strong jump traceability i the computably enumerable case abstract recent investigations in algorithmic randomness have lead to the discovery and analysis of the fundamental class k of reals called the k trivial reals defined as those whose initial segment complexity is identical with that of the sequence of all 1 s there remain many important open questions concerning this class such as whether there is a combinatorial characterization of the class and whether it coincides with possibly smaller subclasses such as the class of reals which are not sufficiently powerful as oracles to cup a turing incomplete martin lof random real to the halting problem hidden here is the question of whether there exist proper natural subclasses of k we show that the combinatorial class of computably enumerable strongly jump traceable reals defined via the jump operator by figueira nies and stephan santiago figueira andre nies frank stephan lowness properties and approximations of the jump electr notes theor comput sci 143 2006 45 57 is such a class and show that like k it is an ideal in the computably enumerable degrees this is the first example of a class of reals defined by a cost function construction which forms a proper subclass of k further we show that every c e strongly jump traceable set is not martin lof cuppable thus giving a combinatorial property which implies non ml cuppability

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

zero one laws for connectivity in inhomogeneous random key graphs we introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks each of the n sensors in the network is classified into r classes according to some probability distribution boldsymbol mu mu 1 ldots mu r before deployment a class i sensor is assigned k i cryptographic keys that are selected uniformly at random from a common pool of p keys once deployed a pair of sensors can communicate securely if and only if they have a key in common we model the communication topology of this network by a newly defined inhomogeneous random key graph we establish scaling conditions on the parameters p and k 1 ldots k r so that this graph 1 has no isolated nodes and 2 is connected both with high probability the results are given in the form of zero one laws with the number of sensors n growing unboundedly large critical scalings are identified and shown to coincide for both graph properties our results are shown to complement and improve those given by godehardt et al and zhao et al for the same model therein referred to as the general random intersection graph

large independent sets in general random intersection graphs we investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs in particular a we propose a new model for random intersection graphs g n m p which includes the model of m karonski e r scheinerman k b singer cohen on random intersection graphs the subgraph problem combinatorics probability and computing journal 8 1999 131 159 the uniform random intersection graph models as an important special case we also define an interesting variation of the model of random intersection graphs similar in spirit to random regular graphs b for this model we derive exact formulae for the mean and variance of the number of independent sets of size k for any k in the graph c we then propose and analyse three algorithms for the efficient construction of large independent sets in this model the first two are variations of the greedy technique while the third is a totally new algorithm our algorithms are analysed for the special case of uniform random intersection graphs our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters

descendant invariants and characteristic numbers on a stack of stable maps the cotangent line classes are modified by subtracting certain boundary divisors these modified cotangent line classes are compatible with forgetful morphisms and are well suited to enumerative geometry tangency conditions allow simple expressions in terms of modified cotangent line classes topological recursion relations are established among their top products in genus 0 yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety in higher genus the obtained numbers are only virtual due to contributions from spurious components of the space of maps for the projective plane the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases

on the geometry of algebraic groups and homogeneous spaces given a connected algebraic group g over an algebraically closed field and a g homogeneous space x we describe the chow ring of g and the rational chow ring of x with special attention to the picard group also we investigate the albanese and the anti affine fibrations of g and x

a change of coordinates on the large phase space of quantum cohomology the gromov witten invariants of a smooth projective variety v when twisted by the tautological classes on the moduli space of stable maps give rise to a family of cohomological field theories and endow the base of the family with coordinates we prove that the potential functions associated to the tautological classes the large phase space and the classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions our result is a generalization of the work of manin zograf who studied the case where v is a point we utilize this change of variables to derive the topological recursion relations associated to the classes from those associated to the classes

on connectivity and robustness in random intersection graphs random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks for these graphs each node is equipped with a set of objects in a random manner and two nodes have an undirected edge in between if they have at least one object in common in this paper we investigate connectivity and robustness in a general random intersection graph model specifically we establish sharp asymptotic zero one laws for k connectivity and k robustness as well as the asymptotically exact probability of k connectivity for any positive integer k the k connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures while k robustness measures the effectiveness of local information based consensus algorithms which do not use global graph topology information in the presence of adversarial nodes in addition to presenting the results under the general random intersection graph model we consider two special cases of the general model a binomial random intersection graph and a uniform random intersection graph which both have numerous applications as well for these two specialized graphs we present asymptotically exact probabilities of k connectivity and asymptotic zero one laws for k robustness

inherent enumerability of strong jump traceability we show that every strongly jump traceable set obeys every be nign cost function moreover we show that every strongly jump traceable set is computable from a computably enumerable strongly jump traceable set this allows us to generalise properties of c e strongly jump traceable sets to all such sets for example the strongly jump traceable sets induce an ideal in the turing degrees the strongly jump traceable sets are precisely those that are computable from all superlow martin lof random sets the strongly jump traceable sets are precisely those that are a base for demuthblr randomness and strong jump traceability is equivalent to strong superlowness

