global holderian error bound for nondegenerate polynomials let f colon mathbb r n to mathbb r be a polynomial and s x in mathbb r n f x le 0 let f x max 0 f x if there exist c 0 alpha 0 beta 0 such that d x s le c f x alpha f x beta for all x in mathbb r n where d x s denotes the distance between x and s then we say that f has a global holderian error bound we prove that if f is convenient and nondegenerate with respect to its newton boundary at infinity in the sense of kouchnirenko invent math 32 1976 pp 1 32 then it has a global holderian error bound since the nondegenerate polynomials form a zariski open subset in a variety of polynomials with a given convenient newton boundary at infinity this result shows that global holderian error bounds hold for a large class of polynomials which can be relatively easy recognized

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

nonlinear stability of traveling waves to a hyperbolic parabolic system modeling chemotaxis we prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic parabolic system modeling repulsive chemotaxis in contrast to the previous related results where various sm

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

numerical methods for singular nonlinear volterra type equations with two point boundary conditions abstract the purpose of this article is to generalize the work in a paper by shaw and garey 1 to include equations with weakly singular kernels an example to illustrate the methods is given in section 4

variational analysis in nonreflexive spaces and applications to control problems with l 1 perturbations we provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is l 1 our underlying technique relies on a recent sequential description of both the generalized gradient of clarke and of the approximate g subdifferential of functions defined on a smooth banach space we also show that the proximal limit formula for the generalized gradient and its l p analogue are direct consequences of these sequential formulas related characterizations of lipschitzness of a function on a smooth space are given

on the fractional fisher information with applications to a hyperbolic parabolic system of chemotaxis we introduce new lower bounds for the fractional fisher information equipped with these bounds we study a hyperbolic parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes

global optimization of polynomials using gradient tentacles and sums of squares we consider the problem of computing the global infimum of a real polynomial f on mathbb r n every global minimizer of f lies on its gradient variety i e the algebraic subset of mathbb r n where the gradient of f vanishes if f attains a minimum on mathbb r n it is therefore equivalent to look for the greatest lower bound of f on its gradient variety nie demmel and sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds based on these certificates they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently we deal here with the more general case when f is bounded from below but does not necessarily attain a minimum in this case the method of nie demmel and sturmfels might yield completely wrong results in order to overcome this problem we replace the gradient variety by larger semialgebraic subsets of mathbb r n which we call gradient tentacles it now gets substantially harder to prove the existence of the necessary sums of squares certificates

ehresmann fibrations and palais smale conditions for morphisms of finsler manifolds the central result of this paper is a generalization of the ehresmann fibration theorem to the infinite dimensional and or non proper setting with this aim we introduce the concepts of strong submersion and of mapping with uniformly split kernels first applications include a global implicit function theorem and a necessary and sufficient version of hadamard s global invertibility criterion in the setting of finsler manifolds another major application makes use of the idea of asymptotic critical value not critical point which helps formulate various generalizations of the palais smale condition for morphisms of finsler manifolds not merely functionals we obtain a critical point theory for morphisms of finsler manifolds extending many results known only for functionals notably ekeland s variational principle these results are used to discuss a new approach to lagrange multiplier and nonlinear eigenvalue problems and to develop an intrinsic critical point theory for complex analytic functionals the latter reveals an intimate connection between conditions of palais smale type and the structure of polynomial automorphisms jacobian conjecture

brouwer s degree without properness a degree deg f y is defined for every continuous function f rn rn which possesses all the properties of brouwer s degree provided that y is restricted to the complement of some closed set a f of asymptotic values sufficient conditions are given for a f to be nowhere dense it is also shown that in the opposite direction a f having nonempty interior has a direct impact on the solutions of f x y which cannot be discovered by degree arguments

growth at infinity and index of polynomial maps abstract let f k n k n be a polynomial map such that f 1 0 is compact where k r or c then we give a condition implying that there is a uniform bound for the ojasiewicz exponent at infinity in certain deformations of f this fact gives a result about the invariance of the global index of f

the dirichlet to neumann map for a nonlinear diffusion convection equation a method for constructing the dirichlet to neumann map for a nonlinear diffusion convection equation is presented the problem is reduced to the solution of a nonlinear integral equation in one independent variable existence and uniqueness of the solution may be proven for small times via a contraction mapping technique

