random attractors for stochastic porous media equations perturbed by space time linear multiplicative noise unique existence of solutions to porous media equations driven by continuous linear multiplicative space time rough signals is proven for initial data in l 1 o on bounded domains o the generation of a continuous order preserving random dynamical system rds on l 1 o and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained the random attractor is shown to be compact and attracting in l o norm uniform l bounds and uniform space time continuity of the solutions is shown general noise including fractional brownian motion for all hurst parameters is treated a pathwise wong zakai result for driving noise given by a continuous semimartingale is obtained for fast diffusion equations driven by continuous linear multiplicative space time rough signals existence of solutions is proven for initial data in l m 1 o

random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

on the stochastic porous medium equation abstract in this paper we discuss an initial boundary value problem and a cauchy problem for the stochastic porous medium equation our basic estimates are based on the known results due to math sbornik 67 1965 609 642 arch rational mech anal 25 1967 64 80 by the procedure developed in trans amer math soc 354 2001 1117 1135 we obtain solutions over the given probability space rather than martingale solutions we will also establish the existence of invariant measures when the space domain is bounded

noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

existence and upper semicontinuity of bi spatial pullback attractors for smoothing cocycles abstract in this work we establish several criteria for the existence as well as the upper semi continuity of bi spatial attractors under a closedness condition which dramatically weakens the usual requirement on the continuity of the cocycle it is also shown that though the continuity plays a less important role in the study of attractors it is impossible to establish an existence criteria for common attractors for systems without any continuity like properties however for such bad systems one can expect a mini attractor which is shown adequate well to depict the asymptotic behavior of non continuous systems finally we study the l 2 h 0 1 pullback attractor for a stochastic complex ginzburg landau equation a spectrum decomposition method is employed to overcome the lack of sobolev compactness embeddings in h 0 1

infimum of a pair of matrices abstract in this paper answers are provided to the following questions given a matrix e1 of order k m does there exist a matrix e2 of order m k m such that the matrix e e 1 e 2 is square idempotent and of the same rank as e1 further when is the choice of e2 unique given a subspace t virtually disjoint with the row span of e1 when does there exist a matrix g g 1 g 2 such that g is square idempotent of rank equal to rank g1 rank e1 and g1 is congruent to e1 mod t these results are then used to determine if a pair of matrices a and b has an infimum in the minus partial order further the existence of the sharp order infimum is studied

on a random scaled porous media equation abstract it is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well posed in l in the fast diffusion case we show existence in l p

stochastic porous media equations and self organized criticality the existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and wiener forcing term is proven the finite time extinction of solutions with high probability is also proven in 1 d the results are relevant for self organized criticality behavior of stochastic nonlinear diffusion equations with critical states

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

random attractors for a class of stochastic partial differential equations driven by general additive noise abstract the existence of random attractors for a large class of stochastic partial differential equations spde driven by general additive noise is established the main results are applied to various types of spde as e g stochastic reaction diffusion equations the stochastic p laplace equation and stochastic porous media equations besides classical brownian motion we also include space time fractional brownian motion and space time levy noise as admissible random perturbations moreover cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

nonlinear fokker planck equations driven by gaussian linear multiplicative noise abstract existence of a strong solution in h 1 r d is proved for the stochastic nonlinear fokker planck equation d x div d x d t x d t x d w in 0 t r d x 0 x respectively for a corresponding random differential equation here d 1 w is a wiener process in h 1 r d d c 1 r d r d and is a continuous monotonically increasing function satisfying some appropriate sublinear growth conditions which are compatible with the physical models arising in statistical mechanics the solution exists for x l 1 l and preserves positivity if is locally lipschitz the solution is unique pathwise lipschitz continuous with respect to initial data in h 1 r d stochastic fokker planck equations with nonlinear drift of the form d x div a x d t x d t x d w are also considered for lipschitzian continuous functions a r r d

strong solutions for stochastic partial differential equations of gradient type unique existence of analytically strong solutions to stochastic partial differential equations spde with drift given by the subdifferential of a quasi convex function and with general multiplicative noise is proven the proof applies a genuinely new method of weighted galerkin approximations based on the distance defined by the quasi convex function spatial regularization of the initial condition analogous to the deterministic case is obtained the results yield a unified framework which is applied to stochastic generalized porous media equations stochastic generalized reaction diffusion equations and stochastic generalized degenerated p laplace equations in particular higher regularity for solutions of such spde is obtained

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

box counting dimensions and upper semicontinuities of bi spatial attractors for stochastic degenerate parabolic equations on an unbounded domain abstract the paper contributes box counting dimensions and upper semicontinuities of random bi spatial attractors for stochastic degenerate parabolic equations on the whole euclid space under some weak assumptions for the force and the nonlinearity we first prove the existence of a unique l 2 d 0 1 2 l q random attractor for any q 2 p 2 i 2 where p 1 is the order of the nonlinearity and i is a given integer such that the force is i 1 times integrable by using truncation and splitting techniques and also induction methods we then prove that a priori estimate is uniform with respect to the density of noise which leads to the upper semicontinuity result of the obtained attractors as the density tends to a constant including zero under the topology of the terminative space furthermore we give a new framework to discuss the bound of l q box counting dimensions of random attractors for spde on an unbounded domain

well posedness of nonlinear diffusion equations with nonlinear conservative noise we prove the pathwise well posedness of stochastic porous media and fast diffusion equations driven by nonlinear conservative noise as a consequence the generation of a random dynamical system is obtained this extends results of the second author and souganidis who considered analogous spatially homogeneous and first order equations and earlier works of lions perthame and souganidis

lattice properties of the order for complex matrices abstract it is shown that the set c m n of complex m n matrices forms a lower semilattice under the partial ordering a b defined by a a a b aa ba where a denotes the conjugate transpose of a as a special case of a result for division rings it is further shown that over any field f for m n 2 and any proper involution of f2 2 the corresponding intersections a b all exist

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

non monotone stochastic generalized porous media equations abstract by using the nash inequality and a monotonicity approximation argument existence and uniqueness of strong solutions are proved for a class of non monotone stochastic generalized porous media equations moreover we prove for a large class of stochastic pde that the solutions stay in the smaller l 2 space provided the initial value does so that some recent results in the literature are considerably strengthened

rough stochastic pdes in this article we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic pdes of burgers type that exhibit too high spatial roughness for classical analytical methods to apply in fact the class of spdes that we consider is genuinely ill posed in the sense that different approximations to the nonlinearity may converge to different limits using rough path theory a pathwise notion of solution to these spdes is formulated and we show that this yields a well posed problem that is stable under a large class of perturbations including the approximation of the rough driving noise by a mollified version and the addition of hyperviscosity we also show that under certain structural assumptions on the coefficients the spdes under consideration generate a reversible markov semigroup with respect to a diffusion measure that can be given explicitly

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

finite time extinction for stochastic sign fast diffusion and self organized criticality we prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space time noise corresponding to the bak tang wiesenfeld model for self organized criticality this solves a problem posed and left open in several works barbu methods appl sci 36 1726 1733 2013 rockner and wang j lond math soc 2 87 545 560 2013 barbu et al j math anal appl 389 147 164 2012 barbu and rockner comm math phys 311 539 555 2012 barbu et al comm math phys 285 901 923 2009 c r math acad sci paris 347 1 2 81 84 2009 the highly singular degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced

adaptive image coding using spectral classification we present a new classification scheme dubbed spectral classification which uses the spectral characteristics of the image blocks to classify them into one of a finite number of classes a vector quantizer with an appropriate distortion measure is designed to perform the classification operation the application of the proposed spectral classification scheme is then demonstrated in the context of adaptive image coding it is shown that the spectral classifier outperforms gain based classifiers while requiring a lower computational complexity

the dynamics of the stochastic shadow gierer meinhardt system we consider the dynamics of the stochastic shadow gierer meinhardt system with one dimensional standard brownian motion we establish the global existence and uniqueness of solutions we also prove a large deviation result

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses

perfect cocycles through stochastic differential equations we prove that if is a random dynamical system cocycle for whicht t x is a semimartingale then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment helix and conversely this relation is succinctly expressed as semimartingale cocycle exp semimartingale helix to implement it we lift stochastic calculus from the traditional one sided time to two sided timet and make this consistent with ergodic theory we also prove a general theorem on the perfection of a crude cocycle thus solving a problem which was open for more than ten years