the stable set problem and the lift and project ranks of graphs we study the lift and project procedures for solving combinatorial optimization problems as described by lovasz and schrijver in the context of the stable set problem on graphs we investigate how the procedures performances change as we apply fundamental graph operations we show that the odd subdivision of an edge and the subdivision of a star operations as well as their common generalization the stretching of a vertex operation cannot decrease the n 0 n or n rank of the graph we also provide graph classes which contain the complete graphs where these operations do not increase the n 0 or the n rank hence we obtain the ranks for these graphs and we also present some graph minor like characterizations for them despite these properties we give examples showing that in general most of these operations can increase these ranks finally we provide improved bounds for n ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the n rank of a graph

rational and polynomial matrix factorizations via recursive pole zero cancellation we develop a recursive algorithm for obtaining factorizations of the type r r1 r2 where all three matrices are rational and r1 is nonsingular moreover the factors r1 and r2 are such that either the poles of r1 1 and r2 are in a prescribed region of the complex plane or their zeros such factorizations cover the specific cases of coprime factorization inner outer factorization gcd extraction and many more the algorithm works on the state space or generalized state space realization of r and derives in a recursive fashion the corresponding realizations of the factors

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

an improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects in this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil and implicitly its kronecker structure the algorithm compares favorably to existing ones in terms of elegance versatility and complexity in particular the algorithm without any modification yields the structural invariants associated with a generalized state space system and its system pencil two related geometric aspects are also discussed we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form we extend the notion of deflating subspace to the singular pencil case c 1997 elsevier science b v

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

on lovasz schrijver lift and project procedures on the dantzig fulkerson johnson relaxation of the tsp we study the lovasz schrijver lift and project procedure n on the linear relaxation of the dantzig fulkerson johnson formulation of the traveling salesman problem tsp a long standing conjecture states that the integrality gap of this relaxation is frac43 in the case of metric costs in this paper we show that the n rank of 2 matching inequalities relative to this relaxation can be arbitrarily high and obtain as a corollary that even after applying n to the relaxation a fixed number of times the integrality gap of the resulting relaxation is at least frac43

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

a simple state space design of an interactor for a non square system via system matrix pencil approach abstract the construction of an interactor cancelling the infinite zeros of a non square proper transfer matrix is discussed in this paper along the line of approach of b r copeland m g safonov int j robust nonlinear control 2 1992 139 in this paper some new properties of the infinite eigenstructure of the system matrix pencil of the transfer matrix are presented a simple state space design of the interactor is proposed by using these properties then some new features about state space relations among the original system the interactor and the compensated system are presented also several conditions about the invariance of the stabilizability or controllability between the state space realizations of the original system and the compensated system are shown furthermore the proposed method of designing the interactor in this paper can be applied to a tall or fat transfer matrix with full normal rank

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

when does the positive semidefiniteness constraint help in lifting procedures we study the lift and project procedures of lovaisz and schrijver for 0 1 integer programming problems we prove that the procedure using the positive semidefiniteness constraint is not better than the one without it in the worst case various examples are considered we also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

on a representation of the matching polytope via semidefinite liftings we consider the relaxation of the matching polytope defined by the non negativity and degree constraints we prove that given an undirected graph on n nodes and the corresponding relaxation of the matching polytope n 2 iterations of the lovasz schrijver semidefinite lifting procedure are needed to obtain the matching polytope in the worst case we show that n 2 iterations of the procedure always suffice

cauchy problem of a nonlocal p laplacian evolution equation with nonlocal convection abstract this paper is concerned with the cauchy problem of a nonlocal equation that takes into account convective and p laplacian diffusive effects u t x t r n j x y u y t u x t p 2 u y t u x t d y g f u f u x t with j radially symmetric and g not necessarily symmetric first we prove the existence and uniqueness of solutions and if the convolution kernels j and g are rescaled appropriately we show that solutions of the nonlocal problem converge to the solution of the usual p laplacian diffusion equation with convection finally as a supplementary result we study the asymptotic behavior of solutions as t and give the decay estimate

