lack of strong completeness for stochastic flows it is well known that a stochastic differential equation sde on a euclidean space driven by a brownian motion with lipschitz coefficients generates a stochastic flow of homeomorphisms when the coefficients are only locally lipschitz then a maximal continuous flow still exists but explosion in finite time may occur if in addition the coefficients grow at most linearly then this flow has the property that for each fixed initial condition x the solution exists for all times almost surely if the exceptional set of measure zero can be chosen independently of x then the maximal flow is called strongly complete the question whether an sde with locally lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years in this paper we construct a two dimensional sde with coefficients which are even bounded and smooth and which is not strongly complete thus answering the question in the negative

random dynamical systems rough paths and rough flows abstract we analyze common lifts of stochastic processes to rough paths rough drivers valued processes and give sufficient conditions for the cocycle property to hold for these lifts we show that random rough differential equations driven by such lifts induce random dynamical systems in particular our results imply that rough differential equations driven by the lift of fractional brownian motion in the sense of friz victoir 26 induce random dynamical systems

density results for proper efficiencies several density results are established for different notions of proper efficiency in vector optimization without the requirement of ordering cones being boundedly based for a compact set the set of henig proper efficient points is shown to be dense in the set of all efficient points density theorems for borwein s proper efficiency and for positive scalarizable efficiency follow immediately with appropriate assumptions

logarithmic residues of analytic banach algebra valued functions possessing a simply meromorphic inverse abstract a logarithmic residue is a contour integral of a logarithmic derivative left or right of an analytic banach algebra valued function for functions possessing a meromorphic inverse with simple poles only the logarithmic residues are identified as the sums of idempotents with the help of this observation the issue of left versus right logarithmic residues is investigated both for connected and nonconnected underlying cauchy domains examples are given to elucidate the subject matter

rough differential equations with unbounded drift term abstract we study controlled differential equations driven by a rough path in the sense of t lyons with an additional possibly unbounded drift term we show that the equation induces a solution flow if the drift grows at most linearly furthermore we show that the semiflow exists assuming only appropriate one sided growth conditions we provide bounds for both the flow and the semiflow applied to stochastic analysis our results imply strong completeness and the existence of a stochastic semi flow for a large class of stochastic differential equations if the driving process is gaussian we can further deduce essentially sharp tail estimates for the semi flow and a freidlin wentzell type large deviation result

the simplest proof of burnside s theorem on matrix algebras a very simple short and self contained proof is presented of burnside s theorem that every proper algebra of matrices over an algebraically closed field has a non trivial invariant subspace

the ekeland variational principle for henig proper minimizers and super minimizers abstract in this paper we consider for the first time approximate henig proper minimizers and approximate super minimizers of a set valued map f with values in a partially ordered vector space and formulate two versions of the ekeland variational principle for these points involving coderivatives in the sense of ioffe clarke and mordukhovich as applications we obtain sufficient conditions for f to have a henig proper minimizer or a super minimizer under the palais smale type conditions the techniques are essentially based on the characterizations of henig proper efficient points and super efficient points by mean of the henig dilating cones and the hiriart urruty signed distance function

global flows for stochastic differential equations without global lipschitz conditions we consider stochastic differential equations driven by wiener processes the vector fields are supposed to satisfy only local lipschitz conditions the lipschitz constants of the drift vector field valid on balls of radius r are supposed to grow not faster than log r while those of the diffusion vector fields are supposed to grow not faster than sqrt log r we regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the wiener process on small intervals by showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation we simultaneously establish the existence of a global flow for the stochastic equation under local lipschitz conditions

a structure theorem for some matrix algebras abstract let f be a field and m n f the algebra of n n matrices over f it is in general a very difficult and tedious problem to determine the structure of the subalgebra x of m n f generated by a given subset s of m n f we show that if x contains a matrix a with n distinct eigenvalues in f then x can be determined up to similarity very quickly by a graph theoretic method as a consequence we show that any such x can be generated by a pair of elements one of which can be taken to be a and the other a 0 1 matrix as an application we obtain a specht type similarity theorem

an integer programming problem and rank decomposition of block upper triangular matrices a necessary and sufficient condition is given for a block upper triangular matrix a to be the sum of block upper rectangular matrices satisfying certain rank constraints the condition is formulated in terms of the ranks of certain submatrices of a the proof goes by reduction to an integer programming problem this integer programming problem has a totally unimodular constraint matrix which makes it possible to utilize farkas lemma

strong p completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds here we discuss the regularity of solutions of sde s and obtain conditions under which a sde on a complete riemannian manifoldm has a global smooth solution flow in particular improving the usual global lipschitz hypothesis whenm r n there are also results on non explosion of diffusions

approximately finite dimensional banach algebras are spectrally regular abstract let b be a unital banach algebra which can in a certain sense be approximated by finite dimensional algebras for instance af c algebras belong to this class further let f be an analytic function on some bounded cauchy domain with values in b and suppose that the contour integral of the logarithmic derivative f f 1 along the positively oriented boundary vanishes or is even only quasinilpotent we prove that then f takes invertible values on all of this means that such banach algebras are spectrally regular

