sensitivity analysis of discrete markov chains via matrix calculus abstract markov chains are often used as mathematical models of natural phenomena with transition probabilities defined in terms of parameters that are of interest in the scientific question at hand sensitivity analysis is an important way to quantify the effects of changes in these parameters on the behavior of the chain many properties of markov chains can be written as simple matrix expressions and hence matrix calculus is a powerful approach to sensitivity analysis using matrix calculus we derive the sensitivity and elasticity of a variety of properties of absorbing and ergodic finite state chains for absorbing chains we present the sensitivities of the moments of the number of visits to each transient state the moments of the time to absorption the mean number of states visited before absorption the quasistationary distribution and the probabilities of absorption in each of several absorbing states for ergodic chains we present the sensitivity of the stationary distribution the mean first passage time matrix the fundamental matrix and the kemeny constant we include two examples of application of the results to demographic and ecological problems

generalization of a fundamental matrix abstract it is shown that for a finite ergodic markov chain basic descriptive quantities such as the stationary vector and mean first passage matrix may be calculated using any one of a class of fundamental matrices new applications of the use of these operators are discussed

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

the computation of the mean first passage times for markov chains abstract a survey of a variety of computational procedures for finding the mean first passage times in markov chains is presented the author recently developed a new accurate computational technique an extended gth procedure hunter 2016 17 similar to that developed by kohlas 1986 20 in addition the author recently developed a variety of new perturbation techniques for finding key properties of markov chains including finding the mean first passage times hunter 2016 18 these recently developed procedures are compared with other procedures including the standard matrix inversion technique using the fundamental matrix kemeny and snell 1960 19 some simple generalized matrix inverse techniques developed by hunter 2007 15 and some modifications to the fund technique of heyman 1995 7 matlab is used to compute errors and estimate computation times when the techniques are used on some test problems that have been used in the literature together with some large sparse state space cases for accuracy a preference for the procedure of the author is exhibited for the test problems however it appears that the procedure as presented requires longer computational times

a multiplicity theorem for problems with the p laplacian abstract we consider nonlinear elliptic equations driven by the p laplacian differential operator using degree theoretic arguments based on the degree map for operators of type s we prove the existence of two nontrivial smooth solutions one of which is of constant sign

computing the phase type renewal and related functions this article presents a procedure for computing the renewal function the renewal density the integral of the renewal function and the variance function of phase type renewal processes the procedure hinges on the computation of the state probability vector of a continuous time markov chain this is accomplished by using a randomization approach that is simple efficient and numerically stable and does not require numerical integration i discuss approximating arbitrary interrenewal distributions by phase type distributions so that the procedure can be used to approximate renewal and related functions

stationary distributions of perturbed markov chains abstract techniques for updating the stationary distribution of a finite irreducible markov chain following a rank one perturbation of its transition matrix are discussed a variety of situations where such perturbations may arise are presented together with suitable procedures for the derivation of the related stationary distributions

new perturbation bounds for denumerable markov chains this paper is devoted to perturbation analysis of denumerable markov chains bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain where the bounds are given by the weighted supremum norm in addition bounds for the perturbed stationary probabilities are established furthermore bounds on the norm of the asymptotic decomposition of the perturbed stationary distribution are provided where the bounds are expressed in terms of the norm of the ergodicity coefficient or the norm of a special residual matrix refinements of our bounds for doeblin markov chains are considered as well our results are illustrated with a number of examples

kemeny s function for markov chains and markov renewal processes abstract extensions of kemeny s constant as derived for irreducible finite markov chains in discrete time to markov renewal processes and markov chains in continuous time are discussed three alternative kemeny s functions and their variants are considered typically they lead to a constant if and only if the mean holding times between the states in the markov renewal process are constant however one particular variant that arises under expected time to hitting of a state chosen according the continuous time stationary distribution leads to a constant analogous to the discrete time markov chain result specifically if the state space s is finite the sum j s of each mean first passage time m i j omitting the mean return time m j j when i j weighted by the stationary probability j associated with the continuous time semi markov process is a constant independent of i for any markov renewal process expressions for the kemeny s functions and the relevant constants are derived for markov renewal processes and special cases involving continuous time markov chains and birth death processes