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

separation of algebraic sets and the łojasiewicz exponent of polynomial mappings we prove a sharp regular separation condition for arbitrary projective algebraic sets as a corollary an extension of kollar s results on the ojasiewicz exponent of polynomial mappings is obtained

injectivity of real polynomial maps and łojasiewicz exponents at infinity in this paper we give a lower bound for the ojasiewicz exponent at infinity of a special class of polynomial maps mathbb r n to mathbb r s s 1 as a consequence we detect a class of polynomial maps mathbb r n to mathbb r n that are global diffeomorphisms if their jacobian determinant never vanishes

on globally diffeomorphic polynomial maps via newton polytopes and circuit numbers in this article we analyze the global diffeomorphism property of polynomial maps f mathbb r n rightarrow mathbb r n by studying the properties of the newton polytopes at infinity corresponding to the sum of squares polynomials vert f vert 2 2 this allows us to identify a class of polynomial maps f for which their global diffeomorphism property on mathbb r n is equivalent to their jacobian determinant det jf vanishing nowhere on mathbb r n in other words we identify a class of polynomial maps for which the real jacobian conjecture which was proven to be false in general still holds

on the existence of pareto solutions for polynomial vector optimization problems we are interested in the existence of pareto solutions to the vector optimization problem begin aligned mathrm min mathbb r m f x x in mathbb r n end aligned where f mathbb r n rightarrow mathbb r m is a polynomial map by using the tangency variety of f we first construct a semi algebraic set of dimension at most m 1 containing the set of pareto values of the problem then we establish connections between the palais smale conditions m tameness and properness for the map f based on these results we provide some sufficient conditions for the existence of pareto solutions of the problem we also introduce a generic class of polynomial vector optimization problems having at least one pareto solution

stability analysis of product t methods for abel integral equations of the second kind stability analysis of product methods for abel integral equations of the second kind based on the test equation y t 1 lambda int limits 0 1 frac y s t s alpha ds t geqq 0 0 0 and we investigate whether this property is inherited by numerical approximations toy

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity abstract this paper establishes the nonlinear stability of traveling wave solutions to a chemotaxis model with singular or logarithmic sensitivity and its transformed parabolic hyperbolic system depending on the parameter signs we discuss the linear instability of traveling wave solutions using the spectral analysis and nonlinear asymptotic stability of traveling wave solutions with zero end state by the weighted energy estimates where the latter result solves the open question left in a previous work li and wang 2009 7

vector valued perturbed equilibrium problems the goal of this paper is to prove some general vector valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi continuity we obtain these results by going through a new concept of approximative equilibrium point

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

superconvergence of collocation methods for volterra and abel integral equations of the second kind this paper deals with the question of the attainable order of convergence in the numerical solution of volterra and abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals the use of a nonlinear variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

a rational basis for second kind abel integral equations abstract a rational basis set is constructed using 1 1 rational interpolation the basis is used in a product integration method for solving second kind abel integral equations criteria are established for determining the amount of bias to be used in the basis set when the integral equation has a nonsmooth solution several test problems are solved to illustrate the performance of the approximation method

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

nonlinear fredholm operators with noncompact fibers and applications to elliptic problems on rn under suitable conditions an equation f x y between banach spaces involving a nonlinear fredholm mapping f of nonnegative index is shown to have a noncompact and hence infinite set of solutions for almost every y for which the equation is solvable the proof of this nonuniqueness but not existence result relies on an entirely new line of arguments in which the concept of generalized critical value plays a central role when f w2 p rn lp rn is associated with a quasilinear elliptic pde on rn with constant coefficient it often happens that a crucial denseness hypothesis in the abstract theorem is equivalent to the existence of a nontrivial solution to the equation f x f 0 and hence can be verified in practice generalizations exist for some classes of problems with nonconstant coefficients and for problems on exterior domains

