<Emphasis Type="Italic">q</Emphasis>-Fibonacci Polynomials and the Rogers-Ramanujan Identities

We derive some formulas for the Carlitz q-Fibonacci polynomials F_n(t) which reduce to the finite version of the Rogers-Ramanujan identities obtained by I. Schur for t = 1. Our starting point is a representation of the q-Fibonacci polynomials as the weight of certain lattice paths in contained in a strip along the x-axis. We give an elementary combinatorial proof by using only the principle of inclusion-exclusion and some standard facts from q-analysis.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

Inclusion-exclusion: Which terms cancel?

We continue our study of the phenomenon of cancellation appearing in the inclusion-exclusion principle. More precisely, we regard the terms occuring in the inclusion-exclusion principle as nodes of a rooted tree and identify each set of cancelling terms with a subtree of this tree.     

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu?0 and for Iu+up?0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.     

严格非匀称线性超树的计数公式

本文应用容斥原理，得到了有n个顶点、m条边的严格非匀称标号线性无圈超图的计数公式。     

Nonempty Intersection Theorems and Generalized Multi-objective Games in Product <Emphasis Type="Italic">FC</Emphasis>-Spaces

A new class of generalized multi-objective games is introduced and studied in FC-spaces where the number of players may be finite or infinite, and all payoff are all set-valued mappings and get their values in a topological space. By using an existence theorems of maximal elements for a family of set-valued mappings in product FC-spaces due to author, some new nonempty intersection theorems for a family of set-valued mappings are first proved in FC-spaces. As applications, some existence theorems of weak Pareto equilibria for the generalized multi-objective games are established in noncompact FC-spaces. These theorems improve, unify and generalize the corresponding results in recent literatures.     

Some aspects of <Emphasis Type="Italic">m</Emphasis>-adic analysis and its applications to <Emphasis Type="Italic">m</Emphasis>-adic stochastic processes

In this paper we consider a generalization of analysis on p-adic numbers field to the m case of m-adic numbers ring. The basic statements, theorems and formulas of p-adic analysis can be used for the case of m-adic analysis without changing. We discuss basic properties of m-adic numbers and consider some properties of m-adic integration and m-adic Fourier analysis. The class of infinitely divisible m-adic distributions and the class of m-adic stochastic Levi processes were introduced. The special class of m-adic CTRW process and fractional-time m-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time m-adic random walk.     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 John Wiley & Sons, Ltd.     

An improvement of the inclusion-exclusion principle

We present an improvement of the inclusion-exclusion principle in which the number of terms is reduced by predicted cancellation. The improvement generalizes a related result of Narushima as well as a graph-theoretic theorem of Whitney. Applications concern chromatic polynomials of graphs and permanents of 0,1-matrices.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.     

Sufficient conditions for functional-limit-theorem versions ofL = λW

The familiar queueing principle expressed by the formulaL=W can be interpreted as a relation among strong laws of large numbers. In a previous paper, we showed that this principle can be extended to include relations among other classical limit theorems such as central limit theorems and laws of the iterated logarithm. Here we provide sufficient conditions for these limit theorems using regenerative structure.Supported by the National Science Foundation under Grant No. ECS-8404809 and by the U.S. Army under Contract No. DAAG29-80-C-0041.     

Maximal Element Theorems with Applications to Generalized Games and Systems of Generalized Vector Quasi-Equilibrium Problems in <Emphasis Type="Italic">G</Emphasis>-Convex Spaces

By applying the maximal element theorems on product of G-convex spaces due to the first author, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in G-convex spaces. As applications, some existence theorems of solutions for the system of generalized vector quasiequilibrium problem are established in noncompact product of G-convex spaces. Our results improve and generalize some recent results in the literature to product of G-convex spaces.The authors thank the referees for valuable comments and suggestionsThe research of this author was supported by the National Science Foundation of China, Sichuan Education Department.The research of this author was supported by the National Science Council of the Republic of China.     

On maximal element theorems,variants of Ekeland’s variational principle and their applications

In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ$\tau$-functions in ?$?$ complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.     

A martingale approach to minimal surfaces

We provide a probabilistic approach to studying minimal surfaces in R³. After a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in minimal surface theory, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.     

Some theorems in the theory of heat conduction with dissipative boundaries

The linear theory of heat conduction in rigid bodies with dissipative boundary conditions is considered. Some basic theorems concerning uniqueness, reciprocity and variational properties are presented. Further, a minimum principle is established.     

Graph theorems for manifolds

Two basic theorems about the graphs of convex polytopes are that the graph of ad-polytope isd-connected and that it contains a refinement of the complete graph ond+1 vertices. We obtain generalizations of these theorems, and others, for manifolds. We also supply some details for a proof of the lower bound inequality for manifolds. Research supported by NSF Grants GP 8470 and GP 19221.     

Fuzzy logics with modalities

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 731–757, November–December, 2006.     

Nilpotency of <Emphasis Type="Italic">n</Emphasis>-tuple Lie algebras and associative <Emphasis Type="Italic">n</Emphasis>-tuple algebras

We obtain conditions for the nilpotency of finite-dimensional n-tuple Lie algebras and finite-dimensional associative n-tuple algebras. The established conditions are analogous to theorems of Engel and Wedderburn for Lie algebras and associative algebras.