on the normalized laplacian spectral radius laplacian incidence energy and kemeny s constant abstract let g v e be a simple connected graph of order n and size m and let 1 2 n 1 n 0 be the normalized laplacian eigenvalues of g the greatest eigenvalue 1 is referred to as the normalized laplacian spectral radius the laplacian incidence energy the randic energy and kemeny s constant of g are defined by l i e g i 1 n 1 i r e g i 1 n i 1 and k g i 1 n 1 1 i respectively new lower bounds for the normalized laplacian spectral radius are obtained the acquired bounds are then used to obtain new bounds for the l i e g in addition some relations between l i e g k g r e g are determined

existence of a non trivial solution for the p laplacian equation with fucik type resonance at infinity abstract we prove the existence of a non trivial weak solution for the equation p u a u p 1 b u p 1 f x u in u 0 on by variational methods we consider the case where a 1 or b 1 1 denotes the first eigenvalue of p or the case where a is close to b while the nonlinearity f belongs to a class that contains for example u q 2 u 1 q p

transition matrices for well conditioned markov chains abstract let t r n n be an irreducible stochastic matrix with stationary distribution vector set a i t and define the quantity 3 t 1 2 max j 1 n j a j 1 where aj j 1 n are the n 1 n 1 principal submatrices of a obtained by deleting the jth row and column of a results of cho and meyer and of kirkland show that 3 provides a sensitive measure of the conditioning of under perturbation of t moreover it is known that 3 t n 1 2 n in this paper we investigate the class of irreducible stochastic matrices t of order n such that 3 t n 1 2 n for such matrices correspond to markov chains with desirable conditioning properties we identify some restrictions on the zero nonzero patterns of such matrices and construct several infinite classes of matrices for which 3 is as small as possible

optimal linear responses for markov chains and stochastically perturbed dynamical systems the linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied we consider the little studied question of selecting an optimal perturbation so as to i maximise the linear response of the equilibrium distribution of the system ii maximise the linear response of the expectation of a specified observable and iii maximise the linear response of the rate of convergence of the system to the equilibrium distribution we also consider the inhomogeneous sequential or time dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time we develop the theory for finite state markov chains provide explicit solutions for some illustrative examples and numerically apply our theory to stochastically perturbed dynamical systems where the markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system

sensitivity of the stationary distribution vector for an ergodic markov chain stationary distribution vectors p sup infinity for markov chains with associated transition matrices t are important in the analysis of many models in the mathematical sciences such as queuing networks input output economic models and compartmental tracer analysis models the purpose of this paper is to provide insight into the sensitivity of p sup infinity to perturbations in the transition probabilities of t and to understand some of the difficulties in computing an accurate p sup infinity the group inverse a sup number of i t is shown to be of fundamental importance in understanding sensitivity or conditioning of p sup infinity the main result shows that if there is a state that is accessible from every other state and the corresponding column of t has no small off diagonal elements then p sup infinity cannot be sensitive to small perturbations in t ecological examples are given a new stable algorithm for calculating a sup number is described

multiple solutions for elliptic problems with asymmetric nonlinearity in this paper we establish the existence of multiple solutions for the semilinear elliptic problem u g x u in u 0on where rn is an open bounded domain with smooth boundary g r r is a function of class c1 such that g x 0 0 and which is asymptotically linear at infinity with jumping nonlinearities we use critical groups to distinguish the critical points

existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero we study the existence of changing sign solutions of an elliptic semilinear boundary value problem which arises as a limiting equation of the two species lotka volterra competing equations system using variational methods and a result of d aujourd hui we find conditions which are both sufficient and necessary for this existence problem

convergence of an operator splitting method on a bounded domain for a convection diffusion reaction system abstract we solve a convection diffusion sorption reaction system on a bounded domain with dominant convection using an operator splitting method the model arises in contaminant transport in groundwater induced by a dual well or in controlled laboratory experiments the operator splitting transforms the original problem to three subproblems nonlinear convection nonlinear diffusion and a reaction problem each with its own boundary conditions the transport equation is solved by a riemann solver the diffusion one by a finite volume method and the reaction equation by an approximation of an integral equation this approach has proved to be very successful in solving the problem but the convergence properties where not fully known we show how the boundary conditions must be taken into account and prove convergence in l 1 loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates generally this is the best convergence obtained for this type of approximation the derivation indicates limitations of the approach being able to consider only some types of boundary conditions a sample numerical experiment of a problem with an analytical solution is given showing the stated efficiency of the method

a brezis nirenberg splitting approach for nonlocal fractional equations in this paper we consider problems modeled by the following nonlocal fractional equation su a x u f u in u 0in rn where s 0 1 is fixed is an open bounded subset of rn n 2s with lipschitz boundary s is the fractional laplace operator and is a real parameter under two different types of conditions on the functions a and f by using a famous critical point theorem in the presence of splitting established by brezis and nirenberg we obtain the existence of at least two nontrivial weak solutions for our problem

generalized inverses of markovian kernels in terms of properties of the markov chain abstract all one condition generalized inverses of the markovian kernel i p where p is the transition matrix of a finite irreducible markov chain can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying markov chain special sub families include the group inverse of i p kemeny and snell s fundamental matrix of the markov chain and the moore penrose g inverse the elements of some sub families of the generalized inverses can also be re expressed involving the second moments of the recurrence time variables some applications to kemeny s constant and perturbations of markov chains are also considered

coupling and mixing times in a markov chain the derivation of the expected time to coupling in a markov chain and its relation to the expected time to mixing as introduced by the author in mixing times with applications to perturbed markov chains linear algebra appl 417 108 123 2006 are explored the two state cases and three state cases are examined in detail

entrywise perturbation theory and error analysis for markov chains grassmann taksar and heyman introduced a variant of gaussian climination for computing the steady state vector of a markov chain in this paper we prove that their algorithm is stable and that the problem itself is well conditioned in the sense of entrywise relative error thus the algorithm computes each entry of the steady state vector with low relative error even the small steady state probabilities are computed accurately the key to our analysis is to focus on entrywise relative error in both the data and the computed solution rather than making the standard assessments of error based on norms our conclusions do not depend on any condition numbers for the problem

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

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

the generalized inverses of tensors and an application to linear models abstract in this paper we recall and extend some tensor operations then the generalized inverse of tensors is established by using tensor equations moreover we investigate the least squares solutions of tensor equations an algorithm to compute the moore penrose inverse of an arbitrary tensor is constructed finally we apply the obtained results to higher order gauss markov theorem

some stochastic properties of semi magic and magic markov chains abstract gustafson and styan gustafson and styan superstochastic matrices and magic markov chains linear algebra appl 430 2009 2705 2715 examined the mathematical properties of superstochastic matrices the transition matrices of magic markov chains formed from scaled magic squares this paper explores the main stochastic properties of such chains as well as semi magic chains with doubly stochastic transition matrices stationary distribution generalized inverses of markovian kernels mean first passage times variances of the first passage times and expected times to mixing are considered some general results are developed some observations from the chains generated by matlab are discussed some conjectures are presented and some special cases involving three and four states are explored in detail

regular markov chains for which the transition matrix has large exponent abstract in this paper we analyze properties of transition matrices t r n n of regular markov chains whose exponent index of primitivity is bounded below by n 1 2 1 2 2 our investigation leads to a proof that for such a t the corresponding condition number k t max 1 i j n q i j where q i t we go on to show that if e r n n is a perturbation of t such that t e is also a transition matrix for a regular markov chain and and are the stationary distribution vectors of t and t e respectively then and 1 1 many of the techniques used to establish these results are combinatorial in nature

applications of paz s inequality to perturbation bounds for markov chains abstract in several papers meyer singly and with coauthors established the usefulness of the group generalized inverse in the study and computations of various aspects of markov chains here we are interested in those results which concern bounds on the condition number of the chain and on the error in the computation of the stationary distribution vector we show that a lemma due to paz can be used to improve sometimes by a factor of 2 some of the constants in the bounds obtained in the aforementioned papers

