how colored environmental noise affects population extinction environmental noise can cause an exponential reduction in the mean time to extinction mte of an isolated population we study this effect on an example of a stochastic birth death process with rates modulated by a colored that is correlated gaussian noise a path integral formulation yields a transparent way of evaluating the mte and finding the optimal realization of the environmental noise that determines the most probable path to extinction the population size dependence of the mte changes from exponential in the absence of the environmental noise to a power law for a short correlated noise and to no dependence for long correlated noise we also establish the validity domains of the white noise limit and adiabatic limit

positive solutions to a system of adjointable operator equations over hilbert c modules abstract we present necessary and sufficient conditions for the existence of a positive solution to the system of adjointable operator equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 over hilbert c modules we also derive a representation for a general positive solution to this system when the solvability conditions are satisfied the results of this paper extend some known results in the literature

the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract in this article we consider common re nnd and re pd solutions of the matrix equations ax c and xb d with respect to x where a b c and d are given matrices we give necessary and sufficient conditions for the existence of common re nnd and re pd solutions to the pair of the matrix equations and derive a representation of the common re nnd and re pd solutions to these two equations when they exist the presented examples show the advantage of the proposed approach

common hermitian solutions to some operator equations on hilber c modules abstract we establish necessary and sufficient conditions for the existence of the general common hermitian solution to the equations a 1 x c 1 xb 1 c 2 a 3 xa 3 c 3 a 4 xa 4 c 4 for adjointable operators between hilbert c modules and present an expression for the common hermitian solution to the equations in terms of moore penrose inverse of operators when the solvability conditions are satisfied the findings of this paper extend some known results in the literature

solutions to operator equations on hilbert c modules abstract let a be a c algebra e f and g be hilbert a modules t l a e f and t l a g f we generalize the douglas theorem about the operator equation tx t from hilbert space to hilbert c module to the equation tx t and to the system of two equations tx t and xs s we get the forms of general solutions in the case that there exists a solution and give some sufficient and necessary conditions for the existence of solutions and the existence of hermitian solutions and positive solutions in the case g e in addition the forms of general hermitian solution and general positive solution in the case that there exists a solution and g e to the equation tx t are given too

extinction rates for fluctuation induced metastabilities a real space wkb approach the extinction of a single species due to demographic stochasticity is analyzed the discrete nature of the individual agents and the poissonian noise related to the birth death processes result in local extinction of a metastable population as the system hits the absorbing state the fokker planck formulation of that problem fails to capture the statistics of large deviations from the metastable state while approximations appropriate close to the absorbing state become in general invalid as the population becomes large to connect these two regimes a real space wkb method based on the master equation is presented and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state the details of the underlying microscopic process smeared out in a mean field treatment are shown to be crucial for an exact determination of the extinction exponent this general scheme is shown to reproduce the known results in the field to yield new corollaries and to fit quite precisely the numerical solutions moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state

reduced solutions of douglas equations and angles between subspaces in this paper we study some particular solutions of douglas type equations by means of generalized inverses and angles we apply this result to characterize positive solutions and some special projections which are symmetrizable for a semidefinited positive bounded linear operator

the common positive solution to adjointable operator equations with an application abstract we give necessary and sufficient conditions for the existence of the common positive solution to the equations a 1 x 1 c 1 x 1 b 1 d 1 a 2 x 2 c 2 x 2 b 2 d 2 a 3 x 1 a 3 a 4 x 2 a 4 c 5 for adjointable operators between hilbert c modules and present an expression of the positive solution to the system when the solvability conditions are satisfied as an application we consider the reflexive positive semidefinite solution to the system of complex matrix equations a x b x c d e x e f

re nnd generalized inverses abstract in this paper we give some necessary and sufficient conditions for the existence of re nnd and nonnegative definite 1 3 and 1 4 inverses of a matrix a c n n and completely described these sets moreover we prove that the existence of nonnegative definite 1 3 inverse of a matrix a is equivalent with the existence of its nonnegative definite 1 2 3 inverse and present the necessary and sufficient conditions for the existence of re nnd 1 3 4 inverse of a

evolution of a fluctuating population in a randomly switching environment environment plays a fundamental role in the competition for resources and hence in the evolution of populations here we study a well mixed finite population consisting of two strains competing for the limited resources provided by an environment that randomly switches between states of abundance and scarcity assuming that one strain grows slightly faster than the other we consider two scenarios one of pure resource competition and one in which one strain provides a public good and investigate how environmental randomness external noise coupled to demographic internal noise determines the population s fixation properties and size distribution by analytical means and simulations we show that these coupled sources of noise can significantly enhance the fixation probability of the slower growing species we also show that the population size distribution can be unimodal bimodal or multimodal and undergoes noise induced transitions between these regimes when the rate of switching matches the population s growth rate

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

equations ax c and xb d in rings and rings with involution with applications to hilbert space operators abstract this paper reviews the equations ax c and xb d from a new perspective by studying them in the setting of associative rings with or without involution results for rectangular matrices and operators between different banach and hilbert spaces are obtained by embedding the rectangles into rings of square matrices or rings of operators acting on the same space necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian skew hermitian reflexive antireflexive positive and real positive solutions and the general solutions are described in terms of the original elements or operators new results are obtained and many results existing in the literature are recovered and corrected

taming the diffusion approximation through a controlling factor wkb method the diffusion approximation da is widely used in the analysis of stochastic population dynamics from population genetics to ecology and evolution the da is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied this failure becomes severe in situations where the direction of selection switches sign here we employ the wkb wentzel kramers brillouin large deviations method which requires only the logarithm of a given quantity to be smooth over its state space combining the wkb scheme with asymptotic matching techniques we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations applicable for much wider regimes of parameters we also introduce a scalable independent of population size wkb based numerical technique the method is applied to a central problem in population genetics and evolution finding the chance of ultimate fixation in a zero sum two types competition

the solutions to linear matrix equations ax b ya d with k involutory symmetries let rcmm and scnn be nontrivial k involutions if their minimal polynomials are both xk1 for some k2 i e rk1 r1i and sk1 s1i we say that acmn is r s symmetric if ras1 a and a is r s symmetric if ras a with 0 1 k1 and 0 let s be one of the subsets of all r s symmetric and r s symmetric matrices given xcnr ycsm bcmr and dcsn we characterize the matrices a in s that minimize axb2 yad2 frobenius norm under the assumption that r and s are unitary moreover among the set s x y b d s of the minimizers of axb2 yad2 min we find the optimal approximate matrix as x y b d that minimizes ag to a given unstructural matrix gcmn we also present the necessary and sufficient conditions such that ax b ya d is consistent in s if the conditions are satisfied we characterize the consistent solution set of all such a finally a numerical algorithm and some numerical examples are given to illustrate the proposed results

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

the re nonnegative definite and re positive definite solutions to the matrix equation axb d in this paper the re nonnegative definite re nnd and re positive definite re pd solutions of the matrix equation axb d are considered the necessary and sufficient conditions for the existence of re nnd and re pd solutions to the equation are provided and the explicit representations of the general re nnd and re pd solutions are given when it is solvable

positive solutions to the equations ax c and xb d for hilbert space operators abstract the paper studies the equation a x c for bounded linear operators between hilbert spaces gives conditions for the existence of hermitian solutions and positive solutions and obtains the formula for the general form of these solutions then the common hermitian and positive solutions to the equations a x c and x b d are studied and new representations of the general solutions are given many results for matrices are recovered as special cases and the results of phadke and thakare s v phadke n k thakare generalized inverses and operator equations linear algebra appl 23 1979 191 199 are corrected

r s conjugate solution to a pair of linear matrix equations abstract let r and s be m m and n n nontrivial real symmetric involutions an m n complex matrix a is termed r s conjugate if a ras where a denotes the conjugate of a in this paper necessary and sufficient conditions are established for the existence of the r s conjugate solution to the system of matrix equations ax c and xb d the expression is also presented for such solution to this system in addition the explicit expression of this solution to the corresponding optimal approximation problem is obtained furthermore the least squares r s conjugate solution with least norm to this system mentioned above is considered the representation of such solution is also derived finally an algorithm and numerical examples are given

application of legendre series to the optimal control of integrodifferential equations this paper extends the application of shifted legendre series expansions to integrodifferential systems with quadratic performance index based on the shifted legendre series a directly computational formulation for evaluating the optimal control and trajectory is developed the formulation is straightforward and convenient for digital computation an example is given to demonstrate the applicability of the proposed method

phase diagram for logistic systems under bounded stochasticity extinction is the ultimate absorbing state of any stochastic birth death process hence the time to extinction is an important characteristic of any natural population here we consider logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity three phases are identified an inactive phase where the mean time to extinction t increases logarithmically with the initial population size an active phase where t grows exponentially with the carrying capacity n and a temporal griffiths phase with a power law relationship between t and n the system supports an exponential phase only when the noise is bounded in which case the continuum diffusion approximation breaks down within the griffiths phase this breakdown is associated with a crossover between qualitatively different survival statistics and decline modes to study the power law phase we present a new wkb scheme which is applicable both in the diffusive and in the nondiffusive regime

the solutions to some operator equations abstract in this paper we study the solvability of the operator equation axb bx a c in the general setting of the adjointable operators between hilbert c modules based on the generalized inverses of the associated operators we propose the necessary and sufficient conditions for the existence of a solution to this equation and obtain the general expression of the solution in the solvable case we apply the results to the study of the real positive and positive solutions to the operator equation axb c in the case that the underlying space is finite dimensional or the range of b is contained in that of a we propose new necessary and sufficient conditions for the existence of a positive solution to the operator equation axb c and derive new formula in each case for the general positive solution to this operator equation

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

on the hermitian solutions to a system of adjointable operator equations we establish necessary and sufficient conditions for the existence of a hermitian solution to the system of equations a1x1 c1 x1b1 d1 a2x2 c2 x2b2 d2 a3x1a3 a4x2a4 c5 for adjointable operators between hilbert c modules and provide an expression for the general hermitian solution to the system we present necessary and sufficient conditions for the existence of a unique hermitian solution to the systems a1x1 c1 x1b1 d1 and a3x1a3 c5 of adjointable operators between hilbert c modules several examples are given to explain the use of the theory some of the findings of this paper extend some known results in the literature

about the von neumann regularity of triangular block matrices abstract necessary and sufficient conditions are given for the von neumann regularity of triangular block matrices with von neumann regular diagonal blocks over arbitrary rings this leads to the characterization of the von neumann regularity of a class of triangular toeplitz matrices over arbitrary rings some special results and a new algorithm are derived for triangular toeplitz matrices over commutative rings finally the drazin invertibility of some companion matrices over arbitrary rings is considered as an application

comments on the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract the purpose of this paper is to present a critical review on the paper the common re nnd and re pd solutions to the matrix equations ax c and xb d by authors xiong and qin 2011 1 we point out that the conditions given by theorem 2 1 are not enough to ensure the existence of common re nnd solution to ax c and xb d and the proofs are unacceptable finally the corrected results are given

on the solutions of operator equation cax c xac abstract in this paper we find the explicit solutions of the equation c a x c x a c for linear bounded operators on hilbert spaces where the unknown operator x is called the inverse of a along c this solution is expressed in terms of the inner inverses of the operators a c and c a from these formulas we derive necessary and sufficient conditions for the solution to be unique

analysis and optimal control of time varying linear systems via shifted legendre polynomials this paper extends the application of shifted legendre polynomial expansion to time varying systems the extension is achieved through representing the product of two shifted legendre series in a new shifted legendre series with this treatment of the product of two time functions the operational properties of the shifted legendre polynomials are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index

analysis of linear time varying systems and bilinear systems via fourier series a method for finding an approximate solution of linear time varying systems and bilinear systems via fourier series is proposed properties of fourier series are first briefly presented and the operational matrix for integration together with the product operational matrix are utilized to reduce the dynamical equation of a linear time varying system or bilinear system to a set of simultaneous linear algebraic equations illustrative examples are also given

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

common hermitian and positive solutions to the adjointable operator equations ax c xb d let a be a c algebra for any hilbert a modules h and k let l h k be the set of adjointable operators from h to k let h k l be hilbert a modules a c l h k and b d l l h in this paper we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions x l h to the equations ax c xb d and obtain the formulae for the general forms of these solutions some results known for finite matrices and hilbert space operators are extended to the adjointable operators acting on hilbert c modules

the reflexive solutions of the matrix equation ax b c in this paper we study the existence of a reflexive with respect to the generalized reflection matrix p solution of the matrix equation ax b c for the special case when b i we get the result of peng and hu 1