density of monodromy actions on non abelian cohomology abstract in this paper we study the monodromy action on the first betti and de rham non abelian cohomology arising from a family of smooth curves we describe sufficient conditions for the existence of a zariski dense monodromy orbit in particular we show that for a lefschetz pencil of sufficiently high degree the monodromy action is dense

arithmetic monodromy groups pinkham 12 showed that the monodromy groups of the 14 exceptional unimodular singularities can be characterized arithmetically using results of m kneser we proved in 4 that this is true for all singularities of arnold s list with certain exceptions see below the object of this note is to show that this is even true for almost all isolated hypersurface singularities more precisely we prove the following theorem

degrees of non monotonicity for restarting automata in the literature various notions of monotonicity for restarting automata have been studied here we introduce two new variants of monotonicity for restarting automata and for two way restarting automata left monotonicity and right left monotonicity it is shown that for the various types of deterministic and nondeterministic two way restarting automata without auxiliary symbols these notions yield infinite hierarchies and we compare these hierarchies to each other further as a tool used to simplify some of the proofs the shrinking restarting automaton is introduced which is a generalization of the standard length reducing restarting automaton to the weight reducing case some of the consequences of this generalization are also discussed

the deformations of the simple space curve singularities in 4 and 5 marc giusti classified the simple isolated complete intersection singularities of positive dimension he showed that besides the simple hypersurface singularities there are only simple singularities of curves in complex 3 space he gave a list of adjacencies between these singularities later v v goryunov see 6 found some additional adjacencies but the complete list remained unknown see 1 p 23 in this note we prove the following theorem theorem the following diagram shows the complete list of adjacencies between the simple space curve singularities which are not hypersurface singularities here giusti s list is completed by the deformations z10 t9 and z10 u9 s5 s6 s7 s8 s9

arithmeticity of certain symplectic hypergeometric groups we give a sufficient condition on a pair of primitive polynomials that the associated hypergeometric group monodromy group of the corresponding hypergeometric differential equation is an arithmetic subgroup of the integral symplectic group

abelianisation of orthogonal groups and the fundamental group of modular varieties abstract we study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms we show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry in particular the moduli of k3 surfaces we give applications to modular forms and to computing the fundamental groups of some moduli spaces

new large rank nichols algebras over nonabelian groups with commutator subgroup z2 abstract in this article we explicitly construct new finite dimensional indecomposable nichols algebras with dynkin diagrams of type a n c n d n e 6 7 8 f 4 over any group g of order 2 l with commutator subgroup isomorphic to z 2 the construction is generic in the sense that the type just depends on the rank and center of g and thus positively answers for all groups of this class a question raised by susan montgomery in 1995 22 4 our construction uses the new notion of a covering nichols algebra as a special case of a covering hopf algebra 18 and produces non faithful nichols algebras we give faithful examples of doi twists for type a 3 c 3 d 4 c 4 f 4 over several nonabelian groups of order 16 and 32 these are hence the first known examples of nondiagonal finite dimensional indecomposable nichols algebras of rank 2 over nonabelian groups

on adjacency distance spectral radius and spread of graphs abstract let g be a connected graph the greatest eigenvalue and the spread of the sum of the adjacency matrix and the distance matrix of g are called the adjacency distance spectral radius and the adjacency distance spread of g respectively both quantities are used as molecular descriptors in chemoinformatics we establish some properties for the adjacency distance spectral radius and the adjacency distance spread by proposing local grafting operations such that the adjacency distance spectral radius is decreased or increased hence we characterize those graphs that uniquely minimize and maximize the adjacency distance spectral radii in several sets of graphs and determine trees with small adjacency distance spreads it transpires that the adjacency distance spectral radius satisfies the requirements of a branching index

on the greatest distance between two permanental roots of a matrix abstract the permanental spread of a complex square matrix a is defined to be the greatest distance between two roots of the equation per zi a 0 a preliminary study of this number as well as of two related quantities is given in particular we derive upper and lower bounds and deal with comparisons of different bounds finally two inequalities involving the permanental spread are treated

more operator versions of the schwarz inequality some new operator versions of the schwarz inequality are obtained one of them is a counterpart of the variance covariance inequality in the context of noncommutative probability

on the accuracy of the gerschgorin circle theorem for bounding the spread ofa symmetric matrix abstract the spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues the gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread for nonsymmetric matrices the gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable this is not true for real symmetric matrices it is shown that for real symmetric matrices or complex hermitian matrices the ratio between the bound and the spread is bounded by p 1 where p is the maximum number of off diagonal nonzeros in any row of the matrix for full matrices this is just n this bound is not quite sharp for n greater than 2 but examples with ratios of n 1 for all n are given for banded matrices with m nonzero bands the maximum ratio is bounded by m independent of the size of n this bound is sharp provided only that n is at least 2m for sparse matrices p may be quite small and the gerschgorin bound may be surprisingly accurate

