an algebraic approach to the time optimal output regulator the time optimal output regulator is considered for a special class of linear time invariant discrete time systems it is shown that a simple state feedback law achieves time optimal output regulator response in one sampling period for a square system the state feedback law always assigns the invariant zeros as closed loop poles for a non square system time optimal regulator response is achieved even if the set of invariant zeros is empty if the set of invariant zeros is not empty the control law always assigns these zeros as closed loop poles the method of this paper for the discrete system described by the state and output equations x k 1 ax k bu k and z k cx k does not require that a be invertible and that the open loop system s a b c be state controllable furthermore the only requirement is the existence of a right inverse for the matrix cb which is sufficient for output controllability

asymptotic expansion of solutions to the drift diffusion equation with large initial data abstract we consider the large time behavior of the solution to the initial value problem for the nernst planck type drift diffusion equation in whole spaces in the l p framework the global existence and the decay of the solution were shown moreover the second order asymptotic expansion of the solution as t was derived we also deduce the higher order asymptotic expansion of the solution especially we discuss the contrast between the odd dimensional case and the even dimensional case

on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

nonlinear aggregation diffusion equations radial symmetry and long time asymptotics we analyze under which conditions equilibration between two competing effects repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction occurs this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses all stationary states with suitable regularity are shown to be radially symmetric by means of continuous steiner symmetrization techniques calculus of variations tools allow us to show the existence of global minimizers among these equilibria finally in the particular case of newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation diffusion equations towards this unique equilibrium profile up to translations as t rightarrow infty

a generalization of wigner s law we present a generalization of wigner s semicircle law we consider a sequence of probability distributions p 1 p 2 dots with mean value zero and take an n n real symmetric matrix with entries independently chosen from p n and analyze the distribution of eigenvalues if we normalize this distribution by its dispersion we show that as n for certain p n the distribution weakly converges to a universal distribution the result is a formula for the moments of the universal distribution in terms of the rate of growth of the k th moment of p n as a function of n and describe what this means in terms of the support of the distribution as a corollary when p n does not depend on n we obtain wigner s law if all moments of a distribution are finite the distribution of eigenvalues is a semicircle

global boundedness in a fully parabolic quasilinear chemotaxis system with singular sensitivity abstract in this paper we study the global boundedness of solutions to the quasilinear fully parabolic chemotaxis system u t d u u s u v v t v v u where bounded domain r n n 2 subject to the non flux boundary conditions the diffusivity fulfills d u a 0 u 1 with a 0 0 and 0 while the density signal governed sensitivity satisfies 0 s u b 0 u 1 and 0 v v k for b 0 0 and k r it is shown that the solution is globally bounded provided 1 and k 1 this result demonstrates the effect of signal dependent sensitivity on the blow up prevention

singularity formation in chemotaxis a conjecture of nagai consider the initial boundary value problem for the system s ut uxx uvx x vt u av on an interval 0 1 for t 0 where a 0 with ux 0 t ux 1 t 0 suppose mu v0 are positive constants the corresponding spatially homogeneous global solution u t mu v t mu a v0 mu a exp at is stable in the sense that if mu v0 are positive constants the corresponding spatially homogeneous solution will be uniformly close to u cdot v cdot we consider in sequence space an approximate system s which is related to s in the following sense the chemotactic term uvx x is replaced by the inverse fourier transform of the finite part of the convolution integral for the fourier transform of uvx x here the finite part of the convolution on the line at a point x of two functions f g is defined as int 0 x f y g y x dy we prove the following if mu a then in every neighborhood of mu v0 there are spatially nonconstant initial data for which the solution of proble

identification of chemotaxis models with volume filling chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant a crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant in this paper we investigate the identification of these nonlinear parameter functions in a chemotaxis model with volume filling we also discuss the numerical realization of tikhonov regularization for the stable solution of the inverse problem our theoretical findings are supported by numerical tests

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

on the limit distributions of random matrices with independent or free entries we state limit distribution results for random matrices with independent or free entries also addressing when we get freeness in the limit and semicircular and circular limits the results generalize some already known results about asymptotic freeness of large random matrices but our goal is to get a more optimal flavour on these results when having matrices with identically distributed entries we show that freeness in the limit is natural when we have free entries but unnatural when the entries are independent and restricted to the case of circular limits

existence of global bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source abstract this paper concerns the global existence of bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source u t d u u u v u w u u 2 in v t v u v in w t w u w in under homogeneous neumann boundary condition with positive parameters and d s c 0 s m 1 for s 0 d 0 0 we prove that in dimension three there exists a unique global bounded classical solution provided that m 6 5 and 0 with an additional assumption d s c 0 s m 1 1 for s 0 we prove that for any m 1 6 5 there exists a constant 0 0 m such that for all 0 the above problem admits a unique global bounded classical solution

analysis for a system of coupled reaction diffusion parabolic equations arising in biology we study a system where the equations are coupled by transport and reaction terms in the general case we prove local existence and uniqueness of the solution and in a particular case for a priori fixed time t global existence on 0 t

blow up in a higher dimensional chemotaxis system despite logistic growth restriction abstract we study radially symmetric solutions of a class of chemotaxis systems generalizing the prototype u t u u v u u x t 0 0 v m t u x t 0 in a ball r n with parameters 0 0 0 and 1 and m t 1 u x t d x it is shown that when n 5 and 3 2 1 2 n 2 then there exist initial data such that the smooth local in time solution of blows up in finite time this indicates that even superlinear growth restrictions may be insufficient to rule out a chemotactic collapse as is known to occur in the corresponding system without any proliferation

explicit lower bound of blow up time for an attraction repulsion chemotaxis system abstract in this paper we study classical solutions to the zero flux attraction repulsion chemotaxis system u t u u v u w in 0 t 0 v u v in 0 t 0 w u w in 0 t where is a smooth and bounded domain of r 2 t is the blow up time and are positive real numbers from the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u x 0 u 0 c 0 system has a unique classical solution which becomes unbounded as t t the main result of this investigation is to provide an explicit lower bound for t estimated in terms of u 0 2 d x and attained by means of well established techniques based on ordinary differential inequalities

global existence versus blow up in a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract in this paper we study the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where m 1 d 3 and b u 2 u with 2 d by analyzing the interaction between the degenerate diffusion and the nonlocal aggregation we determine the conditions for initial data under which weak solutions globally exist or blow up in finite time with m 1 d d d furthermore a sharper criterion is given for global existence and finite time blow up of weak solutions with m in the subrange 2 d 2 d d d 1 d d

dynamics in a parabolic elliptic two species population competition model with cross diffusion for one species abstract this work deals with a simplified parabolic elliptic population competition model with cross diffusion for one species we first claim that this model has at least a non trivial global smooth solution and then we detect explicit parameter ranges under which the coexistence steady state and the two semi trivial steady states are globally and asymptotically stable

time optimal output deadbeat regulators with internal stability for invertible systems abstract hassan and amin 1987 and amin and hassan 1987 have presented a simple algebraic solution for the time optimal output deadbeat tood regulator problem this solution which consists of a state feedback control law has been obtained for right invertible systems these results are generalized here to accom modate all classes of output controllable left invertible systems s a b c e the design of tood regulators with internal stability for all classes of right or output controllable left invertible systems s a b c e is also considered it is shown that the right wiener hopf factorization at infinity of h z the transfer function matrix of s is the key tool for solving all tood regulator problems moreover the internal stability problem which will arise if any invariant zero of s is unstable is solved by designing a tood regulator with internal stability finally numerical examples are worked out to illustrate the generality of the proposed method

application of an adams type inequality to a two chemical substances chemotaxis system abstract this paper deals with positive solutions of the fully parabolic system u t u u v in 0 1 v t v v w in 0 2 w t w w u in 0 under homogeneous neumann boundary conditions or mixed boundary conditions no flux and dirichlet conditions in a smooth bounded domain r n n 4 with positive parameters 1 2 0 and nonnegative smooth initial data u 0 v 0 w 0 in the lower dimensional case n 3 it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded in the case n 4 it is shown that in the radially symmetric setting solutions to the neumann boundary value problem of the system exist globally in time and remain bounded if u 0 l 1 8 2 as to the mixed boundary value problem we will establish global existence and boundedness of solutions if u 0 l 1 8 2 without radial symmetry the key ingredients are a lyapunov functional and an adams type inequality a lyapunov functional of the above problems will be constructed and the constant 8 2 is deduced from the critical constant in the adams type inequality this result is regarded as a generalization of the well known 8 problem in the keller segel system to higher dimensions

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

dynamics of a bio reactor model with chemotaxis long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio reactors with chemotaxis will be studied in particular we show that the dynamical system possesses a global attractor and that it is strongly uniformly persistent if the trivial steady state is unstable using a result of smith and waltman on perturbation of global attractors we also show that the positive steady state is unique and globally attracting

blow up and decay criteria for a model of chemotaxis for a system of equations introduced by jager and luckhaus 1992 6 as a model of chemotaxis the questions of blow up and global existence criteria are investigated specifically for a convex region a lower bound for the blow up time is derived if the solution blows up and explicit criteria to ensure non blow up are determined

spectrum of large random reversible markov chains heavy tailed weights on the complete graph we consider the random reversible markov kernel k obtained by assigning i i d non negative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum the weights are assumed to be in the domain of attraction of an alpha stable law with alpha in 0 2 when 1 alpha 2 we show that for a suitable regularly varying sequence kappa n of index 1 1 alpha the limiting spectral distribution mu alpha of kappa n k coincides with the one of the random symmetric matrix of the un normalized weights levy matrix with i i d entries in contrast when 0 alpha 1 we show that the empirical spectral distribution of k converges without rescaling to a non trivial law wmu alpha supported on 1 1 whose moments are the return probabilities of the random walk on the poisson weighted infinite tree pwit introduced by aldous the limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self adjoint operators defined on the pwit this characterization is used together with recursive relations on the tree to derive some properties of mu alpha and wmu alpha we also study the limiting behavior of the invariant probability measure of k

boundedness of solutions to a quasilinear chemotaxis haptotaxis model we study global solutions of a class of chemotaxis haptotaxis systems generalizing the prototype u t u 1 m 1 u u u 1 q 1 v u u 1 p 1 w h u w 0 v v u w t v w in a bounded domain r n n 1 with smooth boundary h u w u 1 u r 1 w with parameters m 1 r 1 and positive constants p q it is shown that either max q 1 p 2 p m max m 2 n r or max q 1 p 2 p m r and b 0 is large enough then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded the results of this paper improve the results of tao and winkler 2014 46 51

on certain free product factors via an extended matrix model abstract a random matrix model for freeness is extended and used to investigate free products of free group factors with matrix algebras and with the hyperfinite ii 1 factor the latter is shown to be isomorphic to a free group factor having one additional generator

global boundedness in a quasilinear chemotaxis system with general density signal governed sensitivity abstract in this paper we study the global boundedness of solutions to the quasilinear parabolic chemotaxis system u t d u u s u v 0 v v u subject to homogeneous neumann boundary conditions and the initial data u 0 in a bounded and smooth domain r n n 2 where the diffusivity d u is supposed to satisfy d u a 0 u 1 with a 0 0 and r while the density signal governed sensitivity fulfills 0 s u b 0 u 1 and 0 v v k for b 0 0 and k r it is shown that the solution is globally bounded if 1 2 n k 2 n with n 3 and k 1 or 1 for k 1 this implies that the large k benefits the global boundedness of solutions due to the weaker chemotactic migration of the signal dependent sensitivity at high signal concentrations moreover when arrives at the critical value we establish the global boundedness of solutions for the coefficient properly small it should be emphasized that the smallness of under k 1 is positively related to the total cellular mass u 0 d x which is attributed to the stronger singularity of v at v 0 for k 1 and the fact that v can be estimated from below by a multiple of u 0 d x in addition distinctive phenomena concerning this model are observed by comparison with the known results

repulsion effects on boundedness in the higher dimensional fully parabolic attraction repulsion chemotaxis system abstract this paper deals with an attraction repulsion chemotaxis system u t d u u u v u w x t 0 1 v t v u v x t 0 2 w t w u w x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 where parameters i i 1 2 and are positive and diffusion coefficient d u c 2 0 satisfies d u 0 for u 0 d u d u m 1 with d 0 and m 1 for all u 0 it is proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m 2 2 n in particular in the case 1 2 and the solution is globally bounded if m 2 2 n n 2 n 2 n 2 therefore due to the inhibition of repulsion to the attraction the range of m 2 2 n of boundedness is enlarged and the results of 21 is thus extended to the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion

minimal time minimal order deadbeat regulator with internal stability this paper is concerned with the output deadbeat finite settling time regulation with internal stability for linear discrete time multivariable systems the two cases are treated separately the one is the regulation by state feedback and the other is the regulation through function observers for both cases the basic solvability conditions the minimal settling time and the characterizations of the minimal time regulators are derived throughout the paper the well posedness condition plays a fundamental role a deterministic version of the so called separation theorem in lqg problem is derived namely the minimal time state feedback regulator combined with the minimal time minimal order function observer yields the overall minimal time minimal order regulator by output feedback

global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the parabolic chemotaxis system u t u u v u u 2 x t 0 v t v v u x t 0 in bounded convex domains r n n 1 with smooth boundary it is shown that if the ratio is sufficiently large then the unique nontrivial spatially homogeneous equilibrium given by u v 1 is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data u 0 v 0 such that u 0 0 the above problem possesses a uniquely determined global classical solution u v with u v t 0 u 0 v 0 which satisfies u t 1 l 0 and v t 1 l 0 as t

self similar solutions to a parabolic system modeling chemotaxis we study the forward self similar solutions to a parabolic system modeling chemotaxisut u u v vt v u in the whole space r2 where is a positive constant using the liouville type result and the method of moving planes it is proved that self similar solutions u v must be radially symmetric about the origin then the structure of the set of self similar solutions is investigated as a consequence it is shown that there exists a threshold in r2u for the existence of self similar solutions in particular for 0 1 2 there exists a self similar solution u v if and only if r2u 8

levy khintchine random matrices and the poisson weighted infinite skeleton tree we study a class of hermitian random matrices which includes wigner matrices heavy tailed random matrices and sparse random matrices such as adjacency matrices of erd os renyi random graphs with pn 1 our n n random matrices have real entries which are i i d up to symmetry the distribution of entries depends on n and we require sums of rows to converge in distribution it is then well known that the limit distribution must be infinitely divisible we show that a limiting empirical spectral distribution lsd exists and via local weak convergence of associated graphs the lsd corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many poisson weighted infinite trees using a backbone structure of special edges called cords to infinity one example covered by the results are matrices with i i d entries having infinite second moments but normalized to be in the gaussian domain of attraction in this case the limiting graph is n rooted at 1 so the lsd is the semi circle law the results also extend to self adjoint complex matrices and wishart matrices

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

a new approach toward stabilization in a two species chemotaxis model with logistic source abstract this paper aims at providing an alternative approach to study global dynamic properties for a two species chemotaxis model with the main novelty being that both populations mutually compete with the other on account of the lotka volterra dynamics more precisely we consider the following neumann initial boundary value problem u t d 1 u 1 u w 1 u 1 u a 1 v x t 0 v t d 2 v 2 v w 2 v 1 a 2 u v x t 0 0 d 3 w w u v x t 0 in a bounded domain r n n 1 with smooth boundary where d 1 d 2 d 3 1 2 1 2 a 1 a 2 are positive constants when a 1 0 1 and a 2 0 1 it is shown that under some explicit largeness assumptions on the logistic growth coefficients 1 and 2 the corresponding neumann initial boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state u v w of above system in the large time limit the respective decay rate of this convergence is shown to be exponential when a 1 1 and a 2 0 1 if 2 is suitable large for all sufficiently regular nonnegative initial data u 0 and v 0 with u 0 0 and v 0 0 the globally bounded solution of above system will stabilize toward 0 1 1 as t in algebraic

design of precompensators via infinite zero structure for dynamic decoupling of linear invertible systems the problem of designing precompensators necessary for decoupling invertible systems having weak inherent coupling is considered a simple algorithm which is directly applicable to systems characterized by square proper transfer function matrices t s is proposed for such design in the proposed algorithm the concept row reducedness at infinity of t s is utilized as an alternative necessary and sufficient condition for the decouplability of t s in addition to the simple design procedure of the compensator the algorithm yields the infinite zero structure of both t s and the resulting compensated system

