the continuous dependence on nonlinearities of solutions of the neumann problem of a singular parabolic equation abstract in this paper we consider the singular diffusion problem u t u m 1 u x x 0 x 1 t 0 1 m u m x x 0 1 0 t 0 u t 0 u 0 x 0 x 1 with 0 m 1 we will show that there exists a unique global solution u x t m such that 0 0 1 u x t m u x t m 0 2 d x d t c m m 0 where 0 m m 0 1 and c is a explicit constant

the cusped hyperbolic orbifolds of minimal volume in dimensions less than ten abstract we detect the cusped complete hyperbolic orbifolds of minimal volume in dimensions less than ten our method is geometric and uses results from crystallography and the theory of sphere packings in fact such an n orbifold gives rise to a horoball packing whose orthogonal projection to the cusp boundary yields the densest euclidean lattice packing in dimension n 1

an error estimate for the finite difference approximation to degenerate convection diffusion equations we consider semi discrete first order finite difference schemes for a nonlinear degenerate convection diffusion equations in one space dimension and prove an l 1 error estimate precisely we show that the l 1 rm loc difference between the approximate solution and the unique entropy solution converges at a rate mathcal o delta x 1 11 where delta x is the spatial mesh size if the diffusion is linear we get the convergence rate mathcal o delta x 1 2 the point being that the mathcal o is independent of the size of the diffusion

optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

l 1 error estimates for difference approximations of degenerate convection diffusion equations we analyze monotone finite difference schemes for strongly degenerate convection diffusion equations in one spatial dimension these nonlinear equations are well posed within a class of discontinuous entropy solutions we prove that the l1 error between the approximate and exact solutions is o x1 3 where x is the spatial grid parameter this result should be compared with the classical o x1 2 error estimate for conservation laws kuznecov 1976 and a recent estimate of o x1 11 for degenerate convection diffusion equations karlsen koley risebro 2012

information measures the curious case of the binary alphabet four problems related to information divergence measures defined on finite alphabets are considered in three of the cases we consider we illustrate a contrast that arises between the binary alphabet and larger alphabet settings this is surprising in some instances since characterizations for the larger alphabet settings do not generalize their binary alphabet counterparts in particular we show that f divergences are not the unique decomposable divergences on binary alphabets that satisfy the data processing inequality thereby clarifying claims that have previously appeared in the literature we also show that kullback leibler kl divergence is the unique bregman divergence which is also an f divergence for any alphabet size we show that kl divergence is the unique bregman divergence which is invariant to statistically sufficient transformations of the data even when nondecomposable divergences are considered like some of the problems we consider this result holds only when the alphabet size is at least three

alpha divergence is unique belonging to both f divergence and bregman divergence classes a divergence measure between two probability distributions or positive arrays positive measures is a useful tool for solving optimization problems in optimization signal processing machine learning and statistical inference the csiszar f divergence is a unique class of divergences having information monotonicity from which the dual alpha geometrical structure with the fisher metric is derived the bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the alpha structure in general csiszar gave an axiomatic characterization of divergences related to inference problems the kullback leibler divergence is proved to belong to both classes and this is the only such one in the space of probability distributions this paper proves that the alpha divergences constitute a unique class belonging to both classes when the space of positive measures or positive arrays is considered they are the canonical divergences derived from the dually flat geometrical structure of the space of positive measures

a note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations abstract following the lead of carrillo arch ration mech anal 147 1999 269 361 recently several authors have used kru kov s device of doubling the variables to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations in all these results the second order differential operator is not allowed to depend explicitly on the spatial variable which certainly restricts the range of applications of entropy solution theory the purpose of this paper is to extend a version of carrillo s uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator the class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation consolidation processes

entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

continuous dependence on the nonlinearity of viscosity solutions of parabolic equations abstract this paper establishes an upper bound for u v where u is a subsolution of u t f u d x u d 2 x u 0and v is a supersolution of v t g v d x v d 2 x v 0in 0 with neumann boundary conditions and where is an open convex set

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

quasilinear parabolic systems with nonlinear boundary conditions ui x 0 ui0 x 0 x 0 i 1 n where 0 r is a bounded domain with smooth boundary 0 is the unit outward normal vector and the constants mi 0 i 0 mij 0 i j 1 n ui0 x i 1 n are positive c functions and satisfy the compatibility conditions 1 the present address academy of mathematics and systems sciences academia sinica beijing 100080 people s republic of china doi 10 1006 jdeq 2000 3784 available online at http www idealibrary com on

monotone difference approximations of bv solutions to degenerate convection diffusion equations we consider consistent conservative form monotone difference schemes for nonlinear convection diffusion equations in one space dimension since we allow the diffusion term to be strongly degenerate solutions can be discontinuous and in general are not uniquely determined by their data here we choose to work with weak solutions that belong to the bv in space and time class and in addition satisfy an entropy condition a recent result of wu and yin northeastern math j 5 1989 pp 395 422 states that these so called bv entropy weak solutions are unique the class of equations under consideration is very large and contains to mention only a few the heat equation the porous medium equation the two phase flow equation and hyperbolic conservation laws the difference schemes are shown to converge to the unique bv entropy weak solution of the problem in view of the classical theory for monotone difference approximations of conservation laws the main difficulty in obtaining a similar convergence theory in the present context is to show that the strongly degenerate discrete diffusion term is sufficiently smooth we provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes finally we make some concluding remarks about monotone difference schemes for multidimensional equations

periodic solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract this paper is concerned with the existence of maximal and minimal periodic solutions of a class of quasilinear parabolic systems with nonlinear boundary conditions our approach to the problem is based on the method of upper and lower solutions and its associated monotone iterations an application is also made to the reaction diffusion system of lotka volterra competition model

a lower bound for the volume of hyperbolic 3 orbifolds toutes les 3 varietes d orbites orientables completes ont un volume au moins 0 0000013 toutes les 3 varietes d orbites hyperboliques orientables completes cuspides ont un volume au moins 3 1 2 24

covolumes of nonuniform lattices in pu n 1 this paper studies the covolumes of nonuniform arithmetic lattices in pu n 1 we determine the smallest covolume nonuniform arithmetic lattices for each n the number of minimal covolume lattices for each n and study the growth of the minimal covolume as n varies in particular there is a unique lattice up to isomorphism in pu 9 1 of smallest euler poincare characteristic amongst all nonuniform arithmetic lattices in pu n 1 we also show that for each even n there are arbitrarily large families of nonisomorphic maximal nonuniform lattices in pu n 1 of equal covolume

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

l 1 framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations we develop a general l 1 framework for deriving continuous de pendence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the chen perthame kinetic approach 9 we ap ply our l 1 framework to establish an explicit estimate for continuous depen dence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method without imposition of bounded variation of the approximate solutions finally as an example of a direct application of this framework to numerical methods we focus on a linear convection diusi on model equation and derive an l 1 error estimate for an upwind central finite dierence scheme

on a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium the main result of the paper is the uniqueness of nonnegative solutions of the cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut um xx un x where m 1 and n 0 in particular n is allowed to be smaller than one the proof is based on a refined test function argument the condition that u be nonnegative is crucial but the restriction to one space variable is not

determination of all semisymmetric recursive information measures of multiplicative type on n positive discrete probability distributions information measures m entropies divergences inaccuracies information improvements etc depending upon n probability distributions which we unite into a vector distribution are recursive of type if m p1 p2 p3 pm m 1 p1 p2 p3 pm p1 p2 2p1p1 p2 p2p1 p2 if also a similar equation holds with three instead of two distinguished vectors then has to be multiplicative except if all m are identically 0 the information measure is semisymmetric if 3 p1 p2 p3 3 p1 p3 p2 we determine all semisymmetric in particular symmetric recursive information measures of multiplicative type allowing first only positive probabilities previously the cases n 3 have been examined mainly for t 1 2 n 1 1 2 2 n n and some probabilities were allowed to be 0 this has made the proofs easier but permitting certain probabilities to be 0 would exclude most information measures important for applications so the description of appropriate domains became complicated however we show how the measures which we determine here can be extended to the old domains and to more general ones

the densest lattices in pgl3 q2 abstract we find the smallest possible covolume for lattices in pgl 3 q 2 show that there are exactly two lattices with this covolume and describe them explicitly they are commensurable and one of them appeared in mumford s construction of his fake projective plane we also discuss a new 2 adic uniformization of another fake projective plane

global existence and finite time blow up for a degenerate reaction diffusion system abstract this paper investigates the blow up and global existence of solutions of the degenerate reaction diffusion system u t u m u v p v t v n u q v x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 0 and p q 0 it is proved that if m n and pq m n every nonnegative solution is global whereas if m or n or pq m n there exist both global and blow up nonnegative solutions when m n and pq m n we show that there exists 1 which depends on the parameters p q m n such that all positive solutions are global if 1 while if 1 1 all positive solutions blow up in finite time where 1 is the first dirichlet eigenvalue for the laplacian on

a very singular solution of the porous media equation with absorption turns out to play an important role in the description of both the short and the long time behaviour of positive solutions of eq 1 1 for the short time behaviour it provides together with the one parameter family of singular solutions i of problem i a complete description of the possible types of singular behaviour at the point 0 o 121 for the long time behaviour of solutions u x t of the cauchy problem for 1 1 1 2 it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 co t p u x t w x t 0 ast co uniformly in r 7 81 in this paper we shall show that if

blow up solution for the degeneratequasilinear parabolic system with nonlinear boundary condition in this paper we discuss the global existence and blow up of the solution to the quasilinear parabolic system with nonlinear boundary condition we also discuss the blow up set of the blow up solution and obtain the blow up rate of the blow up solution and prove the existence of the self similar blow up solution

lattices of minimum covolume in chevalley groups over local fields of positive characteristic in this article we show that if g is a simply connected chevalley group of either classical type of rank bigger than 1 or type e6 and q 9 is a power of a prime number p 5 then g g fq t 1 up to an automorphism has a unique lattice of minimum covolume which is g fq t 1

the asymptotic behavior of solutions for a class of doubly degenerate nonlinear parabolic equations abstract by using the morse interaction technique supposing that the uniqueness of the barenblatt type solution is true the paper studies the large time asymptotic behavior of solutions for the doubly degenerate parabolic equation u t div d u m p 2 d u m d u m p 1 u q with initial condition u x 0 u 0 x here the exponents m p p 1 q satisfy p p 1 p 1 q m p 1 1 p 1 m 1

arithmeticity discreteness and volume we give an arithmetic criterion which is sufficient to imply the discreteness of various two generator subgroups of p sl 2 c we then examine certain two generator groups which arise as extremals in various geometric problems in the theory of kleinian groups in particular those encountered in efforts to determine the smallest co volume the margulis constant and the minimal distance between elliptic axes we establish the discreteness and arithmeticity of a number of these extremal groups the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

about distances of discrete distributions satisfying the data processing theorem of information theory the distances of discrete probability distributions are considered necessary and sufficient conditions for validity of the data processing theorem of information theory are established these conditions are applied to the burbea rao 1982 divergences and bregman 1967 distances

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

global solutions for quasilinear parabolic systems we present an approach for proving the global existence of classical solutions of certain quasilinear parabolic systems with homogeneous dirichlet boundary conditions in bounded domains with a smooth boundary c 2003 elsevier inc all rights reserved

partial boundary value condition for a nonlinear degenerate parabolic equation abstract in this paper a nonlinear degenerate parabolic equation is considered by proposing a new partial boundary value condition and constructing special test functions we prove the stability of the solutions by means of the kruzkov bi variables method moreover we show that if there is an interaction between the diffusion term and the convection term the stability can be established without any boundary value condition

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

bregman divergence bounds and universality properties of the logarithmic loss a loss function measures the discrepancy between the true values and their estimated fits for a given instance of data in classification problems a loss function is said to be proper if a minimizer of the expected loss is the true underlying probability we show that for binary classification the divergence associated with smooth proper and convex loss functions is upper bounded by the kullback leibler kl divergence to within a normalization constant this implies that by minimizing the logarithmic loss associated with the kl divergence we minimize an upper bound to any choice of loss from this set as such the logarithmic loss is universal in the sense of providing performance guarantees with respect to a broad class of accuracy measures importantly this notion of universality is not problem specific enabling its use in diverse applications including predictive modeling data clustering and sample complexity analysis generalizations to arbitary finite alphabets are also developed the derived inequalities extend several well known f divergence results

on character varieties sets of discrete characters and nonzero degree maps a it knot manifold is a compact connected irreducible orientable 3 manifold whose boundary is an incompressible torus we first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps these results are applied later in the paper where we fix a small knot manifold m and investigate various sets of characters of representations rho pi 1 m to rm psl 2 bbb c whose images are discrete we show that the topology of these sets is intimately related to the algebraic structure of the rm psl 2 bbb c character variety of m as well as dominations of manifolds by m and its dehn fillings we apply our results to the following question of shicheng wang it are nonzero degree maps between infinitely many distinct dehn fillings of two hyperbolic knot manifolds m and n induced by a nonzero degree map m to n we show that the answer is yes generically using this we show that if a small mathcal h minimal hyperbolic knot manifold admits non homeomorphic mathcal h minimal dehn fillings it admits infinitely many such fillings we also construct the first infinite families of small closed connected orientable manifolds which are minimal in the sense that they do not admit nonzero degree maps other than homotopy equivalences to any aspherical manifold

positive solutions of quasilinear parabolic systems with dirichlet boundary condition abstract coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems including systems of parabolic and ordinary differential equations are investigated the aim of this paper is to show the existence uniqueness and asymptotic behavior of time dependent solutions also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i 1 n and the boundary condition is u i 0 using the method of upper and lower solutions we show that a unique global classical time dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a scalar polynomial growth problem a coupled system of polynomial growth problem and a two component competition model in ecology

viscosity solutions of a class of degenerate quasilinear parabolic equations with dirichlet boundary condition abstract this paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous dirichlet boundary condition the equation under consideration arises from a number of practical model problems including reaction diffusion processes in a porous medium the degeneracy of the problem appears on the boundary and possibly in the interior of the domain the goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them an application of the above results is given to a porous medium type of reaction diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem

continuous dependence on the nonlinearities of solutions of degenerate parabolic equations explicit estimates for the continuous dependence inl 0 t l1 rd of solutions of the equation lv v v in 0 rdwith initial conditionv 0 h with respect to the nonlinear continuously differential functions and are established

stability of solutions of quasilinear parabolic equations abstract we bound the difference between solutions u and v of u t a u div x f h and v t b v div x g k with initial data and respectively by u t v t l p e a e t l r n 2 p b t a b x f x g f u g u h k p e p here all functions a f and h are smooth and bounded and may depend on u x r n and t the functions a and h may in addition depend on u identical assumptions hold for the functions that determine the solutions v furthermore e r n is assumed to be a bounded set and p and p are fractions that depend on n and p the diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth

quasilinear reaction diffusion systems with nonlinear boundary conditions this paper deals with the existence and nonexistence of global positive solutions of quasilinear reaction diffusion systems with nonlinear boundary conditions necessary and sufficient conditions on the global existence of all positive solutions are obtained

large time behavior of free boundary for a degenerate parabolic equation with absorption abstract this work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the cauchy problem of a degenerate parabolic partial differential equation with absorption our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously

a note on divergences in many areas of neural computation like learning optimization estimation and inference suitable divergences play a key role in this note we study the conjecture presented by amari 2009 and find a counterexample to show that the conjecture does not hold generally moreover we investigate two classes of divergence zhang 2004 weighted f divergence and weighted divergence and prove that if a divergence is a weighted f divergence as well as a bregman divergence then it is a weighted divergence this result reduces in form to the main theorem established by amari 2009 when

global and nonglobal weak solutions to a degenerate parabolic system this paper deals with a degenerate parabolic system coupled via general reaction terms of power type global weak solutions are obtained by means of energy estimates and the de giorgi s technique in particular the criterion for global nonexistence of weak solutions is proved by introducing suitable weak sub solutions together with a weak comparison principle in summary the critical exponent for weak solutions of the degenerate parabolic system is determined

quasilinear parabolic and elliptic systems with mixed quasimonotone functions abstract this paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions the boundary condition in the system may be dirichlet nonlinear or a combination of these two types the elliptic operators in the system are allowed to be degenerate the aim is to show the existence and uniqueness of a classical solution to the parabolic system the existence of maximal and minimal solutions or quasisolutions of the elliptic system and the asymptotic behavior of the solution of the parabolic system this consideration leads to a global attractor of the parabolic system as well as an one sided stability of the maximal and minimal solutions applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density dependent and are degenerate these applications exhibit quite distinct dynamical behavior of the population species between degenerate density dependent diffusion and constant diffusion

quasilinear parabolic and elliptic equations with nonlinear boundary conditions abstract this paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both dirichlet and nonlinear neumann boundary conditions the equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient the consideration of the class of equations is motivated by some heat transfer problems where the heat capacity and thermal conductivity are both temperature dependent the aim of the paper is to show the existence and uniqueness of a global time dependent solution of the parabolic problem existence of maximal and minimal steady state solutions of the elliptic problem including conditions for the uniqueness of a solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions applications are given to some heat transfer problems and an extended logistic reaction diffusion equation

continuous dependence estimate for conservation laws with levy noise abstract we are concerned with multidimensional stochastic balance laws driven by levy processes using bounded variation bv estimates for vanishing viscosity approximations we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that levy noise only depends on the solution this result is used to show the error estimate for the stochastic vanishing viscosity method in addition we establish fractional bv estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

finite difference reaction diffusion equations with nonlinear diffusion coefficients a monotone iterative method for numerical solutions of a class of finite difference reaction diffusion equations with nonlinear diffusion coefficient is presented it is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system it is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero

existence of positive solutions for elliptic systems degenerate and nondegenerate ecological models with homogeneous dirichlet conditions w o on dd the function s satisfies the conditions rl e c o oo q o and s o for s o problems of this nature are of interest in reaction diffusion processes in biology and chemistry for example the case for u urn m 1 or m e 0 1 for single parabolic equations i e u du f x u has been studied recently for porous medium analysis and population dynamics cf 2 7 15 18 191 as t cc these solutions tend to a solution of the corresponding elliptic scalar equation studies have also been carried out in these and other papers e g 3 4 16 171 with urn replaced by u satisfying the conditions described above in this article however the hypotheses on f are quite different from those in the papers mentioned above for example f is not necessarily lipschitz in u and may depend discontinuously on x thus even for the scalar case the results here are not covered by the other papers although there are some overlaps moreover our emphasis here is on degenerate systems section 3 and their applications section 4 these results are quite different and generalize the papers mentioned above to practical interesting cases other relevant results for the nondegenerate cases can be found in s 9 111 in section 2 we discuss some existence and uniqueness theorems for

well posedness of the cauchy problem for a fourth order thin film equation via regularization approaches abstract this paper is devoted to some aspects of well posedness of the cauchy problem the cp for short for a quasilinear degenerate fourth order parabolic thin film equation the tfe 4 0 1 u t u n u in r n r u x 0 u 0 x in r n where n 0 is a fixed exponent with bounded smooth compactly supported initial data dealing with the cp for at least n 0 3 2 requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces the main goal of the paper is to detect proper solutions of the cp for the degenerate tfe 4 by uniformly parabolic analytic e regularizations at least for values of the parameter n sufficiently close to 0 firstly we study an analytic homotopy approach based on a priori estimates for solutions of uniformly parabolic analytic e regularization problems of the form u t e u u in r n r where e u for e 0 1 is an analytic e regularization of the problem 0 1 such that 0 u u n and 1 u 1 using a more standard classic technique of passing to the limit in integral identities for weak solutions however this argument has been demonstrated to be non conclusive basically due to the lack of a complete optimal estimate regularity theory for these types of problems secondly to resolve that issue more successfully we study a more general similar analytic homotopy transformation in both the parameters as e 0 and n 0 and describe branching of solutions of the tfe 4 from the solutions of the notorious bi harmonic equation u t 2 u in r n r u x 0 u 0 x in r n which describes some qualitative oscillatory properties of cp solutions of 0 1 for small n 0 providing us with the uniqueness of solutions for the problem 0 1 when n is close to 0 finally riemann like problems occurring in a boundary layer construction that occur close to nodal sets of the solutions as e 0 are discussed in other to get uniqueness results for the tfe 4 0 1

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

convergence of a finite volume scheme for nonlinear degenerate parabolic equations one approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t rm div mathbf q f u delta phi u 0 by a piecewise constant function u mathcal d using a discretization mathcal d in space and time and a finite volume scheme the convergence of u mathcal d to u is shown as the size of the space and time steps tend to zero in a first step estimates on u mathcal d are used to prove the convergence up to a subsequence of u mathcal d to a measure valued entropy solution called here an entropy process solution a result of uniqueness of the entropy process solution is proved yielding the strong convergence of u mathcal d to it u some on a model equation are shown

global existence for nonlinear diffusion equations solutions of nonlinear possibly degenerate reaction diffusion models are known to exist for all time if the asymptotic growth of the reaction term is not greater than that of the diffusion term via concavity methods it is also known that negating such a condition results in solutions which blow up in finite time on the other hand two recent studies regarding the case of linear diffusion have reported on the ability for convective terms in a model to create global existence of solutions where no such result is true in their absence in this paper we develop a theory of global existence for a general class of reaction diffusion convection models which only requires a similar balance of diffusion and reaction our result extends previous work to models including convection but it does not reveal the exact role of convection in yielding global solutions further analysis of a slightly simplified model establishes the global existence of all solutions if the reaction term grows asymptotically at a rate less than that of either diffusion or convection