We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
They gave an interpretation of the induction formula in genus 0 as a generalization of the completed cycle formula of Okounkov and Pandharipande (see @cite_19 ). For, now it is not clear if this generalized completed cycle formula has an extension to higher genera.
@cite_19
2086888155
Curves of genus g which admit a map to CP1 with specified ramification profile mu over 0 and nu over infinity define a double ramification cycle DR_g(mu,nu) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR_g(mu,nu) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR_g(mu,nu) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain's formula in the compact type case. When mu and nu are both empty, the formula for double ramification cycles expresses the top Chern class lambda_g of the Hodge bundle of the moduli space of stable genus g curves as a push-forward of tautological classes supported on the divisor of nonseparating nodes. Applications to Hodge integral calculations are given. We give a log-geometric description of the space of twisted canonical divisors constructed by Farkas--Pandharipande. In particular, we introduce the notion of a principal rubber @math -log-canonical divisor, and we study its moduli space. It is a proper Deligne--Mumford stack admitting a perfect obstruction theory whose virtual fundamental cycle is of dimension @math . In the so-called strictly meromorphic case with @math , the moduli space is of the expected dimension and the push-forward of its virtual fundamental cycle to the moduli space of stable curves equals the weighted fundamental class of the moduli space of twisted canonical divisors. Conjecturally, it yields a formula of Pixton generalizing the double ramification cycle in the moduli space of stable curves. We provide a model for reasoning about knowledge and probability together. We allow explicit mention of probabilities in formulas, so that our language has formulas that essentially say «according to agent i, formula φ holds with probability at least b.» The language is powerful enough to allow reasoning about higher-order probabilities, as well as allowing explicit comparisons of the probabilities an agent places on distinct events. We present a general framework for interpreting such formulas, and consider various properties that might hold of the interrelationship between agents' probability assignments at different states. We provide a complete axiomatization for reasoning about knowledge and probability, prove a small model property, and obtain decision procedures We establish the asymptotic expansion of certain integrals of ψ classes on moduli spaces of curves ℳ¯g,n, when either the g or n goes to infinity. Our main tools are cut-join type recursion formulae from the Witten–Kontsevich theorem, as well as asymptotics of solutions to the first Painleve equation. We also raise a conjecture on large genus asymptotics for n-point functions of ψ classes and partially verify the positivity of coefficients in generalized Mirzakhani’s formula of higher Weil–Petersson volumes. A formula is derived for the average delay to vehicles at junctions controlled by fixed-cycle traffic signals. The predicted delays are close to those observed in practice. Optimum signal settings are then derived using the delay formula. A formula is read-once if each variable appears on at most a single input. Previously, Angluin, Hellerstein, and Karpinski gave a polynomial time algorithm hat uses membership and equivalence queries to identify exactly read once boolean formulas over the basis AND, OR, NOT . In this paper we consider natural generalizations of this basis, and develop exact identification algorithms for more powerful classes of read-once formulas. We show that read-once formulas over the basis of arbitrary boolean functions of constant fan-in L (i.e., any ?: 0,1 1 ? c ? k ? 0,1 , where k is a constant) are exactly identifiable i polynomial time using membership and equivalence queries. We also show that read-once formulas over the basis of arbitrary symmetric boolean functions are exactly identifiable in polynomial time in this model. Given standard cryptographic assumptions, there is no polynomial time identification algorithm for read-twice formulas over either of these bases in the model. We further show that for any basis class B meeting certain technical conditions, any polynomial time identification algorithm for read-once formulas over B can be extended to a polynomial time identification algorithm for read-once formulas over the union of B and the arbitrary functions of constant fan-in. As a result, read-once formulas over the union of arbitrary symmetric and arbitrary constant fan-in gates are also exactly identifiable in polynomial time using membership and equivalence queries. An important research area of Spectrum-Based Fault Localization (SBFL) is the effectiveness of risk evaluation formulas. Most previous studies have adopted an empirical approach, which can hardly be considered as sufficiently comprehensive because of the huge number of combinations of various factors in SBFL. Though some studies aimed at overcoming the limitations of the empirical approach, none of them has provided a completely satisfactory solution. Therefore, we provide a theoretical investigation on the effectiveness of risk evaluation formulas. We define two types of relations between formulas, namely, equivalent and better. To identify the relations between formulas, we develop an innovative framework for the theoretical investigation. Our framework is based on the concept that the determinant for the effectiveness of a formula is the number of statements with risk values higher than the risk value of the faulty statement. We group all program statements into three disjoint sets with risk values higher than, equal to, and lower than the risk value of the faulty statement, respectively. For different formulas, the sizes of their sets are compared using the notion of subset. We use this framework to identify the maximal formulas which should be the only formulas to be used in SBFL. In this work, we present new arithmetic formulas for a projective version of the affine point representation ((x,x+y x), ) for (x 0, ) which leads to an efficient computation of the scalar multiplication operation over binary elliptic curves. A software implementation of our formulas applied to a binary Galbraith–Lin–Scott elliptic curve defined over the field ( F _ 2^ 254 ) allows us to achieve speed records for protected unprotected single multi-core random-point elliptic curve scalar multiplication at the 127-bit security level. When executed on a Sandy Bridge 3.4 GHz Intel Xeon processor, our software is able to compute a single multi-core unprotected scalar multiplication in 69,500 and 47,900 clock cycles, respectively, and a protected single-core scalar multiplication in 114,800 cycles. These numbers are improved by around 2 and 46 on the newer Ivy Bridge and Haswell platforms, respectively, achieving in the latter a protected random-point scalar multiplication in 60,000 clock cycles. We describe a correspondence between the Donaldson–Thomas invariants enumerating D0–D6 bound states on a Calabi–Yau 3-fold and certain Gromov–Witten invariants counting rational curves in a family of blowups of weighted projective planes. This is a variation on a correspondence found by Gross–Pandharipande, with D0–D6 bound states replacing representations of generalised Kronecker quivers. We build on a small part of the theories developed by Joyce–Song and Kontsevich–Soibelman for wall-crossing formulae and by Gross–Pandharipande–Siebert for factorisations in the tropical vertex group. Along the way we write down an explicit formula for the BPS state counts which arise up to rank 3 and prove their integrality. We also compare with previous “noncommutative DT invariants” computations in the physics literature. Sarkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with exhaustive and gated service disciplines, some explicit formulas that provide additional insight and characterization of this anomaly. More specifically, we show that, for both of these models, there exist simple formulas that define for each queue a critical value z* of the mean total setup time z per cycle such that, if z < z*, then the expected waiting time at that queue will be minimized if the server is forced to idle for a constant length of time z*- z every cycle; also, for the symmetric polling model, we give a simple explicit formula for the expected waiting time and the critical value z* that minimizes it. Relations among tautological classes on the moduli space of stable curves are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r=4 relations are used to bound the Betti numbers of the tautological ring of the moduli of nonsingular curves. At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class. In the Appendix (with F. Janda), a conjecture relating the r=0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented. We present a large-deviations thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are small.'' One consequence is an immediate derivation of the cavity'' formula for the fraction of vertices in the giant component. As a by-product of our analysis we compute the large-deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only small'' components. We establish foundational results on the computational complexity of deciding entailment in Separation Logic with general inductive predicates whose underlying base language allows for pure formulas, pointers and existentially quantified variables. We show that entailment is in general undecidable, and ExpTime-hard in a fragment recently shown to be decidable by Moreover, entailment in the base language is ( ^ P )-complete, the upper bound even holds in the presence of list predicates. We additionally show that entailment in essentially any fragment of Separation Logic allowing for general inductive predicates is intractable even when strong syntactic restrictions are imposed. In this paper we consider particular equational formulae where equality = AC is the congruence induced by a set of associative-commutative axioms. The formulae we are interested in have the form t ? ACt1 ??t ? ACtn and are usually known as complement problems. To solve a complement problem is to find an instance of t which is not an instance of any ti modulo associativity-commutativity. We give a decision procedure based on tree automata which solves these formulae when all the ti1s are linear. We show that this solution also gives a decision of inductive reducibility modulo associativity and commutativity in the linear case and we give several extensions of this approach. Then, we define a new class of tree automata, called conditional tree automata which recognize sets of generalized terms, i.e terms written as multisets, and where the application of a rule depends on the satisfiability of a formula of Presburger's arithmetic. This class of tree automata allows one to solve non-linear complement problems when all occurrences of a non-linear variable occur under the same node (in flattened terms). This solution also provides a procedure to decide inductive reducibility modulo associativity-commutativity in the same case. We consider the problem of finding edge search strategies of minimum cost. The cost of a search strategy is the sum of searchers used in the clearing steps of the search. One of the natural questions is whether it is possible to find a search strategy that minimizes both the cost and the number of searchers used to clear a given graph G. We call such a strategy ideal. We prove, by an example, that ideal search strategies do not exist in general. On the other hand, we provide a formula for the cost of clearing complete graphs. From our construction it follows that an ideal search strategy of a complete graph does exist and can be calculated efficiently. For general graphs G we give a polynomial-time O(logn)-approximation algorithm for finding minimum cost search strategies. We also prove that recontamination does not help to obtain minimum cost edge search strategies. Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra U_q(X^ (1) _n) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary X_n. This paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines. Often, counting classes, probabilistic classes, and intersection classes lack such enumerations. Concentrating on the counting class UP, we show that there are relativizations for which UPA has no complete languages and other relativizations for which PB ≠ UPB ≠ NPB and UPB has complete languages. Among other results we show that P ≠ UP if and only if there exists a set S in P of Boolean formulas with at most one satisfying assignment such that S ∩ SAT is not in P. P ≠ UP ∩ coUP if and only if there exists a set S in P of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in S. If UP has complete languages then there exists a set R in P of Boolean formulas with at most one satisfying assignment so that SAT ∩ R is complete for UP. Finally, we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP. There is a relativized world where BPPA has no complete languages. If BPP has complete languages then it has a complete language of the form B ∩ MAJORITY, where B ∈ P and MAJORITY = f | f is true for at least half of all assignments is the canonical PP-complete set. We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. Recently, have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and a new enumeration formula for complete simple games with two minimal winning vectors. Given thatα, β are two Lipschitz continuous functions of Ω to ℝ+ and thatf(x, u, p) is a continuous function of ( ) × ℝ × ℝ N to [0, + ∞[ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two pointsα(x) andβ(x), we prove the convergence ofF e (u) = (1 e)∫ Ω f (x, u, eDu) dx when the perturbation parametere goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs. Temporal logic is two-valued: formulas are interpreted as either true or false. When applied to the analysis of stochastic systems, or systems with imprecise formal models, temporal logic is therefore fragile: even small changes in the model can lead to opposite truth values for a specification. We present a generalization of the branching-time logic CTL which achieves robustness with respect to model perturbations by giving a quantitative interpretation to predicates and logical operators, and by discounting the importance of events according to how late they occur. In every state, the value of a formula is a real number in the interval [0,1], where 1 corresponds to truth and 0 to falsehood. The boolean operators and and or are replaced by min and max, the path quantifiers ∃ and ¬ determine sup and inf over all paths from a given state, and the temporal operators ♦ and □ specify sup and inf over a given path; a new operator averages all values along a path. Furthermore, all path operators are discounted by a parameter that can be chosen to give more weight to states that are closer to the beginning of the path.We interpret the resulting logic DCTL over transition systems, Markov chains, and Markov decision processes. We present two semantics for DCTL: a path semantics, inspired by the standard interpretation of state and path formulas in CTL, and a fixpoint semantics, inspired by the µ-calculus evaluation of CTL formulas. We show that, while these semantics coincide for CTL, they differ for DCTL, and we provide model-checking algorithms for both semantics.