on the l r hodge theory in complete non compact riemannian manifolds we study solutions for the hodge laplace equation delta u omega on p forms with displaystyle l r estimates for displaystyle r 1 our main hypothesis is that delta has a spectral gap in displaystyle l 2 we use this to get sl non classical displaystyle l r hodge decomposition theorems an interesting feature is that to prove these decompositions we never use the boundedness of the riesz transforms in displaystyle l s par these results are based on a generalisation of the raising steps method to complete non compact riemannian manifolds

on a conjecture concerning integral real roots of certain cubic polynomials abstract in this paper we determine all the rational pairs x n such that f n x 3 x 3 7 10 n x 2 2 6 n 2 11 n 8 x 4 n 3 6 n 2 10 n 24 0 it follows that for each positive integer n 5 there is no integer solution x for the polynomial this confirms a conjecture of li et al concerning the uniqueness of a class of optimal graphs in the study of an extremal graph theory problem

bounds for the matching number and cyclomatic number of a signed graph in terms of rank abstract a signed graph g is a graph with a sign attached to each of its edges where g is the underlying graph of g let m g c g and r g be the matching number the cyclomatic number and the rank of the adjacency matrix of g respectively in this paper we investigate the relation among the rank the matching number and the cyclomatic number of a signed graph and prove that 2 m g 2 c g r g 2 m g c g furthermore signed graphs reaching the lower bound or the upper bound are respectively characterized

finite group extensions and the atiyah conjecture the atiyah conjecture for a discrete group g states that the l 2 betti numbers of a finite cw complex with fundamental group g are integers if g is torsion free and in general that they are rational numbers with denominators determined by the finite subgroups of g here we establish conditions under which the atiyah conjecture for a torsion free group g implies the atiyah conjecture for every finite extension of g the most important requirement is that h g z p is isomorphic to the cohomology of the p adic completion of g for every prime p an additional assumption is necessary e g that the quotients of the lower central series or of the derived series are torsion free we prove that these conditions are fulfilled for a certain class of groups which contains in particular artin s pure braid groups and more generally fundamental groups of fiber type arrangements free groups fundamental groups of orientable compact surfaces certain knot and link groups primitive positive one relator groups and products of these therefore every finite in fact every elementary amenable extension of these groups satisfies the atiyah conjecture as a consequence if such an extension h is torsion free then the group ring ch contains no non trivial zero divisors i e h fulfills the zero divisor conjecture in the course of the proof we prove that if these extensions are torsionfree then they have plenty of non trivial torsion free quotients which are virtually nilpotent all of this applies in particular to artin s full braid group therefore answering question b6 on www grouptheory org our methods also apply to the baum connes conjecture this is discussed in 47 where for example the baum connes conjecture is proved for the full braid groups msc 55n25 homology with local coecients 16s34 group rings laurent rings 57m25 knots and links

on the sum of laplacian eigenvalues of a signed graph abstract for a signed graph let e denote the number of edges and s k denote the sum of the k largest eigenvalues of the laplacian matrix of we conjecture that for any signed graph with n vertices s k e k 1 2 1 holds for k 1 n we prove the conjecture for any signed graph when k 2 and prove that this conjecture is true for unicyclic and bicyclic signed graphs

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

signed bicyclic graphs minimizing the least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g 1 1 is the sign function on the edges of g for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertices degrees of g and a is the signed adjacency matrix the least laplacian eigenvalue is zero if and only if the signed graph is balanced i e all cycles contain an even number of negative edges here we show that among the unbalanced bicyclic signed graphs of given order n 5 the least laplacian eigenvalue is minimal for signed graphs consisting of two triangles only one of which is unbalanced connected by a path we also identify the signed graphs minimizing the least eigenvalue among those whose unbalanced bicyclic base is a theta graph

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

on the l 2 overline partial cohomology of certain complete kahler metrics let v be a compact and irreducible complex space of complex dimension v whose regular part is endowed with a complete hermitian metric h let pi m rightarrow v be a resolution of v under suitable assumptions on h we prove that begin aligned h v q 2 overline partial mathrm reg v h cong h v q overline partial m quad q 0 ldots v end aligned then we show that the previous isomorphism applies to the case of saper type kahler metrics as introduced by grant melles and milman and to the case of complete kahler metrics with finite volume and pinched negative sectional curvatures

on eigenspaces of some compound signed graphs abstract in the theory of simple graphs the concepts of the line and subdivision graph as compound graphs are well known it is possible to consider them also in the context of edge signed graphs some relations between the laplacian spectrum of signed graphs and adjacency spectra of their associated compound signed graphs have been recently established in the literature in this paper we study the relations between the corresponding eigenspaces

