topological bf theories in 3 and 4 dimensions in this paper we discuss topological bf theories in 3 and 4 dimensions observables are associated to ordinary knots and links in 3 dimensions and to 2 knots in 4 dimensions the vacuum expectation values of such observables give a wide range of invariants here we consider mainly the 3 dimensional case where these invariants include alexander polynomials homfly polynomials and kontsevich integrals

remarks on chern simons invariants the perturbative chern simons theory is studied in a finite dimensional version or assuming that the propagator satisfies certain properties as is the case e g with the propagator defined by axelrod and singer it turns out that the effective bv action is a function on cohomology with shifted degrees that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely out of it one obtains invariants

on certain values of kloosterman sums let k q n a be a kloosterman sum over the finite field f q n of characteristic p in this note so called subfield conjecture is proved if a ne 0 belongs to the proper subfield f q of f q n then k q n a ne 1 this completes recent works on the subfield conjecture by shparlinski and moisio and lisonek the problem is motivated by some applications to bent functions moreover in the course of the proof a large class of translates of dickson polynomials are shown to be irreducible

wilson surfaces and higher dimensional knot invariants an observable for nonabelian higher dimensional forms is introduced its properties are discussed and its expectation value in bf theory is described this is shown to produce potential and genuine invariants of higher dimensional knots

the wilson loop observables of chern simons theory on ℝ3 in axial gauge we define the wilson loop observables wlos for pure chern simons models with base manifold m 3 rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop smearing and framing the values of the wlos can be computed explicitly if the structure group g of the model considered is abelian one obtains well known linking number expressions for the wlos if g is non abelian one obtains expressions which are similar but not identical to the state sum representations for the homfly and kauffman polynomials by jones and turaev

finite volume flows and morse theory in this paper we present a new approach to morse theory based on the de rham federer theory of currents the full classical theory is derived in a transparent way the methods carry over uniformly to the equivariant and the holomorphic settings moreover the methods are substantially stronger than the classical ones and have interesting applications to geometry they lead for example to formulas relating characteristic forms and singularities of bundle maps

automatic classification of protein structure by using gauss integrals we introduce a method of looking at analyzing and comparing protein structures the topology of a protein is captured by 30 numbers inspired by vassiliev knot invariants to illustrate the simplicity and power of this topological approach we construct a measure scaled gauss metric sgm of similarity of protein shapes under this metric protein chains naturally separate into fold clusters we use sgm to construct an automatic classification procedure for the cath2 4 database the method is very fast because it requires neither alignment of the chains nor any chain chain comparison it also has only one adjustable parameter we assign 95 51 of the chains into the proper c class a architecture t topology and h homologous superfamily fold find all new folds and detect no false geometric positives using the sgm we display a map of the space of folds projected onto two dimensions show the relative locations of the major structural classes and zoom into the space of proteins to show architecture topology and fold clusters the existence of a simple measure of a protein fold computed from the chain path will have a major impact on automatic fold classification

remarks on legendrian self linking the thurston bennequin invariant provides one notion of self linking for any homologically trivial legendrian curve in a contact three manifold here we discuss related analytic notions of self linking for legendrian knots in r3 our definition is based upon reformulation of the elementary gauss linking integral and is motivated by ideas from supersymmetric gauge theory we recover the thurston bennequin invariant as a special case the thurston bennequin invariant provides one notion of self linking for any homologically trivial legendrian curve in a contact three manifold here we discuss related analytic notions of self linking for legendrian knots in r3 our definition is based upon reformulation of the elementary gauss linking integral and is motivated by ideas from supersymmetric gauge theory we recover the thurston bennequin invariant as a special case

an efficient deterministic test for kloosterman sum zeros we propose a simple deterministic test for deciding whether or not a non zero element a in mathbb f 2 n or mathbb f 3 n is a zero of the corresponding kloosterman sum over these fields and analyse its complexity the test seems to have been overlooked in the literature for binary fields the test has an expected operation count dominated by just two mathbb f 2 n multiplications when n is odd with a slightly higher cost for even extension degrees making its repeated invocation the most efficient method to date to find a non trivial kloosterman sum zero in these fields the analysis depends on the distribution of sylow p subgroups in two corresponding families of elliptic curves which we prove using a theorem due to howe

