estimates for the extremal sections of lpn balls abstract the problem of finding the maximal hyperplane section of b p n where p 2 has been open for a long time it is known that the answer depends on both p and n in this paper using the well known equivalence between hyperplane sections and the isotropic constant of a body we give an upper bound estimate for the volume of hyperplane sections of normalized l p n balls that does not depend on n and p in addition on the basis of results of meyer pajor and schmuckenschlager we show further the corresponding extremal body and hyperplane section when this volume attains its minimum

building a class of fuzzy equivalence relations in this paper we propose a practical method given a strict triangular norm with a convex additive generator for deriving a fuzzy equivalence relation whose reflexivity condition generalizes ruspini s definition of fuzzy partitions the properties of the relations their comparison their transitivity the construction of fuzzy equivalence relations on cartesian products are presented a large part of the paper is devoted to applications with fuzzy partitions defined on the real line several examples including the fairy tale problem from de cock and kerre on un suitable fuzzy relations to model approximate equality fuzzy sets and systems 133 2 2003 137 153 the comparison of colored objects and comfort situations are proposed

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

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

extremal properties of central half spaces for product measures we deal with the isoperimetric and the shift problem for subsets of measure 1 2 in product probability spaces we prove that the canonical central half spaces are extremal in particular cases products of log concave measures on the real line satisfying precise conditions and products of uniform measures on spheres or balls as a corollary we improve the known log sobolev constants for euclidean balls we also give some new results about the related question of estimating the volume of sections of unit balls of lp sums of minkowski spaces

sections of the unit ball of ipn let bnp xi rn i li n xi p 1 1 p and let ek be a k dimensional subspace of rn it is proved that if p 2 resp then vol bpn ek vol bpk resp we give some applications to linear forms

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

gaussian processes and mixed volumes on demontre la conjecture de dudley si l exposant du volume d un sous ensemble symetrique et convexe compact d un espace de hilbert est 1 alors k est un ensemble g c

computing a t transitive lower approximation or opening of a proximity relation since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation assuming that such a violation is mainly due to decision maker estimation errors in some way the more number of elicitations the more probable inconsistency is this is mostly the case within a fuzzy framework even when the number of alternatives or object to be classified is relatively small in this paper we propose a fast method to compute a t indistinguishability from a reflexive and symmetric fuzzy relation being t any left continuous t norm the computed approximation we propose will take o n 3 time complexity where n is the number of elements under consideration and is expected to produce a t transitive opening to the authors knowledge there are no other proposed algorithm that computes t transitive lower approximations or openings while preserving the reflexivity and symmetry properties

the shattering dimension of sets of linear functionals we evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets the proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates factorization techniques and tail estimates of norms viewed as random variables on euclidean spheres the estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions a fact which was the original motivation of this study although this can probably be done in a more traditional manner we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem

on the relationship between t transitivity and approximate equality the paper discusses the suitability of t transitivity for modelling approximate equality it explains how t transitivity a property suitable for modelling vagueness can still be used to handle the imprecision inherent in approximate equality further remarks on resemblance relations are made to clarify how far they improve traditional fuzzy equivalence relations

a continuous semigroup of notions of independence between the classical and the free one in this paper we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non commutative random variables these notions are related to the liberation process introduced by d voiculescu to each notion of independence correspond new convolutions of probability measures for which we establish formulae and of which we compute simple examples we prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence

a note on the volume of sections of bnp abstract let b p n denote the unit ball in l p n with p 1 we prove that vol n 1 h b p n vol n b p n n 1 n for any n 1 dimensional subspace h of r n this is a consequence of bounding the isotropy constant of b p n above by 1 12 and we show that one can replace 1 12 by a possibly smaller number for n 2

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

weyl numbers in sequence spaces and sections of unit balls using techniques of a geometrical nature the following result is proved denoting by iqp m lpm lqm the natural embedding and by xk the kth weyl number if 0 p q 2 there are positive constants c1 and c2 such that for all k m n with k m2 c1k1q 1p xk iqp m c2k1q 1p the upper estimate being even valid for all k 1 m as a consequence of the approach used some results about sections of unit balls are also derived namely volh h bpm volk bpk for 0 p 2 where bpm bpk are the closed unit balls centred at zero of the spaces lpm and lpk respectively h is a k dimensional subspace of lpm and volk volh denote lebesgue measures in rk and h respectively

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

on the volume of non central sections of a cube abstract let q n be the cube of side length one centered at the origin in r n and let f be an affine n d dimensional subspace of r n having distance to the origin less than or equal to 1 2 where 0 d n we show that the n d dimensional volume of the section q n f is bounded below by a value c d depending only on the codimension d but not on the ambient dimension n or a particular subspace f in the case of hyperplanes d 1 we show that c 1 1 17 is a possible choice we also consider a complex analogue of this problem for a hyperplane section of the polydisc

why fuzzy t equivalence relations do not resolve the poincare paradox and related issues in this paper which is a rejoinder on the comments fuzzy sets and systems 133 2 2003 we will answer the questions whether approximate equality should satisfy transitivity and whether fuzzy t equivalence relations resolve the poincare paradox furthermore we will present an improved definition of resemblance relation and we will investigate its connection with nearness relations

equivalence operators in nilpotent systems a consistent connective system generated by nilpotent operators is not necessarily isomorphic to the ukasiewicz system using more than one generator function consistent nilpotent connective systems so called bounded systems can be obtained with the advantage of three naturally derived negation operators and thresholds in this paper equivalences in bounded systems are examined here three different types of operators are studied and a paradox of the equivalence i e there is no equivalence relation in a non boolean setting which fulfils x e x x 1 and e x n x 0 is resolved by aggregating the implication based equivalence and its dual operator we will also show that the aggregated equivalence has nice properties like associativity threshold transitivity and t transitiviy

isomorphic steiner symmetrization this paper proves that there exist 3n steiner symmetrizations that transform any convex set k n into an isomorphic euclidean ball i e if vol k vol dn where dn is the standard euclidean unit ball then k can be transformed into a body k such that c1dn k c2dn where c1 c2 are numerical constants moreover for any c 2 cn symmetrizations are also enough