riesz transforms and spectral multipliers of the hodge laguerre operator abstract on r d endowed with the laguerre probability measure we define a hodge laguerre operator l acting on differential forms here is the laguerre exterior differentiation operator defined as the classical exterior differential except that the partial derivatives x i are replaced by the laguerre derivatives x i x i and is the adjoint of with respect to inner product on forms defined by the euclidean structure and the laguerre measure we prove dimension free bounds on l p 1 p for the riesz transforms l 1 2 and l 1 2 as applications we prove the strong hodge de rahm kodaira decomposition for forms in l p and deduce existence and regularity results for the solutions of the hodge and de rham equations in l p we also prove that for suitable functions m the operator m l is bounded on l p 1 p

spectra of signed graphs with two eigenvalues abstract in this paper we study connected signed graphs with 2 eigenvalues from several theoretical and computational perspectives we give some basic results concerning the eigenvalues and cyclic structure of such signed graphs in particular we complete the list of those that are 3 or 4 regular there is a natural relation between signed graphs and systems of lines in a euclidean space that are pairwise orthogonal or at fixed angle with a special role of those with 2 eigenvalues in this context we derive a relative bound for the number of such lines an extension of the similar bound related to unsigned graphs we also determine all such graphs whose negative eigenvalue in not less than 2 except for so called exceptional signed graphs using the computer search we determine those with at most 10 vertices several constructions are given and the possible spectra of those with at most 30 vertices are listed

spectral flow and dixmier traces abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi finite von neumann algebra our results have several applications we deduce a formula for the chern character of an odd l 1 summable breuer fredholm module in terms of a hochschild 1 cycle we explain how to derive a wodzicki residue for pseudo differential operators along the orbits of an ergodic r n action on a compact space x finally we give a short proof of an index theorem of lesch for generalised toeplitz operators

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

on the laplacian coefficients of signed graphs abstract let g be a signed graph where g is its underlying graph and its sign function defined on edges of g a signed graph the subgraph of is its signed tu subgraph if the signed graph induced by the vertices of consists of trees and or unbalanced unicyclic signed graphs let l d g a be the laplacian of in this paper we express the coefficient of the laplacian characteristic polynomial of based on the signed tu subgraphs of and establish the relation between the laplacian characteristic polynomial of a signed graph with adjacency characteristic polynomials of its signed line graph and signed subdivision graph as an application we identify the signed unicyclic graphs having extremal coefficients of the laplacian characteristic polynomial

a characterization of oriented hypergraphic laplacian and adjacency matrix coefficients abstract an oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure in this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs additionally we provide bounds on the determinant and permanent of the laplacian matrix characterize the oriented hypergraphs in which the upper bound is sharp and demonstrate that the lower bound is never achieved

growth of maps distortion in groups and symplectic geometry in the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism the uniform norm of the differential of its n th iteration the word length of its n th iteration where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms we find lower bounds for the growth rates of these sequences in a number of situations these bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism they are obtained by using recent results of schwarz on floer homology as an application we prove non existence of certain non linear symplectic representations for finitely generated groups

on the laplacian coefficients of acyclic graphs abstract let g be a graph of order n and let g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix zhou and gutman recently proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for stars a new proof and a strengthening of this result is provided a relation to the wiener index is discussed

symmetric differentials and the fundamental group helene esnault asked whether a smooth complex projective variety x with infinite fundamental group must have a nonzero symmetric differential meaning that h x s x 6 0 for some i 0 this was prompted by the second author s work 23 in fact severi had wondered in 1949 about possible relations between symmetric differentials and the fundamental group 34 p 36 we know from hodge theory that the cotangent bundle x has a nonzero section if and only if the abelianization of 1x is infinite the geometric meaning of other symmetric differentials is more mysterious and it is intriguing that they may have such a direct relation to the fundamental group in this paper we prove the following result on esnault s question in the slightly broader setting of compact kahler manifolds

floer homology for magnetic fields with at most linear growth on the universal cover abstract the floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold as proved by abbondandolo and schwarz salamon and weber and viterbo in this paper we show that in the presence of a dirac magnetic monopole which admits a primitive with at most linear growth on the universal cover the floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology as a consequence we prove that for any atoroidal free homotopy class and any sufficiently small 0 any magnetic flow associated to the dirac magnetic monopole has a closed orbit of period belonging to the given free homotopy class in the case where the dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the conley conjecture for hamiltonians that are quadratic at infinity i e we show that such hamiltonians have infinitely many periodic orbits

oriented gain graphs line graphs and eigenvalues abstract a theory of orientation on gain graphs voltage graphs is developed to generalize the notion of orientation on graphs and signed graphs using this orientation scheme the line graph of a gain graph is studied for a particular family of gain graphs with complex units matrix properties are established as with graphs and signed graphs there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph

characteristic classes and distortion of diffeomorphisms let m and n be manifolds of bounded geometry that is the absolute value of their curvatures as well as their covariant derivatives are bounded from above and their injectivity radii are bounded from below a map f m n has bounded d stortion if there is an upper bound on the norm of the derivative d f a bounded distortion diffeomorphism cb d d is assumed to have an inverse which also has bounded distortion similarly homotopies are assumed to have bounded distortion note that this implies that homotopic maps are a finite distance apart we prove the following theorem that describes when characteristic classes of a compact manifold obstruct the existence of bounded distortion diffeomorphisms of universal covers theorem there are compact manifolds a and b with fundamental group r and a map f a b with jcpcb i pca for some rational pontrjagin class p whose universal cover is b d homotopic to a b d diffeomorphism if and only if r is nonamenable

comparison theorems for forced functional differential equations abstract in this paper we establish comparison results by which oscillating and asymptotic properties of solutions of forced differential equations with diviating arguments of the form d dt 1 a n 1 t d dt d dt 1 a 1 t d dt x t h t x g t q t are inherited from the same properties of solutions of the unforced ordinary equation d dt 1 a n 1 t d dt d dt 1 a 1 t d dt x t h t x t 0 the obtained results extend the oscillation criteria of trench and kusano and naito to the more general equation

geometry of domains with the uniform squeezing property abstract we introduce the notion of domains with uniform squeezing property study various analytic and geometric properties of such domains and show that they cover many interesting examples including teichmuller spaces and hermitian symmetric spaces of non compact type the properties supported by such manifolds include pseudoconvexity hyperconvexity kahler hyperbolicity vanishing of cohomology groups and quasi isometry of various invariant metrics it also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of serre about stein properties of holomorphic fiber bundles

spectral properties of complex unit gain graphs abstract a complex unit gain graph is a graph where each orientation of an edge is given a complex unit which is the inverse of the complex unit assigned to the opposite orientation we extend some fundamental concepts from spectral graph theory to complex unit gain graphs we define the adjacency incidence and laplacian matrices and study each of them the main results of the paper are eigenvalue bounds for the adjacency and laplacian matrices

signed graphs with extremal least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g is the corresponding sign function for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertex degrees of g and a is the signed adjacency matrix it is well known that is balanced that is each cycle contains an even number of negative edges if and only if the least laplacian eigenvalue n 0 therefore if is not balanced then n 0 we show here that among unbalanced connected signed graphs of given order the least eigenvalue is minimal for an unbalanced triangle with a hanging path while the least eigenvalue is maximal for the complete graph with the all negative sign function

topological complexity of symplectic manifolds open access via springer compact agreement the authors wish to thank ay se borat michael farber jarek kedra and john oprea for helpful comments regarding an earlier draft of this paper

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

a vanishing theorem on generalized cartan hartogs domain of the second type abstract in this paper we consider the vanishing theorem of square integrable harmonic forms with respect to the bergman metric on the second type of generalized cartan hartogs domain by using three main tools the properties of bergman kernel function holomorphic invariant and holomorphic automorphism group

a sobolev mapping property of the bergman kernel we prove that if d is a pseudoconvex domain with lipschitz boundary having an exhaustion function rho such that rho eta is plurisubharmonic then the bergman projection maps the sobolev space w s boundedly to itself for any s eta 2

compact kahler manifolds homotopic to negatively curved riemannian manifolds in this paper we show that any compact kahler manifold homotopic to a compact riemannian manifold with negative sectional curvature admits a kahler einstein metric of general type moreover we prove that on a compact symplectic manifold x homotopic to a compact riemannian manifold with negative sectional curvature for any almost complex structure j compatible with the symplectic form there is no non constant j holomorphic entire curve f mathbb c rightarrow x

quantization of multiply connected manifolds the standard berezin toeplitz geometric quantization of a compact kahler manifold is restricted by integrality conditions these restrictions can be circumvented by passing to the universal covering space provided that the lift of the symplectic form is exact i relate this construction to the baum connes assembly map and prove that it gives a strict quantization of the original manifold i also propose a further generalization classify the required structure and provide a means of computing the resulting algebras these constructions involve twisted group c algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form this provides an algebraic counterpart to the morita equivalence of a symplectic manifold with its fundamental group

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

torus fibrations and localization of index ii local index for acyclic compatible system we give a framework of localization for the index of a dirac type operator on an open manifold suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets each of which has a structure of the total space of a torus bundle under an acyclic condition we define the index of the dirac type operator by using the witten type deformation and show that the index has several properties such as excision property and a product formula in particular we show that the index is localized on the compact set

on signed graphs with just two distinct laplacian eigenvalues abstract we investigate signed graph whose laplacian matrix has just two distinct eigenvalues all simple connected signed graphs with maximum degree at most 4 and just two distinct laplacian eigenvalues are characterized in this paper

a note on euler number of locally conformally kahler manifolds let m 2n be a compact riemannian manifold of non positive resp negative sectional curvature we call m j theta a d bounded locally conformally kahler manifold if the lifted lee form tilde theta on the universal covering space of m is d bounded we show that if m 2n is homeomorphic to a d bounded lck manifold then its euler number satisfies the inequality 1 n chi m 2n ge resp 0

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

on euler characteristic and fundamental groups of compact manifolds let m be a compact riemannian manifold pi widetilde m rightarrow m be the universal covering and omega be a smooth 2 form on m with pi omega cohomologous to zero suppose the fundamental group pi 1 m satisfies certain radial quadratic resp linear isoperimetric inequality we show that there exists a smooth 1 form eta on widetilde m of linear resp bounded growth such that pi omega d eta as applications we prove that on a compact kahler manifold m omega with pi omega cohomologous to zero if pi 1 m is mathrm cat 0 or automatic resp hyperbolic then m is kahler non elliptic resp kahler hyperbolic and the euler characteristic 1 frac dim mathbb r m 2 chi m ge 0 resp 0

spectral characterizations of signed lollipop graphs abstract let g be a signed graph where g is the underlying simple graph and e g is the sign function on the edges of g in this paper we consider the spectral characterization problem extended to the adjacency matrix and laplacian matrix of signed graphs after giving some basic results we study the spectral determination of signed lollipop graphs and we show that any signed lollipop graph is determined by the spectrum of its laplacian matrix

the oscillation of higher order differential equations with deviating arguments in this paper some new criteria for the oscillation of higher order functional differential equations of the form lnx t f t x g t 0 n is even are established some of our results are obtained via comparing it with second order ordinary linear and first order delay differential equations the oscillation of this equation when n is odd is also considered then we shall use the obtained results to study the oscillatory behavior of the neutral functional differential equations of the form ln x t c t x t f t x g t 0 n is even and the damped functional differential equations of the type lnx t h t x g t ddtx h t 0 n is even the obtained results extend improve and correlate a number of existing criteria

on the signless laplacian coefficients of unicyclic graphs let g be a graph of order n and let qg x i 0n 1 ipi g xn i be the characteristic polynomial of the signless laplacian of g let eg n respectively cg sn g 1 denote the unicyclic graph of order n obtained by a coalescence of a vertex in the cycle cg with an end vertex respectively the center of the path pn g 1 respectively the star sn g 1 it is proved that for k 2 n 1 as g varies over all unicyclic graphs of order n depending on k and n the maximum value of pk g is attained at g cn or e3 n and the minimum value is attained uniquely at g c4 sn 3 or c3 sn 2 except for the resolution of a conjecture on cubic polynomials the uniqueness issue for the maximization problem is also settled

the first l2 betti number of certain homogeneous spaces including classifying spaces for variations of hodge structures in this paper we prove the vanishing of the first l betti number for certain complex homogeneous spaces d g v where g is a semi simple lie group and v is a compact subgroup l cohomology is the appropriate extension of hodge theory for harmonic forms to the case of a noncompact complete manifold x inasmuch as here every l cohomology class can be represented by an l harmonic form see 1 in particular non existence of non trivial l harmonic p forms means that the p th l betti number vanishes besides offering the possibility to extend hodge theory to the noncompact case it can be used to obtain topological information about compact quotients of x by the lindex theorem of atiyah 1 based on this theorem dodziuk singer 4 12 suggested to use l cohomology to approach the hopf conjecture concerning the euler characteristic of riemannian manifolds of non positive curvature which has been verified in several special but important cases 2 3

on the strong l p hodge decomposition over complete riemannian manifolds we establish the strong lp hodge decomposition theorem and the lp poincare inequalities on differential forms over complete non compact riemannian manifolds as applications we prove some lp estimates and existence theorems for the de rham operator as well as some vanishing theorems for the lp cohomology and the lp torsion over complete non compact riemannian manifolds with suitable geometric conditions

cohomologie l 2 des varietes qale we give a topological interpretation of the space of l 2 harmonis forms of some qale manifolds introduced by d joyce we introduce a analytical criterium which make possible the used of mayer vietoris sequence

on pseudo harmonic maps in conformal geometry we extend harmonic map techniques to the setting of more general differential equations in conformal geometry we discuss existence theorems and obtain an extension of siu s strong rigidity to kahler weyl geometry other applications include topological obstructions to the existence of kahler weyl structures for example we show that no co compact lattice in so 1 n n 2 can be the fundamental group of a compact kahler weyl manifold