A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

Loop coproducts in string topology and triviality of higher genus TQFT operations

Cohen and Godin constructed a positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree d homology group of the connected component of LM consisting of contractible loops, where d=dimM, with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product.     

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.     

Statistical maps and generalized random walks

We continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.     

On the invariant density of the random β-transformation

We construct a Lebesgue measure preserving natural extension of a skew product system related to the random β-transformation K _β . This allows us to give a formula for the density of the absolutely continuous invariant probability measure of K _β , answering a question of Dajani and de Vries, and also to evaluate some estimates on the typical branching rate of the set of β-expansions of a real number.     

Commutative basic algebras and non-associative fuzzy logics

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. ?ukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L _CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers and includes the class of MV-algebras. We show that the logic L _CBA is very close to the ?ukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization.     

(f,ω)-Compatible Pair (B,H) for ω-Smash Coproduct Hopf Algebras

In this paper we introduce the notion of (f,ω)-compatible pair (B,H), by which we construct a Hopf algebra in the category HHYD of Yetter-Drinfeld H-modules by twisting the comultiplication of B. We also study the property of ω-smash coproduct Hopf algebras Bω H.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.     

Khovanov homology, open books, and tight contact structures

We define the reduced Khovanov homology of an open book (S,?), and identify a distinguished “contact element” in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,?). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in Baldwin (2008) [4].     

关于$\omega$-Smash余积Hopf代数的$(f,\omega)$相容对$(B,,H)$

In this paper we introduce the notion of （f,ω）-compatible pair （B,H）, by which we construct a Hopf algebra in the category HHYD of Yetter-Drinfeld H-modules by twisting the comultiplication of B. We also study the property of ω-smash coproduct Hopf algebras Bω H.     

A note on measures of nonconvexity

We define the Hausdorff measure of nonconvexity β(C) of a nonempty bounded subset C of a Banach space X as the Hausdorff distance of C to the family of all the nonempty convex bounded subsets of X. We compare the measure β with the Eisenfeld-Lakshmikantham measure of nonconvexity α and prove that the two measures are equivalent (β≤α≤2β), but in general they are different.     

Bounding Distributions for Stochastic Logic Networks

A stochastic logic network is defined as a connected set of logic and time delay elements. Each of the latter elements has an associated probability distribution describing the nature of that element's delay. When used, for example, in project planning and scheduling, combinations of logic and time delay elements in such networks may represent conditions for the starting of project activities which are themselves represented by time delay elements. It is at present not known how to calculate the probability distributions for the events in such a network. This paper shows how to obtain upper and lower bounds for these probability distributions. The method is not a simulation technique; rather, it is a straightforward computational scheme derived from elementary probability theory. An example is given where the method is applied to a stochastic project scheduling network in which alternative ways exist for carrying out one of the jobs in the network.     

On φ-Families of Probability Distributions

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.     

Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem

We look at the instance distributions used by Goldberg [3] for showing that the Davis Putnam Procedure has polynomial average complexity and show that, in a sense, all these distributions are unreasonable. We then present a ‘reasonable’ family of instance distributions F and show that for each distribution in F a variant of the Davis Putnam Procedure without the pure literal rule requires exponential time with probability 1. In addition, we show that adding subsumption still results in exponential complexity with probability 1.     

Primitive elements in the matroid-minor Hopf algebra

We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis and coproduct given by splitting a matroid into a submatroid and complementary contraction in all possible ways. We introduce two new bases for C; the first of these is related to the distinguished basis by Möbius inversion over the rank-preserving weak order on matroids, the second by Möbius inversion over the suborder excluding matroids that are irreducible with respect to the free product operation. We show that the subset of each of these bases corresponding to the set of irreducible matroids is a basis for the subspace of primitive elements of C. Projecting C onto the matroid-minor Hopf algebra H, we obtain bases for the subspace of primitive elements of H.     

Fusing multiple Bayesian knowledge sources

We address the problem of information fusion in uncertain environments. Imagine there are multiple experts building probabilistic models of the same situation and we wish to aggregate the information they provide. There are several problems we may run into by naively merging the information from each. For example, the experts may disagree on the probability of a certain event or they may disagree on the direction of causality between two events (e.g., one thinks A causes B while another thinks B causes A). They may even disagree on the entire structure of dependencies among a set of variables in a probabilistic network. In our proposed solution to this problem, we represent the probabilistic models as Bayesian Knowledge Bases (BKBs) and propose an algorithm called Bayesian knowledge fusion that allows the fusion of multiple BKBs into a single BKB that retains the information from all input sources. This allows for easy aggregation and de-aggregation of information from multiple expert sources and facilitates multi-expert decision making by providing a framework in which all opinions can be preserved and reasoned over.     

The expected discounted penalty function under a risk model with stochastic income

Quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function, assuming a risk model where both premiums and claims follow compound Poisson processes. Both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established. Some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated. Finally, the case when premiums have Erlang(n,β) distribution and the distribution of the claims is arbitrary is investigated in more depth. Throughout the paper specific examples where claims and premiums have particular distributions are provided.     

On multiplication formulas of affine q-Schur algebras

We provide a simple and conceptual proof of Du-Fu's multiplication formula of affine q-Schur algebras via Lusztig's formula. We use the multiplication formulas to provide a proof of the existence of generic affine Schur algebras, in return, and a formula of the generators under comultiplication.