on the asymptotic behavior for a certain nonlinear evolution equation let x be a real banach space we shall say that an operator a with domain d a and range r a in x is accretive if lx x2 11 0 and yi e axi i 1 2 and that a satisfies the range condition if r z i 1 d a the closure of d a for all a 0 where 1 denotes the identity let g 0 co 0 co be a nonincreasing continuous function such that lim r13 g t 0 let x e x and consider the following initial value problem

introduction to the concept of recursiveness of fuzzy functions abstract the main aim of this paper is to extend church s soft thesis to fuzzy calculus let w 0 1 be an ordered semiring the fuzzy functions are considered as associating an output to an input with membership degree belonging to w for such w functions the concepts of crisp computability and recursiveness are extended obtaining the so called w computability and w recursiveness respectively a few properties of w recursive functions are analized and the equivalence between w recursive and w computable functions is shown

on the gaussian measure of the intersection the gaussian correlation conjecture states that for any two symmetric convex sets in n dimensional space and for any centered gaussian measure on that space the measure of the intersection is greater than or equal to the product of the measures in this paper we obtain several results which substantiate this conjecture for example in the standard gaussian case we show there is a positive constant c such that the conjecture is true if the two sets are in the euclidean ball of radius c n further we show that if for every n the conjecture is true when the sets are in the euclidean ball of radius n then it is true in general our most concrete result is that the conjecture is true if the two sets are arbitrary centered ellipsoids

on diffusion semigroups preserving the log concavity it is proved that if the semigroup with the generator akl xk xl m xm where akl and m are smooth functions sends log concave functions to log concave functions then a akl is constant and m is an affine mapping

isoperimetric and analytic inequalities for log concave probability measures we discuss an approach based on the brunn minkowski inequality to isoperimetric and analytic inequalities for probability measures on euclidean space with logarithmically concave densities in particular we show that such measures have positive isoperimetric constants in the sense of cheeger and thus always share poincare type inequalities we then describe those log concave measures which satisfy isoperimetric inequalities of gaussian type the results are precised in dimension 1

asymptotic behavior of solutions to nonlinear volterra integral equations in section 2 we establish the existence of solutions to 1 1 and to related equations theorems 2 1 and 2 3 in contrast with previous papers e g 7 we do not assume that a is m accretive instead we require a to satisfy a weaker cone condition certain other facts are also included in this section throughout this paper we also assume that b is completely positive 6 in the third section we present two mean ergodic theorems for the solutions of 1 1 in hilbert space the first theorem theorem 3 1 deals with weak almost convergence and the second theorem 3 4 with strong almost convergence these results improve upon those of 1 and include the known results for nonlinear semigroups 191 it is important to establish almost convergence because it then follows that the solution is summed by every strongly regular kernel corollaries 3 3 and 3 5 a weak convergence criterion corollary 3 2 is also obtained if the banach space x is reflexive and strictly convex and 0 e r a then

a particular case of correlation inequality for the gaussian measure our purpose is to prove a particular case of a conjecture concerning the gaussian measure of the intersection of two symmetric convex sets of r n this conjecture states that the measure of the intersection is greater or equal to the product of the measures in this paper we prove the inequality when one of the two convex sets is a symmetric ellipsoid and the other one is simply symmetric the general case is still open

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

isoperimetric constants for product probability measures a dimension free lower bound is found for isoperimetric constants of product probability measures from this some analytic inequalities are derived

entropy jumps in the presence of a spectral gap it is shown that if x is a random variable whose density satisfies a poincare inequality and y is an independent copy of x then the entropy of x y p 2 is greater than that of x by a fixed fraction of the entropy gap between x and the gaussian of the same variance the argument uses a new formula for the fisher information of a marginal which can be viewed as a local reverse form of the brunn minkowski inequality in its functional form due to prekopa and leindler