trace conditions for regular spectral behavior of vector valued analytic functions abstract a new class of banach algebra valued functions is identified for which the logarithmic residue with respect to a cauchy domain vanishes if and only if the functions take invertible values in trace conditions and the extraction of elementary factors of the type e p p play an important role the class contains the fredholm operator valued functions and the banach algebra valued functions possessing a simply meromorphic resolvent as special instances an example is given to show that new ground is covered and a long standing open problem is discussed from a fresh angle

properly maximal points in product spaces we study maximal points in a locally convex space partially ordered by a convex cone with a bounded base properly maximal points are defined and compared with other concepts of efficiency existence and density theorems are given which unify and generalize several results known in recent literature particular attention is paid on properly maximal points in a product space which has an interesting application in obtaining a multiplier rule for convex set valued problems in a general setting

a partial order that is not rank trace complete abstract let p be a preorder relation on a finite set g the algebra c g g g consists of all complex matrices with rows and columns indexed by g which have zeros at those positions i j which are not in p a subset j g is called p convex if the conditions a c j a b p b c p imply b j a matrix m c g g g is said to satisfy the p rank trace conditions if rank m j j trace m j j z holds for the restriction m j j of m to any p convex set j a preorder p is called rank trace complete if any matrix satisfying the p rank trace conditions is a sum of rank one idempotents in c g g g in this note we provide an example of a partial order that is not rank trace complete

some more density results for proper efficiencies in this paper we give sufficient conditions for a set and or a cone ensuring the density of henig s proper efficient points and positive proper efficient points in the set of efficient points in hausdorff locally convex spaces these conditions are based on the concept of the point of continuity an d generalize many existing results

sums of idempotents and logarithmic residues in zero pattern matrix algebras abstract a zero pattern algebra is a matrix subalgebra of c n n determined by a pattern of zeros the issue in this paper is under what conditions is a matrix in a zero pattern algebra a a sum of rank one idempotents in a or a logarithmic residue in a here logarithmic residues are contour integrals of logarithmic derivatives of analytic a valued functions it turns out that there is a necessary condition involving certain rank trace requirements although these requirements are generally not sufficient there are several important cases where they are

unions of rank trace complete preorders abstract a zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder i e a reflexive transitive digraph the issue studied in 6 is under what conditions is a matrix in a zero pattern algebra a a sum of rank one idempotents in a or a logarithmic residue in a here logarithmic residues are contour integrals of logarithmic derivatives of analytic a valued functions as has been established in 6 there is a necessary condition involving certain rank trace requirements algebras for which this necessary condition is also sufficient are said to be rank trace complete several classes of rank trace complete algebras are identified in 6 also operations are described there for producing rank trace complete algebras out of given ones a basic operation of that type is taking a disjoint union in the present article this simple operation is generalized to a more involved one which preserves rank trace completeness too the new operation turns out to be a useful tool for establishing rank trace completeness in situations that up to now could not be handled one of the positive results is concerned with bouquets a special type of rooted trees it provides a partial answer to an issue raised in 7

sums of idempotent matrices abstract we show that any complex square matrix t is a sum of finitely many idempotent matrices if and only if trt is an integer and trt rank t moreover in this case the idempotents may be chosen such that each has rank one and has range contained in that of t we also consider the problem of the minimum number of idempotents needed to sum to t and obtain some partial results

a study of a class of stochastic differential equations with non lipschitzian coefficients we study a class of stochastic differential equations with non lipschitz coefficients a unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved the dependence with respect to the initial values is investigated to obtain a continuous version of solutions the modulus of continuity of coefficients is assumed to be less than x y log open image in new window finally a large deviation principle of freidlin wentzell type is also established in the paper

strong completeness and semi flows for stochastic differential equations with monotone drift abstract it is well known that a stochastic differential equation sde on a euclidean space driven by a possibly infinite dimensional brownian motion with lipschitz coefficients generates a stochastic flow of homeomorphisms if the lipschitz condition is replaced by an appropriate one sided lipschitz condition sometimes called monotonicity condition and the number of driving brownian motions is finite then existence and uniqueness of global solutions for each fixed initial condition is also well known in this paper we show that under a slightly stronger one sided lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables but which is generally neither one to one nor onto we also address the question of strong completeness which means that there exists a modification of the solution which if restricted to any set a r d of dimension is almost surely continuous with respect to the initial condition

on the small time asymptotics of diffusion processes on hilbert spaces in this paper we establish a small time large deviation principle and obtain the following small time asymptotics lim t 0 2t log p x 0 b x t c d 2 b c for diffusion processes on hilbert spaces where d b c is the intrinsic metric between two subsets b and c associated with the diffusions the case of perturbed ornstein uhlenbeck processes is treated separately at the end of the paper