An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK_∞⁺. We show that in these conditions, we can interprete the Kelley-Morse theory (KM) in GPK_∞⁺ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo-Fraenkel set theory (ZF) in GPK_∞⁺. We also see what happens in the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections.     

几何分析中的基本概念与原理

本文是根据作者在中国科学院“中德几何分析伙伴小组”成立仪式上的演讲整理而成，主要目的在于通过回顾几何分析中的基本概念及基本定理，了解现阶段的研究情况和将来的研究线路，主要描述黎曼几何与量子场论中的思想，特别特别是极化物理量与几何量产生的变分问题。     

Research in decision theory: A personal perspective

This essay offers a personal view of developments in two main areas of decision theory — decision making under risk and uncertainty, and social choice theory — with emphasis on the past half century and on likely future directions. Remarks on helpful research practices are also included.     

Finite element analysis of arches undergoing large rotations - I: Theoretical comparison

In this two part paper, the first part deals with five different nonlinear theories applicable to the analysis of arches in the context of solving the large displacement and the large rotation problem. These theories include, classical theory, first-order shear deformation theory, third-order shear deformation theory, modified classical theory and the Donnell-type theory. All the theories are developed using the Total Lagrangian approach. Simplifications and assumptions used in each of the theory are discussed. Explicit strain displacement gradient relations and element independent equilibrium equations in terms of displacement gradients are given for all the theories. Limitations of each of theory are discussed. In the second part of this paper, application of these theories for the classification of arch geometries is considered.     

A Technique for the Search for Vertices of the Components of a k G-Permutation Module

In this paper we present a necessary condition for a p-group V ≤ G to be a vertex of some indecomposable direct summand of the permutation module k _H  ↑ ^G , where H ≤ G, and G is a finite group. We call this condition H-suitability and present a method how to check for it. In an example, we determine all H-suitable groups. In fact, in this example every H-suitable group is the vertex of some indecomposable direct summand of k _H  ↑ ^G .     

On the Ramsey numbers of odd-linked double stars

The linked double star $S_c(n,m)$, where $n\ge m\ge 0$, is the graph consisting of the union of two stars $K_{1,n}$ and $K_{1,m}$ with a path on c vertices joining the centers. Its Ramsey number $r(S_c(n,m))$ is the smallest integer r such that every 2-coloring of the edges of a $K_r$ admits a monochromatic $S_c(n,m)$. In this paper, we study the Ramsey numbers of linked double stars when c is odd. In particular, we establish bounds on the value of $r(S_c(n,m))$ and determine the exact value of $r(S_c(n,m))$ if $n\ge c$, or if $n\le ?\frac{c}{2}??2$ and $m=2$.     

Mutual definability does not imply definitional equivalence,a simple example

We give two theories, Th₁ and Th₂, which are explicitly definable over each other (i.e. the relation symbols of one theory are explicitly definable in the other, and vice versa), but are not definitionally equivalent. The languages of the two theories are disjoint. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy space of each player consists of the bases of a player-specific matroid defined on the set of resources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility location problems, where players have to jointly pay for opened facilities. The goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate two classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium and separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource. We find optimal basic and separable protocols that guarantee the price of stability/price of anarchy to grow logarithmically/linearly in the number of players. These results extend our previous results (cf. von Falkenhausen & Harks, 2013), where optimal basic and separable protocols were given for the case of symmetric matroid games without delays.     

Reverse Mathematics and Recursive Graph Theory

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.     

The canonical Ramsey theorem and computability theory

Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey's Theorem, and König's Lemma.

     

模型论简介

本文将模型论介绍为“可定义集的数学”．首先给出“可定义集”的确切定义，并引入处理可定义集的主要工具——一阶公式的概念．然后给出一些例子．最后介绍一些相当于可定义集的组合性质的稳定性理论．     

New Perturbation Theory for Quantum Field Theory: Convergent Series Instead of Asymptotic Expansions

We present a new approach to perturbation theory for quantum field theory based on convergent series instead of asymptotic expansions. This approach could be considered as the next step after traditional perturbation theory calculations, which allows more comprehensive use of previously obtained information in finding numerical values with greater accuracy.     

A K-theoretical invariant and bifurcation for a parameterized family of functionals

Let F:={fx:x∈X} be a family of functionals defined on a Hilbert manifold and smoothly parameterized by a compact connected orientable n-dimensional manifold X, and let be a smooth section of critical points of F. The aim of this paper is to give a sufficient topological condition on the parameter space X which detects bifurcation of critical points for F from the trivial branch. Finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi-Riemannian manifolds.     

Characteristic function for polynomially contractive commuting tuples

In this note, we develop the theory of characteristic function as an invariant for n-tuples of operators. The operator tuple has a certain contractivity condition put on it. This condition and the class of domains in Cn that we consider are intimately related. A typical example of such a domain is the open Euclidean unit ball. Given a polynomial P in C[z₁,z₂,…,zn] whose constant term is zero, all the coefficients are nonnegative and the coefficients of the linear terms are nonzero, one can naturally associate a Reinhardt domain with it, which we call the P-ball (Definition 1.1). Using the reproducing kernel Hilbert space HP(C) associated with this Reinhardt domain in Cn, S. Pott constructed the dilation for a polynomially contractive commuting tuple (Definition 1.2) [S. Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999) 365-389. MR 2000j:47019]. Given any polynomially contractive commuting tuple T we define its characteristic function θT which is a multiplier. We construct a functional model using the characteristic function. Exploiting the model, we show that the characteristic function is a complete unitary invariant when the tuple is pure. The characteristic function gives newer and simpler proofs of a couple of known results: one of them is the invariance of the curvature invariant and the other is a Beurling theorem for the canonical operator tuple on HP(C). It is natural to study the boundary behaviour of θT in the case when the domain is the Euclidean unit ball. We do that and here essential differences with the single operator situation are brought out.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet     

On smooth foliations with Morse singularities

Let M be a smooth manifold and let F be a codimension one, C^∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S¹, we are able to extend Haefliger?s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.