improved hardy inequalities for grushin operators we show that the hardy inequality for grushin operators like x 1 2 x 2 y on a bounded domain containing the origin can be refined by adding remainder terms such as the improvement of brezis vazquez vazquez zuazua and filippas tertikas

on the invertible cellular automata 150 over f p in this work the reversibility problem for cellular automata with rule number 150 over the finite field f p is tackled it is shown that when null boundary conditions are stated the reversibility is independent of the state set f p and it appears when the number of cells of the cellular space satisfies some conditions furthermore the explicit expressions of the inverse cellular automata are computed

reversibility of general 1d linear cellular automata over the binary field z 2 under null boundary conditions the reversibility of cellular automata ca has long been studied but there are still many problems untouched this paper tackles the reversibility problem of the general case concerning 1d ca under null boundary conditions defined by linear rules over the binary field z 2 although transition matrix has been widely used for some special linear ca lca rules it has severe limitations its complexity depends on the number of cells and determining its reversibility becomes another unsolved tough problem for general rules by constructing deterministic finite automata dfa of the de bruijn graph presentation of ca we conclude the reversibility problem of all 1d linear rules over z 2 under null boundary conditions it turns out that any 1d lca except for some unilateral rules over z 2 under null boundary conditions can be reversible once having a proper number of cells and dfa provides an efficient way to find those numbers moreover the complexity of the dfa solution is independent of the number of cells in addition the relative low complexity of dfa can be further significantly reduced by using only a small part of it

global existence in l1 for the enskog equation and convergence of the solutions to solutions of the boltzmann equation for the enskog equation in a box an existence theorem is proved for initial data with finite mass energy and entropy then by letting the diameter of the molecules go to zero the weak convergence of solutions of the enskog equation to solutions of the boltzmann equation is proved

the enskog boltzmann equation in the whole space r3 some global existence uniqueness and stability results abstract the enskog boltzmann equation a partial differential integral equation of the hyperbolic type is a model which generalizes the classical boltzmann equation to high gas densities and is here considered with an analysis of the well posedness of the initial value problem in unbounded domains some new global existence and uniqueness results for the solutions of the initial value problem and a discussion on the asymptotic stability are provided under suitable assumptions about the size of the norm of the initial data and about the rate of decay to zero at infinity in the phase space

critical phenomena in linear elliptic problems abstract in this article we consider the existence and nonexistence of principal eigenvalues for the linear elliptic equation u v x u in r n and u d 1 2 r n where n 3 and v l 1 loc r n with v 0 in r n we establish that when potential v has strong singularities phenomena which appear under critical exponent nonlinearities brezis and nirenberg are present in linear problems as well

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

on the best constant of hardy sobolev inequalities abstract we obtain the sharp constant for the hardy sobolev inequality involving the distance to the origin this inequality is equivalent to a limiting caffarelli kohn nirenberg inequality in three dimensions in certain cases the sharp constant coincides with the best sobolev constant

global existence inl1 for the generalized enskog equation various existence theorems are given for the generalized enskog equation inr3 and in a bounded spatial domain with periodic boundary conditions a very general form of the geometric factory is allowed including an explicit space velocity and time dependence the method is based on the existence of a liapunov functional an analog of theh function in the boltzmann equation and utilizes a weak compactness argument inl1

stability of global solution for the relativistic enskog equation near vacuum the cauchy problem of the relativistic enskog equation with near vacuum data is considered in this paper under the same assumption as that in jiang j stat phys 127 805 812 2007 for the relativistic enskog equation we obtain the uniform l stability of the solution what s more important is that for two new types of the scattering cross section we give the global existence and l1 x v stability for mild solution when the initial data lies in the space l1 x v as a corollary we have a bv type estimate it is worth mentioning that the stability results in this paper can be applied to the case in jiang j stat phys 127 805 812 2007

matrix methods and local properties of reversible one dimensional cellular automata the local behaviour of reversible one dimensional cellular automata is analysed based on perron frobenius theory it is proved by means of irreducible components that the connectivity matrices of reversible automata have a single eigenvalue equal to 1 the idempotent behaviour of such matrices is also proved by faddeev s algorithm the decomposition of these matrices into triangular factors is used to find the inverse rule for a given reversible automaton

schrodinger equations with critical nonlinearity singular potential and a ground state we study semilinear elliptic equations in a generally unbounded domain rn when the pertinent quadratic form is nonnegative and the potential is generally singular typically a homogeneous function of degree 2 we prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations while the nonlinearity is a perturbation of a self similar possibly oscillating term f of critical growth satisfying f js n 2n 2jf s j z s r this paper focuses on two qualitatively different cases of this problem one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space in the latter case we allow nonlinearities with oscillatory critical growth an important example of such quadratic form is the one on rn with the radial hardy potential x 2 with in the first case in the second case where n 2 24 is the largest constant for which the energy form remains nonnegative

optimizing improved hardy inequalities let be a bounded domain in rn n 3 containing the origin motivated by a question of brezis and vazquez we consider an improved hardy inequality with best constant b that we formally write as n 22 21 x 2 bv x we first give necessary conditions on the potential v under which the previous inequality can or cannot be further improved we show that the best constant b is never achieved in h01 and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in h01 our analysis reveals that the original inequality can be repeatedly improved by adding on the right hand side specific potentials this leads to an infinite series expansion of hardy s inequality the series obtained is in some sense optimal in establishing these results we derive various sharp improved hardy sobolev inequalities

on the linearized relativistic boltzmann equation i existence of solutions the linearized relativistic boltzmann equation inl 2 r p is investigated the detailed analysis of the collision operatorl is carried out for a wide class of scattering cross sections l is proved to have a form of the multiplication operatorv p plus the compact inl 2 p perturbationk the collisional frequencyv p is analysed to discriminate between relativistic soft and hard interactions finally the existence and uniqueness of the solution to the linearized relativistic boltzmann equation is proved

on the size of the inverse neighborhoods for one dimensional reversible cellular automata in this paper we investigate the possible neighborhood size of the inverse automaton of some types of one dimensional reversible cellular automata considering only the case when the local function is a size two map we give a quadratic upper bound for the neighborhood size of the inverse automaton we show that this bound can be lowered in some particular cases and give an algorithm for computing these better bounds

reversibility of linear cellular automata in this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled specifically a criterion for the reversibility in terms of the number of cells is stated and in these cases the inverse cellular automata are explicitly computed

weyl s type estimates on the eigenvalues of critical schrodinger operators using improved hardy sobolev inequalities motivated by the work karachalios n i 2008 lett math phys 83 189 99 we present explicit asymptotic estimates on the eigenvalues of the critical schrodinger operator involving inverse square potential based on improved hardy sobolev type inequalities

small data existence for the enskog equation in l 1 an existence theorem for the enskog equation with small initial data is proved in anl1 setting this type of result is not available for the boltzmann equation

inverse rules of eca with rule number 150 the explicit expressions of local transition functions of the inverse cellular automata of elementary cellular automata with rule number 150 and periodic boundary conditions are given the procedure to obtain such local transition functions is based on the algebraic properties of elementary cellular automata

nonlinear degenerate elliptic equation with hardy potential and critical parameter abstract some embedding inequalities in hardy sobolev spaces with general weight functions were proved and a positive answer to an open problem raised by brezis vazquez was given in the weighted hardy sobolev spaces the existence of nontrivial many solutions to the corresponding nonlinear degenerated elliptic equations with hardy potential and critical parameter under conditions weaker than ambrosetti rabinowitz condition was obtained

uniform stability of the solutions to the relativistic enskog equation abstract we present the uniform stability of classical solutions to the relativistic enskog equation under the locally lipschitz assumption of the collision factors given by polewczak this stability is an extension of the result given by ha and xiao for the relativistic boltzmann equation and it is very useful for the study of the cauchy problem for the relativistic enskog equation

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

optimal hardy rellich inequalities maximum principle and related eigenvalue problem in this paper we deal with three types of problems concerning the hardy rellich s embedding for a bilaplacian operator first we obtain the hardy rellich inequalities in the critical dimension n 4 then we derive a maximum principle for fourth order operators with singular terms then we study the existence non existence simplicity and asymptotic behavior of the first eigenvalue of the hardy rellich operator 2 n 2 n 4 2 16 q x x 4 under various assumptions on the perturbation q

reversibility of 1d cellular automata with periodic boundary over finite fields pmb mathbb z _ p the reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field 2 del rey and rodriguez sanchez in appl math comput 2011 doi 10 1016 j amc 2011 03 033 in this paper we study one dimensional linear cellular automata with periodic boundary conditions over any finite field p for any given p 2 we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one dimensional linear cellular automata more specifically for any given values from any fixed field p of the coefficients of the local rules we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells as an example we give the full criteria for the reversibility of the one dimensional linear cellular automata over the fields 2 and 3

an analytical proof of hardy like inequalities related to the dirac operator we prove some sharp hardy type inequalities related to the dirac operator by elementary direct methods some of these inequalities have been obtained previously using spectral information about the dirac coulomb operator our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity

global existence in l 1 for the modified nonlinear enskog equation in ℝ 3 a global existence theorem with large initial data inl1 is given for the modified enskog equation in 3 the method which is based on the existence of a liapunov functional analog of theh boltzmann theorem utilizes a weak compactness argument inl1 in a similar way to the diperna lions proof for the boltzmann equation the existence theorem is obtained under certain condition on the behavior of the geometric factory the condition ony amounts to the fact that thel1 norm of the collision term grows linearly when the local density tends to infinity

global existence proof for relativistic boltzmann equation the existence and causality of solutions to the relativistic boltzmann equation inl1 and inl loc 1 are proved the solutions are shown to satisfy physically naturala priori bounds time independent inl1 the results rely upon new techniques developed for the nonrelativistic boltzmann equation by diperna and lions

global existence and asymptotic behavior for the nonlinear enskog equation the existence of a classical solution to the nonlinear enskog equation in all mathbb r 3 is proved the solution which is global in time exists if the initial data go to zero fast enough at infinity and the mean path is sufficiently large the solution is smooth in space variable if the initial value is smooth the asymptoticbehavior of solutions is also given it is shown that as t to infty the solution to the enskog equation can be approximated by the solution to the free motion problem key words partial differential equations nonlinear integral equation asymptotic behavior kinetic theory boltzmann equation

on the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term we study questions of existence uniqueness and asymptotic behaviour for the solutions of u x t of the problem begin gathered text u t delta u lambda e u text lambda text 0 t 0 text x text varepsilon b hfill p text u x 0 u 0 x text x text varepsilon b hfill text u x t 0 text on text partial b times 0 infty hfill end gathered where b is the unit ball x varepsilon r n x text leqq text 1 text and n geqq 3 our interest is focused on the parameter 0 2 n 2 for which p admits a singular stationary solution of the form s x 2log x

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

on reversibility of cellular automata with periodic boundary conditions reversibility of one dimensional cellular automata with periodic boundary conditions is discussed it is shown that there exist exactly 16 reversible elementary cellular automaton rules for infinitely many cell sizes by means of a correspondence between elementary cellular automaton and the de bruijn graph in addition a sufficient condition for reversibility of three valued and two neighbour cellular automaton is given

procedures for calculating reversible one dimensional cellular automata we describe two algorithms for calculating reversible one dimensional cellular automata of neighborhood size 2 we explain how this kind of automaton represents all the other cases using two basic properties of reversible automata such as uniform multiplicity of ancestors and welch indices these algorithms only require matrix products and transitive closures of binary relations to classify all the possible reversible automata of neighborhood size 2 we expose the features advantages and differences with other well known methods finally we present results for reversible automata from three to six states and neighborhood size 2

existence and non existence of the first eigenvalue of the perturbed hardy sobolev operator in this paper we study the existence non existence and simplicity of the rst eigenvalue of the perturbed hardy sobolev operator i i 1 4 n i2 2 q jxj2 under various assumptions on the perturbation q we study the asymptotic behaviour of the rst eigenfunction near the origin when the perturbation q is q s 0 s 1 we will also establish the best constant in a hardy sobolev inequality proved by adimurthi e t a l

on the best constants of hardy inequality in rn k r k and related improvements abstract we compute the explicit sharp constants of hardy inequalities in the cone r k n r n k r k x 1 x n x n k 1 0 x n 0 with 1 k n furthermore the spherical harmonic decomposition is given for a function u c 0 r k n using this decomposition and following the idea of tertikas and zographopoulos we obtain the filippas tertikas improvement of the hardy inequality

on the dynamics of a strongly singular parabolic equation abstract in this paper we study the semiflow defined by a semilinear parabolic equation in which both the diffusion and the reaction term present strong order of singularity at the origin we justify the existence of a global branch of nontrivial equilibrium solutions for subcritical nonlinearities the approach based on the critical caffarelli kohn nirenberg inequality follows the arguments of 21 22