on the asymptotic behavior of nonlinear semigroups and the range of accretive operators ii abstract it is known that certain problems in partial differential equations may be interpreted as initial value problems for ordinary differential equations in banach spaces when such an evolution equation is governed by an accretive operator then its solutions give rise to a nonlinear contraction semigroup in this paper we study certain aspects of the asymptotic behavior of nonlinear semigroups and or resolvents of accretive operators we also derive new results on their behavior at the origin it turns out that the behavior of a nonlinear semigroup resembles that of the resolvent of its generator both at infinity and at the origin author

extremal properties of half spaces for log concave distributions n w x n where m s m m is the product measure in r d s y1 1 is the n n dimensional cube in r n and the infimum is then over all borel measurable n n sets a r of measure m a g p 0 p 1 h 0 we are searching for necessary and sufficent conditions under which the n 4 infimum in 1 1 is attained at the half spaces a s x g r x f c of 1 measure p for all p and h the half spaces of this form will be called standard in these notes we give such conditions in the case where m is log concave there exists the following probabilistic consequence of the above property if z z are independent random variables on some probability space 1 n v i p with common distribution m consider a family of random variables s t indexed by a set t and formed by linear combinations n s t s a t z i i is1 where the coefficients a are arbitrary functions on t such that i n s s sup a t f 1 1 i t is1 set m s sup s t then the extremal property of the standard half spaces in t

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

bounds on variance for unimodal distributions we show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy power combining this bound with the well known entropy power lower bound on variance we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the scaled entropy power as the differential entropy decreases the variance is sandwiched between two exponentially decreasing functions in the differential entropy this establishes that for the subclasses of unimodal distributions the differential entropy can be used as a surrogate for concentration of the distribution

analytic inequalities isoperimetric inequalities and logarithmic sobolev inequalities abstract there is a simple equivalence between isoperimetric inequalities in riemannian manifolds and certain analytic inequalities on the same manifold more extensive than the familiar equivalence of the classical isoperimetric inequality in euclidean space and the associated sobolev inequality by an isoperimetric inequality in this connection we mean any inequality involving the riemannian volume and riemannian surface measure of a subset and its boundary respectively we exploit the equivalence to give log sobolev inequalities for riemannian manifolds some applications to schrodinger equations are also given

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

on the unbounded behavior for some non autonomous systems in banach spaces abstract by modifying our previous methods 1992 j nonlinear anal tma 19 741 751 1993 proc amer math soc 117 951 956 and by using the notion of integral solution introduced by ph benilan 1972 equations d evolution dans un espace de banach quelconque et applications thesis universite paris xi orsay we study the asymptotic behaviour of unbounded trajectories for the quasi autonomous dissipative system du dt au where x is a real banach space a an accretive possibly multivalued operator in x x and l p 0 x for some x and 1 p

a functional form of the isoperimetric inequality for the gaussian measure abstract letgbe a smooth function on r nwith values in 0 1 using the isoperimetric property of the gaussian measure itis proved that 1 eg e 1 g e g conversely this inequality implies the isoperimetric property of the gaussian measure

automata theory based on complete residuated lattice valued logic turing machines automata theory based on complete residuated lattice valued logic called l valued finite automata l vfas has been established by the second author in 2001 in view of the importance of turing machines in this paper we establish a theory of turing machines based on complete residuated lattice valued logic which is a continuation of l vfas first we give the definition of l valued nondeterministic turing machines l ntms and observe that the multitape l ntms have the same language recognizing power as the single tape l ntms we give some related properties of l valued turing machines and discuss computing with fuzzy letters via l valued turing machines second we introduce the concepts of l valued recursively enumerable languages and l valued recursive languages and obtain some equivalent relations some results concerning the characterization of n recursively enumerable sets are given and the super computing power of l valued turing machines is investigated we also prove that l valued deterministic turing machines and l ntms are not equivalent in the sense of recognizing or deciding languages finally we show that there is no universal l valued turing machine however a universal l valued turing machine exists if the membership degrees of l valued sets are restricted to a finite complete residuated lattice with universal bounds 0 and 1

a new approach to fuzzy programming abstract a fuzzy program is defined in the usual way as a sequence of statements instruction which are considered as functions possibly fuzzy functions and fuzzy predicates defined on the given input domain the essential difference in the approach presented in this paper is the new interpretation of the execution of fuzzy programs and a new method of evaluating fuzzy predicates the result of the fuzzy program execution is an appropriate fuzzy subset in the output domain

fractional smoothness of images of logarithmically concave measures under polynomials abstract we show that a measure on the real line that is the image of a log concave measure under a polynomial of degree d possesses a density from the nikolskii besov class of fractional order 1 d this result is used to prove an estimate for the total variation distance between such measures in terms of the fortet mourier distance

the asymptotics of nonexpansive iterations abstract let d be a closed subset of a banach space x and t d d a nonexpansive mapping conditions are given on the space x for t to satisfy the following property of ergodic type t n x n converges either weakly or strongly to a vector v rather unexpectedly d is not assumed to be convex nor is i t assumed to satisfy any range condition in addition it is shown that v is the unique point of least norm in the closure of r i t if and only if i t satisfies a certain range condition at infinity several interesting applications to accretive operator and nonlinear semigroup theory are also included

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

a generalization of caffarelli s contraction theorem via reverse heat flow a theorem of l caffarelli implies the existence of a map pushing forward a source gaussian measure to a target measure which is more log concave than the source one which contracts euclidean distance in fact caffarelli showed that the optimal transport brenier map topt is a contraction in this case we generalize this result to more general source and target measures using a condition on the third derivative of the potential by providing two different proofs the first uses a map t whose inverse is constructed as a flow along an advection field associated to an appropriate heat diffusion process the contraction property is then reduced to showing that log concavity is preserved along the corresponding diffusion semi group by using a maximum principle for parabolic pde in particular caffarelli s original result immediately follows by using the ornstein uhlenbeck process and the prekopa leindler theorem the second uses the map topt by generalizing caffarelli s argument employing in addition further results of caffarelli as applications we obtain new correlation and isoperimetric inequalities

the american put is log concave in the log price we show that the american put option price is log concave as a function of the log price of the underlying asset thus the elasticity of the price decreases with increasing stock value we also consider related contracts of american type and we provide an example showing that not all american option prices are log concave in the stock log price

asymptotic properties of some non autonomous systems in banach spaces abstract let x be a reflexive banach space we introduce the notion of weakly almost nonexpansive sequences x n n 0 in x and study their asymptotic behavior by showing that the nonempty weak limit set of the sequence x n n n 1 always lies on a convex subset of a sphere centered at the origin of radius d lim n x n n subsequently we apply our results to study the asymptotic properties of unbounded trajectories for the quasi autonomous dissipative system d u d t a u f where a is an accretive possibly multivalued operator in x x and f f l p 0 x for some f x and 1 p these results extend recent results of j s jung and j s park j s jung j s park asymptotic behavior of nonexpansive sequences and mean points proc amer math soc 124 1996 475 480 and j s jung j s park and e h park j s jung j s park e h park asymptotic behaviour of generalized almost nonexpansive sequences and applications proc nonlinear funct anal 1 1996 65 79 as well as our results cited below containing previous results by several authors

existence and asymptotic properties of solutions of nonlinear multivalued differential inclusions with nonlocal conditions abstract this paper deals with a nonlocal problem governed by the nonlinear differential inclusions with multivalued perturbations in banach spaces first using the approach of geometry of banach space hausdorff metric the measure of noncompactness and fixed point existence results are obtained we have removed from previous papers the crucial restriction on the semigroup then we establish the asymptotic properties of integral solutions when t finally for illustration a partial differential equation is worked out

asymptotic properties of nonexpansive iterations in reflexive spaces abstract let x be a reflexive banach space and xn n 0 a nonexpansive resp firmly nonexpansive sequence in x it is shown that the set of weak limit points of the sequence xn n n 1 resp xn 1 xn n 0 always lies on a convex subset of a sphere centered at the origin of radius d limn xn n this fact quickly yields previous results of b djafari rouhani as well as recent results of j s jung and j s park potential applications are also discussed