higher dimensional bf theories in the batalin vilkovisky formalism the bv action and generalized wilson loops this paper analyzes in detail the batalin vilkovisky quantization procedure for bf theories on an n dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables in particular generalized wilson loops for bf theories with additional polynomial b interactions are discussed in any dimensions the paper also contains the explicit proofs to the theorems stated in 16

integration of singular braid invariants and graph cohomology we prove necessary and sufficient conditions for an arbitrary in variant of braids with m double points to be the mth derivative of a braid invariant we show that the primary obstruction to integration i5 the only obstruction this gives a slight generalization of the existence theorem for vassiliev invariants of braids we give a direct proof by induction on m which works for invariants with values in any abelian group we snd that to prove our theorem we must show that every relation among four term relations satisses a certain geometric condition to snd the relations among relations we show that h1 of a variant of kontsevich s graph complex vanishes we discuss related open questions for invariants of links and other things

further results on generalized bent functions and their complete characterization this paper contributes to increase our knowledge on generalized bent functions including generalized bent boolean functions and generalized p ary bent functions with odd prime p by bringing new results on their characterization and construction in arbitrary characteristic more specifically we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions this enables us to completely characterize generalized bent functions and mathbb z p k bent functions by some affine space associated with the generalized bent functions we also present the relationship between generalized bent boolean functions with an odd number of variables and generalized bent boolean functions with an even number of variables based on the well known maiorana mcfarland class of boolean functions we present some infinite classes of generalized bent boolean functions in addition we introduce a class of generalized hyperbent functions that can be seen as generalized dillon s ps functions finally we solve an open problem related to the description of the dual function of a weakly regular generalized bent boolean function with an odd number of variables via the walsh hadamard transform of their component functions and we generalize these results to the case of odd prime

on the number of injective indecomposable modules for every natural number m there exists a ring r with a completely prime ideal p so that there are exactly m non isomorphic indecomposable injective right r modules with p as associated prime ideal

braids and the jones polynomial an important new invariant of knots and links is the jones polynomia and the subsequent generalized jones polynomial or two variable polynomial we prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends we also relate an exponent in the polynomial to the number of components of this link in j vaughan jones introduced a new polynomial invariant of oriented knots and links in 3 space subsequently a number of researchers freyd and yetter hoste lickorish and millett and ocneanu see fyhlmo independently realized that thlis could be generalized to produce an invariant which is a laurent polynomial of two variables and which specializes to give both the invariant of j and the classical alexander polynomial see r more precisely there is a function p from isotopy classes of oriented links to z x x 1 y y l if l is an oriented link we will write p l j x y by abuse of notation we will also write p b for b an oriented braid meaning of course the invariant associated to the oriented link determined by closing up b in the usual way several equivalent ways of codifying this invariant are described in fyhlmo we have chosen the approach of lickorish and millett hence our p k j x y is precisely their pk with x and y substituted for i and m respectively see fyhlmo and l m there appears as yet to be no consensus on a name for this two variable invariant in order to avoid awkward circumlocutions and being unable to amalgamate the names of the six or eight individuals who actually discovered it we have chosen to refer to it as the generalized jones polynomial or jones polynomial for short when there is no likelihood of confusion with the original jones polynomial this polynomial which we will denote j x y is characterized by figure 1 the interpretation of this is as follows given a regular projection of a link k with a crossing pictured as below one can form two new links k and ko which are identical to k except the one crossing is changed as shown when this is done the jones polynomials of the three links are related as in the formula it is a remarkable fact that this together with the normalization that p unknot 1 un quely determines a laurent polynomial j x y and that this polynomial is a link invariant i e is independent of the projection this result can be found in l m this formula in fact gives a good method for recursively computing j x y one received by the editors january 13 1986 1980 mathematics subject classification 1985 revision primary 57m25 research supported in part by nsf grant mcs 83

integral geometry of plane curves and knot invariants we study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of witten s chern simons path integral associated with a knot one of the integrals involved turns out to be a generalization of the classical crofton integral on convex plane curves and it is related with the invariants of generic plane curves recently defined by arnold with deep motivations in symplectic and contact geometry quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant

uniserial modules over valuation rings this is the first in a series of papers deahng with the theory of modules over valuation rings it has been observed that a variety of concepts and basic results in abelian group theory can be extended mutatis mutandis to modules over valuation rings and several other results which require drastic modification for valuation rings can also be dealt with by suitably generalizing ideas of abelian group theory our purpose is to develop techniques for modules over valuation rings actually these are the simplest kind of commutative non noetherian rings our point of departure is abelian group theory in the local case when the groups are merely modules over z the integers localized at a prime p which is a discrete rank one valuation domain in the process of generalization the most attractive and frequently used properties are sacrilied pleasant properties we are so accustomed to in abelian groups are gone in return we learn new features and discover new phenomena in the behavior of modules several results are known on modules over valuation rings r see the references 3 lo or the survey article 121 this paper is devoted to the study of uniseriul r modules i e those r modules in which the submodules form a chain as far as the simplicity of the structure is concerned these are second only to cyclic modules they have been investigated by shores and lewis 9 we make use of their results especially their description of endomorphism rings we study various aspects of these modules with special emphasis on their quasiand pure injectivity as well as on the existence of pure uniserial submodules in torsion r modules in some cases as expected more explicit results can be established only under the additional hypothesis that r is almost maximal

feynman integrals and motives of configuration spaces we formulate the problem of renormalization of feynman integrals and its relation to periods of motives in configuration space instead of momentum space the algebro geometric setting is provided by the wonderful compactifications overline conf gamma x of arrangements of subvarieties associated to the subgraphs of a feynman graph with x a quasi projective variety the motive and the class in the grothendieck ring are computed explicitly for these wonderful compactifications in terms of the motive of x and the combinatorics of the feynman graph using recent results of li li the pullback to the wonderful compactification of the form defined by the unrenormalized feynman amplitude has singularities along a hypersurface whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification a regularization of the feynman integrals can be obtained by modifying the cycle of integration by replacing the divergent locus with a leray coboundary the ambiguities are then defined by poincare residues while these residues give periods associated to the cohomology of the exceptional divisors and their intersections the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification

associahedra cyclohedra and a topological solution to the a deligne conjecture abstract we give a topological solution to the a deligne conjecture using associahedra and cyclohedra to this end we construct three cw complexes whose cells are indexed by products of polytopes giving new explicit realizations of the polytopes in terms of different types of trees we are able to show that the cw complexes are cell models for the little discs the cellular chains of one complex in particular which is built out of associahedra and cyclohedra naturally act on the hochschild cochains of an a algebra yielding an explicit topological and minimal solution to the a deligne conjecture along the way we obtain new results about the cyclohedra such as new decompositions into products of cubes and simplices which can be used to realize them via a new iterated blow up construction

on polynomial exponential equations let m k be the set of absolute values of k normalized such that they extend the standard or a p adic absolute value of q let s c m k be the subset consisting of the archimedean absolute values as well as of the non archimedean absolute values v e m k having i ol 4 1 for some ej thus s is a finite set an s unit is an element e k having i 1 1 for every v s then in particular the o and therefore every a txe with x e z is an s unit therefore 1 2 is related to the s unit equation

