Hurwitz theory and the double ramification cycle

This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov–Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov–Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several results by many groups of people on the subject, but, perhaps more importantly, collect a number of conjectures and problems which are still open.     

Statistical study of high energy radiation from rotation-powered pulsars

Based on our self-consistent outer gap model for high energy emission from the rotationpowered pulsars, we study the statistical properties of X-ray and y-ray emission from the rotation-powered pulsars, and other statistical properties (e.g. diffuse y-ray background and unidentified y-ray point sources) related to y-ray pulsars in our Galaxy and nearby galaxies are also considered     

A boundary layer theory for diffusively perturbed transport around a heteroclinic cycle

We consider a two‐dimensional transport equation subject to small diffusive perturbations. The transport equation is given by a Hamiltonian flow near a compact and connected heteroclinic cycle. We investigate approximately harmonic functions corresponding to the generator of the perturbed transport equation. In particular, we investigate such functions in the boundary layer near the heteroclinic cycle; the space of these functions gives information about the likelihood of a particle moving a mesoscopic distance into one of the regions where the transport equation corresponds to periodic oscillations (i.e., a “well” of the Hamiltonian). We find that we can construct such approximately harmonic functions (which can be used as “corrector functions” in certain averaging questions) when certain macroscopic “gluing conditions” are satisfied. This provides a different perspective on some previous work of Freidlin and Wentzell on stochastic averaging of Hamiltonian systems. © 2004 Wiley Periodicals, Inc.     

Corrections of rotation of the galaxy to measuredP of pulsars

TheP of pulsars provides important information for studying their physical processes. In an all-round way the effect of the Galactic rotation on the measuredP of pulsars is studied. It is shown that among 706 pulsars discovered so far there are 16 pulsars whose measured values ofP have been affected by the Galactic rotation. So, it is necessary to make the corresponding corrections for those pulsars Project supported by the National Natural Science Foundation of China (Grant No. 19733001).     

Morse theory and Lyusternik-Shnirelman theory in geometric control theory

Questions, related to the application of the ideas of global analysis to optimal control problems, are considered. A theory of Lyusternik-Shnirelman type is constructed for Hilbert manifolds with singularities, the so-called transversally convex subsets. Conditions for the nondegeneracy of the critical points (the extremal controls) are established in the optimal control problem, related to a smooth control system of constant rank, and a formula for their Morse index is given.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 39, pp. 41–117, 1991.     

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph K_m(n₁,n₂,…,n_m) with vertex set V and m independent sets G₁,G₂,…,G_m of n₁,n₂,…,n_m vertices respectively. Let G={G₁,G₂,…,G_m}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?G_i for some G_i∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type g^u. In this paper, we show that there exists a (k,λ)-cycle frame of type g^u for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.     

Model theory via set theory

We develop a technique for applying models of set theory to obtain results in the model theory of infinitary languages. Some results on Δ-logics are also discussed.     

Priority theory and utility theory

The first axiom of the Analytic Hierarchy Process (AHP) states that the pairwise comparison judgements are reciprocal. Here we explore the relationships between the reciprocal property and preference relations with and without the axiom of transitivity and point out some important distinctions between utility theory and the AHP.     

Approach theory meets probability theory

In this paper we reconsider the basic topological and metric structures on spaces of probability measures and random variables, such as e.g. the weak topology and the total variation metric, replacing them with more intrinsic and richer approach structures. We comprehensibly investigate the relationships among, and basic facts about these structures, and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorov?s theorem, can be recaptured in a considerably stronger form in the new setting.     

Classes of cycle bases

In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiries. At present, the complexity status of the MCB problem has been settled only for undirected, directed, and strictly fundamental cycle bases.In this paper, we offer an unitary classification accommodating these three classes and further including the following four relevant classes: 2-bases (or planar bases), weakly fundamental cycle bases, totally unimodular cycle bases, and integral cycle bases. The classification is complete in that, for each ordered pair (A,B) of classes considered, we either prove that A⊆B holds for every graph or provide a counterexample graph for which A?B. The seven notions of cycle bases are distinct (either A?B or B?A is exhibited for each pair (A,B)).All counterexamples proposed have been designed to be ultimately effective in separating the various algorithmic variants of the MCB problem naturally associated to each one of these seven classes. Furthermore, we provide a linear time algorithm for computing a minimum 2-basis of a graph. Finally, notice that the resolution of the complexity status of some of the remaining three classes would have an immediate impact on practical applications, as for instance in periodic railway timetabling, only integral cycle bases are of direct use.     

An application of set theory to model theory

We give a number of simple proofs of results in model theory using the set theoretical result of Lévy thatH(μ) is a Σ-submodel of the Universe. We would like to thank the referee and Taylor Ollmann for their helpful suggestions.     

Completing the spectrum of almost resolvable cycle systems with odd cycle length

In this paper, we construct almost resolvable cycle systems of order $4k+1$ for odd $k\ge 11$. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length. As a by-product, some new solutions to the Hamilton–Waterloo problem are also obtained.     

Short cycle connectivity

Short cycle connectivity is a generalization of ordinary connectivity—two vertices have to be connected by a sequence of short cycles, in which two consecutive cycles have at least one common vertex. If all consecutive cycles in the sequence share at least one edge, we talk about edge short cycle connectivity. Short cycle connectivity can be extended to directed graphs (cyclic and transitive connectivity).It is shown that the short cycle connectivity is an equivalence relation on the set of vertices, while the edge/arc short cycle connectivity components determine an equivalence relation on the set of edges/arcs.     

Embedding intuitionistic-type theory in negationless-type theory

Mathematical Notes -     

Group theory via global semigroup theory

Groups are shown to be special homomorphic images of inverse semigroups that are residually finite (actually: every element has only finitely many elements -above). This also leads to a new approach to the Burnside problem. These results extend an earlier paper ([1.], 249–287), but can be read independently. Our goal here is not so much to prove theorems about inverse semigroups as to demonstrate the usefulness of the constructions of [1.], 249–287.