positive linear maps and spreads of matrices ii abstract it is shown how positive linear maps can be used to obtain lower bounds for spreads of normal matrices some known results and some new ones are obtained in this way

some inequalities for central moments of matrices abstract in this paper we shall study non commutative central moment inequalities with a focus on whether the commutative bounds are tight in the non commutative case as well we prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case as an application we shall present some lower estimates of the spread of hermitian and normal matrices as well

bounds for the spectrum of normal matrices abstract let a be a normal matrix with eigenvalues 1 2 n and let denote the smallest disc containing these eigenvalues we give some inequalities relating the center and radius of to the entries in a when applied to hermitian matrices our results give lower bounds on the spread maxij i j of a when applied to positive definite hermitian matrices they give lower bounds on the kantorovich ratio maxij i j i j

new lower bounds for the randic spread abstract let g v g e g be an n m graph the randic spread of g s r g is defined as the maximum distance of its randic eigenvalues disregarding the randic spectral radius of g in this work we use numerical inequalities and bounds for the matricial spread to obtain relations between this spectral parameter and some structural and algebraic parameters of the underlying graph such as the sequence of vertex degrees the nullity randic index generalized randic indices and its independence number in the last section a comparison is presented for regular graphs

on right equivalence a c m a p g e r m g n xo p yo is an m approximation to f w r t y if g is an eb m approximat ion to f w r t 2 iii let m i or m i 0s for some ideal i c cn we will then write iapp rox ima t ion instead of m approx ima t ion no ment ion of coord ina te choice need be made since as follows at once f rom 4 1 14 this not ion is independent of co ordinate choice for general m or m the not ion of m a p p r o x i m a t i o n or m a p p r o x i m a tion is not independent of the choice of coordinates made however the results of this article will be concerned with m or m where these not ions are independent of co ordinate choice

variance bounds with an application to norm bounds for commutators abstract murthy and sethi m n murthy v k sethi sankhya ser b 27 1965 201 210 gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable we generalise this bound to the complex case and more importantly to the matrix case in doing so we make contact with several geometrical and matrix analytical concepts such as the numerical range and introduce the new concept of radius of a matrix we also give a new and simplified proof for a sharp upper bound on the frobenius norm of commutators recently proven by bottcher and wenzel a bottcher d wenzel the frobenius norm and the commutator linear algebra appl 429 2008 1864 1885 and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced as an immediate application of our variance bounds we obtain stronger versions of bottcher and wenzel s upper bound

laplacian spread of graphs lower bounds and relations with invariant parameters abstract the spread of an n n complex matrix b with eigenvalues 1 2 n is defined by s b max i j i j where the maximum is taken over all pairs of eigenvalues of b let g be a graph on n vertices the concept of laplacian spread of g is defined by the difference between the largest and the second smallest laplacian eigenvalue of g in this work by combining old techniques of interlacing eigenvalues and rank 1 perturbation matrices new lower bounds on the laplacian spread of graphs are deduced some of them involving invariant parameters of graphs as it is the case of the bandwidth independence number and vertex connectivity

church rosser thue systems and formal languages since about 1971 much research has been done on thue systems that have properties that ensure viable and efficient computation the strongest of these is the church rosser property which states that two equivalent strings can each be brought to a unique canonical form by a sequence of length reducing rules in this paper three ways in which formal languages can be defined by thue systems with this property are studied and some general results about the three families of languages so determined are studied

determinacy of sufficiently differentiable maps variants of the algebraic conditions of mather are shown to be sufficient for the k determinacy of cu maps with respect to j flat contact or right cr equivalence relations where u k k the algebraic conditions are seen to be necessary

triangulated categories of matrix factorizations for regular systems of weights with e 1 abstract we construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a nondegenerate regular system of weights whose smallest exponents are equal to 1 in the associated grothendieck group the strongly exceptional collection defines a root basis of a generalized root system of sign l 0 2 and a coxeter element of finite order whose primitive eigenvector is a regular element in the expanded symmetric domain of type iv with respect to the weyl group

upper bounds on the laplacian spread of graphs abstract the laplacian spread of a graph g is defined as the difference between the largest and the second smallest eigenvalue of the laplacian matrix of g in this work an upper bound for this graph invariant that depends on first zagreb index is given moreover another upper bound is obtained and expressed as a function of the nonzero coefficients of the laplacian characteristic polynomial of a graph

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

the diameter and width of numerical ranges abstract we characterize the diameter and width of the numerical range present an algorithm for computing the diameter and width of the numerical range and formulate the diameters of the numerical ranges of unitary bordering matrices for lower dimensions we also determine the boundaries of the numerical ranges of certain nilpotent toeplitz matrices to be curves of constant width

minimizing the least eigenvalues of unicyclic graphs with application to spectral spread the spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph let u k denote the set of connected unicyclic graphs of order n and girth k and let un denote the set of connected unicyclic graphs of order n in this paper we determine the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in u k we finally characterize the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in un

orthogonality of matrices and some distance problems abstract if a and b are matrices such that a zb a for all complex numbers z then a is said to be orthogonal to b we find necessary and sufficient conditions for this to be the case some applications and generalisations are also discussed

on the moduli space of pairs consisting of a cubic threefold and a hyperplane abstract we study the moduli space of pairs x h consisting of a cubic threefold x and a hyperplane h in p 4 the interest in this moduli comes from two sources the study of certain weighted hypersurfaces whose middle cohomology admit hodge structures of k3 type and on the other hand the study of the singularity o 16 the cone over a cubic surface in this paper we give a hodge theoretic construction of the moduli space of cubic pairs by relating x h to certain lattice polarized cubic fourfolds y a period map for the pairs x h is then defined using the periods of the cubic fourfolds y the main result is that the period map induces an isomorphism between a git model for the pairs x h and the baily borel compactification of some locally symmetric domain of type iv

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

lower bounds for the spread of a matrix a characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread comparisons are then made with several known bounds

eigenvalues of a h generalized join graph operation constrained by vertex subsets a generalized h join operation of a family of graphs g1 gp where h has order p constrained by a family of vertex subsets si v gi i 1 p is introduced when each vertex subset si is ki i regular it is deduced that all non main adjacency eigenvalues of gi different from ki i remain as eigenvalues of the graph g obtained by this operation if each gi is ki regular and all the vertex subsets are such that si v gi the h generalized join constrained by these vertex subsets coincides with the h join operation furthermore some applications on the spread of graphs are presented

primitive cohomology and the tube mapping let x be a smooth complex projective variety of dimension d we show that its primitive cohomology in degree d is generated by certain tube classes constructed from the monodromy in the family of all hyperplane sections of x the proof makes use of a result about the group cohomology of certain representations that may be of independent interest

lower bounds for the spread of a hermitian matrix abstract some simple lower bounds for the spread of a hermitian matrix are derived these bounds are explicit functions of the entries of the matrix one of which is sharper than a recent result due to e r barnes and a j hoffman comparisons are made with several known results

positive linear maps and perturbation bounds of matrices abstract we show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices some related bounds for the spread and condition number of hermitian matrices are also discussed here

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

the laplacian spread of quasi tree graphs abstract the laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the laplacian matrix of the graph bao tan and fan y h bao y y tan y z fan the laplacian spread of unicyclic graphs appl math lett 22 2009 1011 1015 characterize the unique unicyclic graph with maximum laplacian spread among all connected unicyclic graphs of fixed order in this paper we characterize the unique quasi tree graph with maximum laplacian spread among all quasi tree graphs in the set q n d with 1 d n 4 2

how big can the commutator of two matrices be and how big is it typically abstract numerical experiments show that the frobenius norm of the commutator of two large matrices typically clusters sharply around a certain value which moreover is much smaller than one would predict the purpose of this paper is to give a rigorous foundation of this phenomenon we also discuss the question how big the frobenius norm of commutator of the two matrices can actually be

new bounds for the signless laplacian spread abstract let g be an undirected simple graph the signless laplacian spread of g is defined as the maximum distance of pairs of its signless laplacian eigenvalues this paper establishes some new bounds both lower and upper for the signless laplacian spread several of these bounds depend on invariant parameters of the graph we also use a minmax principle to find several lower bounds for this spectral invariant

restarting automata with restricted utilization of auxiliary symbols the restarting automaton is a restricted model of computation that was introduced by jancar et al to model the so called analysis by reduction which is a technique used in linguistics to analyse sentences of natural languages the most general models of restarting automata make use of auxiliary symbols in their rewrite operations although this ability does not directly correspond to any aspect of the analysis by reduction here we put restrictions on the way in which restarting automata use auxiliary symbols and we investigate the influence of these restrictions on their expressive power in fact we consider two types of restrictions first we consider the number of auxiliary symbols in the tape alphabet of a restarting automaton as a measure of its descriptional complexity secondly we consider the number of occurrences of auxiliary symbols on the tape as a dynamic complexity measure we establish some lower and upper bounds with respect to these complexity measures concerning the ability of restarting automata to recognize the deterministic context free languages and some of their subclasses

topological equisingularity of hypersurfaces with 1 dimensional critical set abstract we focus on topological equisingularity of families of holomorphic function germs with 1 dimensional critical set we introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular we show that if a family of germs with 1 dimensional critical set has constant generic le numbers then it is equisingular at the critical set and hence topologically equisingular answering a question of d massey 13 it is worth to remark that this does not happen for higher dimensional critical set 5 we use these topological triviality results to modify the definition of singularity stem present in the literature introducing and characterising topological stems being this concept closely related with arnold s series of singularities we provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non flat deformation with singular special fibre and smooth generic fibre

