monotone iterative technique for partial differential equations of first order in view of the fact that first order partial differential equations arise naturally in modelling growth population of cells which constanctly change their properties a study of stable and chaotic solutions of such equations was initiated in 3 in this paper we develop monotone iterative technique for first order partial differential equations and for this purpose we need to discuss existence uniqueness and comparison results we also indicate how the monotone sequences obtained may be employed as candidates for lyapunov functions in stability theory

universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

matrices coupled in a chain ii spacing functions for the eigenvalues of p complex hermitian matrices coupled in a chain we give a method of calculating the spacing functions this is a generalization of the one matrix case which has been known for a long time

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

splines are universal solutions of linear inverse problems with generalized tv regularization splines come in a variety of flavors that can be characterized in terms of some differential operator l the simplest piecewise constant model corresponds to the derivative operator likewise on

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

generalized sampling a variational approach i theory we consider the problem of reconstructing a multidimensional vector function f sub in spl ropf sup m spl rarr spl ropf sup n from a finite set of linear measures these can be irregularly sampled responses of several linear filters traditional approaches reconstruct in an a priori given space e g the space of bandlimited functions instead we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion j first we present the solution of the generalized interpolation problem later we also consider the approximation problem and we show that both lead to the same class of solutions imposing generally desirable properties on the reconstruction largely limits the choice of the criterion j linearity leads to a quadratic criterion based on bilinear forms specifically we show that the requirements of translation rotation and scale invariance restrict the form of the criterion to essentially a one parameter family we show that the solution can be obtained as a linear combination of generating functions we provide analytical techniques to find these functions and the solution itself

ensemble and trajectory statistics in a nonlinear partial differential equation we have examined the influence of parametric noise on the solution behavioru t x of a nonlinear initial value problem arising in cell kinetics in terms of ensemble statistics the eventual limiting solution mean mathop xi limits u and variance mathop sigma u 2 limits are well characterized functions of the noise statistics and mathop xi limits u and mathop sigma u 2 limits depend on when noise is continuously present along the trajectory mathop xi limits u and mathop sigma u 2 limits are independent of the noise statistics and however in their evolution toward mathop xi limits u and mathop sigma u 2 limits both u t x and u2 t x depend on the noise and

mixed problems for hyperbolic functional differential equations with unbounded delay the phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed the set of axioms for generalized solutions of initial boundary value problems is presented a theorem on the existence and continuous dependence upon initial boundary data is given the mixed problem is transformed into a system of integral functional equations the existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities examples of phase spaces are given

strong asymptotics for cauchy biorthogonal polynomials with application to the cauchy two matrix model we apply the nonlinear steepest descent method to a class of 3 3 riemann hilbert problems introduced in connection with the cauchy two matrix random model the general case of two equilibrium measures supported on an arbitrary number of intervals is considered in this case we solve the riemann hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three sheeted riemann surface of arbitrary genus and establish strong asymptotic results for the cauchy biorthogonal polynomials

continuous domain solutions of linear inverse problems with tikhonov versus generalized tv regularization we consider one dimensional 1 d linear inverse problems that are formulated in the continuous domain the object of recovery is a function that is assumed to minimize a convex objective functional the solutions are constrained by imposing a continuous domain regularization we derive the parametric form of the solution representer theorems for tikhonov quadratic and generalized total variation gtv regularizations we show that in both cases the solutions are splines that are intimately related to the regularization operator in the tikhonov case the solution is smooth and constrained to live in a fixed subspace that depends on the measurement operator by contrast the gtv regularization results in a sparse solution composed of only a few dictionary elements that are upper bounded by the number of measurements and independent of the measurement operator our findings for the gtv regularization resonates with the minimization of the ell 1 norm which is its discrete counterpart and also produces sparse solutions finally we find the experimental solutions for some measurement models in one dimension we discuss the special case when the gtv regularization results in multiple solutions and devise an algorithm to find an extreme point of the solution set which is guaranteed to be sparse

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

dimension of measures invariant with respect to the wazewska partial differential equation abstract we study asymptotic properties of a nonlinear first order partial differential equation which describes the reproduction of blood cells this equation under conditions proposed by wazewska generates a semigroup of transformations with highly chaotic behaviour of trajectories we show that this semigroup has invariant measures with arbitrary large dimension

strong mixing gaussian measures for chaotic semigroups abstract in this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable banach spaces we first prove an identity characterizing invariant gaussian measure involving its covariance operator and the infinitesimal generator of the semigroup this gives an answer to a question raised by rudnicki in his inspiring review paper 32 under suitable conditions we use the proved identity to give an invariant mixing gaussian measure as distribution of a wiener integral

generic properties of measures invariant with respect to the wazewska partial differential equation it is shown that measure with the concentration dimension equal to zero is typical in family of measures invariant with respect to the wazewska equation

a nonlinear age and maturity structured model of population dynamics ii chaos instability hypercyclicity and chaos are investigated in a nonlinear model of age and maturity structured population dynamics it is demonstrated that the behavior of solutions depends on the viewpoint of the observer viewed in the direction of age structure the population stabilizes to a regular distribution viewed in the direction of maturity structure there is the possibility of chaotic behavior

a note on the limiting mean distribution of singular values for products of two wishart random matrices the product of m complex random gaussian matrices of size n has recently been studied by akemann kieburg and wei they showed that for fixed m and n the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed we find that in the case m 2 the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with macdonald functions modified bessel functions of the second kind which was first introduced by van assche and yakubovich with known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random gaussian matrices in the limit of large matrix dimensions

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

singular values of products of ginibre random matrices multiple orthogonal polynomials and hard edge scaling limits akemann ipsen and kieburg recently showed that the squared singular values of products of m rectangular random matrices with independent complex gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of meijer g functions we show that this point process can be interpreted as a multiple orthogonal polynomial ensemble we give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel which allows us to find its scaling limits at the origin hard edge the limiting kernels generalize the classical bessel kernels for m 2 they coincide with the scaling limits found by bertola gekhtman and szmigielski in the cauchy laguerre two matrix model which indicates that these kernels represent a new universality class in random matrix theory

generalized smoothing splines and the optimal discretization of the wiener filter we introduce an extended class of cardinal l sup l splines where l is a pseudo differential operator satisfying some admissibility conditions we show that the l sup l spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional spl par ls spl par sub l2 sup 2 subject to the interpolation constraint next we consider the corresponding regularized least squares estimation problem which is more appropriate for dealing with noisy data the criterion to be minimized is the sum of a quadratic data term which forces the solution to be close to the input samples and a smoothness term that privileges solutions with small spline energies here too we find that the optimal solution among all possible functions is a cardinal l sup l spline we show that this smoothing spline estimator has a stable representation in a b spline like basis and that its coefficients can be computed by digital filtering of the input signal we describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of l is rational which corresponds to the case of exponential splines we justify these algorithms statistically by establishing an equivalence between l sup l smoothing splines and the minimum mean square error mmse estimation of a stationary signal corrupted by white gaussian noise in this model based formulation the optimum operator l is the whitening filter of the process and the regularization parameter is proportional to the noise variance thus the proposed formalism yields the optimal discretization of the classical wiener filter together with a fast recursive algorithm it extends the standard wiener solution by providing the optimal interpolation space we also present a bayesian interpretation of the algorithm

large n limit for the product of two coupled random matrices for a pair of coupled rectangular random matrices we consider the squared singular values of their product which form a determinantal point process we show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem this vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure we carry out the steepest descent analysis for a 4 times 4 matrix valued riemann hilbert problem which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified bessel functions a careful study of the vector equilibrium problem combined with this asymptotic analysis ultimately leads to the aforementioned convergence result for the limiting mean distribution an explicit form of the associated spectral curve as well as local sine meijer g and airy universality results for the squared singular values considered

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

the cauchy two matrix model we introduce a new class of two multi matrix models of positive hermitian matrices coupled in a chain the coupling is related to the cauchy kernel and differs from the exponential coupling more commonly used in similar models the correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate riemann hilbert problems thus paving the way to a steepest descent analysis and universality results the interpretation of the formal expansion of the partition function in terms of multicolored ribbon graphs is provided and a connection to the o 1 model a steepest descent analysis of the partition function reveals that the model is related to a trigonal curve three sheeted covering of the plane much in the same way as the hermitian matrix model is related to a hyperelliptic curve

a nonlinear age and maturity structured model of population dynamics i basic theory a nonlinear model of age and maturity structured population dynamics is analyzed the population is structured by age and maturity of the individuals and the nonlinearity in the equations corresponds to density dependent limitation of population growth the existence and asymptotic behavior of solutions are studied

discussion the dantzig selector statistical estimation when p is much larger than n 1 introduction this is a fascinating paper on an important topic the choice of predictor variables in large scale linear models a previous paper in these pages attacked the same problem using the lars algorithm efron hastie johnstone and tibshirani 3 actually three algorithms including the lasso as middle case there are tantalizing similarities between the dantzig selector ds and the lars methods but they are not the same and produce somewhat different models we explore this relationship with the lasso and lars here 2 dantzig selector and the lasso the definition of the dantzig selector ds in 1 7 can be re expressed as

the generalized quasilinearization method for partial differential equations of first order this paper applies the method of generalized quasilinearization to investigating the monotone iterative technique for partial differential equations of first order the rate of convergence is also discussed

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

continuous domain signal reconstruction using l_ p norm regularization we focus on the generalized interpolation problem there one reconstructs continuous domain signals that honor discrete data constraints this problem is infinite dimensional and ill posed we make it well posed by imposing that the solution balances data fidelity and some l p norm regularization more specifically we consider p geq 1 and the multi order derivative regularization operator mathrm l mathrm d n 0 we reformulate the regularized problem exactly as a finite dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid our splines are represented in a b spline basis which results in a well conditioned discretization for a sufficiently fine grid our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted this remarkable property is due to the approximation power of splines we use the alternating direction method of multipliers along with a multiresolution strategy to compute our solution we present numerical results for spatial and fourier interpolation through our experiments we investigate features induced by the l p norm regularization namely sparsity regularity and oscillatory behavior

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

mixed type hermite pade approximation inspired by the degasperis procesi equation abstract in this work we present new results on the convergence of diagonal sequences of certain mixed type hermite pade approximants of a nikishin system the study is motivated by a mixed hermite pade approximation scheme used in the construction of solutions of a degasperis procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string

generalized sampling a variational approach ii applications for pt i see ibid vol 50 no 8 p 1965 76 2000 the variational reconstruction theory from a companion paper finds a solution consistent with some linear constraints and minimizing a quadratic plausibility criterion it is suitable for treating vector and multidimensional signals here we apply the theory to a generalized sampling system consisting of a multichannel filterbank followed by a nonuniform sampling we provide ready made formulas which should permit application of the technique directly to problems at hand we comment on the practical aspects of the method such as numerical stability and speed we show the reconstruction formula and apply it to several practical examples including new variational formulation of derivative sampling landmark warping and tomographic reconstruction

problem of stability and chaotic properties of multidimensional lasota equation in the context of matuszewska orlicz indices abstract we study asymptotic properties of the dynamical system generated by the multidimensional lasota equation we give the conditions of its stability and chaos in the sense of devaney in orlicz spaces we apply matuszewska orlicz indices to a description of asymptotic behavior of the considered semigroup we analyze also the conditions under which asymptotic behavior for more general form of the equation is provided an example illustrates the criteria when the semigroups generated by the equations have not asymptotic behavior

dropping the independence singular values for products of two coupled random matrices we study the singular values of the product of two coupled rectangular random matrices as a determinantal point process each of the two factors is given by a parameter dependent linear combination of two independent complex gaussian random matrices which is equivalent to a coupling of the two factors via an itzykson zuber term we prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability the parameter dependence allows us to interpolate between the singular value statistics of the laguerre ensemble and that of the product of two independent complex ginibre ensembles which are both known we give exact formulae for the correlation kernel in terms of a complex double contour integral suitable for the subsequent asymptotic analysis in particular we derive a christoffel darboux type formula for the correlation kernel based on a five term recurrence relation for our biorthogonal functions it enables us to find its scaling limit at the origin representing a hard edge the resulting limiting kernel coincides with the universal meijer g kernel found by several authors in different ensembles we show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly

compressed sensing off the grid this paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples unlike previous work in compressed sensing the frequencies are not assumed to lie on a grid but can assume any values in the normalized frequency domain 0 1 an atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies which is then reformulated as an exact semidefinite program even with this continuous dictionary it is shown that o slog s log n random samples are sufficient to guarantee exact frequency localization with high probability provided the frequencies are well separated extensive numerical experiments are performed to illustrate the effectiveness of the proposed method

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

strong ergodic properties of a first order partial differential equation on etudie l equation differentielle d ordre 1 u t c x u x f x u pour t x 0 0 1 avec la condition initiale u 0 x v x pour x 0 1 on demontre l existence d une mesure invariante exacte ayant certaines proprietes supplementaires

a cell population model described by impulsive pdes existence and numerical approximation this paper considers a cell population model described by impulsive pdes existence and numerical approximation of solutions are investigated by showing that the related operator a x ddx on c0 generates a strongly conitunous semigroups under general assumption on a x we prove the existence of the solution of the nonlinear problem with impulses and given approximation of the solution of the homogenuous equation

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

cauchy laguerre two matrix model and the meijer g random point field we apply the general theory of cauchy biorthogonal polynomials developed in bertola et al commun math phys 287 3 983 1014 2009 and bertola et al j approx th 162 4 832 867 2010 to the case associated with laguerre measures in particular we obtain explicit formulae in terms of meijer g functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the cauchy two matrix model associated with them the central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists and is given by a new determinantal two level random point field the meijer g random field we conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of laguerre weights we express the joint distributions of the smallest eigenvalues in terms of suitable fredholm determinants and evaluate them numerically we also show that in a suitable limit the meijer g random field converges to the bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the laguerre ensemble

periodic solutions of hyperbolic partial differential equations abstract constructive methods by obtaining multiple solutions of first and second order nonlinear hyperbolic periodic boundary value problems are given appropriate partial differential inequalities are developed and are then employed to derive the iterative schemes which yield the desired solutions the advantage of these methods is that the solutions of the given nonlinear problems are constructed as uniform limits of monotone iterates these iterates being solutions of linear problems are explicitly computable

propagation of population pulses and fronts in a cell replication problem non locality and dependence on the initial function abstract here we study the solution types in a simplified equation arising from a cell population dynamics model formulated as a non linear first order partial differential equation for the cell density u t x in the presence of non local maturational x interactions this can be considered as a reaction convection equation in which the cell density is convected with maturation velocity r the non local action affects only the reaction birth rate an initial function representing a pulse of small amplitude on an almost extinguished population may be dissipated may produce multiple pulses or may remain intact as it travels driven by reaction and convection depending on the magnitude of the non local action alternately for localized initial perturbations the equation also has positive traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution for weak convection r 1 which is the only case treated here reaction driven fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state a classification arises according to whether the magnitude of the maturation delay is larger or smaller than a critical value and scaling arguments are used to determine the critical delay for delays larger than critical the reaction driven fronts are oscillatory and correspond to the case for which the correlation length for non local action is of the same order as the characteristic length of maturation population gradients that yield oscillatory birth rates simulation results are validated and interpreted using scaling arguments and solutions for the local equations