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

time optimal output regulators for linear multivariate discrete time systems part 2 all classes of right invertible systems in part 1 of this paper hassan and amin 1987 a simple algebraic solution for the time optimal output regulator problem was presented this solution which consists of a state feedback control law has been obtained for all classes of right invertible decouplable systems s a b c e the results of part 1 are here extended to all classes of right invertible systems s a b c e a set of optimal output deadbeat indices called the optimal set is defined and related to the observability indices of the optimal closed loop system the time optimal output regulator problem for a right invertible non decouplable system s a b c e is resolved by transforming s into a decouplable system sc a b cc ec having the optimal output deadbeat index of s first an algorithm is presented to construct iteratively in a well defined optimal sense a unimodular left compensator l z and a compensated decouplable system sc a b cc ec from the state space parameters of s then a family of optimal state fee

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

the basin of attraction of the steady states for a chemotaxis model in mathbb r 2 with critical mass we consider the cauchy problem for a parabolic elliptic system in mathbb r 2 which is amathematical model of chemotaxis and also amodel of self attracting particles in the critical mass case we determine the basin of attraction of the steady states for the cauchy problem through a lyapunov functional

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

unboundedness for solutions to a degenerate drift diffusion equation with the l1 supercritical and the energy subcritical exponent abstract we consider large time behavior of weak solutions to a degenerate drift diffusion system related to keller segel system with the l 1 supercritical and the energy subcritical cases under relaxed weight condition it is known that the large time behavior of solutions is classified by the invariant norms of initial data for the l 1 critical case ogawa wakui proved that the negative entropy condition induces the unboundedness of corresponding solutions with the initial data decaying slowly at spacial infinity here the result is a continuous analogy of the l 1 critical case analogous results have been obtained in the theory of nonlinear schrodinger equations

on a parabolic elliptic system with gradient dependent chemotactic coefficient abstract we consider a second order pdes system of parabolic elliptic type with chemotactic terms the system describes the evolution of a biological species u moving towards a higher concentration of a chemical stimuli v in a bounded and open domain of r n in the system considered the chemotaxis sensitivity depends on the gradient of v i e the chemotaxis term has the following expression d i v u v p 2 v where is a positive constant and p satisfies p 1 if n 1 and p 1 n n 1 if n 2 we obtain uniform bounds in l and the existence of global in time solutions for the one dimensional case we prove the existence of infinitely many non constant steady states for p 1 2 for any positive and a given positive mass

on a constrained 2 d navier stokes equation the planar navier stokes equation exhibits in absence of external forces a trivial asymptotics in time nevertheless the appearence of coherent structures suggests non trivial intermediate asymptotics which should be explained in terms of the equation itself motivated by the separation of the different time scales observed in the dynamics of the navier stokes equation we study the well posedness and asymptotic behaviour of a constrained equation which neglects the variation of the energy and moment of inertia

measure valued solutions and the phenomenon of blow down in logarithmic diffusion abstract the existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory a prominent example being the fundamental solutions of the linear theory in nonlinear equations the existence of such solutions may find special obstacles that can be either essential or otherwise they may lead to more general concepts of solution we give a particular review of results in the field of nonlinear diffusion as a new contribution we study in detail the case of logarithmic diffusion associated with ricci flow in the plane where we can prove existence of measure valued solutions the surprising thing is that these solutions become classical after a finite time in that general setting the standard concept of weak solution is not adequate but we can solve the initial value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions we prove that the problem is well posed the phenomenon of blow down in finite time is precisely described initial point masses diffuse into the medium and eventually disappear after a finite time t i m i 4

central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs we consider the adjacency matrix a of a large random graph and study fluctuations of the function fn z u 1 n k 1nexp ugkk z with g z z ia 1 we prove that the moments of fluctuations normalized by n 1 2 in the limit n satisfy the wick relations for the gaussian random variables this allows us to prove central limit theorem for tr g z and then extend the result on the linear eigenvalue statistics tr a of any function r r which increases together with its first two derivatives at infinity not faster than an exponential

the limiting distributions of large heavy wigner and arbitrary random matrices abstract a heavy wigner matrix x n is defined similarly to a classical wigner one it is hermitian with independent sub diagonal entries the diagonal entries and the non diagonal entries are identically distributed nevertheless the moments of the entries of n x n tend to infinity with n as for matrices with truncated heavy tailed entries or adjacency matrices of sparse erdos renyi graphs consider a family x n of independent heavy wigner matrices and an independent family y n of arbitrary random matrices with a bound condition and converging in distribution in the sense of free probability we characterize the possible limiting joint distributions of x n y n giving explicit formulas for joint moments we find that they depend on more than the distribution of y n and that in general x n and y n are not asymptotically free we use the traffic distributions and the associated notion of independence 21 to encode the information on y n and describe the limiting distribution of x n y n we develop this approach for related models and give recurrence relations for the limiting distribution of heavy wigner and independent diagonal matrices

sharp bounds for sums associated to graphs of matrices we provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form s n j1 j2m 1kerj ntj1j2 1 tj3j4 2 tj2m 1j2m m where some of the summation indices are constrained to be equal the upper bound is easily obtained from a graph g associated to the constraints in the sum

spectrum of non hermitian heavy tailed random matrices let x jk j k 1 be i i d complex random variables such that x jk is in the domain of attraction of an alpha stable law with 0 alpha 2 our main result is a heavy tailed counterpart of girko s circular law namely under some additional smoothness assumptions on the law of x jk we prove that there exists a deterministic sequence a n n 1 alpha and a probability measure mu alpha on c depending only on alpha such that with probability one the empirical distribution of the eigenvalues of the rescaled matrix a n 1 x jk 1 j k n converges weakly to mu alpha as n tends to infinity our approach combines aldous steele s objective method with girko s hermitization using logarithmic potentials the underlying limiting object is defined on a bipartized version of aldous poisson weighted infinite tree recursive relations on the tree provide some properties of mu alpha in contrast with the hermitian case we find that mu alpha is not heavy tailed

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

local null controllability for a chemotaxis system of parabolic elliptic type abstract we consider the controllability of a chemotaxis system of parabolic elliptic type by linearizing the nonlinear system into two separate linear equations we can bypass the obstacle caused by the nonlinear drift term and establish local null controllability of the original nonlinear system this approach is different from the usual method for dealing with coupled parabolic systems

eigenvalue distribution of large weighted random graphs we study eigenvalue distribution of the adjacency matrix a n p of weighted random graphs n p we assume that the graphs have n vertices and the average number of edges attached to one vertex is p to each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite in the first part of the paper we consider the moments of normalized eigenvalue counting function n p of a n p assuming all moments of a finite we obtain recurrent relations that determine the moments of the limiting measure p limn n p the method developed is applied to the laplace operator closely related with a n p using the recurrent relations we analyze the form of p for the both of random matrix families in the second part of the paper we consider the resolvents g a z of a n p and of n p and study the functions fn a z u 1 n k 1n exp ugkk a z in the limit n we derive closed equations that uniquely determine the limiting functio

the spectrum of heavy tailed random matrices let x n be an n n random symmetric matrix with independent equidistributed entries if the law p of the entries has a finite second moment it was shown by wigner 14 that the empirical distribution of the eigenvalues of x n once renormalized by sqrt n converges almost surely and in expectation to the so called semicircular distribution as n goes to infinity in this paper we study the same question when p is in the domain of attraction of an stable law we prove that if we renormalize the eigenvalues by a constant a n of order n frac 1 alpha the corresponding spectral distribution converges in expectation towards a law mu alpha which only depends on we characterize mu alpha and study some of its properties it is a heavy tailed probability measure which is absolutely continuous with respect to lebesgue measure except possibly on a compact set of capacity zero

stability of constant steady states and existence of unbounded solutions in time to a reaction diffusion equation modelling chemotaxis abstract we consider the initial boundary value problem to a reaction diffusion equation modelling chemotaxis in one space dimension under neumann boundary conditions we study the stability of constant steady states and show that there exists an unbounded solution in time induced by the instability of constant steady states

global solutions in a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production is considered under suitable assumption on the sensitivity value and the signal production term it is shown that this system admits a global classical solution the proof is based on a series of a priori estimate techniques a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production is considered under suitable assumption on the sensitivity value and the signal production term it is shown that this system admits a global classical solution the proof is based on a series of a priori estimate techniques

parametrization of minimum time output deadbeat controllers for all classes of invertible systems this paper considers the problem of parametrizing through the minimum number of parameters the minimum time output deadbeat controller mtodc for a given linear multivariable output controllable system s a b c e having n states m inputs p outputs and a transfer function matrix h z of full rank r the main result is a new compact linear parametric form for a class of mtodc that zeros the outputs of the system s in the minimum number of control iterations 1 where 1 i 1 2 r assumed to be decreasingly ordered are equal lo the negatives of the right factorization indices at infinity of h z the developed parametric class is a linear function in effective free parameters the principal benefits of this explicit parametrization of mtodcs lie in the ability to accommodate directly various different design criteria

a saddle point variational formulation for projection regularized parameter identification this paper is concerned with the ill posed problem of identifying a parameter in an elliptic equation and its solution applying regularization by projection as the theory has shown the ansatz functions for the parameter have to be sufficiently smooth in this paper we show that these for a practical implementation unrealistic smoothness assumptions can be circumvented by reformulating the problem under consideration as a mixed variational equation we prove convergence as the discretization gets finer in the noise free case and convergence as the data noise level delta goes to zero in the case of noisy data as well as convergence rates under additional smoothness conditions

central limit theorems for linear statistics of heavy tailed random matrices we show central limit theorems clt for the stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries including entries in the domain of attraction of alpha stable laws and entries with moments exploding with the dimension as in the adjacency matrices of erdos renyi graphs for the second model we also prove a central limit theorem of the moments of its empirical eigenvalues distribution the limit laws are gaussian but unlike to the case of standard wigner matrices the normalization is the one of the classical clt for independent random variables

time optimal output regulators for linear multivariable discrete time systems part 1 right invertible decouplable systems corsetti and houpis 1985 presented a simple algebraic solution for the timeoptimal output regulator problem this solution has been obtained for a special class of right invertible decouplable systems s a b c satisfying the condition that cb is of full rank in this paper a general method is presented to extend the results of corsetti and houpis to all classes of right invertible decouplable systems s a b c e it is shown that a simple state feedback control law drives each output of s a b c e in a minimum number of steps equal to the order of the associated infinite zero this state feedback represents the optimal solution for the considered time optimal output regulator problem a class of optimal solutions parametrized by a free parameter matrix k is obtained for non square systems the properties of the resulting optimal closed loop system is given and a numerical example is worked out to illustrate the generality and feasibility of the proposed method

eventual smoothness and stabilization of large data solutions in a three dimensional chemotaxis system with consumption of chemoattractant abstract this paper deals with positive solutions of u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in bounded convex domains r 3 with smooth boundary it is shown that for arbitrarily large initial data this problem admits at least one global weak solution for which there exists t 0 such that u v is bounded and smooth in t moreover it is asserted that such solutions approach spatially constant equilibria in the large time limit

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

energy type estimates and global solvability in a two dimensional chemotaxis haptotaxis model with remodeling of non diffusible attractant this paper deals with the coupled chemotaxis haptotaxis model ut u u v u w u 1 u w x t 0 0 v u v x t 0 wt vw w 1 w u x t 0 which was initially proposed by chaplain and lolas 2006 10 to describe the interactions between cancer cells the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue extracellular matrix here r2 is a bounded domain with smooth boundary and and are positive parameters as compared to previous mathematical studies the novelty here consists of allowing for positive values of reflecting processes of self remodeling of the extracellular matrix under zero flux boundary conditions it is shown that for any sufficiently smooth initial data u0 w0 satisfying first order compatibility conditions the model admits a unique global smooth solution a crucial ingredient in the proof is an energy like inequality which given t 0 yields boundedness of u t in llogl this serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions

preventing blow up in a chemotaxis model abstract this paper deals with a chemotactic model of keller segel type the main feature of the keller segel model is the possibility of blow up of solutions in a finite time to eliminate the possibility of blow up a modified version of keller segel model is introduced the blow up control relies on the presence of a pressure function which increases faster than a logarithm for high enough cells densities for such a pressure function the solutions cannot blow up in a finite time some other conditions are introduced to ensure the global boundedness