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

on bent and semi bent quadratic boolean functions the maximum length sequences also called m sequences have received a lot of attention since the late 1960s in terms of linear feedback shift register lfsr synthesis they are usually generated by certain power polynomials over a finite field and in addition are characterized by a low cross correlation and high nonlinearity we say that such a sequence is generated by a semi bent function some new families of such function represented by f x spl sigma sub i 1 sup n 1 2 c sub i tr x 2 sup i 1 n odd and c sub i spl isin f sub 2 have recently 2002 been introduced by khoo et al we first generalize their results to even n we further investigate the conditions on the choice of c sub i for explicit definitions of new infinite families having three and four trace terms also a class of nonpermutation polynomials whose composition with a quadratic function yields again a quadratic semi bent function is specified the treatment of semi bent functions is then presented in a much wider framework we show how bent and semi bent functions are interlinked that is the concatenation of two suitably chosen semi bent functions will yield a bent function and vice versa finally this approach is generalized so that the construction of both bent and semi bent functions of any degree in certain range for any n spl ges 7 is presented n being the number of input variables

bennequin s inequality and the positivity of the signature we use an algorithm for special diagrams to prove a bennequin type inequality tor the signature of an arbitrary link diagram related to its murasugi sum decomposition we apply this inequality to show that the signature of a non trivial positive 3 braid knot is greater than its genus and that the signature of a positive braid link is minorated by an increasing function of its negated euler characteristic the latter property is conjectured to extend to positive links

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

on the crossing number of positive knots and braids and braid index criteria of jones and morton williams franks we give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations in particular we show that positive braid knots may not have positive minimal strand number braid representations giving a counterpart to results of franks williams and murasugi other examples answer partially questions of cromwell on homogeneous and of adams on almost alternating knots we give a counterexample to and a corrected version of a theorem of jones on the alexander polynomial of 4 braid knots we also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply from below the braid index a relation between generalizations of such examples and a conjecture of jones that a minimal braid representation has unique writhe is discussed

a new family of p ary sequences with low correlation constructed from decimated sequences in this paper for an odd prime p and an integer n geq 3 a new family of p ary sequences of period p n 1 over 2 with low correlation is constructed the new family is constructed by shifts and additions of two decimated sequences of a p ary m sequence and its family size is 2 p n 1 the complete correlation distribution of this new family is derived it is also shown that the family is optimal with respect to the parameter theta rm rms which denotes the root mean square of all nontrivial correlations compared with the known sequence families our sequence family is new and has a larger family size which is four times of its period

nonbinary kasami sequences over gf p the correlation values and the distribution of these correlation values for the small set of nonbinary kasami sequences over gf p p prime are presented the correlation results are an extension of the binary results and have p 2 correlation levels this nonbinary kasami set is asymptotically optimum with respect to its correlation properties these sequences are obtained as in the binary case from a large primitive polynomial of degree n 2 m and a small primitive polynomial of degree m that yields a sequence length of p sup n 1 and maximum nontrivial correlation value of 1 p sup m nonbinary kasami sequences are directly implemented using shift registers and are applicable for code division multiple access systems

knot theory and the heuristics of functional integration abstract this paper is an exposition of the relationship between the heuristics of witten s functional integral and the theory of knots and links in three dimensional space

generalized plateaued functions and admissible plateaued functions plateaued functions are very important crypto graphic functions due to their various desirable cryptographic characteristics we point out that plateaued functions are more general than bent functions that is functions with maximum nonlinearity some boolean plateaued functions have large nonlinearity which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers and contributes when they are the component functions of the substitution boxes in block ciphers to protection against linear cryptanalysis p ary plateaued functions have attracted recently some attention in the literature and many activities on generalized p ary functions have been carried out this paper increases our knowledge on plateaued functions in the general context of generalized p ary functions we first introduce two new versions of plateaued functions which we shall call generalized plateaued functions and admissible plateaued functions the generalized plateaued functions extend the standard notion of plateaued p ary functions to those whose outputs are in the ring mathbb z p k next we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components finally we provide for the first time two constructions of generalized plateaued functions in particular we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes

the topology of spaces of knots cosimplicial models we present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial these models build on the calculus of isotopy functors and are weakly homotopy equivalent to knot spaces when the ambient dimension is greater than three the mapping space model and the evaluation map on which it builds is suitable for analysis through differential topology the cosimplicial model gives rise to spectral sequences which converge to ohomology and homotopy groups of spaces of knots when they are connected we explicitly identify and establish vanishing lines in these spectral sequences

prime phase sequences with periodic correlation properties better than binary sequences for the case where p is an odd prime n or 2 is an integer and omega is a complex primitive pth root of unity a construction is presented for a family of p sup n p phase sequences symbols of the form omega sup i where each sequence has length p sup n 1 and where the maximum nontrivial correlation value c sub max does not exceed 1 square root p sup n a complete distribution of correlation values is provided as a special case of this construction a previous construction due to sidelnikov 1971 is obtained the family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs the present design has the additional advantage of not requiring an alphabet of size larger than three the new sequences are suitable for achieving code division multiple access and are easily implemented using shift registers they wee discovered through an application of deligne s bound 1974 on exponential sums of the weil type in several variables the sequences are also shown to have strong identification with certain bent functions

vectorial bent functions and their duals abstract motivated by the observation that for two weakly regular bent functions f g for which also f g is bent the sum f g of their duals f and g is sometimes but not always bent we initiate the study of duality for vectorial bent functions we propose and investigate two concepts of self duality for vectorial bent functions self duality and weak self duality

proofs of two conjectures on ternary weakly regular bent functions in this paper we study ternary monomial functions of the form f x trn axd where x isin bbf 3 n and trn bbf 3 nrarr bbf 3 is the absolute trace function using a lemma of hou stickelberger s theorem on gauss sums and certain ternary weight inequalities we show that certain ternary monomial functions arising in the 2006 ieee transactions on information theory paper vol 52 pp 2018 2032 2006 are weakly regular bent thus settling a conjecture of helleseth and kholosha we also prove that the coulter matthews bent functions are weakly regular

enumeration of quadratic functions with prescribed walsh spectrum the walsh transform f over cap of a quadratic function f f p n f p satisfies vertical bar f over cap vertical bar epsilon 0 p n s 2 for an integer 0 s n 1 depending on f in this paper quadratic functions of the form f p f n x tr n sigma k i 0 a i x pt 1 are studied with the restriction that a i is an element of f p 0 i k three methods for enumeration of such functions are presented when the value for s is prescribed this paper extends earlier enumeration results significantly for instance the generating function for the counting function is obtained when n is odd and relatively prime to p or when n 2m for odd m and p 2 the number of bent and semibent functions for various classes of n is also obtained

value distributions of exponential sums from perfect nonlinear functions and their applications in this paper we present a unified way to determine the values and their multiplicities of the exponential sums sigmaxisinf q zetap tr af x bx a bisinfq q pm pges3 for all perfect nonlinear functions f which is a dembowski ostrom polynomial or p 3 f x 3 k 1 2 where k is odd and k m 1 as applications we determine 1 the correlation distribution of the m sequence alambda tr gammalambda lambda 0 1 and the sequence blambda tr f gammalambda lambda 0 1 over fp where gamma is a primitive element of fq and 2 the weight distributions of the linear codes over fp defined by f

bent functions of maximal degree in this paper a technique for constructing p ary bent functions from plateaued functions is presented this generalizes earlier techniques of constructing bent from near bent functions the fourier spectrum of quadratic monomials is analyzed and examples of quadratic functions with highest possible absolute values in their fourier spectrum are given applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when p 3 5 until now no construction of bent functions attaining these bounds was known

integral invariants of 3 manifolds ii this note is a sequel to our earlier paper of the same title 4 and describes invariants of rational homology 3 spheres associated to acyclic orthogonal local systems our work is in the spirit of the axelrod singer papers 1 generalizes some of their results and furnishes a new setting for the purely topological implications of their work

new perspectives on self linking abstract we initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot which is the induced map on the compactified n point configuration space sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots we compute the homotopy types of the first three mapping space models showing that the third model gives rise to an integer valued invariant we realize this invariant in two ways in terms of collinearities of three or four points on the knot and give some explicit computations we show this invariant coincides with the second coefficient of the conway polynomial thus giving a new geometric definition of the simplest finite type invariant finally using this geometric definition we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

permutahedra and generalized associahedra given a finite coxeter system w s w s and a coxeter element c or equivalently an orientation of the coxeter graph of w we construct a simple polytope whose outer normal fan is n reading s cambrian fan fc settling a conjecture of reading that this is possible we call this polytope the c generalized associahedron our approach generalizes loday s realization of the associahedron a type a c generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the coxeter fan arising from the tamari lattice to any finite coxeter group a crucial role in the construction is played by the c singleton cones the cones in the c cambrian fan which consist of a single maximal cone from the coxeter fan moreover if w is a weyl group and the vertices of the permutahedron are chosen in a lattice associated to w then we show that our realizations have integer coordinates in this lattice

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

on the braid index of alternating links we show that at least for an alternating fibered link or 2 bridge link l there is an exact formula which expresses the braid index b l of l as a function of the 2 variable generalization pl i m of the jones polynomial

surface holonomy for non abelian 2 bundles via double groupoids in the context of non abelian gerbes we define a cubical version of categorical group 2 bundles with connection over a smooth manifold we address their two dimensional parallel transport study its properties and construct non abelian wilson surface functionals

on the dual of certain ternary weakly regular bent functions in 2006 helleseth and kholosha conjectured and partially proved the existence of a class of ternary weakly regular monomial bent functions and also the expression for the dual bent function up to the sign value the bentness was finally proved later in 2009 using a complicated technique that employs stickelberger s theorem extensively using the previously found results and approaches in this paper a surprisingly short proof for the conjectured expression of the dual is given but without resolving the sign ambiguity furthermore we resolve the sign by finding the trace representation of the dual function

coxeter submodular functions and deformations of coxeter permutahedra abstract we describe the cone of deformations of a coxeter permutahedron or equivalently the nef cone of the toric variety associated to a coxeter complex this class of polytopes contains important families such as weight polytopes signed graphic zonotopes coxeter matroids root cones and coxeter associahedra our description extends the known correspondence between generalized permutahedra polymatroids and submodular functions to any finite reflection group

the cyclic bar construction on a h spaces abstract we set up a general framework for enriching a category a over a symmetric monoidal category c using a non operad p in c to do this we require a to come with a functor u a to the category of noncommutative sets by viewing the simplicial indexing category as a category over in two different ways we obtain two generalizations of simplicial objects for the operad given by the stasheff associahedra we obtain a model for the 2 sided bar construction in the first case and the cyclic bar and cobar construction in the second case using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects

on integer values of kloosterman sums this paper considers rational integer values of kloosterman sums over finite fields of characteristic p 3 we shall prove two main results the first one is a congruence relation satisfied by possible integer values one consequence is that there are no kloosterman zeroes in the case of characteristic p 3 which generalizes recent works by shparlinski moisio and lisonek on this subject this in turn implies that there are no dillon type bent functions in the case p 3 thus answering a question posed recently by helleseth and kholosha our other main result states that the kloosterman sum obtains an integer value at a point if and only if the same sum lifted to any extension field remains an integer

linear codes with two or three weights from weakly regular bent functions linear codes with a few weights have applications in consumer electronics communication data storage system secret sharing authentication codes association schemes and strongly regular graphs this paper first generalizes the method of constructing two weight and three weight linear codes of ding et al and zhou et al to general weakly regular bent functions and determines the weight distributions of these linear codes it solves an open problem proposed by ding et al furthermore this paper constructs new linear codes with two or three weights and presents their weight distributions they contain some optimal codes meeting certain bound on linear codes

on cobordisms between knots braid index and the upsilon invariant we use ozsvath stipsicz and szabo s upsilon invariant to provide bounds on cobordisms between knots that contain full twists in particular we recover and generalize a classical consequence of the morton franks williams inequality for knots positive braids that contain a positive full twist realize the braid index of their closure we also establish that quasi positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure finally we provide inductive formulas for the upsilon invariant of torus knots and compare it to the levine tristram signature profile

counting alternating knots by genus it is shown that the number of alternating knots of given genus g 1 grows as a polynomial of degree 6g 4 in the crossing number the leading coefficient of the polynomial which depends on the parity of the crossing number is related to planar trivalent graphs with a bieulerian path the rate of growth of the number of such graphs is estimated

monomial and quadratic bent functions over the finite fields of odd characteristic considered are p ary bent functions having the form f x tr sub n spl sigma sub i 0 sup s a sub i x sup di a new class of ternary monomial regular bent function with the dillon exponent is discovered the existence of dillon bent functions in the general case is an open problem of deciding whether a certain kloosterman sum can take on the value 1 also described is the general gold like form of a bent function that covers all the previously known monomial quadratic cases the weak regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given finally some criteria for an arbitrary quadratic function to be bent are proven

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling

several new classes of bent functions from dillon exponents several new classes of binary and p ary regular bent functions are obtained in this paper the bentness of all these functions is determined by some exponential sums over finite fields most of which have close relations with the well known kloosterman sums

constructions of quadratic and cubic rotation symmetric bent functions in this paper we consider constructions of rotation symmetric bent functions which are of the forms fc x i 1m 1 ci j 0n 1 xjxi j cm j 0m 1 xjxm j and ft x i 0n 1 xixt ixm i xixt i i 0m 1 xixm i where n 2m ci 0 1 the subscript u of xu in the previous expressions is taken as u modulo n for each case a necessary and sufficient condition is obtained to the best of our knowledge this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions

valuation domains without pathological modules abstract let r be a valuation domain an r module is said to be pathological if it does not contain nonzero uniserial pure submodules it is shown that there are no pathological r modules if and only if r is a totally branched discrete valuation domain three characterizations of these domains are given jr j is principal for all j e spec r every ideal is isomorphic to a prime ideal the value group of r is discrete and the set of convex subgroups is well ordered by inclusion

kontsevich integral for vassiliev invariants from chern simons perturbation theory in the light cone gauge we analyze the structure of the perturbative series expansion of chern simons gauge theory in the light cone gauge after introducing a regularization prescription that entails the consideration of framed knots we present the general form of the vacuum expectation value of a wilson loop the resulting expression turns out to give the same framing dependence as the one obtained using nonperturbative methods and perturbative methods in covariant gauges it also contains the kontsevich integral for vassiliev invariants of framed knots

abelian bf theories and knot invariants in the context of the batalin vilkovisky formalism a new observable for the abelian bf theory is proposed whose vacuum expectation value is related to the alexander conway polynomial the three dimensional case is analyzed explicitly and it is proved to be anomaly free moreover at the second order in perturbation theory a new formula for the second coefficient of the alexander conway polynomial is obtained an account on the higher dimensional generalizations is also given

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

a three valued walsh transform from decimations of helleseth gong sequences the walsh transform of two level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal for p ary sequences there are only two basic classes of two level autocorrelation sequences with no subfield structures for an arbitrary odd prime p one is the class of p ary m sequences and the other is the class of p ary helleseth gong sequences in this paper the walsh transform of a subclass of p ary helleseth gong sequences and the walsh transform of their particular decimations are completely determined and are shown to be three valued

generating functions fibonacci numbers and rational knots abstract we describe rational knots with any of the possible combinations of the properties a chirality non positivity non fiberedness and unknotting number one or higher and determine exactly their number for a given number of crossings in terms of their generating functions we show in particular how fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots then we show how to enumerate rational knots of given crossing number depending on genus and or signature this allows to determine the asymptotical average value of these invariants among rational knots we give also an application to the enumeration of lens spaces