Avoidable algebraic subsets of Euclidean space

Fix an integer and consider real -dimensional . A partition of avoids the polynomial , where each is an -tuple of variables, if there is no set of the partition which contains distinct such that . The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over , and (2) a characterization of the avoidable polynomials. An important feature is that both the ``master' partition in (1) and the characterization in (2) depend on the cardinality of .

     

Ascoli-Arzelà type theorem for rational iterations on the complex projective plane

For a dominant algebraically stable rational self-map of the complex projective plane of degree at least 2, we will consider three different definitions of the Fatou set and show the equivalence of them (Ascoli-Arzelà type theorem). As a corollary, it follows that all Fatou components are Stein. This is an improvement of an early result by Fornæss and Sibony.

     

Lot-sizing on a tree

For the problem of lot-sizing on a tree with constant capacities, or stochastic lot-sizing with a scenario tree, we present various reformulations based on mixing sets. We also show how earlier results for uncapacitated problems involving (Q,S_Q) inequalities can be simplified and extended. Finally some limited computational results are presented.     

Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.     

Duality Theorem for a Three-Phase Partition Problem

In some nonlinear diffusive phenomena, the systems have three or more stable states. Sternberg and Zeimer established the existence of minimal solutions for the problem of partitioning a certain domain Ω⊂ℝ² into three subdomains having least interfacial area. Ikota and Yanagida investigated stability and instability for stationary curves with one triple junction and for stationary binary-tree type interfaces. In this paper, we introduce a new concept of separation of three convex sets by a triangle, define a dual problem to the three-phase partition problem, and present a duality theorem. The author thanks Professor F. Giannessi for valuable comments, especially on Gale and Klee-type separation theorems. This research was partially supported by Kyushu University 21st Century COE Program (Development of Dynamic Mathematics with High Functionality) and by the Grant-in-Aid for General Scientific Research from the Japan Society for the Promotion of Science 14340037.     

Self-Affine Sets and Graph-Directed Systems

   Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ⊂ Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ≠  ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

Vacuum condensates of QCD

We study the properties of QCD vacuum state in this paper. The values of various local quark vacuum condensates, quark-gluon mixed vacuum condensates, and the structure of non-local quark vacuum condensate are predicted by the solution of Dyson-Schwinger Equations in "rainbow" approximation with three sets of different parameters for effective gluon propagator. The light quark virtuality is also obtained in a consistent way. Our all theoretical results here are in good agreement with the empirical values used widely in literature and many other theoretical calculations.     

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．     

An interdependency index for the outputs of uncertain systems

The study of mechanical systems with uncertain parameters is gaining increasing interest in the field of system analysis to provide an expedient model for the prediction of the system behavior. Making use of the Transformation Method, the uncertain parameters of the system are modeled by fuzzy numbers in contrast to random numbers used in stochastic approaches. As a result of this analysis, a quantification of the overall uncertainty of the system outputs, including a worst-case scenario, is obtained. The inputs of the resulting fuzzy-valued model are a priori uncorrelated but after the uncertainties are propagated through the model, interdependency (or interaction) between the outputs may arise. If such interdependency is neglected, a misinterpretation of the results may occur. For example, in the case of applying uncertainty analysis in the early design phase of a product to determine the relevant design-parameter space, the interdependency between the design variables may reduce significantly the available part of the design space. This paper proposes a measure of interdependency between the uncertain system outputs. The interdependency index can be derived by a postprocessing of the data gained by the analysis with the Transformation Method. Such information can be obtained by a negligible amount of extra computation time.     

On the sharpness of two-sided bounds for the Perron Root and of the related eigenvalue inclusion sets

The paper considers the sharpness problem for certain two-sided bounds for the Perron root of an irreducible nonnegative matrix. The results obtained are applied to prove the sharpness of the related eigenvalue inclusion sets in classes of matrices with fixed diagonal entries, bounded above deleted absolute row sums, and a partly specified irreducible sparsity pattern.