the group inverse

a note on comparison theorems for nonnegative matrices in a recent paper 4 csordas and varga have unified and extended earlier theorems of varga in 10 and wo nicki in 11 on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings the main purpose of this note is to show a connection between the csordas varga paper and a paper by beauwens 1 in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular monotonic norms already used in 1 play an important role in our work here

resolvent positive linear operators exhibit the reduction phenomenon abstract the spectral bound s a v of a combination of a resolvent positive linear operator a and an operator of multiplication v was shown by kato to be convex in kato s result is shown here to imply through an elementary dual convexity lemma that s a v is also convex in 0 and notably s a v s a diffusions typically have s a 0 so that for diffusions with spatially heterogeneous growth or decay rates greater mixing reduces growth models of the evolution of dispersal in particular have found this result when a is a laplacian or second order elliptic operator or a nonlocal diffusion operator implying selection for reduced dispersal these cases are shown here to be part of a single broadly general reduction phenomenon

optimization of the spectral radius of a product for nonnegative matrices abstract let a be an n n irreducible nonnegative matrix we show that over the set n of all n n doubly stochastic matrices s the multiplicative spectral radius sa attains a minimum and a maximum at a permutation matrix for the case when a is a symmetric nonnegative matrix a by product of our technique of proof yields a result allowing us to show that s 1 a s 2 a when s 1 and s 2 are two symmetric matrices such that both s 1 a and s 2 a are nonnegative matrices and s 1 s 2 is a positive semidefinite matrix this result has several corollaries one corollary is that s 1 a s 2 a when s 1 1 n j and s 2 1 n 1 j i where j is the matrix of all 1 s a second corollary is a comparison theorem for weak regular splittings of two monotone matrices

dominant eigenvalue minimization with trace preserving diagonal perturbation subset design problem abstract motivated by network resource allocation needs we study the problem of minimizing the dominant eigenvalue of an essentially nonnegative matrix with respect to a trace preserving or fixed trace diagonal perturbation in the case where only a subset of the diagonal entries can be perturbed graph theoretic characterizations of the optimal subset design are obtained in particular the design is connected to the structure of a reduced effective graph defined from the essentially nonnegative matrix also the change in the optimum is studied when additional diagonal entries are constrained to be undesignable from both an algebraic and graph theoretic perspective these results are developed in part using properties of the perron complement of nonnegative matrices and the concept of line sum symmetry some results apply to general essentially nonnegative matrices while others are specialized for sub classes e g diagonally symmetrizable or having a single node cut

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

perturbation of null spaces with application to the eigenvalue problem and generalized inverses we consider properties of a null space of an analytically perturbed matrix in particular we obtain taylor expansions for the eigenvectors which constitute a basis for the perturbed null space furthermore we apply these results to the calculation of puiseux expansion of the perturbed eigenvectors in the case of general eigenvalue problem as well as to the calculation of laurent series expansions for the perturbed group inverse and pseudoinverse matrices

a trace inequality for m matrices and the symmetrizability of a real matrix by a positive diagonal matrix abstract the question of whether a real matrix is symmetrizable via multiplication by a diagonal matrix with positive diagonal entries is reduced to the corresponding question for m matrices and related to hadamard products in the process for a nonsingular m matrix a it is shown that tr a 1 a t n with equality if and only if a is symmetric and that the minimum eigenvalue of a 1 a is 1 with equality in the irreducible case if and only if a is positive diagonally symmetrizable

genetic code on the diadic plane abstract we introduce the simple parametrization for the space of codons triples of nucleotides by 8 8 table this table which we call the diadic plane possesses the natural 2 adic ultrametric we show that after this parametrization the genetic code will be a locally constant map of the simple form the local constancy of this map will describe degeneracy of the genetic code the map of the genetic code defines 2 adic ultrametric on the space of amino acids we show that hydrophobic amino acids will be clustered in two balls with respect to this ultrametric therefore the introduced parametrization of space of codons exhibits the hidden regularity of the genetic code

on uniqueness of gibbs measure for p adic countable state potts model on the cayley tree abstract in the present paper we consider countable state of p adic potts model on the tree under some condition on weights we establish uniqueness of gibbs measures for the model note that this condition does not depend on values of the prime p an analogous fact is not true when the number of spins is finite

dynamics of the eigen and the crow kimura models for molecular evolution we introduce an alternative way to study molecular evolution within well established hamilton jacobi formalism showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1n accuracy where n is the genome length for a smooth and monotonic fitness function this approach gives two dynamical phases smooth dynamics and discontinuous dynamics the latter phase arises naturally with no explicite singular fitness function counterintuitively the hamilton jacobi method yields straightforward analytical results for the models that utilize fitness as a function of hamming distance from a reference genome sequence we also show the way in which this method gives dynamical phase structure for multipeak fitness

harmonic analysis on gf pp part ii abstract harmonic analysis on gf p p is studied the heisenberg weyl hw g gf p p g group of displacements is shown to be a locally compact and totally disconnected topological group the formalism introduces algebraic concepts from the theory of galois fields into harmonic analysis for example a galois group of frobenius transformations on functions analogous to the galois group of frobenius transformations in galois theory is introduced into harmonic analysis a larger group which we call heisenberg weyl galois group is also discussed

eigen model as a quantum spin chain exact dynamics we map the eigen model of biological evolution naturwissenschaften 58 465 1971 into a quantum spin model with non hermitian hamiltonian based on such a connection we derive exact relaxation periods for the eigen model to approach static energy landscape from various initial conditions we also study a simple case of dynamic fitness function

nonnegative alternating circulants leading to m matrix group inverses abstract let c be the set of all an n n nonnegative irreducible alternating circulant matrices we characterize a subset c of c such that if b c and the perron root of b is r then the group inverse ri b of the singular and irreducible m matrixri b is also an m matrix this is equivalent to the fact that for each such b the perron root at b is a concave function in each of the off diagonal entries the characterization for the case when n is odd presents more difficulties than for the case when n is even so the two cases are treated separately

nonnegativity of principal minors of generalized inverses of p0 matrices abstract the result of principal interest established in this paper is that if a is an n n singular irreducible m matrix then a large class of generalized inverses of a possesses the property that each of its elements has all its principal minors nonnegative the class contains both the group and the moore penrose generalized inverses of a in an application of our results it is shown that the fundamental matrix of a continuous in time ergodic markov chain on a finite state space has all its principal minors nonnegative

reaction diffusion equations on complex networks and turing patterns via p adic analysis abstract nakao and mikhailov proposed using continuous models mean field models to study reaction diffusion systems on networks and the corresponding turing patterns this work aims to show that p adic analysis is the natural tool to carry out this program this is possible due to the fact that the discrete laplacian attached to a network turns out to be a particular case of a p adic laplacian by embedding the graph attached to the system into the field of p adic numbers we construct a continuous p adic version of a reaction diffusion system on a network the existence and uniqueness of the cauchy problems for these systems are established we show that turing criteria for the p adic continuous reaction diffusion system remain essentially the same as in the classical case however the properties of the emergent patterns are very different the classical turing patterns consisting of alternating domains do not occur in the p adic continuous case instead of this several domains clusters occur multistability that is coexistence of a number of different patterns with the same parameter values occurs clustering and multistability have been observed in the computer simulations of large reaction diffusion systems on networks such behavior can be naturally explained in the p adic continuous model in a rigorous mathematical way but not in the original discrete model

multiple positive solutions for a critical elliptic system with concave convex nonlinearities we consider a semilinear elliptic system with both concave convex nonlinearities and critical growth terms in bounded domains the existence and multiplicity results of positive solutions are obtained by variational methods

harmonic analysis on rational numbers abstract harmonic analysis on q is studied the fourier transforms of complex functions f y y q in the schwartz bruhat space s are functions f u where u a q q it is shown that the corresponding heisenberg weyl group is a locally compact topological group various time frequency analysis methods properties of displacement operators wigner and ambiguity functions etc are discussed in this context

on the spectral radius of positive operators on banach sequence spaces abstract let k 1 k n be infinite non negative matrices that define operators on a banach sequence space given a function f 0 0 0 of n variables we define a non negative matrix f k 1 k n and consider the inequality r f k 1 k n 1 n r k 1 r k n where r denotes the spectral radius we find the largest function f for which this inequality holds for all k 1 k n we also obtain an infinite dimensional extension of the result of cohen asserting that the spectral radius is a convex function of the diagonal entries of a non negative matrix

maximizing the spectral radius of fixed trace diagonal perturbations of nonnegative matrices abstract let a be an n by n irreducible entrywise nonnegative matrix for a given t 0 we consider the problem of maximizing the perron root of a nonnegative diagonal trace t perturbation of a because of the convexity of the perron root as a function of diagonal entries the maximum occurs for some teii such an index i which is called a winner may depend on t we show how to determine the nonempty set of indices i that are winners for all sufficiently small t and the possibly different nonempty set of indices that are winners for all sufficiently large t we also show how to determine if there are indices that are winners for all t

taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices let a and b be two n n complex matrices and let be an eigenvalue of a the purpose of this paper is to derive under certain conditions taylor power series expansions of the form k 1 kek and k 0 kek respectively for eigenvalues and corresponding eigenvectors of the perturbed matrices a eb for e that has sufficiently small absolute value our results apply to the case where is a simple eigenvalue of a e g when a is nonnegative and irreducible and is the spectral radius of a in particular if a eb is nonnegative for sufficiently small nonnegative e and a is irreducible we obtain power series expansions for the spectral radii of the perturbed matrices a eb and for corresponding eigenvectors the coefficients of the expansions yield explicit expressions for the regular and mixed derivatives of the spectral radius and of a corresponding eigenvector of a nonnegative irreducible matrix when viewed as a function of the elements of the matrix our approach is constructive and we present a recursive algorithm that will compute the coefficients of the above series

dominant eigenvalues under trace preserving diagonal perturbations abstract for an essentially nonnegative matrix a we consider the problem of minimizing the dominant eigenvalue of a d over real diagonal matrices d with zero trace the solution is closely related to the unique line sum symmetric diagonal similarity of a in the irreducible case and we describe the solution for general essentially nonnegative a the minimizer d is always unique and we characterize those matrices a for which the minimizer d is 0 we solve the problem for several classes of matrices by finding the line sum symmetric diagonal similarity as an explicit function of the entries of a in some cases and in terms of the zeros of polynomials with coefficients constructed from the entries of a in others

replica symmetry breaking related to a general ultrametric space ii rsb solutions and the n 0 limit we study the replica symmetry breaking related to a general ultrametric space ii in particular the rsb solutions and the n to0 limit

replica symmetry breaking related to a general ultrametric space iii the case of general measure we study replica symmetry breaking related to a general ultrametric space in the case of general measure