an algorithm for simplex tableau reduction the push to pull solution strategy the simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point we present a new general purpose solution algorithm called push to pull which obviates the use of artificial variables the algorithm consists of preliminaries for setting up the initialization followed by two main phases the push phase develops a basic variable set bvs which may or may not be feasible unlike simplex and dual simplex this approach starts with an incomplete bvs initially and then variables are brought into the basis one by one if the bvs is complete but the optimality condition is not satisfied then push phase pushes until this condition is satisfied using the rules similar to the ordinary simplex since the proposed strategic solution process pushes towards an optimal solution it may generate an infeasible bvs the pull phase pulls the solution back to feasibility using pivoting rules similar to the dual simplex method all phases use the usual gauss pivoting row operation and it is shown to terminate successfully or indicates unboundedness or infeasibility of the problem a computer implementation which further reduces the size of simplex tableau to the dimensions of the original decision variables is provided this enhances considerably the storage space and the computational complexity of the proposed solution algorithm illustrative numerical examples are also presented

large deviations upper bounds for the laws of matrix valued processes and non communicative entropies using ito s calculus we study the large deviations properties of the law of the spectral measure of the hermitian brownian motion we extend this strategy to the symmetric unitary and wishart processes this dynamical approach is generalized to the study of the large deviations of the non commutative laws of several independent hermitian brownian motions as a consequence we can bound from above entropies defined in the spirit of the microstates entropy introduced by voiculescu

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

finite element modelling of distribution problems a direct solution for the distribution problem is given being much more concise and efficient than those used at present it is then shown that with the constraint equations in the form used for this new method basis transformation can be applied to them to obtain a simple finite element network model of the optimal distribution problem more general fem network and continuum models are then discussed these having wide application in engineering and management science

the reconstruction problem for the euler poisson system in cosmology the motion of a continuum of matter subject to gravitational interaction is classically described by the euler poisson system prescribing the density of matter at initial and final times we are able to obtain weak solutions for this equation by minimizing the action of the lagrangian which is a convex functional through this variational formulation the reconstruction problem becomes very similar to an optimal transportation problem then we see that such minimizing solutions are consistent with smooth solutions of the euler poisson system and enjoy some special regularity properties

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 stability analysis of multiple objective dynamic input output model the popularity of multiple objective dynamic input output modio models in real application was much restricted by their instabilities in this paper we investigate the mechanism of the instability of the original modio models theoretically and develop a modified modio model in virtue of the idea of goal programming it is shown that the proposed model is stable while remaining the characteristics of the original modio models

existence of c 1 critical subsolutions of the hamilton jacobi equation let m be a c second countable manifold without boundary we denote by tm the tangent bundle and by tm m the canonical projection a point in tm will be denoted by x v with x m and v tx m 1 x in the same way a point of the cotangent space t m will be denoted by x p with x m and p t x m a linear form on the vector space tx m we will suppose that g is a complete riemannian metric on m for v tx m the norm v is g v v 1 2 we will denote by the dual norm on t x m we will also use the notations v x for v tx m and p x for v t x m we will assume in the whole paper that h t m r is a function of class at least c2 which satisfies the following three conditions 1 uniform superlinearity for every k 0 there exists c k r such that

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

the dilute potts model on random surfaces we present a new solution of the asymmetric two matrix model in the large n limit which only involves a saddle point analysis the model can be interpreted as ising in the presence of a magnetic field on random dynamical lattices with the topology of the sphere resp the disk for closed resp open surfaces we elaborate on the resulting phase diagram the method can be equally well applied to a more general q 1 matrix model which represents the dilute potts model on random dynamical lattices we discuss in particular duality of boundary conditions for open random surfaces

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

a method of integration over matrix variables ii the integral over twon n hermitan matricesz g c dadbexp tr a2 b2 2cab g n a4 b4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

a computer implementation of the push and pull algorithm and its computational comparison with lp simplex method the simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point we present a new general purpose solution algorithm called push and pull which obviates the use of artificial variables the algorithm consists of preliminaries for setting up the initialization followed by two main phases the push phase develops a basic variable set bvs which may or may not be feasible unlike simplex and dual simplex this approach starts with an incomplete bvs initially and then variables are brought into the basis one by one if the bvs is complete but the optimality condition is not satisfied then push phase pushes until this condition is satisfied using the rules similar to the ordinary simplex since the proposed strategic solution process pushes towards an optimal solution it may generate an infeasible bvs the pull phase pulls the solution back to feasibility using pivoting rules similar to the dual simplex method all phases use the usual gauss pivoting row operation and it is shown to terminate successfully or indicates unboundedness or infeasibility of the problem a computer implementation which enables the user to run either push and pull or ordinary simplex algorithms is provided the fully coded version of the algorithm is available from the authors upon request a comparison analysis to test the efficiency of push and pull algorithm comparing to ordinary simplex is accomplished illustrative numerical examples are also presented

optimization of primal and dual network models of distribution abstract a direct one pass method of solution of the distribution problem is developed basis transformation is then applied to the original constraint equations of this to obtain a quadratic problem which can also be obtained by summation of finite element matrices in which the element constitutive parameter is respectively the route cost or inverse cost for the dual and primal problems using the method of steepest descent extrema for both these are obtained which are equal or close to the exact solutions

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

effective hamiltonians and averaging for hamiltonian dynamics i this paper building upon ideas of mather moser fathi e and others applies pde partial differential equation methods to understand the structure of certain hamiltonian flows the main point is that the cell or corrector pde introduced and solved in a weak sense by lions papanicolaou and varadhan in their study of periodic homogenization for hamilton jacobi equations formally induces a canonical change of variables in terms of which the dynamics are trivial we investigate to what extent this observation can be made rigorous in the case that the hamiltonian is strictly convex in the momenta given that the relevant pde does not usually in fact admit a smooth solution

nonradial solvability structure of super diffusive nonlinear parabolic equations we study the solvability of the cauchy problem for the nonlinear parabolic equation u div u m 1 u at when m 0 then there is no local solution of the initial value problem on the other hand if such an interval does not exist then the initial value problem is locally solvable and the time of existence can be estimated explicitly

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

optimal transport by omni potential flow and cosmological reconstruction one of the simplest models used in studying the dynamics of large scale structure in cosmology known as the zeldovich approximation is equivalent to the three dimensional inviscid burgers equation for potential flow for smooth initial data and sufficiently short times it has the property that the mapping of the positions of fluid particles at any time t1 to their positions at any time t2 t1 is the gradient of a convex potential a property we call omni potentiality are there other flows with this property that are not straightforward generalizations of zeldovich flows this is answered in the affirmative in both two and three dimensions how general are such flows using a wkb technique we show that in two dimensions for sufficiently short times there are omni potential flows with arbitrary smooth initial velocity mappings with a convex potential are known to be associated with the quadratic cost optimal transport problem this has important implications for the problem of reconstructing the dyna

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

addendum to large deviations asymptotics for spherical integrals abstract we correct an omission in guionnet and zeitouni j funct anal 188 2002 461 and improve the main result there to a full large deviations trajectorial result

numerical procedures for input output analysis input output analysis is one of the most important tools of management science one ideally suited to today s ubiquitous pc the present paper introduces the method with a miniscule solution routine with this lagrange multiplier constraints are easily included and the resulting formulation is then condensed to an equivalent penalty factor result simple cases of this being especially easy to use in practice finally a general purpose band solution routine of the type much used in the finite element method is given it being useful to structure ioa models in an element fashion and indeed to understand the continuum character that large ioa models may possess

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

large deviation bounds for matrix brownian motion we prove large deviation bounds for the convergence of hermitian matrix valued brownian motion towards free brownian motion as a consequence we obtain upper and lower bounds on the microstates entropy introduced by voiculescu 24

discussions around voiculescu s free entropies abstract in this article we set two analogous definitions of the free entropies and introduced by voiculescu invent math 118 1994 411 132 1998 189 we discuss their relations improving the preceding results obtained in cabanal duvillard and guionnet ann probab 2001 to appear where a bound on the microstates entropy was established

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

on singular diffusion equations with applications to self organized criticality we consider solutions of the singular diffusion equation t um 1ux x m 0 associated with the flux boundary condition limx um 1ux x 0 the evolutions defined by this problem depend on both m and we prove existence and stability of traveling wave solutions parameterized by each traveling wave is stable in its appropriate evolution these traveling waves are in l1 for 1 m 0 but have non integrable tails for m 1 we also show that these traveling waves are the same as those for the scalar conservation law ut f u x uxx for a particular nonlinear convection term f u f u m 1993 john wiley sons inc

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

character expansion method for the first order asymptotics of a matrix integral the estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics geometry and free probability questions on a rigorous ground only integrals of one matrix or of several matrices with simple quadratic interaction called ab interaction could be evaluated so far see e g 19 17 or 9 in this article we follow an idea widely developed in the physics literature which is based on character expansion to study more complex interaction in this context we derive a large deviation principle for the empirical measure of young tableaux we then use it to study a matrix model defined in the spirit of the dually weighted graph model introduced in 13 but with a cutoff function such that the matrix integral and its character expansion converge we prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit

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

volumes of restricted minkowski sums and the free analogue of the entropy power inequality in noncommutative probability theory independence can be based on free products instead of tensor products this yields a highly noncommutative theory free probability theory for an introduction see 9 the analogue of entropy in the free context was introduced by the second named author in 8 here we show that shannon s entropy power inequality 6 1 has an analogue for the free entropy x theorem 2 1

universality of correlation functions of hermitian random matrices in an external field the behavior of correlation functions is studied in a class of matrix models characterized by a measure exp s containing a potential term and an external source term s n tr v m ma in the large n limit the short distance behavior is found to be identical to the one obtained in previously studied matrix models thus extending the universality of the level spacing distribution the calculation of correlation functions involves finite n determinant formulae reducing the problem to the large n asymptotic analysis of a single kernel k this is performed by an appropriate matrix integral formulation of k multi matrix generalizations of these results are discussed

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

logarithmically singular parabolic equations as limits of the porous medium equation abstract let u m be a local weak solution to the porous medium equation u m t w m 0 where w m u m m 1 m it is shown that if u m is locally in l loc r for r 1 2 n uniformly in m and if w m is in l loc p for p n 2 in the space variables uniformly in time then u m contains a subsequence converging in c loc 1 2 to a local weak solution to the logarithmically singular equation u t ln u the result is based on local upper and lower bounds on u m uniform in m the uniform local lower bounds are realized by a harnack type inequality