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

negation limited formulas understanding the power of negation gates is crucial to bridge the exponential gap between monotone and non monotone computation we focus on the model of formulas over the de morgan basis and study the connection of negation limited formulas with negation limited circuits and with monotone formulas we show the following results we give an efficient transformation of formulas with t negation gates to circuits with log t negation gates this transformation provides a generic way to cast results for negation limited circuits to the setting of negation limited formulas for example using a result of rossman ccc 15 we obtain an average case lower bound for formulas of polynomial size on n variables with n 1 2 e negations we prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size o t with the shrinkage exponent of monotone formulas as a result the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas the above results follow from an efficient structural decomposition theorem for formulas that depends on their negation complexity which may be of independent interest

maximal sensitivity of boolean nested canalizing functions abstract boolean nested canalizing functions ncf have important applications in molecular regulatory networks engineering and computer science in the literature there are two sensitivities to measure the complexity of a boolean function one is average sensitivity the other one is maximal sensitivity we follow the tradition and omit the word maximal in other words in this paper sensitivity is always maximal sensitivity using the past work of the authors and their coauthors on a characterization of ncf we obtain the formula of the sensitivity of any ncf we find that the sensitivity of any ncf is between n 2 2 and n both lower and upper bounds are tight we prove that the block sensitivity hence the l block sensitivity is the same as the sensitivity for ncf it is well known that monotone boolean functions mbf also have this property we characterize all functions which are both monotone and nested canalizing mncf the closed formula of the cardinality of the set of mncfs is also provided

levy area for the free brownian motion existence and non existence abstract this note extends the work of capitaine j funct anal 179 1 2001 153 on the levy area process for the free brownian motion in two directions first we reprove that a levy area for the free brownian motion exists in the von neumann tensor product by exhibiting a non commutative burkholder davis gundy type inequality then we show that there does not exist a levy area in the projective tensor product

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

on stochastic calculus with respect to q brownian motion abstract following the approach and the terminology introduced in deya and schott 2013 6 we construct a product levy area above the q brownian motion for q 0 1 and use this object to study differential equations driven by the process we also provide a detailed comparison between the resulting rough integral and the stochastic ito integral exhibited by donati martin 2003 7

the first variation of the total mass of log concave functions and related inequalities abstract on the class of log concave functions on r n endowed with a suitable algebraic structure we study the first variation of the total mass functional which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions we prove some integral representation formulae for such a first variation which suggest to define in a natural way the notion of area measure for a log concave function in the same framework we obtain a functional counterpart of minkowski s first inequality for convex bodies as corollaries we derive a functional form of the isoperimetric inequality and a family of logarithmic type sobolev inequalities with respect to log concave probability measures finally we propose a suitable functional version of the classical minkowski s problem for convex bodies and prove some partial results towards its solution

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

rogers shephard inequality for log concave functions abstract in this paper we prove different functional inequalities extending the classical rogers shephard inequalities for convex bodies the original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body such as its difference body we characterize the equality cases in all these inequalities our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log concave functions and the study of some geometrical properties of these new sets

correlation bounds and sat algorithms for small linear size circuits we revisit the gate elimination method generalize it to prove correlation bounds of boolean circuits with parity and also derive deterministic satisfiability counting algorithms for small linear size circuits let b2 be the full binary basis and let u2 b2 we prove that for circuits over u2 of size 3nn for any constant 0 5 the correlation with parity is at most 2n 1 and there is a sat algorithm which counts the number of satisfying assignments running in time 2nn 1 for circuit size 3nn for 0 the correlation with parity is at most 2 n and there is a sat algorithm running in time 2n n similar correlation bounds and algorithms are also proved for circuits over b2 of size almost 2 5n

stochastic integration with respect to q brownian motion we develop a stochastic integration with respect to a q brownian motion for i e a non commutative process where the operator at and its adjoint fulfill the q commutation relation under the vacuum state expectation we show that this process enjoys a predictable representation type property

on the rough paths approach to non commutative stochastic calculus abstract we study different possibilities to apply the principles of rough paths theory in a non commutative probability setting first we extend previous results obtained by capitaine donati martin and victoir in lyons s original formulation of rough paths theory then we settle the bases of an alternative non commutative integration procedure in the spirit of gubinelli s controlled paths theory and which allows us to revisit the constructions of biane and speicher in the free brownian case new approximation results are also derived from the strategy

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

limiting negations in constant depth circuits it follows from a theorem of markov that the minimum number of negation gates in a circuit sufficient to compute any boolean function on n variables is l lfloor log n rfloor 1 it can be shown that for functions computed by families of polynomial size o log n depth and bounded fan in circuits nc 1 the same result holds on such circuits l negations are necessary and sufficient in this paper it is proven that this situation changes when polynomial size circuit families of constant depth are considered l negations are no longer sufficient for threshold circuits it is proven that there are boolean functions computable in constant depth tc 0 such that no such threshold circuit containing o n epsilon for all epsilon 0 negations can compute them there is a matching upper bound for any epsilon 0 everything computable by constant depth threshold circuits can be computed by constant depth threshold circuits using n epsilon negations asymptotically there are a

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

a homogeneous decomposition theorem for valuations on convex functions abstract the existence of a homogeneous decomposition for continuous and epi translation invariant valuations on super coercive functions is established continuous and epi translation invariant valuations that are epi homogeneous of degree n are classified by duality corresponding results are obtained for valuations on finite valued convex functions

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

fourth moment theorem and q brownian chaos in 2005 nualart and peccati ann probab 33 1 177 193 2005 proved the so called fourth moment theorem asserting that for a sequence of normalized multiple wiener ito integrals to converge to the standard gaussian law it is necessary and sufficient that its fourth moment tends to 3 a few years later kemp et al ann probab 40 4 1577 1635 2011 extended this theorem to a sequence of normalized multiple wigner integrals in the context of the free brownian motion the q brownian motion q in 1 1 introduced by the physicists frisch and bourret j math phys 11 364 390 1970 in 1970 and mathematically studied by bozejko and speicher commun math phys 137 519 531 1991 interpolates between the classical brownian motion q 1 and the free brownian motion q 0 and is one of the nicest examples of non commutative processes the question we shall solve in this paper is the following what does the fourth moment theorem become when dealing with a q brownian motion

volume polar volume and euler characteristic for convex functions abstract functional analogs of the euler characteristic and volume together with a new analog of the polar volume are characterized as non negative continuous sl n and translation invariant valuations on the space of finite convex and coercive functions on r n

mixed integrals and related inequalities abstract in this paper we define an addition operation on the class of quasi concave functions while the new operation is similar to the well known sup convolution it has the property that it polarizes the lebesgue integral this allows us to define mixed integrals which are the functional analogs of the classic mixed volumes we extend various classic inequalities such as the brunn minkowski and the alexandrov fenchel inequalities to the functional setting for general quasi concave functions this is done by restating those results in the language of rearrangement inequalities restricting ourselves to log concave functions we prove generalizations of the alexandrov inequalities in a more familiar form

lyz ellipsoid and petty projection body for log concave functions abstract the aims of this paper are to develop the lyz ellipsoid and petty projection body for log concave functions which correspond to the lyz ellipsoid and petty projection body for convex bodies when restricted to the subclass of characteristic functions moreover the continuous sl n contravariant valuation on a subclass of log concave functions is classified

limiting negations in non deterministic circuits the minimum number of not gates in a boolean circuit computing a boolean function f is called the inversion complexity of f in 1958 markov determined the inversion complexity of every boolean function and in particular proved that log 2 n 1 not gates are sufficient to compute any boolean function on n variables in this paper we consider circuits computing non deterministically and determine the inversion complexity of every boolean function in particular we prove that one not gate is sufficient to compute any boolean function in non deterministic circuits if we can use an arbitrary number of guess inputs

on the complexity of negation limited boolean networks a theorem of markov precisely determines the number r of negation gates necessary and sufficient to compute a system of boolean functions f for a system of boolean functions on n variables r leq b n lceil log 2 n 1 rceil we call a circuit using b n negation gates negation limited we continue recent investigations into negation limited circuit complexity giving both upper and lower bounds a circuit with inputs x1 xn and outputs neg x 1 ldots neg x n is called an inverter for which r lceil log 2 n 1 rceil fischer has constructed negation limited inverters of size o n2 log n and depth o log n recently tanaka and nishino have reduced the circuit size to o n log2 n at the expense of increasing the depth to log2 n we construct negation limited inverters of size o n log n with depth only o log n and we conjecture that this is optimal we also improve a technique of valiant for constructing monotone circuits for slice functions introduced by berkowitz next we introduce some lower bound techniques for negation limited circuits we provide a 5n 3 log n 1 c lower bound for the size of a negation limited inverter in addition we show that for two different restricted classes of circuit negation limited inverters require superlinear size

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

reductions for monotone boolean circuits the large class say nlog of boolean functions including 0 1 sort and 0 1 merge have an upper bound of o nlogn for their monotone circuit size i e they have circuits with o nlogn and or gates of fan in two suppose that we can use besides such normal and or gates any number of more powerful f gates which realize a monotone boolean function f with r 2 inputs and r 1 outputs note that the cost of each and or gate is one and we assume that the cost of each f gate is r now we define a boolean function f in nlog is said to be f easy if f can be constructed by a circuit with and or f gates whose total cost is o nlogn in this paper we show that 0 1 merge is not f easy for an arbitrary monotone function f such that r r logr

some functional forms of blaschke santalo inequality we establish new functional versions of the blaschke santalo inequality on the volume product of a convex body which generalize to the non symmetric setting an inequality of ball isometric problems in l p and sections of convex sets phd dissertation cambridge 1986 and we give a simple proof of the case of equality as a corollary we get some inequalities for log concave functions and legendre transforms which extend the recent result of artstein et al mathematika 51 33 48 2004 with its equality case

a superpolynomial lower bound for a circuit computing the clique function with at most 1 6 log log n negation gates in this paper we investigate the lower bound on the number of gates in a boolean circuit that computes the clique function with a limited number of negation gates to derive strong lower bounds on the size of such a circuit we develop a new approach by combining three approaches the restriction applied to constant depth circuits due to hastad the approximation method applied to monotone circuits due to razborov and the boundary covering developed in the present paper we prove that if a circuit c with at most floor 1 6 log log m negation gates detects cliques of size log m 3 log m 1 2 in a graph with m vertices then c contains at least 2 1 5 log m log m 1 2 gates no nontrivial lower bounds on the size of such circuits were previously known even if we restrict the number of negation gates to be a constant moreover it follows from a result of fischer it lect notes comput sci 33 1974 pp 71 82 that if one can improve the number of negation gates from floor 1 6 log log m to floor 2 log m in the statement then we have p neq np we also show that the problem of lower bounding the negation limited circuit complexity can be reduced to the one of lower bounding the maximum of the monotone circuit complexity of the functions in a certain class of monotone functions

a 4 n lower bound on the combinational complexity of certain symmetric boolean functions over the basis of unate dyadic boolean functions a simple and easy to check property of a symmetric boolean function is shown to imply a 4n o 1 lower bound on the circuit complexity of the function over u 2 b 2 oplus equiv the basis of unate dyadic boolean functions among the functions to which this lower bound applies are the modular functions operatorname mod k n for any fixed k geqq 3 operatorname mod k n is the function which returns 1 if and only if sum x i bmod k 0 finally a 5n upper bound is obtained on the circuit complexity over u 2 of the function operatorname mod 4 n