surgery in cusp neighborhoods and the geography of irreducible 4 manifolds since its inception the theory of smooth 4 manifolds has relied upon complex surface theory to provide its basic examples this is especially true for simply connected 4 manifolds a longstanding conjecture held that every smooth simply connected closed 4 manifold is the connected sum of manifolds each admitting a complex structure with one of its orientations for this conjecture the 4 sphere was considered as the empty connected sum quite recently this was disproved by gompf and mrowka gm1 who produced infinite families of examples of 4 manifolds homeomorphic to the k3 surface and which admit no complex structure with either orientation the discovery of these dramatic counterexamples to the h cobordism theorem in dimension 4 was the most recent success in a long line of applications of donaldson invariants to the study of smooth structures on 4 manifolds beginning with donaldson s d4 and followed by fm1 ko ov1 ov2 which depended on donaldson s f invariant or a suitable generalization and fmm dk e2 eo1 mn2 sa using donaldson s polynomial invariant a common thread running through many of these counterexamples to the h cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface in controlled situations this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed taking advantage of the complex structure on an elliptic surface this change can often be detected by using the relationship between anti self dual connections over a complex surface and stable holomorphic bundles d2 an alternative and more differential geometric

antidiagonals of matrices in a unitary similarity orbit abstract let mn be the algebra of n n complex matrices for a aij mn the kth antidiagonal of a is the vector ak 1 ak 1 2 a1 k t and u a u au u is unitary is the unitary similarity orbit of a we study the sets 1 d k a the collection of all kth antidiagonals of matrices in u a 2 i k a the collection of z k i 1 v i c such that v v 1 v k t d k a 3 p k a the collection of z k i 1 v i c such that v v 1 v k t d k a particular attention is paid to the relation between the geometrical properties of the sets and the algebraic properties of the matrix a when k 1 all three sets reduce to w a the numerical range of a which has been studied extensively we also consider the following related concepts 1 d k a max x x d k a where x x x 1 2 2 s k a max z z i k a 3 p k a max z z p k a

bounds on the q spread of a graph the spread s m of an n n complex matrix m is s m maxij i j where the maximum is taken over all pairs of eigenvalues of m i 1 i n 9 and 11 based on this concept gregory et al 7 determined some bounds for the spread of the adjacency matrix a g of a simple graph g and made a conjecture regarding the graph on n vertices yielding the maximum value of the spread of the corresponding adjacency matrix the signless laplacian matrix of a graph g q g d g a g where d g is the diagonal matrix of degrees of g and a g is its adjacency matrix has been recently studied 4 5 the main goal of this paper is to determine some bounds on s q g we prove that for any graph on n 5 vertices 2 s q g 2n 4 and we characterize the equality cases in both bounds further we prove that for any connected graph g with n 5 vertices s q g 2n 4 we conjecture that for n 5 sq g 4n 2 20n 33 and that in this case the upper bound is attained if and only if g is a certain pathcomplete graph

deformations of singularities and variation of git quotients we study the deformations of the minimally elliptic surface sin gularity n16 a standard argument reduces the study of the deformations of n16 to the study of the moduli space of pairs c l consisting of a plane quintic curve and a line we construct this moduli space in two ways via the periods of k3 surfaces and by using geometric invariant theory git the git construction depends on the choice of the linearization in particular for one choice of linearization we recover the space constructed via k3 surfaces and for another we obtain the full deformation space of n16 the two spaces are related by a series of explicit flips in conclusion by using the flexibil ity given by git and the standard tools of hodge theory we obtain a good understanding of the deformations of n16

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined

lower bounds for the numerical radius abstract let a be a complex n n matrix we find lower bounds for its numerical radius r a max x ax x c n x x 1 first we choose x satisfying x x 1 and compute x ax we also improve the simple bound so obtained second we apply r a max z a z a 2 z c z 1 where denotes the largest eigenvalue

monodromy of the sl n and gl n hitchin fibrations we compute the monodromy of the hitchin fibration for the moduli space of l twisted sl n mathbb c and gl n mathbb c higgs bundles for any n on a compact riemann surface of genus g 1 we require the line bundle l to either be the canonical bundle or satisfy deg l 2g 2 the monodromy group is generated by picard lefschetz transformations associated to vanishing cycles of singular spectral curves we construct such vanishing cycles explicitly and use this to show that the sl n mathbb c monodromy group is a skew symmetric vanishing lattice in the sense of janssen using the classification of vanishing lattices over mathbb z we completely determine the structure of the monodromy groups of the sl n mathbb c and gl n mathbb c hitchin fibrations as an application we determine the image of the restriction map from the cohomology of the moduli space of higgs bundles to the cohomology of a non singular fibre of the hitchin fibration