lois de repartition des diviseurs des entiers friables according to a general probabilistic principle the natural divisors of friable integers i e integers free of large prime factors should normally present a gaussian distribution we show that this indeed is the case with conditional density tending to 1 as soon as the standard necessary conditions are met furthermore we provide explicit essentially optimal estimates for the decay of the involved error terms the size of the exceptional set is sufficiently small to enable recovery of the average behaviour in the same optimal range our argument combines the saddle point method with new large deviations estimates for the distribution of certain additive functions

independence properties in subalgebras of ultraproduct ii1 factors let mn be a sequence of finite factors with dim mn and denote by m mn their ultraproduct over a free ultrafilter we prove that if q m is either an ultraproduct q qn of subalgebras qn mn with qn mnqn mn n or the centralizer q b m of a separable amenable subalgebra b m then for any separable subspace x m q m there exists a diffuse abelian von neumann subalgebra in q which is free independent to x relative to q m some related independence properties for subalgebras in ultraproduct ii1 factors are also discussed

a group of paths in ℝ we define a group structure on the set of compact minimal paths in r2 we classify all finitely generated subgroups of this group g they are free products of free abelian groups and surface groups moreover each such group occurs in g the subgroups of g isomorphic to surface groups arise from certain topological 1 forms on the corresponding surfaces we construct examples of such 1 forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo anosov diffeomorphism using g we construct a non polygonal tiling problem in 1r that is a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles the group g has applications to combinatorial tiling problems of the type given a set of tiles p and a region r can r be tiled by translated copies of tiles in p

ergodicity of foliations with singularities abstract this paper deals with the question of ergodicity of foliations defined by smooth closed one forms on a compact manifold without boundary

concavification of free entropy abstract we introduce a modification of voiculescu s free entropy which coincides with the lim inf variant of voiculescu s free entropy on extremal states but is a concave upper semi continuous function on the trace state space we also extend the orbital free entropy of hiai et al 2009 8 to non hyperfinite multivariables and prove freeness in the case of additivity of voiculescu s entropy or vanishing of our extended orbital entropy

asymptotics of unitary and orthogonal matrix integrals abstract in this paper we prove that in small parameter regions arbitrary unitary matrix integrals converge in the large n limit and match their formal expansion secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the hciz integral our convergence result also leads us to new results of smoothness of microstates we finally generalize our approach to integrals over the orthogonal group

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

asymptotics of unitary multimatrix models the schwinger dyson lattice and topological recursion abstract we prove the existence of a 1 n 2 expansion in unitary multimatrix models which are gibbs perturbations of the haar measure and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem the recursion obtained is closely related to the topological recursion which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems our approach consists of two main ingredients an asymptotic study of the schwinger dyson lattice over noncommutative laurent polynomials and uniform control on the cumulants of gibbs measures on product unitary groups the required cumulant bounds are obtained by concentration of measure arguments and change of variables techniques

entiers friables inegalite de turan kubilius et applications we generalize the classical turan kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains several applications are derived in probabilitic number theory

free transportation cost inequalities via random matrix approximation by means of random matrix approximation procedure we re prove biane and voiculescu s free analog of talagrand s transportation cost inequality for measures on r in a more general setup furthermore we prove the free transportation cost inequality for measures on t as well by extending the method to special unitary random matrices

the non microstates free entropy dimension of dt operators abstract dykema and haagerup introduced the class of dt operators amer j math 126 2004 121 189 and also showed that every dt operator generate l f 2 j funct anal 209 2004 332 366 the von neumann algebra generated by the free group on two generators in this paper we prove that voiculescu s non microstates free entropy dimension is 2 for all dt operators

stable equivalence of the weak closures of free groups convolution algebras we prove in this paper that the von neumann algebras associated to the free non commutative groups are stably isomorphic i e that they are isomorphic when tensorized by the algebra of all linear bounded operators on a separable infinite dimensional hilbert space this gives positive evidence for an old question due to r v kadison see also s sakai s book on w algebras whether the von neumann algebras associated to free groups are isomorphic or not

sur l unique ergodicite des 1 formes fermees singulieres we study the relations between the de rham cohomology class of a closed differential 1 form with morse singularities on a manifoldm of dimensionn 3 and the ergodic properties of the foliationf it defines we show by examples that if the fundamental group is large enough very different behaviours can occur in the same class in contrast if the fundamental group admits no surjective homomorphism onto the free group on 3 generators thenf is always uniquely ergodic provided it has no compact leaf if the natural homomorphism 1 m h1 m z torsion does not factor through a free group then the same result is true in almost every cohomology class we also give results about the existence of noncompact leaves and the number of ergodic measures