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis abstract the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper by a cole hopf type transformation the singular chemotaxis is converted into a non singular hyperbolic system then we study the transformed system and establish the global existence asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates the main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

a global smooth version of the classical łojasiewicz inequality abstract let f r n r be a function of class c d d 1 such that d f x 1 d 0 on r n then the following global ojasiewicz inequality holds true c d dist x f 0 f x 1 0 d f x for all x r n where c d d 2 2 d 1 and dist x a denotes the euclidean distance from x to a as applications of this inequality we have the following statements if the sets f 0 and f x 1 0 are non asymptotic at infinity then there exist positive constants e and r such that c d dist x f 0 d f x whenever dist x f 0 e and x r if f is a polynomial of degree d with an isolated critical point at the origin the following effective ojasiewicz inequality holds true c dist x f 0 d 2 d 3 n 1 2 f x for all x r for some c 0 and r 0 finally we establish a relation between the above global ojasiewicz inequality and the phenomenon of singularities at infinity

global solutions for a hyperbolic parabolic system of chemotaxis abstract we study a hyperbolic parabolic model of chemotaxis related to tumor angiogenesis in dimensions one and two we consider diffusions given by the fractional laplacian and in particular we prove the global existence of classical solutions in certain dissipation regimes

global existence and asymptotic stability of the fractional chemotaxis fluid system in r3 abstract in this paper we consider a model arising from biology consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in r 3 we establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument moreover the decay rates of the solutions and their higher order spatial derivatives are established towards the equilibrium by introducing the negative sobolev space h s 0 s 3 2

on the limit set at infinity of a gradient trajectory of a semialgebraic function given any c2 semialgebraic function f defined on a non bounded open set of rn we prove that the limit of the secants at infinity of a non bounded trajectory of the gradient of f does exist as a corollary we find a new sufficient condition to ensure the trivialisation by the gradient flow of f nearby a regular asymptotic critical value at infinity

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence

on smooth variational principles in banach spaces abstract a new smooth variational principle for spaces admitting frechet differentiable bump functions is proved further it is shown that each proper lower semicontinuous bounded below function can be supported by a smooth function with locally holder derivative if and only if the space is superreflexive some geometrical refinements of the borwein preiss smooth variational principle using deville s techniques are obtained

initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis abstract this paper is devoted to the analytical study of initial boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity in the first part of the paper we show that subject to the dirichlet boundary conditions classical solutions exist globally in time for large initial data asymptotically in time the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity numerical simulations are supplied to corroborate the analytical results the analytic approach developed herein can be utilized to handle a family of initial and initial boundary value problems of the model and related models with similar mathematical structure as a demonstration of the effectiveness of our approach in the second part of the paper we show that subject to the neumann dirichlet boundary conditions classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient this improves a previous result obtained in 30 where the smallness of the chemical diffusion coefficient was required

detecting bifurcation values at infinity of real polynomials we present a new approach for estimating the set of bifurcation values at infinity this yields a significant shrinking of the number of coefficients in the recent algorithm introduced by jelonek and kurdyka for reaching critical values at infinity by rational arcs

the discriminant of a system of equations abstract we introduce and study the discriminant of a system of polynomial equations with indeterminate coefficients generalizing a number of known special cases such as the sparse resultant and the a determinant and inheriting their nice properties one such property is the fact that the discriminant is always a hypersurface in the space of systems of equations which seems unexpected when compared to the existence of dual defect toric varieties this fact reflects the combinatorics of a certain partial order relation on the set of tuples of faces of a tuple of polytopes the newton polytopes of the equations in our case generalizing the poset of faces of one polytope this combinatorics is the main challenge in our work and can be reduced to a certain fact of tropical geometry which resembles the geometry of the discriminants that we study as a sample application of our version of the multivariate discriminant we prove that atypical fibers of a generic polynomial map are distinguished by their euler characteristics and the bifurcation set of such a map is a hypersurface whose degree we explicitly compute

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

a sard theorem for tame set valued mappings abstract if f is a set valued mapping from r n into r m with closed graph then y r m is a critical value of f if for some x with y f x f is not metrically regular at x y we prove that the set of critical values of a set valued mapping whose graph is a definable tame set in an o minimal structure containing additions and multiplications is a set of dimension not greater than m 1 respectively a porous set as a corollary of this result we get that the collection of asymptotically critical values of a set valued mapping with a semialgebraic graph has dimension not greater than m 1 we also give an independent proof of the fact that a definable continuous real valued function is constant on components of the set of its subdifferentiably critical points

convergence analysis of an augmented lagrangian method for mathematical programs with complementarity constraints abstract in this paper a mathematical program with complementarity constraints mpcc is reformulated as a nonsmooth constrained optimization problem by using the fischer burmeister function an augmented proximal lagrangian method is applied to tackle the resulting constrained optimization problem the augmented lagrangian problems are in general nonsmooth we derive first and second order optimality conditions for the augmented lagrangian problems using an approximate smooth variational principle and establish that the limit point of a sequence of points that satisfy the second order necessary optimality conditions of the augmented lagrangian problems is a strongly stationary point of the original mpcc if the limit point is feasible to mpcc and the linear independence constraint qualification for mpcc and the upper level strict complementarity condition hold at the limit point

proximal normal cone analysis on smooth banach spaces and applications unlike the frechet penot clarke and other order one notions of normal cones the notion of proximal normal cone describes variational behavior of order two and most of results on proximal normal cone are established in the hilbert space framework a possible reason is that in a general banach space such as in the classical banach spaces l p and l p with 1 leq p 2 the proximal normal cone of a smooth subset a at every point of a may consist of the zero element only nevertheless on the other hand we show in this paper that for a fairly big class of smooth banach spaces including the classical banach spaces l p and l p with 2 leq p infty the consideration of proximal normal cones provides a lot of useful information especially for leading to satisfactory treatment for constrained optimization problems some of our results are new even in the hilbert space case

on accurate product integration rules for linear fractional differential equations this paper addresses the numerical solution of linear fractional differential equations with a forcing term competitive and highly accurate product integration rules are derived by starting from an equivalent formulation in terms of a volterra integral equation with a generalized mittag leffler function in the kernel the error analysis is reported and aspects related to the computational complexity are treated numerical tests confirming the theoretical findings are presented

semialgebraic sard theorem for generalized critical values we prove that a semialgebraic differentiable mapping has a generalized critical values set of measure zero moreover if the mapping is c 2 we obtain bya generalisation of ehresmann s fibration theorem due to p j rabier 20 a locallytrivial fibration over the complement of this set in the complex case we prove that the set of generalized critical values of a polynomial mapping is a proper algebraic set

reconstruction of the clarke subdifferential by the lasry lions regularizations abstract we prove that the clarke subdifferential of a locally lipschitz function with a growth condition defined on a hilbert space can be represented by the derivatives of its lasry lions regularizations we complete a result of j benoist showing a similar representation for the clarke subdifferential of a lipschitz function in an arbitrary banach space by the clarke subdifferentials of its lasry lions regularizations

global łojasiewicz type inequality for non degenerate polynomial maps abstract let f f 1 f p r n r p be a polynomial map this paper studies the existence of the following global ojasiewicz type inequality f x f x c d x f 1 0 for all x r n for some constants c 0 0 and 0 we show that the above inequality holds if one of the following conditions is satisfied i f is convenient and khovanskii non degenerate at infinity ii f is convenient and non degenerate at infinity iii f is mikhailov gindikin non degenerate further in cases ii and iii the exponents and can be determined explicitly

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

global dynamics of a hyperbolic parabolic model arising from chemotaxis we prove global existence and qualitative behavior of classical solutions for a hyperbolic parabolic system describing chemotaxis on bounded domains it is shown that classical solutions to the initial boundary value problem of the one dimensional model exist globally in time for large initial data and the solutions converge to constant equilibrium states exponentially in time which rigorously demonstrates the collapsing of cell populations in chemotaxis moreover similar results are established for the multidimensional model when the initial data are small

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

łojasiewicz inequality at infinity for polynomials in two real variables we propose different types of ojasiewicz inequality at infinity for polynomials in two real variables the formulas for the ojasiewicz exponents are given

asymptotic profile of a parabolic hyperbolic system with boundary effect arising from tumor angiogenesis this paper concerns a parabolic hyperbolic system on the half space r with boundary effect the system is derived from a singular chemotaxis model describing the initiation of tumor angiogenesis we show that the solution of the system subject to appropriate boundary conditions converges to a traveling wave profile as time tends to infinity if the initial data is a small perturbation around the wave which is shifted far away from the boundary but its amplitude can be arbitrarily large

blow up criterion for a hyperbolic parabolic system arising from chemotaxis abstract in this paper we investigate a hyperbolic parabolic system arising from chemotaxis a blow up criterion of local strong solutions to the model in r d is established based on the scaling invariance of the system as a by product we prove a global in time well posedness result when d 1 for large initial data

a nonsmooth morse sard theorem for subanalytic functions abstract according to the morse sard theorem any sufficiently smooth function on a euclidean space remains constant along any arc of critical points we prove here a theorem of morse sard type suitable as a tool in variational analysis we broaden the definition of a critical point to the standard notion in nonsmooth optimization while we restrict the functions under consideration to be semialgebraic or subanalytic we make no assumption of subdifferential regularity ojasiewicz type inequalities for nonsmooth functions follow quickly from tools of the kind we develop leading to convergence theory for subgradient dynamical systems

testing sets for properness of polynomial mappings in this paper we characterize testing sets for properness of polynomial mappings f mathbb c n rightarrow mathbb c m we also study the set of points at which such mappings are not proper as the first application we give a proof of the complementary conjecture of mckay and wang conjecture 12 in 16 the second application is an answer to the problem of kraft russell about a characterization of mathbb c n the third application is given in 13 a solution of the problem of van den essen and shpilrain about endomorphism of the ring mathbb c x 1 x n the fourth application is a theorem about extensions of affine varieties

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

a family of methods for abel integral equations of the second kind abstract a class of methods depending on some parameters are introduced for the numerical solution of the abel integral equations of the second kind some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals

second order sufficient conditions for error bounds in banach spaces recently huang and ng presented second order sufficient conditions for error bounds of continuous and ga teaux differentiable functions in banach spaces wu and ye dropped the assumption of huang and ng on ga teaux differentiability but required the space to be a hilbert space we carry on this research in two directions first we extend wu and ye s result to some non hilbert spaces second same as huang and ng we work on banach spaces but provide different second order sufficient conditions that may allow the function to be non ga teaux differentiable

collocation methods for second kind volterra integral equations with weakly singular kernels in this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of volterra integral equations vies are smooth if the exact solutions are not smooth however suitable transformations can be made so that the new vies possess smooth solutions spline collocation methods with uniform meshes applied to these new vies are then shown to be able to yield optimal global convergence rates the general theory is applied to a typical case i e the integral kernels consisting of the singular term t s

smooth approximation of convex functions in banach spaces abstract this paper shows that every extended real valued lower semi continuous proper respectively lipschitzian convex function defined on an asplund space can be represented as the point wise limit respectively uniform limit on every bounded set of a sequence of lipschitzian convex functions which are locally affine hence c at all points of a dense open subset and shows an analogous for w lower semi continuous proper respectively lipschitzian convex functions defined on dual spaces whose pre duals have the radon nikodym property

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

tame nonsmooth inverse mapping theorems we give several versions of local and global inverse mapping theorems for tame nonnecessarily smooth mappings here tame mapping means a continuous mapping which is locally definable in some o minimal structure our sufficient conditions are formulated in terms of various properties convexity positivity of some principal minors of the space of jacobian matrices at smooth points

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method