abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

linear preservers of weak majorization on l1 i when i is an infinite set abstract the necessary and sufficient conditions that a bounded linear map on the banach space l 1 i may be considered as a linear preserver of weak majorization on l 1 i where i is an arbitrary infinite set are given also we prove that the set of all linear preservers of weak majorization is closed under the norm topology

linear preservers of weak majorization on lp i when p 1 abstract we present a necessary and sufficient conditions that a bounded linear map on l p i when p 1 may be considered as a linear preserver of weak majorization on l p i where i is an arbitrary infinite set we also prove that the class of all linear preservers of weak majorization is closed under the norm topology

group invariant analogues of hadamard s inequality a wide class of inequalities for the determinant and other real valued functions of an n n complex hermitian or real symmetric matrix h hjk may be obtained by generalizing marshall and olkin s proof of hadamard s inequality deth j 1nhjj for positive definite pd h we shall see that each subgroup g of the group un of n x n unitary matrices not only determines an analogue of 1 for det h but also provides inequalities for a large family of unitarily invariant functions of h not necessarily pd

inequalities theory of majorization and its applications springer series in statistics by albert w marshall ingram olkin and barry c arnold 2nd edition springer 2011 xxvii 909 pp hardback isbn 978 0 387 40087 7 e isbn 978 0 387 68276 1

inter and extrapolatory properties of matrix partial orderings in this paper we study some matrix partial orderings with respect to the preserving the relation a b for original a or and b and the linear combinations of a and b the property that between above respectively a and below respectively b one can locate convex affine combinations of original matrices will be called interpolatory extrapolatory respectively besides classical orderings we introduce and discuss in the mentioned aspect the notion of f orderings

linear maps preserving group majorization abstract a necessary and sufficient condition for a linear map to preserve group majorizations is given the condition is applied to prove some preservation results

linear operators preserving directional majorization abstract a characterization of linear operators preserving directional majorization is given the result extends a theorem of t ando linear algebra appl 118 1989 163 and the proof also gives the characterization of linear operators preserving multivariate majorization recently obtained by beasley and lee

monotonicity for entrywise functions of matrices abstract we characterize real functions f on an interval for which the entrywise matrix function a ij f a ij is positive monotone and convex respectively in the positive semidefiniteness order fractional power functions are exemplified and related weak majorizations are shown

a new method of analysis of the effect of weak colored noise in nonlinear dynamical systems a systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed it is based on the search of wkb solutions to the fokker planck equation for the joint probability density of the system and noise in which the perturbation expansion is continued to the first correction beyond the hamilton jacobi limit the method can be applied to noise with correlation time of order unity it is illustrated on the normal form of a pitchfork bifurcation where it is pointed out that additive noise can induce a shift of the most probable value this prediction is confirmed by numerical simulation of the stochastic differential equations

g majorization with applications to matrix orderings abstract a vector x is said to g majorize a vector y if y lies in the convex hull of the orbit of x under a group g the present paper contains a straightforward account with two important statements equivalent to g majorization in certain cases for example when g is a finite reflection group one equivalent condition reduces to a finite set of linear inequalities representing a cone ordering in the fundamental region of the group the other condition is that every convex g invariant function of y is less than the same function of x upper and lower weak majorizations gw majorizations are introduced by combining g with a second ordering compatible with it the results are applied where possible to matrix orderings where a g b is g majorization when g is a subgroup of the orthogonal group o n acting by congruence i e g b qbq t with q the matrix representation for an element of o n when g is the symmetric group of permutation matrices this defines a new ordering and generalizes a proposition of kiefer in the design of experiments

weak matrix majorization abstract given x y r n m we introduce the following notion of matrix majorization called weak matrix majorization x w y if there exists a row stochastic matrix a r n n such that ax y and consider the relations between this concept strong majorization s and directional majorization it is verified that s w but none of the reciprocal implications is true nevertheless we study the implications w s and s under additional hypotheses we give characterizations of strong directional and weak matrix majorization in terms of convexity we also introduce definitions for majorization between abelian families of selfadjoint matrices called joint majorizations they are induced by the previously mentioned matrix majorizations we obtain descriptions of these relations using convexity arguments

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

a reaction diffusion advection model of harmful algae growth with toxin degradation abstract this paper is devoted to the study of a reaction diffusion advection system modeling the dynamics of a single nutrient harmful algae and algal toxin in a flowing water habitat with a hydraulic storage zone we introduce the basic reproduction ratio r 0 for algae and show that r 0 serves as a threshold value for persistence and extinction of the algae more precisely we prove that the washout steady state is globally attractive if r 0 1 while there exists a positive steady state and the algae is uniformly persistent if r 0 1 with an additional assumption we obtain the uniqueness and global attractivity of the positive steady state in the case where r 0 1

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

a characterization of strong preservers of matrix majorization abstract a matrix a is said to be matrix majorized by a matrix b written a b if there exists an n n row stochastic matrix x such that a bx this is a generalization of multivariate majorization in this paper we characterize the linear operators that strongly preserve the matrix majorization

a new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

majorization and singular values iii abstract we obtain a majorization inequality which relates the singular values of a complex square matrix a and those of t a where t is a linear map the inequality takes a particularly simple form if t is positive the results generalize several known majorization inequalities for eigenvalues and singular values

some inequalities for sum and product of positive semidefinite matrices abstract the purpose of this paper is to present some inequalities on majorization unitarily invariant norm trace and eigenvalue for sum and product of positive semidefinite hermitian matrices some open questions proposed by marshall and olkin are resolved

stochastic analysis of symmetry breaking bifurcations master equation approach the multivariate master equation for a general reaction diffusion system is solved perturbatively in the vicinity of a bifurcation point leading to symmetry breaking transitions the possibility to express the result through a brazovskii type of potential is examined and a comparison with the langevin analysis of walgraefet al adv chem phys 49 311 1982 is performed

sensitivity of a hopf bifurcation to multiplicative colored noise the location of a hopf bifurcation in parameter space may be postponed or advanced by multiplicative colored noise depending on the interplay between the time scale of the noise and the rotation period of the phase variable

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

on the stationary distribution of self sustained oscillators around bifurcation points a double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones systems undergoing hopf bifurcations belong to this class as an illustrative example the generalized van der pol oscillator is considered around its bifurcation point a calculation is carried out up to third order in both the noise intensity and the bifurcation parameter damping coefficient

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

adaptive subgradient methods for online learning and stochastic optimization we present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient based learning metaphorically the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm we describe and analyze an apparatus for adaptively modifying the proximal function which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight we give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints we experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state of the art yet non adaptive subgradient algorithms

the solution of van der waerden s problem for permanents in this paper a solution to van der waerden s problem is presented it is hoped that the solution of this problem will stimulate new and vigorous interest in the theory of permanents and related topics permanents are ideal objects of study having both practical applications and abstract mathematical problems problems involving permanents tend to be easily stated but only solvable using deep methods the theory of permanents as an independent branch of mathematics which has developed parallel to the theory of determinants is far from exhausted many difficult questions must still be resolved in particular the problem of calculation and estimation of permanents for specific classes of matrices is of prime importance the solution of van der waerden s problem makes use of the penetrating force of the method of mixed volumes from combinatorial analysis even hilbert noted that there were connections to be uncovered between the theory of mixed volumes and other areas of mathematics the method of mixed volumes is ideally suited to solving extremal problems and problems of uniqueness and obtaining deep new inequalities it is reasonable to assume that in the future the method of mixed volumes will stand with that of generating functions as one of the basic analytic tools of computational combinatorial analysis

linear operators preserving multivariate majorization abstract in this paper we characterize linear operators on real matrices that preserve multivariate majorization we place no additional restrictions on the linear operator

weak noise expansions through functional integrals for colored noise we use path integral methods to obtain expansions for the correlation functions of the non markovian stochastic processes generated by stochastic differential equations with colored noise

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

a nonlocal and time delayed reaction diffusion model of dengue transmission a nonlocal and time delayed reaction diffusion model of dengue fever is first proposed that incorporates the aquatic stage the winged stage and the incubation periods of the dengue virus within mosquitos and hosts then the basic reproduction number r 0 is established for the model system and an explicit formula of r 0 is obtained in the case of spatially homogeneous infections it is shown that this r 0 gives the threshold dynamics in the sense that the disease free equilibrium is asymptotically stable if r 0 1 the influences of diffusion coefficients time delays and infection heterogeneity on the spread of the disease are also studied via numerical simulations it turns out that the infection risk may be underestimated if the spatially averaged parameters are used to compute the basic reproduction number for spatially heterogeneous infections

an approximate master equation for systems driven by linear ornstein uhlenbeck noise abstract by use of an operator method we construct a novel approximate evolution equation for a one dimensional probability distribution of a single degree of freedom system driven by ornstein uhlenbeck noise this equation is an integro differential equation of the time convolutionless type and its steady state solution is presented a class of non linear equations with multiplicative noise is considered

a shorter proof of kanter s bessel function concentration bound we give a shorter proof of kanter s j multivariate anal 6 222 236 1976 sharp bessel function bound for concentrations of sums of independent symmetric random vectors we provide sharp upper bounds for the sum of modified bessel functions i0 x i1 x which might be of independent interest corollaries improve concentration or smoothness bounds for sums of independent random variables due to cekanavicius roos lith math j 46 54 91 2006 roos bernoulli 11 533 557 2005 barbour xia esaim probab stat 3 131 150 1999 and le cam asymptotic methods in statistical decision theory springer berlin heidelberg new york 1986