order statistics and concentration of lr norms for log concave vectors abstract we establish upper bounds for tails of order statistics of isotropic log concave vectors and apply them to derive a concentration of l r norms of such vectors

cone volume measure of general centered convex bodies abstract we show that the cone volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition this extends former results obtained in the discrete as well as in the symmetric case and implies among others a conjectured best possible inequality for the u functional of a convex body

transference principles for log sobolev and spectral gap with applications to conservative spin systems we obtain new principles for transferring log sobolev and spectral gap inequalities from a source metric measure space to a target one when the curvature of the target space is bounded from below as our main application we obtain explicit estimates for the log sobolev and spectral gap constants of various conservative spin system models consisting of non interacting and weakly interacting particles constrained to conserve the mean spin when the self interaction is a perturbation of a strongly convex potential this partially recovers and partially extends previous results of caputo chafai grunewald landim lu menz otto panizo villani westdickenberg and yau when the self interaction is only assumed to be non strongly convex as in the case of the two sided exponential measure we obtain sharp estimates on the system s spectral gap as a function of the mean spin independently of the size of the system

some kinds of view the fuzzy filters of bl algebras the concepts of view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras are introduced and some related properties are investigated some characterizations of these generalized fuzzy filters are derived in particular we describe the relationships among ordinary fuzzy implicative positive implicative and fantastic filters view the mathml source fuzzy implicative positive implicative and fantastic filters and view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras finally we prove that a fuzzy set f of a bl algebra l is an view the mathml source fuzzy implicative filter of l if and only if it is both an view the mathml source fuzzy positive implicative filter and an view the mathml source fuzzy fantastic filter

quantitative stability of the entropy power inequality we establish quantitative stability results for the entropy power inequality epi specifically we show that if uniformly log concave densities nearly saturate the epi then they must be close to gaussian densities in the quadratic kantorovich wasserstein distance furthermore if one of the densities is gaussian and the other is log concave or more generally has positive spectral gap then the deficit in the epi can be controlled in terms of the l 1 kantorovich wasserstein distance or relative entropy respectively as a counterpoint an example shows that the epi can be unstable with respect to the quadratic kantorovich wasserstein distance when densities are uniformly log concave on sets of measure arbitrarily close to one our stability results can be extended to non log concave densities provided certain regularity conditions are met the proofs are based on mass transportation

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

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

new types of fuzzy filters of bl algebras this paper introduces the concepts of a b fuzzy implicative positive implicative fantastic filters and b a fuzzy implicative positive implicative fantastic filters of bl algebras where a b c q d c q d c q d a b c q d c q d c q d a c q d and b c q d and some related properties are investigated special attention is paid to c c q d fuzzy implicative positive implicative fantastic filters and c c q d fuzzy implicative positive implicative fantastic filters which are generalizations of q fuzzy implicative positive implicative fantastic filters and q fuzzy implicative positive implicative fantastic filters studied in ma et al 2008 2009 34 35 it also points out that examples 3 1 2 3 1 9 3 2 2 3 2 9 and 3 3 9 in ma et al 2009 35 are incorrect