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 sofic actions and equivalence relations abstract the notion of sofic equivalence relation was introduced by gabor elek and gabor lippner their technics employ some graph theory here we define this notion in a more operator algebraic context starting from connes embedding problem and prove the equivalence of these two definitions we introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic we also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic

mismatched estimation and relative entropy a random variable with distribution p is observed in gaussian noise and is estimated by a mismatched minimum mean square estimator that assumes that the distribution is q instead of p this paper shows that the integral over all signal to noise ratios snrs of the excess mean square estimation error incurred by the mismatched estimator is twice the relative entropy d p q in nats this representation of relative entropy can be generalized to nonreal valued random variables and can be particularized to give new general representations of mutual information in terms of conditional means inspired by the new representation we also propose a definition of free relative entropy which fills a gap in and is consistent with the literature on free probability

approximating the number of integers without large prime factors x y denotes the number of positive integers y hildebrand and tenenbaum gave a smooth approximation formula for x y in the range logx 1 y x where e is a fixed positive number 1 2 in this paper by modifying their approximation formula we provide a fast algorithm to approximate x y the computational complexity of this algorithm is o log x log y we give numerical results which show that this algorithm provides accurate estimates for x y and is faster than conventional methods such as algorithms exploiting dickman s function

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

amenability tubularity and embeddings into mathcal r omega suppose m is a tracial von neumann algebra embeddable into mathcal r omega the ultraproduct of the hyperfinite ii1 factor and x is an n tuple of selfadjoint generators for m denote by x m k the microstate space of x of order m k we say that x is tubular if for any e 0 there exist m in mathbb n and 0 such that if x 1 ldots x n y 1 ldots y n in gamma x m k gamma then there exists a k k unitary u satisfying ux iu y i 2 epsilon for each 1 i n we show that the following conditions are equivalent m is amenable i e injective x is tubular any two embeddings of m into mathcal r omega are conjugate by a unitary u in mathcal r omega

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

an estimate for the number of integers without large prime factors x y denotes the number of positive integers y hildebrand and tenenbaum provided a good approximation of x y however their method requires the solution a x y to the equation p y log p p 1 log x and therefore it needs a large amount of time for the numerical solution of the above equation for large y hildebrand also showed 1 u log y approximates for 1 0 we show e m approximates u and i e m log y also approximates where m log u log log y log log u 1 using these approximations we give a simple method which approximates x y within a factor 1 o 1 u 1 log y in the range log log x 5 3 logy log x e where e is any positive constant

moyennes de certaines fonctions multiplicatives sur les entiers friables 4 we provide two generic applications of previously established estimates for friable mean values of non negative multiplicative functions

on kirchberg s embedding problem kirchberg s embedding problem kep asks whether every separable c algebra embeds into an ultrapower of the cuntz algebra o 2 o 2 in this paper we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses in order to prove this result we study general properties of existentially closed c algebras along the way we establish a connection between existentially closed c algebras the weak expectation property of lance and the local lifting property of kirchberg the paper concludes with a discussion of the model theory of o 2 o 2 several results in this last section are proven using some technical results concerning tubular embeddings a notion first introduced by jung for studying embeddings of tracial von neumann algebras into the ultrapower of the hyperfinite ii 1 factor

convex sets associated to c algebras abstract for a separable unital c algebra a and a separable mcduff ii 1 factor m we show that the space h om w a m of weak approximate unitary equivalence classes of unital homomorphisms a m may be considered as a closed bounded convex subset of a separable banach space a variation on n brown s convex structure h om n r u many separable unital c algebras including all separable unital nuclear c algebras have the property that for any mcduff ii 1 factor m h om w a m is affinely homeomorphic to the trace space of a in general h om w a m and the trace space of a do not share the same data several examples are provided we characterize extreme points of h om w a m in many cases and we give two different conditions one necessary and the other sufficient for extremality in general the universality of c f is reflected in the fact that for any unital separable a h om w a m may be embedded as a face in h om w c f m we also extend brown s construction to apply more generally to h om a m u

absence of algebraic relations and of zero divisors under the assumption of full non microstates free entropy dimension abstract we show that in a tracial and finitely generated w probability space existence of conjugate variables excludes algebraic relations for the generators moreover under the assumption of maximal non microstates free entropy dimension we prove that there are no zero divisors in the sense that the product of any non commutative polynomial in the generators with any element from the von neumann algebra is zero if and only if at least one of those factors is zero in particular this shows that in this case the distribution of any non constant self adjoint non commutative polynomial in the generators does not have atoms questions on the absence of atoms for polynomials in non commuting random variables or for polynomials in random matrices have been an open problem for quite a while we solve this general problem by showing that maximality of free entropy dimension excludes atoms

la dynamique des pseudogroupes de rotations we study dynamical systems on the circle generated by a finite number of partially defined rotations we construct new examples with all orbits dense this leads to non simplicial free actions of free groups on trees we study the generic dynamics for these pseudogroups and their 1 parameter families we show that in suitable 2 parameter families the set of pseudogroups having a dense orbit is a sierpinski curve we generalize results on interval exchange transformations obtained by boshernitzan veech rips

an alternating moment condition for bi freeness abstract in this note we demonstrate an equivalent condition for bi freeness inspired by the well known vanishing of alternating centred moments condition from free probability we show that all products satisfying a centred condition on maximal monochromatic intervals have vanishing moments if and only if the family of pairs of faces they come from is bi free and show that similar characterisations hold for the amalgamated and conditional settings in addition we construct a bi free unitary brownian motion and show that conjugation by this process asymptotically creates bi freeness these considerations lead to another characterisation of bi free independence

numerical solution of some classical differential difference equations for differential difference equations we provide a method that gives numerical solutions accurate to hundreds or even thousands of digits we illustrate with numerical solutions to three classical problems with a few exceptions previous claims of extended accuracy for these problems are found to be wrong

liberation of projections abstract we study the liberation process for projections p q p t q u t p u t q where u t is a free unitary brownian motion freely independent from p q its action on the operator valued angle q p t q between the projections induces a flow on the corresponding spectral measures t we prove that the cauchy transform of the measure satisfies a holomorphic pde we develop a theory of subordination for the boundary values of this pde and use it to show that the spectral measure t possesses a piecewise analytic density for any t 0 and any initial projections of trace 1 2 we us this to prove the unification conjecture for free entropy and information in this trace 1 2 setting

floer novikov homology and applications to lagrangian submanifolds we use a non hamiltonian version of lagrangian floer homology to prove that an exact lagrangian submanifold in the cotangent bundle of the 3 torus t 3 must be diffeomorphic to t 3 this improves a previous result of fukaya seidel and smith

approximating the number of integers free of large prime factors define x y to be the number of positive integers n x such that n has no prime divisor larger than y we present a simple algorithm that approximates x y in o y loglogx logy 1 loglogy floating point operations this algorithm is based directly on a theorem of hildebrand and tenenbaum we also present data which indicate that this algorithm is more accurate in practice than other known approximations including the well known approximation x y xp log x log y where p u is dickman s function

on certain free product factors via an extended matrix model abstract a random matrix model for freeness is extended and used to investigate free products of free group factors with matrix algebras and with the hyperfinite ii 1 factor the latter is shown to be isomorphic to a free group factor having one additional generator

prime sieves using binary quadratic forms we introduce an algorithm that computes the prime numbers up to n using o n log log n additions and n 1 2 o 1 bits of memory tie algorithm enumerates representations of integers by certain binary quadratic forms we present implementation results for this algorithm and one of the best previous algorithms

topological dynamical systems associated to ii1 factors abstract if n r is a separable ii1 factor the space h om n r of unitary equivalence classes of unital homomorphisms n r is shown to have a surprisingly rich structure if n is not hyperfinite h om n r is an infinite dimensional complete metrizable topological space with convex like structure and the outer automorphism group out n acts on it by affine homeomorphisms if n r then h om n r is just a point property t is reflected in the extreme points they re discrete in this case for certain free products n r every countable group acts nontrivially on h om n r and we show the extreme points are not discrete for these examples finally we prove that the dynamical systems associated to free group factors are isomorphic

on the local behavior of ψ x y etude du comportement local de la fonction x y qui designe le nombre d entiers positifs x depourvus de facteurs premiers y

the free entropy dimension of hyperfinite von neumann algebras suppose m is a hyperfinite von neumann algebra with a normal tracial state and 1 n is a set of selfadjoint generators for m we calculate 0 1 n the modified free entropy dimension of 1 n moreover we show that 0 1 n depends only on m and consequently 0 1 n is independent of the choice of generators for m in the course of the argument we show that if b 1 b n is a set of selfadjoint generators for a von neumann algebra r with a normal tracial state and b 1 b n has finite dimensional approximants then 0 n 0 b 1 b n for any hyperfinite von neumann subalgebra n of r combined with a result by voiculescu this implies that if r has a regular diffuse hyperfinite von neumann subalgebra then 0 b 1 b n 1

explicit tauberian estimates for functions with positive coefficients abstract if x a n x n has an 0 for all n then for each x 0 for which the series converges we have n n x n x for each n by choosing that x which minimizes the upper bound one obtains a saddle point estimate for each an that has been known to be close to best possible in several cases this paper presents a lower bound for summatory functions of the coefficients that is derived by elementary methods it is not as sharp as the estimates that one obtains from most modern tauberian theorems however this method can be used when tauberian theorems are not applicable for example when one is dealing not with a single generating function but a sequence of them applications to partitions integers without large prime factors and other problems are presented

local limit theorems for multiplicative free convolutions abstract this paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution we show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law superconvergence to the marginal law of free multiplicative brownian motion at a specified time is also studied in the unitary case the superconvergence to free brownian motion and that to the haar measure are shown to be uniform over the entire unit circle implying further a free entropic limit theorem and a universality result for unitary free levy processes finally the method of proofs on the positive half line gives rise to a new multiplicative boolean to free bercovici pata bijection

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

analogues of entropy in bi free probability theory non microstate abstract in this paper we extend the notion of non microstate free entropy to the bi free setting using a diagrammatic approach involving bi non crossing diagrams bi free difference quotients are constructed as analogues of the free partial derivations adjoints of bi free difference quotients are discussed and used to define bi free conjugate variables notions of bi free fisher information non microstate bi free entropy and non microstate bi free entropy dimension are defined and known properties in the free setting are extended to the bi free setting

a note about proving non γ under a finite non microstates free fisher information assumption we prove that if x1 xn n 1 are self adjoints in a w probability space with finite non microstates free fisher information then the von neumann algebra w x1 xn they generate doesn t have property especially is not amenable this is an analog of a well known result of voiculescu for microstates free entropy we also prove factoriality under finite non microstates entropy

on sharp transitions in making squares in the fastest performing integer factoring algorithms one creates a sequence of integers in a pseudo random way and wishes to rapidly determine a subsequence whose product is a square in 1994 pomerance stated the following problem which encapsulates all of the key issues select integers a1 a2 at random from the interval 1 x until some nonempty subsequence has product equal to a square find a good estimate for the expected stopping time of this process a good solution should allow one to determine the optimal choice of parameters in many factoring algorithms pomerance 1994 using an idea of schroeppel 1985 showed that with probability 1 o 1 the rst subsequence whose product equals a square

the spectral edge of unitary brownian motion the brownian motion u n t t ge 0 on the unitary group converges as a process to the free unitary brownian motion u t t ge 0 as n rightarrow infty in this paper we prove that it converges strongly as a process not only in distribution but also in operator norm in particular for a fixed time t 0 we prove that the unitary brownian motion has a spectral edge there are no outlier eigenvalues in the limit we also prove an extension theorem any strongly convergent collection of random matrix ensembles independent from a unitary brownian motion also converge strongly jointly with the brownian motion we give an application of this strong convergence to the jacobi process

a differential delay equation arising from the sieve of eratosthenes consideration is given to the differential delay equation introduced by buchstab 1937 in connection with an asymptotic formula for the uncanceled terms in the sieve of eratosthenes maier 1985 used this result to show there is unexpected irreqularity in the distribution of primes in short intervals the function omega u is studied in this paper using numerical and analytical techniques the results are applied to give some numerical constants in maier s theorem

on classical analogues of free entropy dimension abstract we define a classical probability analogue of voiculescu s free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on r n we show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension first it can be viewed as a kind of fractal dimension second if one extends bochner s inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical bochner s inequalities then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in bochner s inequality for a measure regularized by convolution with the gaussian law as the regularization is removed we introduce a free analogue of the bochner inequality and study the related free entropy dimension quantity we show that it is greater or equal to the non microstates free entropy dimension

some applications of free fisher information on frame theory we introduce the notion of operator valued free fisher information with respect to a positive map of a random variable in an operator valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations then we can apply this notion on the study of frame theory especially on the disjointness problem of modular frames arising from conditional expectations

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

global solvability of real analytic involutive systems on compact manifolds the focus of this work is the smooth global solvability of a linear partial differential operator mathbb l associated to a real analytic closed non exact 1 form b defined on a real analytic closed n manifold that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co rank one we define an appropriate covering projection widetilde m rightarrow m such that the pullback of b has a primitive widetilde b and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of widetilde b are connected as a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator when m is orientable we prove a connection between the global solvability of mathbb l and that of mathbb l n 1 which is the last non trivial operator of the complex in particular we prove that mathbb l is globally solvable if and only if mathbb l n 1 is globally solvable in the smooth category we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a morse 1 form i e when the structure is of mizohata type

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

topological free entropy dimension in unital c algebras abstract the notion of topological free entropy dimension of n tuple of elements in a unital c algebra was introduced by voiculescu in the paper we compute topological free entropy dimension of one self adjoint element and topological free orbit dimension of one self adjoint element in a unital c algebra we also calculate the values of topological free entropy dimensions of any families of self adjoint generators of some unital c algebras including irrational rotation c algebra uhf algebra and minimal tensor product of two reduced c algebras of free groups

on integers free of large prime factors the number l x y of integers y has been given satisfactory estimates in the regions y exp log log x 5 3 6 in the intermediate range only very crude estimates have been obtained so far we close this gap and give an expression which approximates l x y uniformly for x y 2 within a factor 1 o log y log x log y y as an application we derive a simple formula for ti cx y fl x y where 1 c y we also prove a short interval estimate for l x y

1 formes fermees singulieres et groupe fondamental we study the influence of the 1 of a closed manifoldm n n 3 on the foliations ofm defined by closed differential 1 forms with morse singularities of index 0 n every nonexact form is cohomologous to a weakly complete one that is one whose leaf space is of the same type as that of a nonsingular form generically a form has compact leaves or is weakly complete if 1 m has no quotient isomorphic to then every nonexact form onm is weakly complete we also say a form is complete if every path inm is homotopic to either a path transverse to or a path contained in a leaf of completeness of depends only on its de r ham cohomology class the set of complete cohomology classes depends only on 1 m and is related to finitely generated normal subgroups of 1 m with quotient if 1 m is nilpotent or even polycyclic every nonexact form onm is complete on irreducible 3 manifolds a form is complete iff it is cohomologous to a nonsingular one

holder continuity of cumulative distribution functions for noncommutative polynomials under finite free fisher information abstract this paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory more precisely we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non microstates free fisher information highlighting the special case of lipschitz conjugate variables for the first time in this generality it is shown that the analytic distributions of those evaluations have holder continuous cumulative distribution functions with an explicit holder exponent that depends only on the degree of the considered polynomial for linear polynomials we reach in the case of finite non microstates free fisher information the optimal holder exponent 2 3 and get lipschitz continuity in the case of lipschitz conjugate variables in particular our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite non microstates free entropy which partially settles a conjecture of charlesworth and shlyakhtenko 8 we further provide a very general criterion that gives for weak approximations of measures having holder continuous cumulative distribution functions explicit rates of convergence in terms of the kolmogorov distance finally we combine these results to study the asymptotic eigenvalue distributions of polynomials in gues or matrices with more general gibbs laws for gibbs laws this extends the corresponding result obtained in 21 from convergence in distribution to convergence in kolmogorov distance in the gue case we even provide explicit rates which quantify results of 23 24 in terms of the kolmogorov distance

asymptotic semismoothness probabilities we call an integer semismooth with respect to y and z if each of its prime factors is y and all but one are z such numbers are useful in various factoring algorithms including the quadratic sieve let g be the asymptotic probability that a random integer n is semismooth with respect to n and n we present new recurrence relations for g and related functions we then give numerical methods for computing g tables of g and estimates for the error incurred by this asymptotic approximation

sur l inegalite de turan kubilius friable the exact value of the friable turan kubilius constant is determined when u log x log y tends to a finite limit for the set of y friable integers not exceeding x

free analog of pressure and its legendre transform the free analog of the pressure is introduced for multivariate noncommutative random variables and its legendre transform is compared with voiculescu s microstate free entropy

construction of symplectic structures on 4 manifolds with a free circle action let m be a closed 4 manifold with a free circle action if the orbit manifold n satisfies an appropriate fibering condition then we show how to represent a cone in h m r by symplectic forms this generalizes earlier constructions by thurston bouyakoub and fernandez gray morgan in the case that m is the product 4 manifold s n our construction complements the results of fv08 and allows us to completely determine the symplectic cone of such 4 manifolds

intersections of schubert varieties and eigenvalue inequalities in an arbitrary finite factor the intersection ring of a complex grassmann manifold is generated by schubert varieties and its structure is governed by the littlewood richardson rule given three schubert varieties s1 s2 s3 with intersection number equal to one we show how to construct an explicit element in their intersection this element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags and is therefore well defined over an arbitrary field of scalars moreover this result also applies to appropriately defined analogues of schubert varieties in the grassmann manifolds associated with a finite von neumann algebra the arguments require the combinatorial structure of honeycombs particularly the structure of the rigid extremal honeycombs it is known that the eigenvalue distributions of self adjoint elements a b c with a b c 0 in the factor r are characterized by a system of inequalities analogous to the classical horn inequalities of linear algebra we prove that these inequalities are in fact true for elements of an arbitrary finite factor in particular if x y z are self adjoint elements of such a factor and x y z 0 then there exist self adjoint a b c r such that a b c 0 and a respectively b c has the same eigenvalue distribution as x respectively y z a complete matricial form of this result is known to imply an affirmative answer to an embedding question formulated by connes the critical point in the proof of this result is the production of elements in the intersection of three schubert varieties when the factor under consideration is the algebra of n n complex matrices our arguments provide new and elementary proofs of the horn inequalities which do not require knowledge of the structure of the cohomology of the grassmann manifolds

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

compact and locally dense leaves of a closed one form foliation abstract we study a foliation defined by a possibly singular smooth closed one form on a connected smooth closed orientable manifold we prove two bounds on the total number of homologically independent compact leaves and of connected components of the union of all locally dense leaves which we call minimal components in particular we generalize the notion of minimal components previously used in the context of morse form foliations to general foliations finally we give a condition for the form foliation to have only closed leaves closed in the complement of the singular set

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group

matricially free random variables we show that the framework developed by voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication the associated notion of independence called matricial freeness can be viewed as a concept which not only leads to a natural generalization of freeness but also underlies other fundamental types of noncommutative independence such as monotone independence and boolean independence at the same time the sums of matricially free random variables called random pseudomatrices are closely related to random matrices the main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws

quadratic class numbers and character sums we present an algorithm for computing the class number of the quadratic number field of discriminant d the algorithm terminates unconditionally with the correct answer and under the grh executes in o d 1 4 steps the technique used combines algebraic methods with burgess theorem on character sums to estimate l 1 x d we give an explicit version of burgess theorem valid for prime discriminants and as an application we compute the class number of a 32 digit discriminant