chow groups of quotient varieties let g be a reductive algebraic group operating on a scheme x with quotient y x g let ch x be the chow group of x ch y the chow group of y as defined in f in many concrete cases e g binary quantics or certain subspaces of moduli spaces of curves x has a simple structure and we have a lot of information on ch x while ch y is more interesting and much harder to calculate so it is natural to ask what relation is there between ch x and ch y in this article we give a partial answer to this question with the following restrictions we assume that the stabilizer of each point of x is finite and we only consider chow groups with rational coefficients furthermore we assume that x is an algebraic scheme over an algebraically closed field of characteristic 0 set a ch q where q is the field of rational numbers if g is a finite group then a y a x see f example 1 7 61 thus it is only necessary to consider the case in which g is connected theorem 1 says that in this case there is a natural surjective homomorphism a y a x this means that a y is at least as large as a x to be able to bound the kernel of the homomorphism a y a x we make two further assumptions that x is normal and that g cl n or sl n the case g sl n includes the case g pgl n which the one usually found in applications the case of a principal gl n bundle is treated in sga6 exposi ix proposition 4 11 considering the k group of coherent sheaves instead of the chow group what is proved there amounts in the present contest to the following let c c be the chern classes of the rank n vector bundle

the gromov witten potential of a point hurwitz numbers and hodge integrals hurwitz numbers which count certain covers of the projective line or equivalently factorizations of permutations into transpositions have been extensively studied for over a century the gromov witten potential f of a point the generating series for descendent integrals on the moduli space of curves is a central object of study in gromov witten theory we define a slightly enriched gromov witten potential g including integrals involving one lambda class and show that after a non trivial change of variables g h in positive genus where h is a generating series for hurwitz numbers we prove a conjecture of goulden and jackson on higher genus hurwitz numbers which turns out to be an analogue of a genus expansion ansatz of itzykson and zuber as consequences we have new combinatorial constraints on f and a much more direct proof of the ansatz of itzykson and zuber we can produce recursions and explicit formulas for hurwitz numbers the algorithm presented proves all such recursions as examples we present surprisingly simple new recursions in genus 0 to 3 similar recursions should exist for all genera as we expect this paper also to be of interest to combinatorialists we have tried to make it as self contained as possible including reviewing some results and definitions well known in algebraic and symplectic geometry and mathematical physics 2000 mathematical subject classification primary 14h10 81t40 secondary 05c30 58d29

characterizing the strongly jump traceable sets via randomness abstract we show that if a set a is computable from every superlow 1 random set then a is strongly jump traceable together with a result of greenberg and nies noam greenberg andre nies benign cost functions and lowness properties j symbolic logic 76 1 2011 289 312 this theorem shows that the computably enumerable c e strongly jump traceable sets are exactly the c e sets computable from every superlow 1 random set we also prove the analogous result for superhighness a c e set is strongly jump traceable if and only if it is computable from every superhigh 1 random set finally we show that for each cost function c with the limit condition there is a 1 random 2 0 set y such that every c e set a t y obeys c to do so we connect cost function strength and the strength of randomness notions together with a theorem of greenberg and nies ibd this result gives a full correspondence between obedience of cost functions and being computable from 2 0 1 random sets

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

on the independence number and hamiltonicity of uniform random intersection graphs in the uniform random intersection graphs model denoted by g n m l to each vertex v we assign exactly l randomly chosen labels of some label set m of m labels and we connect every pair of vertices that has at least one label in common in this model we estimate the independence number a g n m l for the wide range m n a a 1 and l o m 1 4 we also prove the hamiltonicity of this model by an interesting combinatorial construction finally we give a brief note concerning the independence number of g n m p random intersection graphs in which each vertex chooses labels with probability p

from zwiebach invariants to getzler relation we introduce the notion of zwiebach invariants that generalize gromov witten invariants and homotopical algebra structures we outline the induction procedure that induces the structure of zwiebach invariants on the sub bicomplex that gives the structure of gromov witten invariants on sub bicomplex with zero differentials we propose to treat hodge dgbv with 1 12 axiom as the simplest set of zwiebach invariants and explicitly prove that it induces wdvv and getzler equations in genera 0 and 1 respectively

lowness properties and randomness abstract the set a is low for martin lof randomness if each random set is already random relative to a a is k trivial if the prefix complexity k of each initial segment of a is minimal namely n k a n k n o 1 we show that these classes coincide this answers a question of ambos spies and kucera in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 each low for martin lof random set is 2 0 our class induces a natural intermediate 3 0 ideal in the r e turing degrees which generates the whole class under downward closure answering a further question in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 we prove that each low for computably random set is computable

the semi classical approximation for modular operads we study the contribution of one loop graphs the semi classical expansion problem in the setting of modular operads as an application we calculate the betti numbers of the deligne mumford knudsen moduli spaces of stable curves of genus 1 with n marked smooth points

zero one laws for connectivity in random key graphs the random key graph is a random graph naturally associated with the random key predistribution scheme introduced by eschenauer and gligor in the context of wireless sensor networks wsns for this class of random graphs we establish a new version of a conjectured zero one law for graph connectivity as the number of nodes becomes unboundedly large the results reported here complement and strengthen recent work on this conjecture by blackburn and gerke in particular the results are given under conditions which are more realistic for applications to wsns

constructions of independent sets in random intersection graphs this paper concerns constructing independent sets in a random intersection graph we concentrate on two cases of the model a binomial and a uniform random intersection graph for both models we analyse two greedy algorithms and prove that they find asymptotically almost optimal independent sets we provide detailed analysis of the presented algorithms and give tight bounds on the independence number for the studied models moreover we determine the range of parameters for which greedy algorithms give better results for a random intersection graph than this is in the case of an erd s renyi random graph g n p

extremal higher codimension cycles on moduli spaces of curves we show that certain geometrically defined higher codimension cycles are ex tremal in the effective cone of the moduli space mg n of stable genus g curves with n ordered marked points in particular we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension we also show that the locus of hyperelliptic curves with a marked weierstrass point in m3 1 and the locus of hyperel liptic curves in m4 are extremal cycles in addition we exhibit infinitely many extremal codimension two cycles in m1 n for n 5 and in m2 n for n 2

k connectivity in random k out graphs intersecting erdős renyi graphs we investigate k connectivity in secure wireless sensor networks under the random pairwise key predistribution scheme with unreliable links when wireless communication links are modeled as independent on off channels this amounts to analyzing a random graph model formed by intersecting a random k out graph and an erd s renyi graph we present conditions on how to scale the parameters of this intersection model so that the resulting graph is k connected with probability approaching to one resp zero as the number of nodes gets large the resulting zero one law is shown to improve and sharpen the previous result on the 1 connectivity of the same model we also provide numerical results to support our analysis

k connectivity in random key graphs with unreliable links random key graphs form a class of random intersection graphs that are naturally induced by the random key predistribution scheme of eschenauer and gligor for securing wireless sensor network wsn communications random key graphs have received much attention recently owing in part to their wide applicability in various domains including recommender systems social networks secure sensor networks clustering and classification analysis and cryptanalysis to name a few in this paper we study connectivity properties of random key graphs in the presence of unreliable links unreliability of graph links is captured by independent bernoulli random variables rendering them to be on or off independently from each other the resulting model is an intersection of a random key graph and an erd s renyi graph and is expected to be useful in capturing various real world networks e g with secure wsn applications in mind link unreliability can be attributed to harsh environmental conditions severely impairing transmissions we present conditions on how to scale this model s parameters so that 1 the minimum node degree in the graph is at least k and 2 the graph is k connected both with high probability as the number of nodes becomes large the results are given in the form of zero one laws with critical thresholds identified and shown to coincide for both graph properties these findings improve the previous results by rybarczyk on k connectivity of random key graphs with reliable links as well as the zero one laws by yagan on one connectivity of random key graphs with unreliable links

pseudo jump inversion upper cone avoidance and strong jump traceability abstract there are noncomputable c e sets computable from every c e set relative to which is strongly jump traceable this yields a natural pseudo jump operator increasing on all sets which cannot be inverted back to a minimal pair or even avoiding an upper cone

pseudojump operators i the r e case call an operator j on the power set of co a pseudo jump operator if j a is uniformly recursively enumerable in a and a is recursive in j a for all subsets a of c thus the turing jump operator is a pseudo jump operator and any existence proof in the theory of r e degrees yields when relativized one or more pseudo jump operators extending well known results about the jump we show that for any pseudo jump operator j every degree 0 has a representative in the range of j and that there is a nonrecursive r e set a with j a of degree 0 the latter result yields a finite injury proof in two steps that there is an incomplete high r e degree and by iteration analogous results for other levels of the h ln hierarchy of r e degrees we also establish a result on pairs of pseudo jump operators this is combined with lachlan s result on the impossibility of combining splitting and density for r e degrees to yield a new proof of harrington s result that 0 does not split over all lower r e degrees 1 we prove some generalizations of theorems about the turing jump operator denoted a h a to theorems about operators of the form a a ed wa for an arbitrary fixed godel number e here a ed b is the recursive join of a and b and wa is the eth set r e in a in a fixed standard enumeration for instance friedberg see 25 theorem 4 1 showed that there is a nonrecursive r e set a with a tk where k is a complete r e set we prove by a finite injury priority argument similar to friedberg s that for every e there is a nonrecursive r e set a with a ed wa t k now friedberg s argument relative to an arbitrary oracle b yields a fixed godel

every sequence is reducible to a random one every infinite sequence is turing reducible to an infinite sequence which is random in the sense of martin lof

chow rings of moduli spaces of curves ii some results on the chow ring of overline mathscr m _4 ring of the moduli space of stable curves of genus 4 these results are not complete we find generators for the chow ring of 4 and for the chow groups in codimension 1 and 2 of w4 for a2 g 4 we find fourteen generators using test surfaces we prove that the dimension of a 2 4 4 is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist finally we have two proofs that this relation does indeed hold so that the dimension of a2 4 4 equals 13 this enables us to determine the chow ring of 4 4 our original proof is based on a rather delicate argument the second proof uses a result of ran see r and is much simpler

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence