loewner matrices and operator convexity let f be a function from mathbb r into itself a classic theorem of k lowner says that f is operator monotone if and only if all matrices of the form left frac f p i f p j p i p j right vphantom x x 1 are positive semidefinite we show that f is operator convex if and only if all such matrices are conditionally negative definite and that f t t g t for some operator convex function g if and only if these matrices are conditionally positive definite elementary proofs are given for the most interesting special cases f t t r and f t t log t several consequences are derived

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

a weak majorization involving the matrices a b and ab abstract let a and b be n n positive definite matrices and let the eigenvalues of a b and ab be arranged in decreasing order then for all r 0 i k n i r ab i k n i r a b for k 1 n furthermore it is shown that i k n i r a b i k n i r ab for k 1 n is a limiting case of the weak majorization above which settles a conjecture of bapat and johnson

parabolic minkowski convolutions of solutions to parabolic boundary value problems abstract we introduce a new kind of convolution which is a sort of parabolic version of the classical supremal convolution of convex analysis this operation allows us to compare solutions of different parabolic problems in different domains as examples of applications of our main result we study the parabolic concavity of solutions to parabolic boundary value problems analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term we also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time proving some optimality of the ball

on dsmc calculations of rarefied gas flows with small number of particles in cells the direct simulation monte carlo dsmc analysis of two and three dimensional rarefied gas flows requires computational resources of very large proportions one of the major causes for this is that along with the multidimensional computational mesh the standard dsmc approach also requires a large number of particles in each cell of the mesh in order to obtain sufficiently accurate results this paper presents the development and validation of a modified simulation procedure which allows more accurate calculations with a smaller mean number of particles langle n rangle sim1 in the grid cells in the new algorithm the standard dsmc collision scheme is replaced by a two step collision procedure based on bernoulli trials scheme or its simplified version proposed by the author which is applied twice to the cells or subcells of a dual grid within a time step the modified algorithm uses a symmetric strang splitting scheme that improves the accuracy of the splitting scheme to o tau 2 with respect to the time step tau making the modified dsmc method an effective numerical tool for both steady and unsteady gas flow calculations on fine multidimensional grids the latter is particularly important for simulation of vortical and unstable rarefied gas flows the modified simulation scheme might also be useful for dsmc calculations within the subcell areas of a multilevel computational grid

some results on kwong functions and related inequalities abstract we investigate some relations between kwong functions and operator monotone functions as an application we present an arithmetic geometric mean type inequality by showing that for two continuous functions f g on 0 such that h t f t g t is a kwong function and f t g t t any positive matrices a b and any matrix x it holds that f a x g b g a x f b a x x b for each unitarily invariant norm

mathematical analysis of a chemical reaction with lumped temperature and strong absorption abstract we consider a mathematical model for the evolution of a single reactant and of the temperature in an isothermal catalyst the temperature is assumed to be spatially homogeneous and the absorption term is supposed to be nonlipschitzian existence and uniqueness of a regular solution is proved together with some bounds the steady state problem is also investigated and information about the dead core of the reactant are given

the finite element approximation of a coupled reaction diffusion problem with non lipschitz nonlinearities a coupled semilinear elliptic problem modelling an irreversible isothermal chemical reaction is introduced and discretised using the usual piecewise linear galerkin finite element approximation an interesting feature of the problem is that a reaction order of less than one gives rise to a dead core region initially one reactant is assumed to be acting as a catalyst and is kept constant it is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term this technique is then applied to the full coupled problem and optimal h 1 and l infty error bounds are proved in the absence of quadrature for a scheme involving numerical integration bounds similar to those obtained for the catalyst problem are shown to hold

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

a singular reaction diffusion system modelling prey predator interactions invasion and co extinction waves abstract we consider a singular reaction diffusion system arising in modelling prey predator interactions in a fragile environment since the underlying odes system exhibits a complex dynamics including possible finite time quenching one first provides a suitable notion of global travelling wave weak solution then our study focusses on the existence of travelling waves solutions for predator invasion in such environments we devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co extinction tail for both species under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere such a property arising for every admissible wave speeds

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

dead cores in nonlinear reaction diffusion systems on considere le systeme elliptique semilineaire u f u g v x u a x v k f u g v x v b x ou est un domaine borne dans r n 0 k 0 a x 0 b x 0 et max x a x max x b x 1

study of the blow up set by transformation abstract we obtain some new results on blow up for the heat equation with nonlinear sources by transformation into an absorption problem our work is related to that of kawohl and peletier who first developed such a transformation in some special cases here we provide a systematic method for constructing a suitable transformation by using the class of regularly varying functions through regularization we obtain a precise characterization of the blow up set as zeros of a certain steady state

positive semidefinite solutions of the operator equation j 1nan jxaj 1 b abstract let a be a positive definite operator and b be a self adjoint operator we discuss the existence of positive semidefinite solutions of the operator equation j 1 n a n j xa j 1 b via an order preserving operator inequality by using the solutions we also give concrete examples of positive semidefinite matrices

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

on the time relaxed monte carlo computations for the lid driven micro cavity flow abstract in the present study the first second and third orders of the trmc scheme trmc1 trmc2 and trmc3 schemes are employed to numerically investigate the lid driven micro cavity flow with different lid velocities and knudsen numbers the simulations cover flow regimes from early slip to early transition regimes 0 005 k n 0 1 higher order terms in the wild sum expansion are considered to obtain higher order collisions the results are compared with those from the standard dsmc method comparisons show that among the studied schemes the results obtained from the trmc3 scheme have excellent agreement with the ones from the dsmc method on the other hand the results of trmc1 and trmc2 schemes show deviations compared to those from the dsmc method the deviations are more pronounced for the velocity distributions moreover the present investigation illustrates that as the knudsen number increases the accuracy of lower orders of the trmc scheme improves it is observed that truncating the wild sum expansion up to the third order approximation of the trmc scheme may be a proper alternative method for the dsmc method in simulating the lid driven micro cavity problem for knudsen numbers 0 005 k n 0 1 with reasonable accuracy and simplicity in mathematics

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed

convexity of the inverse and moore penrose inverse abstract convexity properties of the inverse of positive definite matrices and the moore penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied for the positive definite case null space characterizations are derived and lead naturally to a concept of strong convexity of a matrix function extending the conventional concept of strict convexity the positive definite results are shown to allow for a unified analysis of problems in reproducing kernel hilbert space theory and inequalities involving matrix means the main results comprise a detailed study of the convexity properties of the moore penrose inverse providing extensions and generalizations of all the earlier work in this area

improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices

compact solitary waves in linearly elastic chains with non smooth on site potential it was recently observed by saccomandi and sgura that one dimensional chains with nonlinear elastic interaction and regular on site potential can support compact solitary waves i e travelling solitary waves with strictly compact support in this paper we show that the same applies to chains with linear elastic interaction and an on site potential which is continuous but non smooth at minima some different features arise in particular the speed of compact solitary waves is not uniquely fixed by the equation we also discuss several generalizations of our findings

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

on infinite divisibility of positive definite functions arising from operator means abstract for a point wisely non negative positive definite function a certain criterion for its infinite divisibility i e all its fractional powers are also positive definite is obtained this criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means in particular we determine when the function cosh x s cosh x s 0 1 s s 1 1 is infinitely divisible

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

moving boundaries in reaction diffusion systems with absorption consider the reaction diffusion system t u d 0 xx u ku 1 v 0 0 t u d 1 xx v ku 1 v with boundary and initial conditions u 0 t c 0 u t 0 t 0 x v 0 t 0 v t 0 u x 0 u 0 x v x 0 v 0 x where 0 u 0 x c 0 and 0 v 0 x 1 for x 0

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

parametric extensions of shannon inequality and its reverse one in hilbert space operators we shall state the following parametric extensions of shannon inequality and its reverse one in hilbert space operators let p 0 1 and also let a1 a2 an and b1 b2 bn be two sequences of strictly positive operators on a hilbert space h such that j 1naj pbj i then j 1nsp 1 aj bj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp aj bj log j 1n aj p 1bj i j 1naj pbj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp 1 aj bj where sq a b a12a 12ba 12qloga 12ba 12a12 for a 0 b 0 and any real number q and a qb a12a 12ba 12qa12 for a 0 b 0 and any real number q in particular if j 1naj j 1nbj i then j 1ns2 aj bj j 1nbjaj 1bjlog j 1nbjaj 1bj log j 1nbjaj 1bj j 1ns1 aj bj 0 j 1ns aj bj log j 1najbj 1aj j 1najbj 1ajlog j 1najbj 1aj j 1ns 1 aj bj where s a b s0 a b a12loga 12ba 12a12 which is the relative operator entropy of a 0 and b 0 our results can be considered as parametric extensions of the following celebrated shannon inequality ann math statistics 22 1951 79 bull syst tech j 27 1948 379 pitman monographs and surveys in pure and applied mathematics 97 addison wesley longman 1998 p 233 which is very useful and so famous in information theory let a1 a2 an and b1 b2 bn be two probability vectors then 0 j 1najlogbj j 1najlogaj see inequalities 2 4 of corollary 2 4

on the definiteness of the solutions of certain matrix equations abstract it is shown that for each dimension n 2 there is a real number t n such that for all t 2 t n the unique n n solution x of the matrix equation a 2 x xa 2 taxa b must be positive semidefinite for any positive definite a and positive semidefinite b this generalizes the classical lyapunov result it is established that t n 8 4 for n 2 3 4 respectively and that t n 2 for all n upper bounds are given for n up to 6 related equations are also discussed

operator inequalities related to the corach porta recht inequality abstract we prove some refinements of an inequality due to x zhan in an arbitrary complex hilbert space by using some results on the heinz inequality we present several related inequalities as well as new variants of the corach porta recht inequality we also characterize the class of operators satisfying sxs 1 s 1 xs kx k 2 x under certain conditions

soc monotone and soc convex functions vs matrix monotone and matrix convex functions abstract the soc monotone function respectively soc convex function is a scalar valued function that induces a map to preserve the monotone order respectively the convex order when imposed on the spectral factorization of vectors associated with second order cones socs in general hilbert spaces in this paper we provide the sufficient and necessary characterizations for the two classes of functions and particularly establish that the set of continuous soc monotone respectively soc convex functions coincides with that of continuous matrix monotone respectively matrix convex functions of order 2

a characterization of the class of partial isometries abstract let b h be the set of all bounded linear operators on a hilbert space h in this paper we show that if s is a closed range operator with r s r s then s s s s ast 2 if and only if s is a non zero real scalar of a normal partial isometry also we find some other characterizations of this space using some inequalities related to corach porta recht inequality

a high order monte carlo algorithm for the direct simulation of boltzmann equation a high order algorithm of the direct simulation monte carlo dsmc method h dsmc has been developed to simulate rarefied flow regimes the mth order taylor series expansion has been employed to obtain a more generalized form of the time discretization for the collision part of boltzmann equation in the purposed algorithm the higher order collision terms are introduced as well as higher order terms in the time step of the probabilistic coefficients these newly implemented higher order terms improve the accuracy and efficiency of the solution and enhance the convergence rate quite significantly comparison between results of the classic dsmc method and the h dsmc method shows the promising performance of the introduced technique

on the lid driven problem in a porous cavity a theoretical and numerical approach the purpose of this paper is to analyze the motion of an incompressible viscous fluid through a porous medium located in a two dimensional square cavity i e the lid driven flow problem described by a generalized darcy brinkman model first we study the dirichlet boundary value problem for a generalized darcy brinkman system in a n dimensional bounded lipschitz domain n 2 3 when the given data belongs to some l 2 based sobolev spaces we obtain an existence and uniqueness result for this problem under the assumption of sufficiently small given data an existence and uniqueness result for the robin boundary value problem associated to the same nonlinear system is also formulated further we consider a special dirichlet boundary value problem for the generalized darcy brinkman system i e the lid driven problem associated with such a nonlinear system and we obtain the streamlines of the fluid flow for different reynolds and darcy numbers moreover we consider an additional sliding parameter that corresponds to a mixed dirichlet robin boundary value condition imposed on the upper moving wall and analyze the behavior of the flow with respect to such a parameter

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

adaptive and recursive time relaxed monte carlo methods for rarefied gas dynamics recently a new class of monte carlo methods called time relaxed monte carlo trmc designed for the simulation of the boltzmann equation close to fluid regimes has been introduced l pareschi and g russo siam j sci comput 23 2001 pp 1253 1273 a generalized wild sum expansion of the solution is the basis of the simulation schemes after a splitting of the equation the time discretization of the collision step is obtained from the wild sum expansion of the solution by replacing high order terms in the expansion with the equilibrium maxwellian distribution in this way speed up of the methods close to fluid regimes is obtained by efficiently thermalizing particles close to the equilibrium state in this work we present an improvement of such methods which allows us to obtain an effective uniform accuracy in time without any restriction on the time step and subsequent increase of the computational cost the main ingredient of the new algorithms is recursivity l pareschi and b wennberg monte carlo methods appl 7 2001 pp 349 358 several techniques can be used to truncate the recursive trees generated by the schemes without degrading the accuracy of the numerical solution techniques based on adaptive strategies are presented numerical results emphasize the gain of efficiency of the present simulation schemes with respect to standard dsmc direct simulation monte carlo methods

critical exponents and lower bounds of blow up rate for a reaction diffusion system abstract this paper is concerned with a reaction diffusion system with absorption terms under dirichlet boundary conditions modelling the cooperative interaction of two diffusion biological species by constructing blow up sub solutions and bounded super solutions we obtain the optimal conditions on the exponent of reaction and absorption terms for the existence or nonexistence of global solutions moreover for a special case we derive the lower bound estimates of blow up rate

a robust numerical method for the r13 equations of rarefied gas dynamics application to lid driven cavity in this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment r13 equations of rarefied gas dynamics the scheme allows fast solutions for 2d and 3d boundary value problems bvps with velocity slip and temperature jump boundary conditions the scheme is applied to the lid driven cavity problem for knudsen numbers up to 0 7 the results compare well with those obtained from more costly solvers for rarefied gas dynamics such as the integro moment method imm and the direct simulation monte carlo dsmc method the r13 equations yield better results than the classical navier stokes fourier equations for this boundary value problem since they give an approximate description of knudsen boundary layers at moderate knudsen numbers the r13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate knudsen numbers

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

asymptotic behavior of the solutions of the dead core problem u u pup 0 1 1 this equation originates from the study of a class of models for the reaction diffusion processes of a gas inside a chemical reactor 2 81 from the mathematical point of view the major interest of equation 1 1 lies in the competition of the linear diffusion and the nonlinear fast p 0 t 0 with initial and boundary conditions

the evolution of travelling waves in fractional order autocatalysis with decay ii the initial boundary value problem in this paper we examine an initial boundary value problem for a singular reaction diffusion equation arising from a model of isothermal chemical autocatalysis with termination we consider the case where the reaction and termination orders m and n respectively are fractional that is 0 0 k being a parameter which measures the relative strength of the termination step to that of the autocatalysis we show that solutions approach the equilibrium state 1 k for k 1 we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t finally for m n we show that traveling waves with semi infinite support occ

time relaxed monte carlo methods for the boltzmann equation a new family of monte carlo schemes is introduced for the numerical solution of the boltzmann equation of rarefied gas dynamics the schemes are inspired by the wild sum expansion of the solution of the boltzmann equation for maxwellian molecules and consist of a novel time discretization of the equation in particular high order terms in the expansion are replaced by the equilibrium maxwellian distribution the two main features of the schemes are high order accuracy in time and asymptotic preservation the first property allows to recover accurate solutions with time steps larger than those required by direct simulation monte carlo dsmc while the latter guarantees that for the vanishing knudsen number the numerical solution relaxes to the local maxwellian conservation of mass momentum and energy are preserved by the scheme numerical results on several space homogeneous problems show the improvement of the new schemes over standard dsmc applications to a one dimensional shock wave problem are also presented

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

approaching an extinction point in one dimensional semilinear heat equations with strong absorption abstract this paper deals with the cauchy problem ut uxx up 0 0 u x 0 u0 x 0 and u x t 0 appear when t is close to t we describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached we also give conditions under which some of these behaviours actually occur

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

high order numerical methods for the space non homogeneous boltzmann equation in this paper we present accurate methods for the numerical solution of the boltzmann equation of rarefied gas the methods are based on a time splitting technique the transport is solved by a third order accurate in space positive and flux conservative pfc method the collision step is treated by a fourier approximation of the collision integral which guarantees spectral accuracy in velocity coupled with several high order integrators in time strang splitting is used to achieve second order accuracy in space and time several numerical tests illustrate the properties of the methods

symmetric function means of positive operators abstract many authors have studied means involving ratios of the elementary symmetric functions this work involves generalizing these means to an operator theoretic setting and extending many of the classical scalar inequalities to operators it is shown that these operator means are related to electrical networks

zhan s inequality on a g mean inequalities abstract in this paper we show some generalizations of zhan s inequality by virtue of hadamard product as follows let a b x be n n matrices such that a b are positive definite and f g be two continuous functions on 0 k max f g a b where a is the spectrum of a if f x g x is kwong i e the matrix f g i f g j i j is positive semidefinite for any positive numbers 1 n then for any 0 i a 1 2 f a x g b g a x f b b 1 2 k 2 1 2 2 r 0 a x b 4 1 r 0 t 2 a 2 x t a x b x b 2 holds for 2 t 2 2 where r 0 min 1 2 1 r 1 1 r i i a 1 2 f a x g b g a x f b b 1 2 2 k t 2 a 2 x t a x b x b 2 holds for any 2 t 2 where is any unitarily invariant norm

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

regularity for degenerate evolution equations with strong absorption abstract in this manuscript we study geometric regularity estimates for degenerate parabolic equations of p laplacian type 2 p under a strong absorption condition p u u t 0 u q in t 0 t where 0 q 1 this model is interesting because it yields the formation of dead core sets i e regions where non negative solutions vanish identically we shall prove sharp and improved parabolic c regularity estimates along the set f 0 u t u 0 t the free boundary where p p 1 q 1 1 p 1 some weak geometric and measure theoretical properties as non degeneracy positive density porosity and finite speed of propagation are proved as an application we prove a liouville type result for entire solutions a specific analysis for blow up type solutions will be done as well the results are new even for dead core problems driven by the heat operator

some convex and monotone matrix functions we prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections as a consequence we prove that a certain product of matrix monotone functions is a matrix monotone function similar results are proved for matrix convex functions these results include some known results for matrix monotone and matrix convex functions

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

concrete examples of operator monotone functions obtained by an elementary method without appealing to lowner integral representation we study concrete examples of operator monotone functions obtained by an elementary method without appealing to lowner integral representation

sublinear elliptic problems with a hardy potential abstract in this paper we study the positive solutions of sub linear elliptic equations with a hardy potential which is singular at the boundary by means of ode techniques a fairly complete picture of the class of radial solutions is given local solutions with a prescribed growth at the boundary are constructed by means of contraction operators some of those radial solutions are then used to construct ordered upper and lower solutions in general domains by standard iteration arguments the existence of positive solutions is proved an important tool is the hardy constant

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

on the injective norm and characterization of some subclasses of normal operators by inequalities or equalities abstract let b h be the c algebra of all bounded linear operators acting on a complex hilbert space h in this note we shall show that if s is an invertible normal operator in b h the following estimation holds s s 1 s 1 s s s 1 1 s s 1 where is the injective norm on the tensor product b h b h this last inequality becomes an equality when s is invertible self adjoint on the other hand we shall characterize the set of all invertible normal operators s in b h satisfying the relation s s 1 s 1 s s s 1 1 s s 1 and also we shall give some characterizations of some subclasses of normal operators in b h by inequalities or equalities

cauchy schwarz norm inequalities for weak integrals of operator valued functions for a finite measures on and weakly measurable families at t and bt t of hilbert space operators we have the non commutative cauchy schwarz inequalities in schatten p ideals axbd p a aa d q 1ad 2qx b b bd r 1b d 2rp for all x p h and for all p q r 1 such that 1q 1r 2p if both at t and bt t consists of commuting normal operators then axbd a ad x b bd for all unitarily invariant norms and all x h if additionally a ad i and b bd i then i a ad xi b bd h and i a ad xi b bd x axbd applications include young s and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions

an operator monotone function tlogt t 1log2t and strictly chaotic order we consider the function f t defined by f t tlogt t 1log2t firstly we shall show direct and simplified proofs of operator monotonicity of f t and its dual function f t tlog2ttlogt t 1 without appealing of lowner theorem then we shall show the following loga logb thereexists 0 1 suchthatf a f b holds forall 0 by using the result we summarize the following interesting contrast between a b 0 and a b 0 a b 0 thereexists 0 1 suchthatf a f b holds i forall 0 ii a b 0 f a f b forall 0 1 finally we cite a counterexample related to as follows thereexiststritlypositiveoperatorsaandbsuchthatloga logb c butf a f b doesnotholdforany 0

on singular value inequalities for matrix means abstract for positive semidefinite n n matrices a and b the singular value inequality 2 t s j a r b 2 r a 2 r b r 2 s j a 2 t a b b 2 is shown to hold for r 1 2 1 3 2 and all 2 t 2

short time behaviour of a singular solution to the heat equation with absorption on considere l evolution aux temps petits de l interface qui apparait dans le probleme aux limites u t u xx u p u o t 1 t 0 u x o 0 x 0 pour p 1 on construit une solution interieure d ordre zero consistante avec une interface mobile ou u est zero

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations