lifting and separation procedures for the cut polytope the max cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years however in comparison only little research has been conducted for the cut polytope on arbitrary graphs in particular separation algorithms have received only little attention in this study we describe new separation and lifting procedures for the cut polytope on general graphs these procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid and sometimes facet defining inequalities for the cut polytope on arbitrary graphs in a cutting plane framework we report computational results on a set of well established benchmark problems

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

an efficient numerical method for singular perturbation problems in this paper we propose a method for the numerical solution of singularly perturbed two point boundary value problems bvps first we develop two schemes to integrate initial value problem ivp for system of two first order differential equations and then by using these schemes we solve the bvp precisely we convert the second order bvp into a system of first order differential equations and then apply the numerical schemes to obtain the solution in order to get an initial condition for the system we use the asymptotic approximate solution error estimates are derived and numerical examples are provided to illustrate the present method

free probability for pairs of faces i we consider a notion of bi freeness for systems of non commutative random variables with two faces one of left variables and another of right variables this includes bi free convolution operations bi free cumulants the bi free central limit and bi freeness with amalgamation over an algebra b

collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers a numerical scheme is proposed to solve singularly perturbed two point boundary value problems with a turning point exhibiting twin boundary layers the scheme comprises a b spline collocation method on a uniform mesh which leads to a tridiagonal linear system asymptotic bounds are established for the derivative of the analytical solution of a turning point problem the analysis is done on a uniform mesh which permits its extension to the case of adaptive meshes which may be used to improve the solution the design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem some relevant numerical examples are also illustrated to verify computationally the theoretical aspects

parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers this article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers classical finite difference schemes do not yield parameter uniform convergent results on a uniform mesh in general robust computational techniques for boundary layers chapman hall london crc press boca raton fl 2000 in order to overcome this difficulty we propose an appropriate piecewise uniform shishkin mesh and apply the classical finite difference schemes on this mesh error estimates are derived by decomposing the solution into smooth and singular components the present method is layer resolving as well as parameter uniform convergent numerical examples are presented to show the applicability and efficiency of the method

a novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method abstract in this paper a novel method is presented for solving a class of singularly perturbed boundary value problems firstly the original problem is reformulated as a new boundary value problem whose solution does not change rapidly via a proper transformation then the reproducing kernel method is employed to solve the boundary value new problem numerical results show that the present method can provide very accurate analytical approximate solutions

a numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location a class of nonlinear singularly perturbed interior layer problems is examined in this paper solutions exhibit an interior layer at an a priori unknown location a numerical method is presented that uses a piecewise uniform mesh refined around numerical approximations to successive terms of the asymptotic expansion of the interior layer location the first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered denote the perturbation parameter as e and the number of mesh intervals to be used as n the method is shown to converge point wise to the true solution with a first order convergence rate overlooking a logarithmic factor for sufficiently small e n 1 a numerical experiment is presented to demonstrate the convergence rate established

comparative study of singularly perturbed two point bvps via fitted mesh finite difference method b spline collocation method and finite element method abstract the objective of this paper is to present a comparative study of fitted mesh finite difference method b spline collocation method and finite element method for general singularly perturbed two point boundary value problems due to the small parameter the boundary layer arises we have taken a piecewise uniform fitted mesh to resolve the boundary layer and we have shown that fitted mesh finite difference method has uniform first order convergence b spline collocation method has almost second order uniform convergence and ritz galerkin method also has almost second order uniform convergence two test examples have been solved to compare the maximum absolute error and rate of convergence of the methods

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

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

a numerical method for singularly perturbed turning point problems with an interior layer the objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer the method is based on the asymptotic expansion technique and the reproducing kernel method rkm the original problem is reduced to interior layer and regular domain problems the regular domain problems are solved by using the asymptotic expansion method the interior layer problem is treated by the method of stretching variable and the rkm four numerical examples are provided to illustrate the effectiveness of the present method the results of numerical examples show that the present method can provide very accurate approximate solutions

a new tension b spline method for third order self adjoint singularly perturbed boundary value problems abstract the aim of this paper is to present a new kind of tension b spline called hyperbolic trigonometric tension b spline of order n which is generated over the space span sin t cos t sinh t cosh t 1 t t n 5 n 5 where is tension parameter this kind of b spline has most properties of usual polynomial b splines and enjoys some other advantageous properties as well also hyperbolic trigonometric tension b spline method is applied for solving self adjoint singularly perturbed problems we use some numerical examples to illustrate the accuracy and implementation of the method

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

analytic aspects of the bi free partial r transform abstract it is shown that the bi freely infinitely divisible laws and only these laws can be used to approximate the distributions of sums of identically distributed bi free pairs of commuting faces furthermore the necessary and sufficient conditions for this approximation are found bi free convolution semigroups of measures and their levy khintchine representations are also studied here from an infinitesimal point of view the proofs depend on the harmonic analysis machinery we developed for integral transforms of two variables without reference to the combinatorics of moments and bi free cumulants

harmonic analysis for the bi free partial s transform abstract we develop an analytic machinery to study voiculescu s bi free partial s transform and then use the results to characterize the multiplicative bi free infinite divisibility it is shown that the class of infinitely divisible distributions coincides with the class of limit distributions for products of bi free pairs of left and right infinitesimal unitaries where the pairs are not required to be identically distributed but all left variables are assumed to commute with all right variables furthermore necessary and sufficient conditions for convergence to a given infinitely divisible distribution are found

a modified graded mesh and higher order finite element approximation for singular perturbation problems abstract a singularly perturbed convection diffusion problem is solved numerically using finite element method based on higher order polynomials more precisely we introduce a modified graded mesh generated using some implicitly defined functions higher order parameter uniform convergence is obtained in weighted energy norm moreover the error estimates obtained are optimal in the sense that they are free from logarithmic factor a number of test examples are taken into account and a rigorous comparative analysis is presented moreover we compare the proposed method with others found in the literature

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

quintic b spline method for solving second order linear and nonlinear singularly perturbed two point boundary value problems in this paper we have studied a numerical scheme to solve second order singularly perturbed two point linear and nonlinear boundary value problems the boundary layer of this type of problems exhibits at one end left or right point of the domain due to the presence of perturbation parameter the quintic b spline method and suitable piecewise uniform shishkin mesh have been used linear and nonlinear second order singularly perturbed boundary value problems have been solved by the present method the convergence analysis is also provided and the method is shown to have uniform convergence of fourth order numerical results have demonstrated the efficiency of the present method

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 q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

numerical solution of singularly perturbed convection diffusion problem using parameter uniform b spline collocation method abstract this paper is concerned with a numerical scheme to solve a singularly perturbed convection diffusion problem the solution of this problem exhibits the boundary layer on the right hand side of the domain due to the presence of singular perturbation parameter the scheme involves b spline collocation method and appropriate piecewise uniform shishkin mesh bounds are established for the derivative of the analytical solution moreover the present method is boundary layer resolving as well as second order uniformly convergent in the maximum norm a comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter several numerical examples are also given to demonstrate the efficiency of b spline collocation method and to validate the theoretical aspects

existence and asymptotic stability of periodic solutions with an interior layer of reaction advection diffusion equations abstract we consider a singularly perturbed parabolic periodic boundary value problem for a reaction advection diffusion equation we construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions by using sufficiently precise lower and upper solutions we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics moreover we are able to establish the asymptotic stability of this solution

primal separation for 0 1 polytopes the 0 1 primal separation problem is given an extreme point x of a 0 1 polytope p and some point x find an inequality which is tight at x violated by x and valid for p or assert that no such inequality exists it is known that this separation variant can be reduced to the standard separation problem for p

a numerical algorithm for singular perturbation problems exhibiting weak boundary layers abstract a class of singularly perturbed two point boundary value problems bvp for second order ordinary differential equations is considered here to avoid the numerical difficulties in the solution to these problems we divide the domain into two subdomains the first bvp is a layer domain problem and the second bvp is a regular domain problem error estimates are derived for the numerical solution numerical examples are provided in support of the proposed method

testing the odd bicycle wheel inequalities for the bipartite subgraph polytope in their paper on facets of the bipartite subgraph polytope barahona grotschel and mahjoub pose the question is there a polynomial time algorithm testing the odd bicycle wheel inequalities this note describes such an algorithm

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

dual lukacs regressions for non commutative variables abstract dual lukacs type characterizations of random variables in free probability are studied here first we develop a freeness property satisfied by lukacs type transformations of free poisson and free binomial non commutative variables which are free second we give a characterization of non commutative free poisson and free binomial variables by properties of first two conditional moments which mimic lukacs type assumptions known from classical probability more precisely our result is a non commutative version of the following result known in classical probability if u v are independent real random variables such that e v 1 u u v and e v 2 1 u 2 u v are non random then v has a gamma distribution and u has a beta distribution

an initial value approach for solving singularly perturbed two point boundary value problems in this paper an initial value approach is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point this approach is based on the boundary layer behavior of the solution the method is distinguished by the following fact the given singularly perturbed two point boundary value problem is replaced by three first order initial value problems several linear and non linear problems are solved to demonstrate the applicability of the method it is observed that the present method approximates the exact solution very well

clique web facets for multicut polytopes let g v e be a graph an edge set uv e u si v sj i j where s1 sk is a partition of v is called a multicut with k shores we investigate the polytopes mck n and mck n that are defined as the convex hulls of the incidence vectors of all multicuts with at most k shores and at least k shores respectively of the complete graph kn we introduce a large class of inequalities called clique web inequalities valid for these polytopes and provide a quite complete characterization of those clique web inequalities that define facets of mck n and mck n using general facet manipulation techniques like collapsing and node splitting we construct further new classes of facets for these multicut polytopes we also exhibit a class of clique web facets for which the separation problem can be solved in polynomial time

on operator valued bi free distributions abstract in this paper operator valued bi free distributions are investigated given a subalgebra d of a unital algebra b it is established that a two faced family z is bi free from b b op over d if and only if certain conditions relating the b valued and d valued bi free cumulants of z are satisfied using this we verify that a two faced family of matrices is r cyclic if and only if they are bi free from the scalar matrices over the scalar diagonal matrices furthermore the operator valued bi free partial r s and t transforms are constructed new proofs of results from free probability are developed in order to facilitate many of these bi free results

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

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

convolution operations arising from vandermonde matrices different types of convolution operations involving large vandermonde matrices are considered the convolutions parallel those of large gaussian matrices and additive and multiplicative free convolution and include additive and multiplicative convolution of vandermonde matrices and deterministic diagonal matrices and cases where two independent vandermonde matrices are involved it is also shown that the convergence of any combination of vandermonde matrices is almost sure the convolutions are divided into two types those which depend on the phase distribution of the vandermonde matrices and those which depend only on the spectra of the matrices a general criterion is presented to find which type applies for any given convolution a simulation is presented verifying the results implementations of the presented convolutions are provided and discussed the implementation is based on the technique of fourier motzkin elimination and is quite general as it can be applied to virtually any combination of vandermonde matrices connections with related matrices such as toeplitz and hankel matrices are also discussed

free probability for pairs of faces iii 2 variables bi free partial s and t transforms abstract we introduce two 2 variables transforms the partial bi free s transform and the partial bi free t transform these transforms are the analogues for the bi multiplicative and respectively for the additive multiplicative bi free convolution of the 2 variables partial bi free r transform in our previous paper in this series

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools