raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

asymptotic form of the density profile for gaussian and laguerre random matrix ensembles with orthogonal and symplectic symmetry in a recent study we have obtained correction terms to the large n asymptotic expansions of the eigenvalue density for the gaussian unitary and laguerre unitary ensembles of random n n matrices both in the bulk and at the soft edge of the spectrum in the present study these results are used to similarly analyze the eigenvalue density for gaussian and laguerre random matrix ensembles with orthogonal and symplectic symmetry as in the case of unitary symmetry a matching is exhibited between the asymptotic expansion of the bulk density expanded about the edge and the asymptotic expansion of the edge density expanded into the bulk in addition aspects of the asymptotic expansion of the smoothed density which involves delta functions at the endpoints of the support are interpreted microscopically

applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

boundedness and decay behavior in a higher dimensional quasilinear chemotaxis system with nonlinear logistic source this paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype 0 1 ut um uv uur x t 0 vt vv u x t 0 u v 0 x t 0 u x 0 u0 x v x 0 v0 x x in a smooth bounded domain rn n2 with parameters m r1 and 0 the pde system in 0 1 is used in mathematical biology to model the mechanism of chemotaxis that is the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space it is shown that if m 22nif11 n 22r n 2ifn 22rn 2n 1ifr n 22 and the nonnegative initial data u0 v0 c w1 0 then 0 1 possesses at least one global bounded weak solution apart from this it is proved that if 0 then both u t and v t decay to zero with respect to the norm in l as t

global existence and boundedness in a keller segel stokes system involving a tensor valued sensitivity with saturation the 3d case abstract in this paper we continue to deal with the initial boundary value problem for the coupled keller segel stokes system n t u n n n s x n c c x t 0 t c t u c c c n x t 0 t u t p u n x t 0 t u 0 x t 0 t where r d is a bounded domain with smooth boundary and the chemotactic sensitivity s is not a scalar function but rather attains values in r d d and satisfies s x n c c s 1 n with some c s 0 and 0 when d 2 our previous work j differential equations 2015 has established the existence of global bounded classical solutions under the subcritical assumption 0 which is consistent with the corresponding results of the fluid free system but the method seems to be invalid in the three dimensional setting in this paper for the case d 3 we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption 1 2 which is slightly stronger than the corresponding subcritical assumption 1 3 on the fluid free system where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions the key idea here is to establish the general l p regularity of u from a rather low l p regularity of n which will be obtained by a new combinational functional

on probabilistic and quantum reaction systems the notion of reaction systems was introduced by ehrenfeucht and rozenberg 2004 2007 5 6 in this article we discuss the general properties of probabilistic reaction systems and introduce their quantum extension

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

finite time blowup and global in time unbounded solutions to a parabolic parabolic quasilinear keller segel system in higher dimensions abstract in this paper we consider quasilinear keller segel type systems of two kinds in higher dimensions in the case of a nonlinear diffusion system we prove an optimal with respect to possible nonlinear diffusions generating explosion in finite time of solutions finite time blowup result in the case of a cross diffusion system we give results which are optimal provided one assumes some proper non decay of a nonlinear chemical sensitivity moreover we show that once we do not assume the above mentioned non decay our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross diffusion terms to this end we provide an example interesting by itself of global in time unbounded solutions to the nonlinear cross diffusion keller segel system with chemical sensitivity decaying fast enough in a range of parameters in which there is a finite time blowup result in a corresponding case without nonlinear cross diffusion

exact density of states for finite gaussian random matrix ensembles via supersymmetry we calculate the exact density of states dos for the three classical and two non classical random matrix ensembles for finite matrix size n using supersymmetric integrals the 1 n expansion yields already in lowest order good approximations to the exact result even for small values of n 5 we conjecture a connection between the n dependence of the oscillating part of the dos and the short distance behaviour of the two level correlation function

density of states for gaussian unitary ensemble gaussian orthogonal ensemble and interpolating ensembles through supersymmetric approach we use the supersymmetric formalism to derive an integral formula for the density of states of the gaussian orthogonal ensemble and then apply saddle point analysis to give a new derivation of the 1 n correction to wigner s law this extends the work of disertori on the gaussian unitary ensemble we also apply our method to the interpolating ensembles of mehta pandey

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

boundedness in a three dimensional chemotaxis fluid system involving tensor valued sensitivity with saturation abstract we consider the chemotaxis fluid system cf n t u n d n n n s x n c c a n b n 2 x t 0 c t u c c c n x t 0 u t p u n g x t x t 0 u 0 x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r 3 where w 1 a 0 and b 0 here g c 1 0 l 0 d n u m 1 s x n c 1 n and the parameter 0 if m max 6 5 1 3 then for all reasonably regular initial data a corresponding initial boundary value problem for cf possesses a globally defined weak solution through the moser type iteration

global weak solutions in a three dimensional keller segel navier stokes system with nonlinear diffusion abstract the coupled quasilinear keller segel navier stokes system ksnf n t u n n m n c x t 0 c t u c c c n x t 0 u t u u p u n x t 0 u 0 x t 0 is considered under neumann boundary conditions for n and c and no slip boundary conditions for u in three dimensional bounded domains r 3 with smooth boundary where m 0 r are given constants w 1 if m 2 then for all reasonably regular initial data a corresponding initial boundary value problem for ksnf possesses a globally defined weak solution

on the asymptotics of some large hankel determinants generated by fisher hartwig symbols defined on the real line we investigate the asymptotics of hankel determinants of the form detj k 0n 1 dx n x i 1m i x 2qixj k as n with q and fixed where is an infinite subinterval of r and n x is a positive weight on such objects are natural analogs of toeplitz determinants generated by fisher hartwig symbols and arise in random matrix theory in the investigation of certain expectations involving random characteristic polynomials the reduced density matrices of certain one dimensional systems of trapped impenetrable bosons can also be expressed in terms of hankel determinants of this form we focus on the specific cases of scaled hermite and laguerre weights we compute the asymptotics by using a duality formula expressing the n n hankel determinant as a 2 q1 qm fold integral which is valid when each qi is natural we thus verify for such q a recent conjecture of forrester and frankel derived using a log gas argument

painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons the recent experimental realisation of a one dimensional bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable bose gas of primary concern is the ground state occupation of effective single particle states in the finite system and thus the tendency for bose einstein condensation this requires the computation of the density matrix for the impenetrable bose gas on a circle we evaluate the density matrix in terms of a particular painleve vi transcendent in form and furthermore show that the density matrix satisfies a recurrence relation in the number of particles for the impenetrable bose gas in a harmonic trap and with dirichlet or neumann boundary conditions we give a determinant form for the density matrix a form as an average over the eigenvalues of an ensemble of random matrices and in special cases an evaluation in terms of a transcendent related to painleve v and vi we discuss how our results can be used to compute the ground state occupations

an optimal result for global existence and boundedness in a three dimensional keller segel stokes system with nonlinear diffusion abstract this paper investigates the following keller segel stokes system with nonlinear diffusion k s f n t u n n m n c x t 0 c t u c c c n x t 0 u t p u n x t 0 u 0 x t 0 under homogeneous boundary conditions of neumann type for n and c and of dirichlet type for u in a three dimensional bounded domains r 3 with smooth boundary where w 2 m 0 it is proved that if m 4 3 then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system k s f which in view of the known results for the fluid free system mentioned below see introduction is an optimal restriction on m

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

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

large time behavior of solution to a fully parabolic chemotaxis haptotaxis model in higher dimensions abstract this paper deals with the chemotaxis haptotaxis model of cancer invasion u t u u v u w u 1 u w x t 0 v t v v u x t 0 w t v w x t 0 in a bounded smooth domain r n with zero flux boundary conditions where and are positive parameters it is shown that if is suitably large then for all sufficiently smooth initial data the associated initial boundary value problem possesses a unique global in time classical solution that is bounded in 0 and if the initial data w 0 is small w becomes asymptotically negligible moreover we prove that when domain is convex 1 1 0 is globally asymptotically stable provided that u 0 0 and thereby extends the result of hillen et al 2013 18 to the higher space dimensions

a new result for global solvability and boundedness in the n dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant abstract we consider the following chemotaxis model u t d u u u v u u 2 x t 0 v t v u v x t 0 d u u u v v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x on a bounded domain r n n 1 with smooth boundary and are positive constants here d is supposed to be smooth positive function satisfying d u c d u 1 m 1 for all u 0 with some c d m 0 besides appropriate smoothness assumptions in this paper it is only required that m 1 2 1 max 1 s 2 0 s v 0 l 2 3 if n 2 1 if n 3 then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded where 0 is a positive constant which is corresponding to the maximal sobolev regularity the results of this paper extends the results of jin j differential equations 263 9 2017 5759 5772 who proved the possibility of boundness of weak solutions in the case m 1 and n 3 to the best of our knowledge this is the first result which gives the relationship between m and that yields to the boundedness of the solution

universality limits in the bulk for varying measures abstract universality limits are a central topic in the theory of random matrices we establish universality limits in the bulk of the spectrum for varying measures using the theory of entire functions of exponential type in particular we consider measures that are of the form w n 2 n x d x in the region where universality is desired w n does not need to be analytic nor possess more than one derivative and then only in the region where universality is desired we deduce universality in the bulk for a large class of weights of the form w 2 n x d x for example when w e q where q is convex and q satisfies a lipschitz condition of some positive order we also deduce universality for a class of fixed exponential weights on a real interval

asymptotic corrections to the eigenvalue density of the gue and lue we obtain correction terms to the large n asymptotic expansions of the eigenvalue density for the gaussian unitary and laguerre unitary ensembles of random n n matrices both in the bulk of the spectrum and near the spectral edge this is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density to which we apply the saddle point method the main correction to the bulk density is oscillatory in n and depends on the distribution function of the limiting density while the corrections to the airy kernel at the soft edge are again expressed in terms of the airy function and its first derivative we demonstrate numerically that these expansions are very accurate a matching is exhibited between the asymptotic expansion of the bulk density expanded about the edge and the asymptotic expansion of the edge density expanded into the bulk

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

functional form for the leading correction to the distribution of the largest eigenvalue in the gue and lue the neighbourhood of the largest eigenvalue max in the gaussian unitary ensemble gue and laguerre unitary ensemble lue is referred to as the soft edge it is known that there exists a particular centring and scaling such that the distribution of max tends to a universal form with an error term bounded by 1 n2 3 we take up the problem of computing the exact functional form of the leading error term in a large n asymptotic expansion for both the gue and lue two versions of the lue are considered one with the parameter a fixed and the other with a proportional to n both settings in the lue case allow for an interpretation in terms of the distribution of a particular weighted path length in a model involving exponential variables on a rectangular grid as the grid size gets large we give operator theoretic forms of the corrections which are corollaries of knowledge of the first two terms in the large n expansion of the scaled kernel and are readily computed using a method due to bornemann we also give expressions in terms of the solutions of particular systems of coupled differential equations which provide an alternative method of computation both characterisations are well suited to a thinned generalisation of the original ensemble whereby each eigenvalue is deleted independently with probability 1 in sec v we investigate using simulation the question of whether upon an appropriate centring and scaling a wider class of complex hermitian random matrix ensembles have their leading correction to the distribution of max proportional to 1 n2 3 the neighbourhood of the largest eigenvalue max in the gaussian unitary ensemble gue and laguerre unitary ensemble lue is referred to as the soft edge it is known that there exists a particular centring and scaling such that the distribution of max tends to a universal form with an error term bounded by 1 n2 3 we take up the problem of computing the exact functional form of the leading error term in a large n asymptotic expansion for both the gue and lue two versions of the lue are considered one with the parameter a fixed and the other with a proportional to n both settings in the lue case allow for an interpretation in terms of the distribution of a particular weighted path length in a model involving exponential variables on a rectangular grid as the grid size gets large we give operator theoretic forms of the corrections which are corollaries of knowledge of the first two terms in the large n expansion of the scaled kernel and are readily computed using a method due to bornemann we also

some recent methods for establishing universality limits abstract we survey some recent methods for establishing universality limits for random matrices in the unitary case these include levin s method using a markov bernstein inequality a comparison inequality of the author and a method based on complex analysis and reproducing kernels we focus on the bulk of the spectrum for measures with compact support but the methods may also be used at the soft or hard edge and for measures with unbounded support

new critical exponents in a fully parabolic quasilinear keller segel system and applications to volume filling models abstract we carry on our studies related to the fully parabolic quasilinear keller segel system started in 6 and continued in 7 in the above mentioned papers we proved finite time blowup of some radially symmetric solutions to the quasilinear keller segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds on the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time the present paper is devoted to looking for critical exponents distinguishing between those two behaviors moreover we apply our results to the so called volume filling models with a power type jump probability function the most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2 namely we find a value m such that when the solution to the two dimensional volume filling keller segel system starts with mass smaller than m then it is bounded while for some initial data with mass exceeding m solutions are unbounded though being defined for any time t 0

boundedness of solutions to a quasilinear parabolic elliptic keller segel system with logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u 1 m 1 u u u 1 q 1 v a u b u r x t 0 0 v v u x t 0 in a bounded domain r n n 1 with smooth boundary with parameters m 1 r 1 a 0 b q 0 it is shown that when q 1 max r m 2 n or b b 0 n r m 2 r m n 2 r 2 if q 1 r then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded the results improve the results of wang et al 2014 37 and cao and zheng 2014 6

large n expansions for the laguerre and jacobi β ensembles from the loop equations the ensembles of random matrix theory with classical weights have many special properties one is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at the general order of the correlator via aomoto s method from the theory of the selberg integral we use aomoto s method to derive the full hierarchy of loop equations for laguerre and jacobi ensembles and use these to systematically construct the explicit form of the 1 n expansion at low orders this allows us to give the explicit form of corrections to the global density and allows various moments to be computed complementing results available in the literature motivated by problems in quantum transport

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

boundedness in quasilinear keller segel systems of parabolic parabolic type on non convex bounded domains abstract this paper deals with the quasilinear fully 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 bounded domain r n with smooth boundary n n the diffusivity d u is assumed to satisfy some further technical conditions such as algebraic growth and d 0 0 which says that the diffusion is allowed to be not only non degenerate but also degenerate the global in time existence and uniform in time boundedness of solutions are established under the subcritical condition that s u d u k u e for u 0 with 2 n k 0 and e 0 when d 0 0 this paper represents an improvement of tao and winkler 17 because the domain does not necessarily need to be convex in this paper in the case r n and d 0 0 uniform in time boundedness is an open problem left in a previous paper 7 this paper also gives an answer to it in bounded domains

boundedness of solutions to a quasilinear parabolic parabolic keller segel system with a logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u u u v a u b u r x t 0 v t v v u x t 0 in a bounded domain r n n 1 with smooth boundary u u 1 u u u 1 1 and the parameters r 1 a 0 b 0 and r it is proved that if 0 max r 1 2 n or b is big enough if r 1 then the classical solutions to the above system are uniformly in time bounded our results improve the results of wang et al 2014 28 and cao 2014 3 and also enlarge the range of the results of tao and winkler 2012 25 and ishida et al 2014 14

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

scaled limit and rate of convergence for the largest eigenvalue from the generalized cauchy random matrix ensemble in this paper we are interested in the asymptotic properties for the largest eigenvalue of the hermitian random matrix ensemble called the generalized cauchy ensemble gcye whose eigenvalues pdf is given by textrm const cdot prod 1 leq j k leq n x j x k 2 prod j 1 n 1 ix j s n 1 ix j overline s n dx j where s is a complex number such that s 1 2 and where n is the size of the matrix ensemble using results by borodin and olshanski commun math phys 223 1 87 123 2001 we first prove that for this ensemble the law of the largest eigenvalue divided by n converges to some probability distribution for all s such that s 1 2 using results by forrester and witte nagoya math j 174 29 114 2002 on the distribution of the largest eigenvalue for fixed n we also express the limiting probability distribution in terms of some non linear second order differential equation eventually we show that the convergence of the probability distribution function of the re scaled largest eigenvalue to the limiting one is at least of order 1 n