minimal indices cancellation and rank revealing factorizations for rational matrix functions we study the problem of eliminating the minimal indices the indices of a minimal polynomial basis of the null space of a rational matrix function by multiplication with a suitably chosen invertible rational matrix function we derive the class of all invertible factors that dislocate the minimal indices to certain zero locations and feature minimal mcmillan degree we impose additional conditions on the factor like being j unitary or j inner either with respect to the imaginary axis or to the unit circle and characterize the classes of solutions en route we extend the well known rank revealing factorization of a constant matrix to rational matrix functions the results are completely general and apply in particular to matrices which are polynomial strictly proper or improper rank deficient with arbitrary poles and zeros including at infinity all characterization are made by using descriptor realizations while the associated computations are performed by employing constant unitary transformations and standard reliable procedures for eigenvalue assignment or for solving riccati equations

zero cancellation for general rational matrix functions the problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints firstly the class of all factors that dislocate the zeros and feature minimal mcmillan degree are derived further necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being j unitary or j inner either with respect to the imaginary axis or to the unit circle the main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial proper or improper rank deficient with arbitrary poles and zeros including at infinity a particular attention is paid to the numerically sound construction of solutions by employing at each stage unitary transformations reliable numerical algorithms for eigenvalue assignment and efficient lyapunov equation solvers

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

on a new class of realization formulas and their application abstract a new set of realization formulas is derived for a class of matrix valued functions w these include the standard realization formulas for rational w observability controllability and minimality are defined and characterized conditions for w to be j 1 j 2 isometric and or j 1 j 2 coisometric with respect to a pair of signature matrices j 1 and j 2 are given in terms of the realizations minimal factorizations for square w are considered and formulas for the factors are deduced associated reproducing kernel spaces of pairs too are discussed

tighter linear and semidefinite relaxations for max cut based on the lovasz schrijver lift and project procedure we study how the lift and project method introduced by lovasz and schrijver siam j optim 1 1991 pp 166 190 applies to the cut polytope we show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no k5 minor therefore for a graph g with n ge 4 nodes with stability number alpha g n 4 iterations suffice instead of the m number of edges iterations required in general and under some assumption n alpha g 3 iterations suffice the exact number of needed iterations is determined for small n le 7 by a detailed analysis of the new relaxations if positive semidefiniteness is added to the construction then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by anjos and wolkowicz discrete appl math to appear we also show how the lovasz schrijver relaxations for the stable set polytope of g can be strengthened using the corresponding relaxations for the cut polytope of the graph g nabla obtained from g by adding a node adjacent to all nodes of g

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

lovasz schrijver sdp operator near perfect graphs and near bipartite graphs we study the lovasz schrijver lift and project operator mathrm text ls ls based on the cone of symmetric positive semidefinite matrices applied to the fractional stable set polytope of graphs the problem of obtaining a combinatorial characterization of graphs for which the mathrm text ls ls operator generates the stable set polytope in one step has been open since 1990 we call these graphs mathrm text ls ls perfect in the current contribution we pursue a full combinatorial characterization of mathrm text ls ls perfect graphs and make progress towards such a characterization by establishing a new close relationship among mathrm text ls ls perfect graphs near bipartite graphs and a newly introduced concept of full support perfect graphs

on the matrix cut rank of polyhedra lovasz and schrijver 1991 described a semidefinite operator for generating strong valid inequalities for the 0 1 vectors in a prescribed polyhedron among their results they showed that n iterations of the operator are sufficient to generate the convex hull of 0 1 vectors contained in a polyhedron in n space we give a simple example having chvatal rank 1 that meets this worst case bound of n we describe another example requiring n iterations even when combining the semidefinite and gomory chvatal operators this second example is used to show that the standard linear programming relaxation of a k city traveling salesman problem requires at least k 8 iterations of the combined operator this bound is best possible up to a constant factor as k 1 iterations suffice