the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

on the uniqueness problems of entire functions and their derivatives in this paper we obtain some uniqueness theorems for entire functions and their derivatives sharing the same fixed points with the same multiplicities

the inverse eigenvalue problem for nonnegative matrices abstract let a be a nonnegative matrix with spectrum 1 2 m and b be a nonnegative matrix with spectrum 1 2 n where 1 is the perron eigenvalue of b furthermore let a maximal diagonal element of a be greater than or equal to 1 in the article we construct a nonnegative matrix c with spectrum 1 2 m 2 n this construction enables us to obtain several results on how to determine new realizable lists from known realizable lists

correction to meromorphic functions that share four values it has been brought to my attention by norbert steinmetz that the symmetry reasoning that is used in the proof of theorem 2 in 1 to conclude that d1 is symmetric in a and b in the third line from the bottom on page 564 and also to conclude that 123 holds on page 565 is not valid we note that the symmetry reasoning that is used in the sentence in lines 8 and 9 from the bottom on page 564 is valid because this sentence is meant in the sense of the following sentence by considering a b point that is simple for f and double for 9 a similar argument to the above argument with 61 instead of 60 will produce

on the graph complement conjecture for minimum rank abstract the minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so one of the many unresolved questions on this topic is the so called graph complement conjecture which grew out of a workshop in 2006 this conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement and may be classified as a nordhaus gaddum type problem involving the graph parameter minimum rank the conjectured bound is the order of the graph plus two other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the colin de verdiere type parameter we show that if the graph complement conjecture is true for two graphs then it is true for the join of these graphs related results for the graph complement conjecture and the positive semidefinite version for joins of graphs are also established we also report on the use of recent results on partial k trees to establish the graph complement conjecture for graphs of low minimum rank

the strong spectral property for graphs abstract we introduce the set g ssp of all simple graphs g with the property that each symmetric matrix corresponding to a graph g g ssp has the strong spectral property we find several families of graphs in g ssp and in particular characterise the trees in g ssp

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

meromorphic functions that share four values an old theorem of r nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities then the functions are mbbius transformations of each other two of the shared values are picard values for both functions and the cross ratio of a particular permutation of the shared values equals 1 in this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities then the functions share all four values counting multiplicities

on a result of ozawa concerning uniqueness of meromorphic functions we prove a uniqueness theorem for meromorphic functions sharing three values with some finite weight which improves a recent result of the author

meromorphic functions sharing three weighted values in this paper we deal with the relation between the characteristics of two meromorphic functions that share three weighted values which improve the results of q c zhang and through the results two conjectures which proposed by mues and osgood yang are solved in a weaker condition

a nordhaus gaddum conjecture for the minimum number of distinct eigenvalues of a graph abstract we propose a nordhaus gaddum conjecture for q g the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph g for every graph g excluding four exceptions we conjecture that q g q g c g 2 where g c is the complement of g we compute q g c for all trees and all graphs g with q g g 1 and hence we verify the conjecture for trees unicyclic graphs graphs with q g 4 and for graphs with g 7

a recursive procedure to generate all cuts for 0 1 mixed integer programs we study several ways of obtaining valid inequalities for mixed integer programs we show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to gomory s mixed integer cuts we also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities

on meromorphic functions that share three values of finite weights abstract a uniqueness theorem for two distinct non constant meromorphic functions that share three values of finite weights is proved which generalizes two previous results by h x yi and x m li and h x yi as applications of it many known results by h x yi and p li etc could be improved furthermore with the concept of finite weight sharing extensions on osgood yang s conjecture and mues conjecture and a generalization of some prevenient results by m ozawa and h ueda ect could be obtained

the second main theorem for small functions and related problems we shall establish the following three results in more general forms 1 the second main theorem for small functions let f be a meromorphic function on the complex plane c let a1 aq be distinct meromorphic functions on c assume that ai are small with respect to f i e t r a 0 corollary 1 here as usual in nevanlinna theory the terms t r f and n r ai f denote for the characteristic function and the truncated counting function respectively 2 application to functional equations let kc be the field of meromorphic functions on c for a function r 0 r put k c a kc t r a o r which is a subfield of kc then the following holds let f x y k c x y be a polynomial in two variables over k c assume that the curve f x y 0 over k c has genus greater than one if 1 2 kc satisfy the functional equation f 1 2 0 then both 1 and 2 are contained in k c corollary 2 3 height inequality for curves over function fields let k be a function field of one variable over c let x be a smooth projective curve over k let d x be a reduced divisor let l be an ample line bundle on x and let e 0 then we have hk kx d p n 1 k s d p dk p ehk l p oe 1 for all p x k d theorem 5 here the notations are introduced in v1 v3 see also section 9 our proof uses ahlfors theory of covering surfaces and the geometry of the moduli space of q pointed stable curves of genus 0

reduce and split cuts improving the performance of mixed integer gomory cuts mixed integer gomory cuts have become an integral part of state of the art software for solving mixed integer linear programming problems therefore improvements in the performance of these cutting planes can be of great practical value in this paper we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed integer gomory cuts this is motivated by the fact that in a mixed integer gomory cut the coefficient of an integer variable lies between 0 and 1 whereas for a continuous variable there is no upper bound the heuristic tries to reduce the coefficients of the continuous variables we call the resulting cuts reduce and split cuts we found that on several test problems reduce and split cuts can substantially enhance the performance of a branch and bound algorithm

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

a connection between cutting plane theory and the geometry of numbers in this paper we relate several questions about cutting planes to a fundamental problem in the geometry of numbers namely the closest vector problem using this connection we show that the dominance membership and validity problems are np complete for chvatal and split cuts

uniqueness of meromorphic functions and a question of k tohge we prove a uniqueness theorem for meromorphic functions sharing three values which answers a question asked by k tohge

on the separation of split cuts and related inequalities the split cuts of cook kannan and schrijver are general purpose valid inequalities for integer programming which include a variety of other well known cuts as special cases to detect violated split cuts one has to solve the associated separation problem the complexity of split cut separation was recently cited as an open problem by cornuejols li cl01 in this paper we settle this question by proving strong completeness of separation for split cuts as a by product we also show completeness of separation for several other classes of inequalities including the mir inequalities of nemhauser and wolsey and some new inequalities which we call balanced split cuts and binary split cuts we also strengthen completeness results of caprara fischetti cf96 for cuts and eisenbrand e99 for chvatal gomory cuts to compensate for this bleak picture we also give a positive result for the symmetric travelling salesman problem we show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle

the nowhere zero eigenbasis problem for a graph using the implicit function theorem it is shown that for any n distinct real numbers 1 2 n and for each connected graph g of order n there is a real symmetric matrix a whose graph is g the eigenvalues of a are 1 n and every entry in each eigenvector of a is nonzero

on the rank of mixed 0 1 polyhedra for a polytope in the 0 1 n cube eisenbrand and schulz showed recently that the maximum chvatal rank is bounded above by o n 2logn and bounded below by 1 e n for some e 0 chvatal cuts are equivalent to gomory fractional cuts which are themselves dominated by gomory mixed integer cuts what do these upper and lower bounds become when the rank is defined relative to gomory mixed integer cuts an upper bound of n follows from existing results in the literature in this note we show that the lower bound is also equal to n this result still holds for mixed 0 1 polyhedra with n binary variables

on the characteristics of meromorphic functions that share three values cm in this paper by studying the counting functions of the common 1 points of meromorphic functions a more precise relation between the characteristics of meromorphic functions that share three values cm has been obtained as applications of this many known results can be improved

projected chvatal gomory cuts for mixed integer linear programs recent experiments by fischetti and lodi show that the first chvatal closure of a pure integer linear program ilp often gives a surprisingly tight approximation of the integer hull they optimize over the first chvatal closure by modeling the chvatal gomory cg separation problem as a mixed integer linear program milp which is then solved by a general purpose milp solver unfortunately this approach does not extend immediately to the gomory mixed integer gmi closure of an milp since the gmi separation problem involves the solution of a nonlinear mixed integer program or a parametric milp in this paper we introduce a projected version of the cg cuts and study their practical effectiveness for milp problems the idea is to project first the linear programming relaxation of the milp at hand onto the space of the integer variables and then to derive chvatal gomory cuts for the projected polyhedron though theoretically dominated by gmi cuts projected cg cuts have the advantage of producing a separation model very similar to the one introduced by fischetti and lodi which can typically be solved in a reasonable amount of computing time

some further results on weighted sharing three values and brosch s theorem using the notion of weighted sharing of values we prove two uniqueness theorems which improve the results proved by t c alzahary t c alzahary weighted sharing three values and brosch s theorem j math anal appl 323 2006 8 25 t c alzahary on a result of ueda concerning unicity of meromorphic functions kodai math j 30 2007 140 146 under a weaker hypothesis

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

approximate fixed rank closures of covering problems consider a 0 1 integer program min ctx ax b x 0 1 n where a is nonnegative we show that if the number of minimal covers of ax b is polynomially bounded then for any e 0 and any fixed q there is a polynomially large lift and project relaxation whose value is at least 1 e times the value of the rank q relaxation a special case of this result is that given by set covering problems or generally problems where the coefficients in a and b are bounded

exponential lower bounds on the lengths of some classes of branch and cut proofs we examine the complexity of branch and cut proofs in the context of 0 1 integer programs we establish an exponential lower bound on the length of branch and cut proofs that use 0 1 branching and lift and project cuts called simple disjunctive cuts by some authors gomory chvatal cuts and cuts arising from the n0 matrix cut operator of lovasz and schrijver a consequence of the lower bound result in this paper is that branch and cut methods of the type described above have exponential running time in the worst case

small functions and uniqueness of meromorphic functions we discuss the uniqueness of meromorphic functions sharing three weighted values and provide a complete answer to a question of t c alzahary

an improvement of the nevanlinna gundersen theorem a well known result of nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane c and for four distinct values aj c if f aj g aj for all 1 j 4 then g is a mobius transformation of f in 1983 gundersen generalized the result of nevanlinna to the case where the above condition is replaced by min f aj 1 min g aj 1 for j 1 2 and f aj g aj for j 3 4 in this paper we prove that the theorem of gundersen remains valid to the case where min f aj 1 min g aj 1 for j 1 2 and min f aj 2 min g aj 2 for j 3 4 furthermore we work on the case where aj are small functions

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

on uniqueness of meromorphic functions sharing finite sets in this article we first study polynomials p w such that p f cp g implies f g for any nonzero constant c and nonconstant meromorphic functions f and g on next we give some sufficient conditions for a finite set s to be a uniqueness range set namely to satisfy the condition that f 1 s g 1 s implies f g for any nonconstant meromorphic functions f and g on

correction to meromorphic functions that share four values trans amer math soc 277 1983 no 2 545 567 mr0694375 84g 30028