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

analytic formulas for the regulator of a number field letr rk andw wk be the regulator and the number of roots of unity in the number fieldk we determine allk for whichr w 1 8 there are nine such fields sincew 2 we have allk withr 1 4 in particular we find the first three minima ofr over allk there are three main ingredients to the proof i a new analytic formula theorem a below ii a refinement of remak s geometric approach iii a refinement of zimmert s analytic method

linear systems with signed solutions linear systems ax b for which the sign patterns a and b completely determine the set of sign patterns of vectors in the solution set are studied these systems generalize the existing notion of sign solvable linear systems linear systems with signed solutions are characterized in terms of two classes of matrices the first class l matrices plays a central role in the study of sign solvable linear systems the second class is new and generalizes the class of s matrices that arise in sign solvable linear systems several constructions for the new class of matrices are given

locally conformal kahler manifolds with potential a locally conformally kahler lck manifold m is one which is covered by a kahler manifold widetilde m with the deck transformation group acting conformally on widetilde m if m admits a holomorphic flow acting on widetilde m conformally it is called a vaisman manifold neither the class of lck manifolds nor that of vaisman manifolds is stable under small deformations we define a new class of lck manifolds called lck manifolds with potential which is closed under small deformations all vaisman manifolds are lck with potential we show that an lck manifold with potential admits a covering which can be compactified to a stein variety by adding one point this is used to show that any lck manifold m with potential dim m 3 can be embedded into a hopf manifold thus improving similar results for vaisman manifolds ornea and verbitsky math ann 332 121 143 2005

oeljeklaus toma manifolds and arithmetic invariants we consider oeljeklaus toma manifolds coming from number fields with precisely one complex place our general theme is to relate the geometry to the arithmetic we show that just knowing the fundamental group allows us to recover the number field we also show that this fails if there are more complex places the first homology turns out to be related to an interesting ideal we compute the volume in terms of the discriminant and regulator of the number field is there a conceptual reason for this we explore this and see what happens if we entirely experimentally regard them as baby siblings of hyperbolic manifolds coming from number fields

on the digraphs of sign solvable linear systems abstract a linear system of equations ax b is sign solvable if it is solvable and both its solvability and the sign pattern of its solution vector x are uniquely determined by the sign patterns of a and b it was proved in l bassett j maybee j quirk economitrica 36 1968 544 563 r a bruali b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 that the sign solvability of a linear system ax b in standard form and with a square coefficient matrix a can be described in terms of the signed digraph s a of the matrix a with a specified vertex subset depending on the sign pattern of the vector b in this paper we define the notion of w sign solvable signed digraphs which represent precisely the sign solvable linear systems in standard form we give necessary and sufficient conditions for the underlying unsigned digraphs of w sign solvable signed digraphs in terms of forbidden subdigraphs we also give several necessary and sufficient conditions for strongly connected w sign solvable signed digraphs

nonlinear discrete inequalities with two variables and their applications abstract an error in the proof of theorem 1 by zheng et al k zheng y wu s zhong discrete nonlinear integral inequalities in two variables and their applications appl math comput 207 2009 140 147 is reported this paper gives the right proof under some additional condition application examples to show boundedness and uniqueness of solutions of a volterra type difference equation are also given

predecessor property full combinatorial column rank and the height characteristics of an m matrix abstract for a loopless acyclic transitive directed graph we give a new result about the relation between the t predecessor property of subsets of its levels and the property of having full combinatorial column rank for certain blocks in its adjacency matrix as a corollary we give new lower bounds for the i th height characteristic number of a singular m matrix an open question about determining the height characteristics of singular m matrices with a prescribed singular graph is also posed

an immersion theorem for vaisman manifolds a locally conformally kahler lck manifold is a complex manifold admitting a kahler covering open image in new window with monodromy acting on open image in new window by kahler homotheties a compact lck manifold is vaisman if it admits a holomorphic flow acting by non trivial homotheties on open image in new window we prove that any compact vaisman manifold admits a natural holomorphic immersion to a hopf manifold n 0 as an application we obtain that any sasakian manifold has a contact immersion to an odd dimensional sphere

complex geometry of moment angle manifolds moment angle manifolds provide a wide class of examples of non kahler compact complex manifolds a complex moment angle manifold mathcal z is constructed via certain combinatorial data called a complete simplicial fan in the case of rational fans the manifold mathcal z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori in general a complex moment angle manifold mathcal z is equipped with a canonical holomorphic foliation mathcal f which is equivariant with respect to the mathbb c times m action examples of moment angle manifolds include hopf manifolds of vaisman type calabi eckmann manifolds and their deformations we construct transversely kahler metrics on moment angle manifolds under some restriction on the combinatorial data we prove that any kahler submanifold or more generally a fujiki class mathcal c subvariety in such a moment angle manifold is contained in a leaf of the foliation mathcal f for a generic moment angle manifold mathcal z in its combinatorial class we prove that all subvarieties are moment angle manifolds of smaller dimension and there are only finitely many of them this implies in particular that the algebraic dimension of mathcal z is zero

systems of multidimensional discrete inequalities on considere des systemes lineaires de m inegalites discretes comprenant des fonctions a n variables independantes

about some new integral inequalities of wendroff type for discontinuous functions abstract in this article we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables wendroff type by including also inequalities with delay we deduce new generalizations of earlier results given by r p agarwal r bellman i bihari b k bondge v lakshmikantham s leela b g pachpatte for continuous and discrete functions furthermore generalizations of some results for integro sum inequalities are obtained as well

zero forcing for sign patterns we introduce a new variant of zero forcing signed zero forcing the classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph i e zero nonzero pattern our new variant provides an analogous bound for the maximum nullity of a matrix with a given sign pattern this allows us to compute for instance the maximum nullity of a z matrix whose graph is l kn the line graph of a clique

on the metric structure of non kahler complex surfaces we give a characterization of a locally conformally kahler l c k metric with parallel lee form on a compact complex surface using the kodaira classification of surfaces we classify the compact complex surfaces admitting such structures this gives a classification of sasakian structures on compact three manifolds a weak version of the above mentioned characterization leads to an explicit construction of l c k metrics on all hopf surfaces we characterize the locally homogeneous l c k metrics on geometric complex surfaces and we prove that some inoue surfaces do not admit any l c k metric

on digraphs and forbidden configurations of strong sign nonsingular matrices a square real matrix a called a strong sign nonsingular matrix or s2ns matrix if all matrices with the same sign pattern as a nonsingular and the inverses of these matrices all have the same sign pattern a digraph which is the underlying digraph of the signed digraph of an s2ns matrix with a negative main diagonal is called an s2ns digraph in 9 thomassen gave a characterization of strongly connected s2ns digraphs in terms of the forbidden subdigraphs in 2 brualdi and shader constructed minimal forbidden configurations for s2ns digraphs for the general cases where the digraphs considered are not necessarily strongly connected they also proposed the problem about the existence of new minimal forbidden configurations other than those found in 2 9 in this paper we construct infinitely many new basic minimal forbidden configurations and thus obtain the answer this problem we also obtain several necessary conditions for minimal forbidden configurations and give a generalization of thomassen s theorem

a new positivity condition for the curvature of hermitian manifolds in this note we introduce a new type of positivity condition for the curvature of a hermitian manifold which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non kahler case we derive a bochner formula for closed 1 1 forms from which this condition appears naturally and prove that if a hermitian manifold satisfies our positivity condition then any class alpha in h 1 1 bc x can be represented by a closed 1 1 form which is parallel with respect to the bismut connection lastly we show that such a curvature positivity condition holds on certain generalized hopf manifolds and on certain vaisman manifolds

sharp conditions for existence of nontrivial invariant cones of nonnegative initial values of difference equations it is proved that an arbitrary scalar difference equation has a nontrivial invariant cone in the set of the nonnegative initial values if and only if the associated characteristic equation has a positive root by using this fact there is given a construction of the maximal invariant cone in the set of nonnegative initial values via a positive root of the associated characteristic equation finally the present results are compared with some earlier ones which were obtained for delay differential equations in the same direction

linear operators which preserve sign nonsingular matrices abstract we characterize the linear operators that preserve the set of sign nonsingular matrices we show that if a linear operator t preserves the set of sign nonsingular matrices then t preserves the set of matrices of term rank 1 then by a result of beasley and pullman on preservers of matrices of term rank 1 we can obtain the structure of t

linear operators preserving l matrices abstract we investigate the linear operators that preserve the set of l matrices we show that a linear operator t preserves the set of l matrices and t is also one to one on the set of cells if and only if t strongly preserves the set of l matrices in each case t preserves the set of matrices of term rank 1 then by a result of beasley and pullman on preservers of matrices of term rank 1 we can obtain the structure of t

on the compactification of hyperconcave ends and the theorems of siu yau and nadel we show that the pseudoconcave holes of some naturally arising class of manifolds called hyperconcave ends can be filled in including the case of complex dimension two as a consequence we obtain a stronger version of the compactification theorem of siu yau and extend nadel s theorems to dimension two

nearly l matrices and generalized row sign balanced matrices abstract a real matrix a is called an l matrix if every matrix with the same sign pattern as a has linearly independent columns a nearly l matrix is a matrix which is not an l matrix but each matrix obtained by deleting one of its columns is an l matrix a generalized row sign balanced grsb matrix is a matrix which can be transformed to a matrix having both positive and negative entries in each row by multiplying some of its columns by 1 in this paper we study the relations between l matrices nearly l matrices and grsb matrices we obtain a complete characterization of nearly l matrices in terms of grsb matrices by comparing this result with a well known theorem about l indecomposable barely l matrices we find an interesting duality relation between l matrices and grsb matrices we also use grsb matrices to characterize the conditional s matrices which are closely related to nearly l matrices and the conditionally sign solvable linear systems finally we propose some unsolved problems for further research

locally conformally kahler manifolds admitting a holomorphic conformal flow a manifold m is locally conformally kahler lck if it admits a kahler covering tilde m with monodromy acting by holomorphic homotheties let m be an lck manifold admitting a holomorphic conformal flow of diffeomorphisms lifted to a non isometric homothetic flow on tilde m we show that m admits an automorphic potential and the monodromy group of its conformal weight bundle is mathbb z

zero forcing sets and the minimum rank of graphs 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 6 j is nonzero whenever i j is an edge in g and is zero otherwise this paper introduces a new graph parameter z g that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs often enabling computation of the minimum rank

on minimum rank and zero forcing sets of a graph abstract for a graph g on n vertices and a field f the minimum rank of g over f written as mr f g is the smallest possible rank over all n n symmetric matrices over f whose i j th entry for i j is nonzero whenever ij is an edge in g and is zero otherwise the maximum nullity of g over f is m f g n mr f g the minimum rank problem of a graph g is to determine mr f g or equivalently m f g this problem has received considerable attention over the years in f barioli w barrett s butler s m cioab d cvetkovic s m fallat c godsil w haemers l hogben r mikkelson s narayan o pryporova i sciriha w so d stevanovic h van der holst k v meulen a w wehe aim minimum rank special graphs work group zero forcing sets and the minimum rank of graphs linear algebra appl 428 2008 1628 1648 a new graph parameter z g the zero forcing number was introduced to bound m f g from above the authors posted an attractive question what is the class of graphs g for which z g m f g for some field f this paper focuses on exploring the above question

nonsingular sign patterns and the orthogonal group abstract a nonsingular sign pattern h is said to require property p if every matrix a in its sign class q h has property p h is said to allow property p if there is a q h which has the property a new infinite family of maximal nonsingular sign patterns h w n associated with wheels is shown to require not p 0 where p 0 a a 1 t a second infinite family h t n is shown to be associated with binary trees and to allow p 0

minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns abstract the set of real matrices described by a sign pattern a matrix whose entries are elements of 0 has been studied extensively but only loose bounds were available for the minimum rank of a tree sign pattern a simple graph has been associated with the set of symmetric matrices having a zero nonzero pattern of off diagonal entries described by the graph and the minimum rank maximum eigenvalue multiplicity among matrices in this set is readily computable for a tree in this paper we extend techniques for trees to tree sign patterns and trees allowing loops with the presence or absence of loops describing the zero nonzero pattern of the diagonal allowing precise computation of the minimum rank of a tree sign pattern and a tree allowing loops for a symmetric tree sign pattern or a tree that allows loops we provide an algorithm that allows exact computation of maximum multiplicity and minimum rank and can be used to obtain a symmetric integer matrix realizing minimum rank

on maximal sign nonsingular matrices abstract we derive several new results on maximal sign nonsingular sns matrices these include two methods for constructing new sns matrices from smaller sns matrices the results are used to produce a matrix definition for the c 4 cockades and to introduce new families of sns matrices not previously defined all of our results were deduced with the help of tables of maximal irreducible sns matrices of orders three through nine which we previously computed the interesting matrices in these tables for orders three through eight are reproduced at the end of an appendix in which we describe the computations

rectangular l matrices abstract an l matrix is an m by n 0 1 1 matrix a such that every m by n real matrix with the same sign pattern as a has linearly independent rows the most widely studied l matrices are the square l matrices also called sign nonsingular matrices in this paper we investigate rectangular l matrices we obtain and study a decomposition theorem for l matrices we introduce two new classes of l matrices which for square matrices reduce to sign nonsingular matrices the maximum number of columns for matrices in each of these classes is obtained and those matrices attaining the maximum are characterized

examples of non trivial rank in locally conformal kahler geometry we consider locally conformal kahler geometry as an equivariant homothetic kahler geometry k we show that the de rham class of the lee form can be naturally identified with the homomorphism projecting to its dilation factors thus completing the description of locally conformal kahler geometry in this equivariant setting the rank r m of a locally conformal kahler manifold is the rank of the image of this homomorphism using algebraic number theory we show that r m is non trivial providing explicit examples of locally conformal kahler manifolds with 1 nless text upshape rmfamily r m nless b 1 as far as we know these are the first examples of this kind moreover we prove that locally conformal kahler oeljeklaus toma manifolds have either r m b 1 or r m b 1 2

embeddability of some strongly pseudoconvex cr manifolds we obtain an embedding theorem for compact strongly pseudoconvex cr manifolds which are boundaries of some complete hermitian manifolds we use this to compactify some negatively curved kahler manifolds with compact strongly pseudoconvex boundary an embedding theorem for sasakian manifolds is also derived

classes of sign nonsingular matrices with a specified number of zero entries abstract an n by n matrix bn is sign nonsingular sns if every matrix with the same sign pattern as bn is nonsingular a given sns matrix determines an equivalence class with respect to transposition and multiplication by permutation and signature matrices of sns matrices all of which have the same number of zero entries such a matrix is maximal if no zero entry can be set nonzero so that the resulting matrix is sns and is fully indecomposable if it does not have an n k by k zero submatrix for some k where 1 k n 1 for fixed n the hessenberg matrix is known to represent the unique equivalence class with the minimum number of zero entries namely n 1 2 we prove that for n 5 there is exactly one equivalence class of fully indecomposable maximal sns matrices with n 1 2 1 zero entries similarly for n 5 we prove that there are exactly two such equivalence classes having n 1 2 2 zero entries for these proofs we identify two new infinite classes of fully indecomposable maximal sns matrices which can be obtained by stretching known sns matrices

reduction of sasakian manifolds we show that the contact reduction can be specialized to sasakian manifolds we prove that the sasakian reduction is compatible with the kahler reduction both in the cone construction and in the boothby wang fibration in particular applying futaki s results we obtain a sufficient condition for the reduced space of a regular sasakian einstein manifold to be sasakian einstein we present examples of sasakian einstein manifolds obtained by s1 reduction of standard sasakian spheres

recursive structure of s matrices and an o m2 algorithm for recognizing strong sign solvability abstract an s matrix is an m m 1 real matrix a such that for each matrix a of the same sign pattern as a a s columns are the vertices of an m simplex in r m that has the origin in its interior an s matrix is one that can be made into an s matrix by replacing some columns with their negatives such matrices are of interest because of their fundamental role in the study of sign solvability new results on the recursive structure of these classes of matrices are presented here and are used as the basis of algorithms of time complexity o m 2 for recognizing members of the classes and for testing the strong sign solvability of linear systems

on compact complex surfaces of kahler rank one the kahler rank of compact complex surfaces was introduced by harvey and lawson in their 1983 paper on kahler manifolds as a measure of kahlerianity here we give a partial classification of compact complex surfaces of kahler rank 1 these are either elliptic surfaces or hopf surfaces or they admit a holomorphic foliation of a very special type as a consequence we give an affirmative answer to the question raised by harvey and lawson whether the kahler rank is a bimeromorphic invariant

dedekind zeta motives for totally real number fields let k be a totally real number field for every odd n 3 we construct an element in the category mt k of mixed tate motives over k out of the quotient of a product of hyperbolic spaces by an arithmetic group by a volume calculation we prove that its period is a rational multiple of pi n k mathbb q zeta k 1 n where zeta k 1 n denotes the special value of the dedekind zeta function of k we deduce that the group mathrm ext 1 mathrm mt k mathbb q 0 mathbb q n is generated by the cohomology of a quadric relative to hyperplanes and that zeta k 1 n is a determinant of volumes of geodesic hyperbolic simplices defined over k

a comparison theorem for difference inequalities in the study of growth and decay properties of nonlinear evolution equations in particular hyperbolic and parabolic partial differential equations as t the following difference inequality which is satisfied by an energy expression arises mathop sup limits t leq s leq t 1 u s 1 alpha leq c 1 t r u t u t 1 g t

the inertia set of a signed graph abstract a signed graph is a pair g where g v e is a graph in which parallel edges are permitted but loops are not with v 1 n and e the edges in are called odd edges and the other edges of e even by s g we denote the set of all symmetric v v matrices a a i j with a i j 0 if i and j are adjacent and all edges between i and j are even a i j 0 if i and j are adjacent and all edges between i and j are odd a i j r if i and j are connected by even and odd edges a i j 0 if i j and i and j are non adjacent and a i i r for all vertices i the stable inertia set of a signed graph g is the set of all pairs p q for which there exists a matrix a s g with p positive and q negative eigenvalues which has the strong arnold property in this paper we study the stable inertia set of signed graphs

systems of multidimensional volterra integral equations and inequalities on obtient des bornes superieures explicites pour les solutions u x d equations integrales de volterra lineaires ou non lineaires on demontre des theoremes d existence globale

mixed dominating matrices abstract we characterize the class of matrices for which the set of supports of nonnegative vectors in the null space can be determined by the signs of the entries of the matrix this characterization is in terms of mixed dominating matrices which are defined by the nonexistence of square submatrices that have nonzeros of opposite sign in each row the class of mixed dominating matrices is contained in the class of l matrices from the theory of sign solvability and generalizes the class of s matrices we give a polynomial time algorithm to decide if a matrix is mixed dominating we derive combinatorial conditions on the face lattice of a gale transform of a matrix in this class

sign nonsingular matrices and even cycles in directed graphs a sign nonsingular matrix or l mahix a is a real m x n matrix such that the columns of any real m x n matrix with the same sign pattern as a are linearly independent the problem of recognizing square l matrices is equivalent to that of finding an even cycle in a directed graph in this paper we use graph theoretic methods to investigate l matrices in particular we determine the maximum number of nonzero elements in square l matrices and we characterize completely the semicomplete l matrices i e the square l matrices a j such that at least one of a and ajl is nonzero for any i j and those square l matrices which are combinatorially symmetric i e the main diagonal has only nonzero entries and a j 0 iff a jr 0 we also show that for any n x n l matrix there is an i such that the total number of nonzero entries in the i th row and i th column is less than n unless the matrix has a completely specified structure finally we discuss the algorithmic aspects

flat affine subvarieties in oeljeklaus toma manifolds the oeljeklaus toma ot manifolds are compact complex non kahler manifolds constructed by oeljeklaus and toma and generalizing the inoue surfaces their construction uses the number theoretic data a number field k and a torsion free subgroup u in the group of units of the ring of integers of k with rank of u equal to the number of real embeddings of k ot manifolds are equipped with a torsion free flat affine connection preserving the complex structure this structure is known as flat affine structure we prove that any complex subvariety of smallest possible positive dimension in an ot manifold is also flat affine this is used to show that if all elements in u setminus 1 are primitive in k then x contains no proper analytic subvarieties

comparison results for multidimensional difference equations on compare les solutions u x de l equation aux differences non lineaire x u u x f x u x avec les solutions x et x des inegalites aux differences non lineaires correspondantes x n x f x x et x n x f x x

discrete inequalities of the gronwall bellman type in n independent variables ii soit r 0 n l ensemble des entiers non negatifs et n un entier positif soit u x k x n n r et f x s n 2n r avec s x si u x k x s 0 x 1 f x s u s pour x n n alors u x k x sd 1 0 xu 1 1 1 s 0 x 1 f x s 1 s ou k x sup k x 0 s x f x s sup f t s 0 t x