a precise numerical scheme for contaminant transport in dual well flow 1 a precise numerical method is developed for the simulation of contaminant transport in dual well flow our main motivation is the determination of model parameters by means of dual well tests the method presented here guarantees a very low level of numerical dispersion we have applied a bipolar transformation to the transport equation so that the convective term is parallel to one of the axes in the time stepping we split the transport and diffusion parts the transport part is semianalytically solved thanks to the bipolar transformation via a multiple riemann problem and the diffusion part is solved using a finite volume method examples are provided to show that the method is suitable for parameter identification

blockwise perturbation theory for block p cyclic stochastic matrices summary let p be a block p cyclic stochastic matrix with stationary distribution bf pi rm t which is partitioned conformally in the form bf pi 1 rm t ldots bf pi p rm t this paper establishes the relative error bound for bf pi i rm t when each block of p gets a small relative perturbation

response and sensitivity using markov chains dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties a prominent real life example of this class of problems is the investigation of climate response to perturbations in this respect it is crucial to determine what the linear response of a system is as a quantification of sensitivity alongside previous work here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations by projecting the transfer operator onto a suitable finite dimensional vector space one is able to obtain matrix representations which determine finite markov processes further using perturbation theory for markov matrices it is possible to determine the linear and nonlinear response of the system given a prescribed forcing here we suggest a methodology which puts the scope on the evolution law of densities the liouville fokker planck equation allowing to effectively calculate the sensitivity and response of two representative dynamical systems

solution of convection diffusion problems with nonequilibrium adsorption in this paper an efficient method for solving contaminant transport problem with diffusion and adsorption in nonequilibrium mode is introduced the method is based on operator splitting approach where the solution of the transport problem is found in an analytical form and the diffusion problem is solved by finite volume method finally the system corresponding to nonequilibrium adsorption problem is transformed to one integral equation which is solved numerically using time discretization some numerical aspects of the problem are discussed and various results of numerical experiments are shown as well as a comparison to precise semi analytical solution for special cases

generalized inverses and their application to applied probability problems the main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a markov chain is present either directly or indirectly through some form of imbedding by characterizing all generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain we are able to obtain general procedures for finding stationary distributions moments of the first passage time distributions and asymp totic forms for the moments of the occupation time random variables it is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework more generally in the context of a markov renewal process setting the aforementioned problems are also examined using generalized inverses of i p as a special case markov chains in continuous time are considered and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process instead of to i p where p is the transition matrix of the discrete time jump markov chain

a direct projection method for markov chains a direct method based on oblique projections is adapted to compute the stationary distribution vector of a finite markov chain the algorithm can also be used to compute the group inverse of the corresponding generator matrix it is shown how to update the stationary vector and other quantities of interest when one row of the transition probability matrix is modified a gth like variant that appears to compute the stationary probabilities to high relative accuracy is developed

robotic surveillance and markov chains with minimal weighted kemeny constant this article provides analysis and optimization results for the mean first passage time also known as the kemeny constant of a markov chain first we generalize the notion of the kemeny constant to environments with heterogeneous travel and service times denote this generalization as the weighted kemeny constant and we characterize its properties second for reversible markov chains we show that the minimization of the kemeny constant and its weighted counterpart can be formulated as convex optimization problems and moreover as semidefinite programs third we apply these results to the design of stochastic surveillance strategies for quickest detection of anomalies in network environments we numerically illustrate the proposed design compared with other well known markov chains the performance of our kemeny based strategies are always better and in many cases substantially so

critical points and positive solutions of singular elliptic boundary value problems usually we do not think there is variational structure for singular elliptic boundary value problems so it cannot be considered by using critical points theory in this paper we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems especially use the ordinary differential equation theory of banach spaces to obtain new results on the existence of multiple positive solutions the method is useful for other singular problems

on bifurcation critical groups and exact multiplicity of solutions to semilinear elliptic boundary value problems in this paper we use a bifurcation approach morse theory to study asymptotically linear and superlinear elliptic boundary value problems and obtain new results on the exact multiplicity of solutions and sign changing solutions to asymptotically linear and superlinear problems and the properties of the solutions

multiple solutions of impulsive sturm liouville boundary value problem via lower and upper solutions and variational methods abstract in this paper we prove the existence of multiple solutions for second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is a sturm liouville operator r 1 u u 0 u 0 r 2 u u 1 u 1 the technical approach is fully based on lower and upper solutions and variational methods the interesting point is that the existence of four solutions and seven solutions is given

kemeny s constant and the effective graph resistance kemeny s constant and its relation to the effective graph resistance has been established for regular graphs by palacios et al 1 based on the moore penrose pseudo inverse of the laplacian matrix we derive a new closed form formula and deduce upper and lower bounds for the kemeny constant furthermore we generalize the relation between the kemeny constant and the effective graph resistance for a general connected undirected graph

dancer fucik spectrum for fractional schrodinger operators with a steep potential well on rn abstract in this paper we study dancer fucik spectrum of the fractional schrodinger operators which is defined as the set of r 2 such that s u v x u u u in r n has a nontrivial solution u where the potential v has a steep potential well for sufficiently large parameter 0 it is allowed that s v has essential spectrum with finitely many eigenvalues below the infimum of e s s s v many difficulties are caused by general nonlocal operators we develop new techniques to overcome them to construct the first nontrivial curve of dancer fucik point spectrum by minimax methods to show some qualitative properties of the curve and to prove that the corresponding eigenfunctions are foliated schwartz symmetric as applications we obtain the existence of nontrivial solutions for nonlinear schrodinger equations with nonresonant nonlinearity

stationary distributions and mean first passage times of perturbed markov chains stationary distributions of perturbed finite irreducible discrete time markov chains are intimately connected with the behaviour of associated mean first passage times this interconnection is explored through the use of generalized matrix inverses some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are obtained together with some improved bounds ams classification 15a51 60j10 60j20 65f20 65f35

solution of contaminant transport with equilibrium and non equilibrium adsorption abstract in this paper we present a numerical approximation scheme for the solution of contaminant transport problems with diffusion and adsorption in equilibrium and non equilibrium mode the method is based on time stepping and operator splitting the non linear transport is solved semi analytically via the multiple riemann problem the non linear diffusion by a finite volume method and by newton s type of linearisation and finally the reaction part incorporating the non equilibrium adsorption is transformed to an integral equation which is solved numerically using time discretization various results of numerical experiments are shown and the method is applied to the practical dual well problem

strong stability and perturbation bounds for discrete markov chains we consider the stability problem of discrete markov chains when their transition matrices are perturbed for finite irreducible markov chains many perturbation bounds for the stationary vector are available in this paper we identify a condition under which these bounds are carried over to discrete irreducible markov chains which are not finite

analytical perturbations in markov chains abstract the problem of recomputing the steady state probabilities in a markov chain is considered after a small deviation is introduced to the original infinitesimal generator approximate expressions are developed to calculate the corresponding perturbed vector of probabilities computational stability and accuracy issues associated with the approximation are discussed as an application the perturbation approach is incorporated into an optimization scheme to identify workloads that maximize the output rate in a production line where qualitative and quantitative measures are introduced to control the accuracy of the perturbation

the intercept term of the asymptotic variance curve for some queueing output processes we consider the output processes of some elementary queueing models such as the m m 1 k queue and the m g 1 queue an important performance measure for these counting processes is their variance curve v t which gives the variance of the number of customers in the time interval 0 t recent work has revealed some non trivial properties dealing with the asymptotic rate at which the variance curve grows in this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote for m m 1 k queues our results are based on the deviation matrix of the generator it turns out that by viewing output processes as markovian point processes and considering the deviation matrix one can obtain explicit expressions for the intercept term together with some further insight regarding the bravo balancing reduces asymptotic variance of outputs effect for m g 1 queues our results are based on a classic transform of d j daley in this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution in addition we shed light on a conjecture of daley dealing with characterization of stationary m m 1 queues within the class of stationary m g 1 queues based on the variance curve

on optimal condition numbers for markov chains let t and tilde t t e be arbitrary nonnegative irreducible stochastic matrices corresponding to two ergodic markov chains on n states a function is called a condition number for markov chains with respect to the norm pair if pi tilde pi alpha leq kappa t e beta here and tilde pi are the stationary distribution vectors of the two chains respectively various condition numbers particularly with respect to the 1 and norm pairs have been suggested in the literature they were ranked according to their size by cho and meyer in a paper from 2001 in this paper we first of all show that what we call the generalized ergodicity coefficient tau p a rm sup y t e 0 frac y ta p y 1 where e is the n vector of all 1 s and a is the group generalized inverse of a i t is the smallest condition number of markov chains with respect to the p norm pair we use this result to identify the smallest condition number of markov chains among the and 1 norm pairs these are respectively 3 and 6 in the cho meyer list of 8 condition numbers kirkland has studied 3 t he has shown that kappa 3 t geq frac n 1 2n and he has characterized transition matrices for which equality holds we prove here again that 2 3 t 6 which appears in the cho meyer paper and we characterize the transition matrices t for which kappa 6 t frac n 1 n there is actually only one such matrix t j n i n 1 where j n is the n n matrix of all 1 s

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

cycle frequency in standard rock paper scissors games evidence from experimental economics the rock paper scissors rps game is a widely used model system in game theory evolutionary game theory predicts the existence of persistent cycles in the evolutionary trajectories of the rps game but experimental evidence has remained to be rather weak in this work we performed laboratory experiments on the rps game and analyzed the social state evolutionary trajectories of twelve populations of n 6 players we found strong evidence supporting the existence of persistent cycles the mean cycling frequency was measured to be 0 029 0 009 period per experimental round our experimental observations can be quantitatively explained by a simple non equilibrium model namely the discrete time logit dynamical process with a noise parameter our work therefore favors the evolutionary game theory over the classical game theory for describing the dynamical behavior of the rps game c 2013 the authors published by elsevier b v all rights reserved

parametric forms for generalized inverses of markovian kernels and their applications parametric forms for multicondition generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain are derived these are utilized in techniques for obtaining moments of first passage time distributions

on sign changing and multiple solutions of the p laplacian abstract in this paper we construct the pseudo gradient vector field in w 0 1 p by which we obtain the positive and negative cones of w 0 1 p are both invariant sets of the descent flow of the corresponding functional then we use differential equations theory in banach spaces and dynamics theory to study p laplacian boundary value problems with jumping nonlinearities at zero or infinity and get new multiple solutions and sign changing solutions theorems of p laplacian

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

blockwise perturbation theory for nearly uncoupled markov chains and its application abstract let p be the transition matrix of a nearly uncoupled markov chain the states can be grouped into aggregates such that p has the block form p p ij i j 1 k where p ii is square and p ij is small for i j let t be the stationary distribution partitioned conformally as t 1 t k t in this paper we bound the relative error in each aggregate distribution i t caused by small relative perturbations in p ij the error bounds demonstrate that nearly uncoupled markov chains usually lead to well conditioned problems in the sense of blockwise relative error as an application we show that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve such structured backward errors and compute each aggregate distribution with high relative accuracy

on the existence of signed and sign changing solutions for a class of superlinear schrodinger equations this paper deals with a semilinear schrodinger equation whose nonlinear term involves a positive parameter and a real function f u which satisfies a superlinear growth condition just in a neighborhood of zero by proving an a priori estimate for a suitable class of solutions we are able to avoid further restrictions on the behavior of f u at infinity in order to prove for sufficiently large the existence of one sign and sign changing solutions minimax methods are employed to establish this result

a parametric uncertainty analysis method for queues with vacations in this paper we propose a numerical method based on taylor series expansion for computing the uncertainty in performance measures of vacation queueing model due to epistemic uncertainties in its parameters specifically we consider the m g 1 n queue with vacations where we assume that vacation parameter is derived from finite number of observations they themselves will have epistemic uncertainty associated with them using this method we compute the expected value and the variance of the stationary distribution associated with the m g 1 n queue with vacations in order to illustrate the applicability of the proposed method a series of numerical examples are discussed and the obtained results are compared to the corresponding monte carlo simulations ones

construction of pseudo gradient vector field and sign changing multiple solutions involving p laplacian existence of multiple and sign changing solutions for a problem involving p laplacian and jumping nonlinearities are considered via the construction of descent flow in c01 sign changing and multiple solutions are obtained under additional assumption on the nonlinearity the uniqueness of positive negative solution theorem is included too

fucik spectrum sign changing and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity in this paper we use fucik spectrum ordinary differential equation theory of banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity especially with resonance at infinity and obtain new results on the existence of multiple solutions and sign changing solutions with a weakening of the p s condition in one case we get up to seven nontrivial solutions the techniques have independent interest

non uniformly asymptotically linear p laplacian problems abstract we relax conditions on uniform convergences of f x s s p 2 s which are essential in many results on multiplicity for asymptotically linear p laplacian problems

the use of positive matrices for the analysis of the large time behavior of the numerical solution of reaction diffusion systems in this paper we study the numerical solution of nonlinear reaction diffusion systems with homogeneous neumann boundary conditions via the known o method we show that if conditions for the positivity of solutions are imposed then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices in this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one in particular we obtain the decay of the solution to its mean value an analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of m matrices finally we consider the case in which the nonlinear term satisfies a condition of quasi monotonicity

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

stability and criticality analysis for integer linear programs with markovian problem data this paper presents the stability and criticality analysis of integer linear programs with respect to perturbations in stochastic data given as markov chains these perturbations affect the initial distribution the transition matrix or the stationary distribution of markov chains stability analysis is concerned with obtaining the set of all perturbations for which a solution remains optimal this paper gives expressions for stability regions for perturbations in the initial distribution the transition matrix the stationary distribution and the product of elements of the transition matrix and the stationary distribution furthermore criticality measures that describe the sensitivity of the objective function with respect to an element of the problem data are derived stability regions that preserve the stochasticity of the problem data are given finally stability regions for perturbations of elements of the transition matrix given that the problem is not linear in the initial distribution or the transition matrix are obtained using a small perturbation analysis the results are applied to sensor placement problems and numerical examples are given

comparison of perturbation bounds for the stationary distribution of a markov chain abstract the purpose of this paper is to review and compare the existing perturbation bounds for the stationary distribution of a finite irreducible homogeneous markov chain

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

mixing times with applications to perturbed markov chains abstract a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain is considered the statistic i j 1 m m ij j where j is the stationary distribution and mij is the mean first passage time from state i to state j of the markov chain is shown to be independent of the initial state i so that i for all i is minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations an application considering the effects perturbations of the transition probabilities have on the stationary distributions of markov chains leads to a new bound involving for the 1 norm of the difference between the stationary probability vectors of the original and the perturbed chain when is large the stationary distribution of the markov chain is very sensitive to perturbations of the transition probabilities

multiple and nodal solutions for parametric neumann problems with nonhomogeneous differential operator and critical growth abstract we consider a parametric neumann problem with nonhomogeneous differential operator and critical growth combining variational methods based on critical point theory with suitable truncation techniques and flow invariance arguments we show that for all large the problem has at least three nontrivial smooth solutions two of constant sign one positive the other negative and the third nodal we also study the asymptotic behavior of all solutions obtained when converges to infinity the interesting point is that we do not impose any restrictions to the behavior of the nonlinear term f at infinity our work unifies and sharply improves several recent papers

a note on entrywise perturbation theory for markov chains abstract o cinneide presented an entrywise perturbation theorem for markov chains the error bound he obtained is nearly best possible we give the best possible error bound

effect of function perturbation on the steady state distribution of genetic regulatory networks optimal structural intervention the dynamics of a rule based gene regulatory network are determined by the regulatory functions in conjunction with whatever probability distributions are involved in network transitions in the case of boolean networks bns and more generally probabilistic boolean networks pbns there has been a significant amount of investigation into the effect of perturbing gene states in particular the design of intervention strategies based on finite or infinite horizon control polices this paper considers the less investigated issue of function perturbations a single function perturbation affects network dynamics and alters the long run distribution whereas any individual gene perturbation has only transient effects and does not change the long run distribution we derive analytic results for changes in the steady state distributions of pbns resulting from modifications to the underlying regulatory rules and apply the derived results to find optimal structural interventions to avoid undesirable states the results are applied to a wnt5a network and a mammalian cell cycle related network respectively to achieve more favorable steady state distributions and reduce the risk of getting into aberrant phenotypes

sign changing solutions for resonant neumann problems we consider semilinear neumann problems at resonance we are concerned with two distinct cases in the first one the potential function is unbounded and indefinite in the second case the potential function is bounded and the reaction exhibits a concave parametric term near the origin we prove two existence theorems producing sign changing solutions the first main result is a complement of the results obtained by papageorgiou and r dulescu 2015 22 the second theorem gives an answer to the open question raised by candito et al 2016 4 our approach uses variational methods together with flow invariance arguments

the computation of key properties of markov chains via perturbations abstract computational procedures for the stationary probability distribution the group inverse of the markovian kernel and the mean first passage times of a finite irreducible markov chain are developed using perturbations the derivation of these expressions involves the solution of systems of linear equations and structurally inevitably the inverses of matrices by using a perturbation technique starting from a simple base where no such derivations are formally required we update a sequence of matrices formed by linking the solution procedures via generalised matrix inverses and utilising matrix and vector multiplications four different algorithms are given some modifications are discussed and numerical comparisons are made using a test example the derivations are based upon the ideas outlined by hunter 14

sensitivity to perturbation of the stationary distribution some refinements abstract the sensitivity of the unique stationary distribution of a finite markov chain which has a state accessible in one step from every other state is examined using ordinary matrix inversion involving an arbitrary vector a result of funderlic and meyer is sharpened the presence of the arbitrary vector allows extension of the result to stochastic matrices with other structures the examples given by funderlic and meyer are reconsidered

characterizations of generalized inverses associated with markovian kernels abstract a systematic investigation of the various multicondition generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain is presented the characterizations are based upon the result that i p tu is nonsingular if and only if t and u are chosen so that t 0 where is the stationary probability vector of p and u e 0 where e is the vector of ones partitioned forms for the g inverses are also presented based on a full rank factorization of i p special well known cases such as the group inverses and the moore penrose inverse are given

sign changing and multiple solutions of the sturm liouville boundary value problem via invariant sets of descending flow abstract in this paper we prove the existence of sign changing and multiple solutions for the second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is the sturm liouville operator r 1 u u 0 u 0 and r 2 u u 1 u 1 the technical approach is fully based on minimax methods and invariant sets of descending flow

parameter estimation in convection dominated nonlinear convection diffusion problems by the relaxation method and the adjoint equation the development of numerical methods for strongly nonlinear convection diffusion problems with dominant convection is an ongoing topic in numerical analysis for inverse problems in this setting there is a need of fast and accurate solvers here we present operator splitting with a riemann solver for the convective part and a relaxation method for the diffusive part as a means to achieve this goal combined with the adjoint equation method this allows us to solve inverse problems within reasonable time frames and with modest computing power as an example the dual well experiment is considered and the adjoint method is compared with a conjugate gradient algorithm and a levenberg marquardt type of iteration method

sign changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems with a reaction term nonzero at zero abstract in this paper we use the ordinary differential equation theory of banach spaces and minimax theory in particular the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero and get new multiple solutions and sign changing solutions theorems moreover the sign changing solutions change sign exactly once under some conditions

variances of first passage times in a markov chain with applications to mixing times jm m 1 as a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain with stationary distribution pj and mij as the mean first passage time from state i to state j of the markov chain this was shown to be independent of the initial state i with i for all i minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations in this paper we explore the variance of the mixing time v i starting in state i the v i are shown to depend on i and an exploration of recommended starting states given knowledge of the transition probabilities is considered as a preamble a study of the computation of second moments of the mixing times m ij 2 and the variance of the first passage times in a discrete time markov chain is carried out leading to some new results