pull backs of algebras and crossed products by certain diagonal actions let g be a locally compact group and p q t a principal g bundle if a is a c algebra with primitive ideal space t the pull back p a of a along p is the balanced tensor product co q c t a if 3 g aut a consists of c t module automorphisms and y g aut co q is the natural action then the automorphism group y 0x 3 of co q 0 a respects the balancing and induces the diagonal action p 3 of g on p a we discuss some examples of such actions and study the crossed product p a x p g we suggest a substitute d for the fixed point algebra prove p a x g is strongly morita equivalent to d and investigate the structure of d in various cases in particular we ask when d is strongly morita equivalent to a sometimes but by no means always and investigate the case where a has continuous trace let b be a c algebra and g a locally compact group acting on b as a strongly continuous automorphism group a our goal here is to study the crossed product c algebra b x g for two classes of diagonal actions for which the induced action of g on b is free the first class includes actions of the form y 0x3 on b co q 0 a where p q t is a principal g bundle y is the dual action of g on co q and 3 g aut a is an action of g on another c algebra a we also consider diagonal actions on algebras which are the pull backs of another algebra a along a principal bundle p q t if a is a c algebra with primitive ideal space t then the pull back p a is the balanced tensor product co q cb t a when 3 g aut a consists of c t module automorphisms the product action ty 0 3 preserves the balancing and the diagonal action p 3 is by definition the induced action on p a in general if f x y and q prima y are continuous then cb y acts on co x by composition with f and on a by composition with q and the daunshofmann theorem we can therefore define the pull back f a of a along f as the c algebraic tensor product co x cb y a the reason for the name is that when a is the algebra of sections of some c bundle e over y there is a natural isomorphism of f a onto the algebra of sections of the pull back f e in 1 we discuss this and other basic properties of pull backs and give some evidence to show they are likely to be of interest in particular we show that if g is abelian and a g aut a is locally unitary in the sense of 18 then the crossed product received by the editors october 20 1983 and in revised form february 15 1984 1980 mathematics subject classification primary 46l40 46l55

the pseudo linear semantics of interval valued fuzzy logics triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals which are called interval valued residuated lattices ivrls triangle algebras have been used to construct triangle logic tl a formal fuzzy logic that is sound and complete w r t the class of ivrls in this paper we prove that the so called pseudo prelinear triangle algebras are subdirect products of pseudo linear triangle algebras this can be compared with mtl algebras prelinear residuated lattices being subdirect products of linear residuated lattices as a consequence we are able to prove the pseudo chain completeness of pseudo linear triangle logic ptl an axiomatic extension of tl introduced in this paper this kind of completeness is the analogue of the chain completeness of monoidal t norm based logic mtl this result also provides a better insight in the structure of triangle algebras it enables us amongst others to prove properties of pseudo prelinear triangle algebras more easily it is known that there is a one to one correspondence between triangle algebras and couples l a in which l is a residuated lattice and a an element in that residuated lattice we give a schematic overview of some properties of pseudo prelinear triangle algebras and a number of others that can be imposed on a triangle algebra and the according necessary and sufficient conditions on l and a

the radical of a perfect residuated structure in this paper we extend some properties of the radical of an mtl algebra to the non commutative case of a more general residuated structure namely the fl w algebra for the particular case of pseudo mtl algebras some specific results are presented we introduce the notion of a local additive measure on a perfect pseudo mtl algebra and we prove that with some additional conditions every local additive measure can be extended to a riecan state a necessary and sufficient condition is given for such an extension to be a bosbach state

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

on generic chaining and the smallest singular value of random matrices with heavy tails we present a very general chaining method which allows one to control the supremum of the empirical process suph h n 1 i 1nh2 xi eh2 in rather general situations we use this method to establish two main results first a quantitative non asymptotic version of the celebrated bai yin theorem on the singular values of a random matrix with i i d entries that have heavy tails and second a sharp estimate on the quadratic empirical process when h t t t t rn and is an isotropic unconditional log concave measure

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

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

extremal slabs in the cube and the laplace transform abstract we study the volume of symmetric slabs in the unit cube we show that for t 3 4 the slab parallel to a face has the minimal volume among all symmetric slabs with width t for large width we prove the asymptotic extremality of the slab orthogonal to the main diagonal the proof is based on certain concavity properties of the laplace transform and on several limit theorems from probability the central limit theorem and classical principles of moderate and large deviations finally we extend some of the results to more general classes of bodies

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

a gradient estimate on a manifold with convex boundary we present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact riemannian manifold with convex boundary or on a noncompact complete manifold our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation and an abstract gradient estimate for a markov semigroup

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

a berry esseen type inequality for convex bodies with an unconditional basis suppose x x1 xn is a random vector distributed uniformly in a convex body k subset mathbb r n we assume the normalization mathbb e x i 2 1 for i 1 n the body k is further required to be invariant under coordinate reflections that is we assume that x1 xn has the same distribution as x1 xn for any choice of signs then we show that mathbb e left x sqrt n right 2 leq c 2 where c 4 is a positive universal constant and is the standard euclidean norm in mathbb r n the estimate is tight up to the value of the constant it leads to a berry esseen type bound in the central limit theorem for unconditional convex bodies

power law estimates for the central limit theorem for convex sets abstract we investigate the rate of convergence in the central limit theorem for convex sets established in b klartag a central limit theorem for convex sets invent math in press 8 we obtain bounds with a power law dependence on the dimension these bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets

filters of residuated lattices and triangle algebras an important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic is that of a filter filters can be used amongst others to define congruence relations specific kinds of filters include boolean filters and prime filters in this paper we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections triangle algebras characterize interval valued residuated lattices

an equivariant brauer semigroup and the symmetric imprimitivity theorem suppose that x g is a second countable locally compact transformation group we let sg x denote the set of morita equivalence classes of separable dynamical systems a g where a is a c0 x algebra and is compatible with the given g action on x we prove that sg x is a commutative semigroup with identity with respect to the binary operation a g b g a x b g x for an appropriately defined balanced tensor product on c0 x algebras if g and h act freely and properly on the left and right of a space x then we prove that sg x h and sh g x are isomorphic as semigroups if the isomorphism maps the class of a g to the class of b h then a o g is morita equivalent to b o h

crossed products by actions which are locally unitary on the stabilisers let a g be a c dynamical system with g abelian and hausdorff we investigate the ideal structure of the crossed product a g under the hypothesis that the stabiliser subgroups for the action of g on vary continuously we discuss a new notion of locally trivial g space for such actions and dually actions which are locally unitarily implemented on the stabiliser groups our main result asserts that when is locally unitary in this sense and is a locally trivial g space a g is a locally trivial space

isotropic constants and mahler volumes abstract this paper contains a number of results related to volumes of projective perturbations of convex bodies and the laplace transform on convex cones first it is shown that a sharp version of bourgain s slicing conjecture implies the mahler conjecture for convex bodies that are not necessarily centrally symmetric second we find that by slightly translating the polar of a centered convex body we may obtain another body with a bounded isotropic constant third we provide a counter example to a conjecture by kuperberg on the distribution of volume in a body and in its polar

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

concentration inequalities for non lipschitz functions with bounded derivatives of higher order building on the inequalities for homogeneous tetrahedral polynomials in independent gaussian variables due to r latala we provide a concentration inequality for not necessarily lipschitz functions f mathbb r n rightarrow mathbb r with bounded derivatives of higher orders which holds when the underlying measure satisfies a family of sobolev type inequalities begin aligned vert g mathbb eg vert p le c p vert nabla g vert p end aligned such sobolev type inequalities hold e g if the underlying measure satisfies the log sobolev inequality in which case c p le c sqrt p or the poincare inequality then c p le cp our concentration estimates are expressed in terms of tensor product norms of the derivatives of f when the underlying measure is gaussian and f is a polynomial not necessarily tetrahedral or homogeneous our estimates can be reversed up to a constant depending only on the degree of the polynomial we also show that for polynomial functions analogous estimates hold for arbitrary random vectors with independent sub gaussian coordinates we apply our inequalities to general additive functionals of random vectors in particular linear eigenvalue statistics of random matrices and the problem of counting cycles of fixed length in erd s renyi random graphs obtaining new estimates optimal in a certain range of parameters

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

fuzzy implicative and boolean ideals of mv algebras abstract we show how to obtain results in mv algebras by considering their fuzzy ideals in particular we prove the following theorem let be a fuzzy ideal of an mv algebra x then the folllowing are equivalent 1 x x 0 for all x x 2 x n x for all x x and all n 1 3 x x is a boolean algebra 4 is fuzzy implicative if i is an ideal of x and we take i then we obtain corresponding results for the ideal i and x we show how to extend fuzzy prime and fuzzy implicative ideals various fuzzy radicals are considered and related to the corresponding radicals of the algebra and we show how they live in the spectrum of the algebra

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

amenability of groupoids arising from partial semigroup actions and topological higher rank graphs we consider the amenability of groupoids g equipped with a group valued cocycle c g q with amenable kernel c 1 e we prove a general result which implies in particular that g is amenable whenever q is amenable and if there is countable set d g such that c g u d q for all u g 0 we show that our result is applicable to groupoids arising from partial semigroup actions we explore these actions in detail and show that these groupoids include those arising from directed graphs higher rank graphs and even topological higher rank graphs we believe our methods yield a nice alternative groupoid approach to these important constructions

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

the variance conjecture on projections of the cube abstract we prove that the uniform probability measure on every n k dimensional projection of the n dimensional unit cube verifies the variance conjecture with an absolute constant c var x 2 c sup s n 1 e x 2 e x 2 provided that 1 k n we also prove that if 1 k n 2 3 log n 1 3 the conjecture is true for the family of uniform probabilities on its projections on random n k dimensional subspaces

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

tl filters of integral residuated l monoids in this paper we introduce and discuss the concept of tl filters of integral residuated l monoids first we study some basic properties of tl filters and give the formula for calculating the tl filters generated by l subsets then we discuss some properties of tl filters under homomorphisms and study the relationship between tl filters and t congruence l relations on integral commutative residuated l monoids

the b conjecture for the gaussian measure of dilates of symmetric convex sets and related problems we show that given a symmetric convex set k rd the function t etk is log concave on r where denotes the standard d dimensional gaussian measure we also comment on the extension of this property to unconditional log concave measures and sets and on the complex case

pointwise unitary automorphism groups abstract let a g be a separable c dynamical system in which g is abelian and a has continuous trace the action is said to pointwise unitary if for each in a there is a covariant representation u of the system a g it is shown that for pointwise unitary actions the spectrum a g of the crossed product is a locally compact hausdorff space and the restriction map res which takes u in a g to maps a g onto a furthermore res induces a homeomorphism of a g g onto a these results lead to a characterization of crossed products a g with continuous trace when a has continuous trace and g acts freely on a this happens precisely when the action of g on a is also proper this extends green s characterization of free and proper transformation groups finally it is shown that the results have applications to the problem of determining the dual topology on a locally compact group

on fuzzy filters of pseudo bl algebras abstract we introduce the concept of fuzzy filters of pseudo bl algebras and obtain some of their important properties we further give several kinds of fuzzy filters such as fuzzy boolean filter fuzzy normal filter fuzzy prime filter fuzzy ultrafilter and fuzzy obstinate filter and discuss the relations among the above fuzzy filters we establish the prime fuzzy filter theorem in pseudo bl algebras by studying the relation between fuzzy boolean filter and fuzzy normal filter in pseudo bl algebras we solve an open problem that whether or not every boolean filter is normal in pseudo bl algebras

on the role of logical connectives for primality and functional completeness of algebras of logics we consider certain finite universal algebras arising from algebraic semantics in implicational logics they contain a binary operation and two constants 0 and 1 satisfying the axioms 0 x x 1 x x 1 and 1 x x valid in most implicational logics we characterize the completeness also called primality of such algebras i e the property that every finitary operation on their universe is a term operation of the algebra in other words it is a composition of the basic operations of the algebra using clone theory and the knowledge of maximal clones we describe completeness functional completeness in terms of nonpreservation of three types of specific relations if has a simple property and the algebra contains a binary operation with a neutral element 1 and a unary operation satisfying 1 0 and x 1 otherwise the algebra is functionally complete

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

complemented brunn minkowski inequalities and isoperimetry for homogeneous and non homogeneous measures abstract elementary proofs of sharp isoperimetric inequalities on a normed space r n equipped with a measure w x d x so that w p is homogeneous are provided along with a characterization of the corresponding equality cases when p 0 and in addition w p is assumed concave the result is an immediate corollary of the borell brascamp lieb extension of the classical brunn minkowski inequality providing a new elementary proof of a recent cabre ros oton serra result when p 1 n 0 the relevant property turns out to be a novel q complemented brunn minkowski inequality 0 1 borel sets a b r n such that r n a r n b r n a 1 b r n a q 1 r n b q 1 q which we show is always satisfied by when w p is homogeneous with 1 q 1 p n in particular this is satisfied by the lebesgue measure with q 1 n this gives rise to a new class of measures which are complemented analogues of the class of convex measures introduced by borell but which have vastly different properties the resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of canete rosales and howe the isoperimetric and brunn minkowski type inequalities also extend to the non homogeneous setting under a certain log convexity assumption on the density finally we obtain functional sobolev and nash type versions of the studied inequalities

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

moser inequalities in gauss space the sharp constants in a family of exponential sobolev type inequalities in gauss space are exhibited they constitute the gaussian analogues of the moser inequality in the borderline case of the sobolev embedding in the euclidean space interestingly the gaussian results have features in common with the euclidean ones but also reveal marked diversities

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

isoperimetric inequalities in euclidean convex bodies in this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body i e a compact convex set in euclidean space with interior points we shall not impose any regularity assumption on the boundary of the convex set amongst other results we shall prove the equivalence between hausdorff and lipschitz convergence the continuity of the isoperimetric profile with respect to the hausdorff distance and the convergence in hausdorff distance of sequences of isoperimetric regions and their free boundaries we shall also describe the behavior of the isoperimetric profile for small volume and the behavior of isoperimetric regions for small volume

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

redefined fuzzy implicative filters using the belongs to relation and quasi coincidence with relation q between fuzzy points and fuzzy sets the concept of q j fuzzy implicative filters where q j are any two of q email protected q email protected q with q email protected q is introduced and related properties are discussed relations between email protected q email protected q fuzzy implicative filters and email protected q fuzzy implicative filters are investigated and conditions for an email protected q fuzzy implicative filter to be an fuzzy implicative filter are provided characterizations of email protected q fuzzy implicative filters are given and conditions for a fuzzy set to be a q email protected q fuzzy implicative filter are provided

regularity of isoperimetric hypersurfaces in riemannian manifolds we add to the literature the well known fact that an isoperimetric hypersurface s of dimension at most six in a smooth riemannian manifold m is a smooth submanifold if the metric is merely lipschitz then s is still holder differentiable

sharp convex lorentz sobolev inequalities new sharp lorentz sobolev inequalities are obtained by convexifying level sets in lorentz integrals via the lp minkowski problem new lp isocapacitary and isoperimetric inequalities are proved for lipschitz star bodies it is shown that the sharp convex lorentz sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities

fuzzy implicative and boolean filters of r 0 algebras the notions of fuzzy filters fuzzy implicative filters and fuzzy boolean filters are introduced in r0 algebras respectively some of their properties are investigated the relations between fuzzy implicative filters and fuzzy boolean filters are discussed the characterizations of fuzzy implicative filters are studied and the following is proved to be equivalent 1 a fuzzy filter f is implicative 2 f is a fuzzy boolean filter 3 x y z l f x z f x z y f y z 4 x y l f x y f x y y 5 x l n 1 f nx f x

bci implicative ideals of bci algebras abstract this paper introduces the notion of bci implicative ideals and characterizes bci implicative ideals and closed bci implicative ideals using these characterizations the connections between bci implicative ideals and other ideals in bci bck algebras are investigated additionally the extension property of bci implicative ideals is established finally the implicative bci algebras are completely described using bci implicative ideals the above work generalizes the corresponding results in bck algebras

tail estimates for norms of sums of log concave random vectors we establish new tail estimates for order statistics and for the euclidean norms of projections of an isotropic log concave random vector more generally we prove tail estimates for the norms of projections of sums of independent log concave random vectors and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub matrices in the setting of a log concave ensemble this is used to study a quantity ak m that controls uniformly the operator norm of the sub matrices with k rows and m columns of a matrix a with independent isotropic log concave random rows we apply our tail estimates of ak m to the study of restricted isometry property that plays a major role in the compressive sensing theory ams classification 46b06 15b52 60e15 60b20

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

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

fuzzy ideals of bci and mv algebras abstract using fuzzy ideals it is shown that every ideal of a bci algebra is weakly implicative for a fuzzy ideal of an mv algebra it is shown that it satisfies x v y x y x y for all x y if it is fuzzy implicative and fuzzy prime then it also satisfies x y x y x x 1 and x x 0 for all x and y and in fact im 0 1

nontrivial bck bci algebras do not satisfy the fuzzy ascending chain condition in j meng x guo on fuzzy ideals in bck bci algebras fuzzy sets and systems 149 2005 509 525 meng and guo gave some characterizations of noetherian bck bci algebras by fuzzy ideals they introduced see definition 6 2 the notion of the fuzzy ascending chain condition briefly facc and stated that a bck bci algebra x is noetherian if and only if x satisfies the facc theorem 6 5 but it is not true

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

fuzzy aggregation with artificial color filters artificial color filters are designed to attenuate some pixels and pass others the pass attenuate decision is made on the basis of the learned association of spectral components with user defined concepts in earlier work it has been shown that there are various ways to design artificial color filters using multiple user designated classes and those filters are subjected to useful manipulations such as image processing and boolean aggregation the artificial color filtering has always been binary therefore the boolean logic was the only choice for aggregating filters this paper shows how to fuzzify artificial color filters fuzzy logic subsumes boolean logic and can do so in many ways several different fuzzy t norms are applied to artificial color filters to illustrate the richness in aggregation margin setting a supervised statistical pattern recognition method to train the filters is very conservative in what is definitely assigned to a class m 1 while allowing a useful gradation of membership m 1 for other cases a parametric exploration of these effects for an image is presented

fuzzy filters and fuzzy prime filters of bounded r l monoids and pseudo bl algebras the logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics the sets of provable formulas in corresponding inference systems from the point of view of uncertain information can be described by fuzzy filters of those algebraic semantics bounded residuated lattice ordered monoids email protected monoids are a common generalization of pseudo bl algebras and consequently of pseudo mv algebras and heyting algebras i e algebras behind fuzzy and intuitionistic reasoning in the paper we introduce and investigate fuzzy filters of bounded email protected monoids and fuzzy prime filters of pseudo bl algebras

isoperimetric rigidity and distributions of 1 lipschitz functions abstract we prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal then the space is foliated by minimal geodesics where the observable variance is defined to be the supremum of the variance of 1 lipschitz functions on the space our result can be considered as a variant of cheeger gromoll s splitting theorem and also of cheng s maximal diameter theorem as an application we obtain a new isometric splitting theorem for a complete weighted riemannian manifold with a positive bakry emery ricci curvature

the hot spots conjecture for domains with two axes of symmetry consider a convex planar domain with two axes of symmetry we show that the maximum and minimum of a neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only we deduce j rauch s hot spots conjecture in the following form if the initial temperature distribution is not orthogonal to the first nonzero eigenspace then the point at which the temperature achieves its maximum tends to the boundary in fact the maximum point reaches the boundary in finite time if the boundary has positive curvature results of this type have already been proved by bafiuelos and burdzy bb using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions we introduce here a new technique based on deformation of the domain an advantage of our method is that it works even in the case of multiple eigenvalues on the way toward our results we prove monotonicity properties for neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the dirichlet problem of independent interest

fuzzy hyperrings hv rings based on fuzzy universal sets the aim of this paper is to define and study the concept of a fuzzy hyperring which depends on the concept of a fuzzy universal set a correspondence relation between a fuzzy hyperring based on a fuzzy universal set and both ordinary and fuzzy hyperideal in the sense of davvaz is obtained