The Degree of Unsolvability of Models

The purpose of this paper is to describe the complexity of models by theirdegrees of unsolvability,J.Richter defined the degree of a structure to be deg ()=sup{deg(),deg(R_i),i=1,…,n},Where is a model for afinite language L={R_i,i=1,…,n}and the universe of is a subset of ω. Shepointed out that, according to her definition, there can be models which areisomorphic but their degrees are different.Also,her discussions are restricted tofinite languages and models whose universes are subsets of ω.     

The Categories of m-semilatices

§１．ｍ┐ｓｅｍｉｌａｔｉｃｅｓＧｉｖｅｎａｃａｒｄｉｎａｌｍ．ＡｐｏｓｅｔＰｉｓｃａｌｅｄａｎｍ－ｓｅｍｉｌａｔｉｃｅ，ｉｆｆｏｒａｎｙｎｏｎ－ｅｍｐｔｙｓｕｂｓｅｔＢｏｆＰｗｉｔｈ｜Ｂ｜＜ｍ，ｔｈｅｉｎｆｉｍｕｍ∧ＢｅｘｉｓｔｓｉｎＰ．Ａｓｅｔｗ...     

The Continuity of Epimorphisms of Banach Algebras

ＬｅｔｆｂｅａｌｉｎｅａｒｍａｐｐｉｎｇｆｒｏｍａＢａｎａｃｈｓｐａｃｅＸｔｏａＢａｎａｃｈｓｐａｃｅＹ．Ｔｈｅｎｔｈｅｓｅｐａｒａｔｉｎｇｓｐａｃｅｏｆｆｉｓ（ｆ）＝｛ｙ∈Ｙ：ｔｈｅｒｅｅｘｉｓｔｓａｓｅｑｕｅｎｃｅ（ｘｎ）ｉｎＸｗｉｔｈｘｎ０ａ...     

The distribution of sublattices of ℤ m

Lattices , are similar if one can be transformed into the other by an angle-preserving linear map. Similarity classes of lattices of rankn may be parametrized by a fundamental domain of the action ofGL _n () on the generalized upper half-plane _n . Given 1<nm and, letN(D,T) be the number of sublattices of _n which have rankn, similarity class inD, and determinant T. Our most basic result will be thatN(D,T)c ₁(m, n)(D)T ^m asT for suitable setsD, where is the invariant measure on _n . The casen=2 had been dealt with by Roelcke and by Maass using the theory of modular forms.Herrn Professor Hlawka zum achtzigsten Geburtstag gewidmetSupported in part by NSF-DMS-9401426     

The Semimodularity of Cardinal Power of Posets

LetX,Ybeanyposets,Y'denotethecardinalpowerwithbaseYandexponentX.Fortherelatedconceptslsymbolsorterminologies,seeF1],[2jor[3].lnL1],someexamplesaregaven,whichshowthatthesemimodularityofX,YandYxareindepentent.AndaquestionisaskedthatwhenthecardinalpowerYxwithbaseYandexpo-nentXissemimodular.Wenowanswerthequestioninthispaper.LemmaLetX,Ybeanyposets,f,geyx,thenthefollowingconditionsareequivalent:(l)fiscoveredbyg(denotedbyf,g);(2)Thereexists'anelementxoeX,suchthatf(x,),g(x,),andf(x)=g(x)(foran…     

The Linear Arboricity of Graphs

Inthispaper,allgraphsarefinite,simpleandundirected.Forarealnumberx,[x]istheleastintegernotlessthanx.LetG＝(V(G),E(G))beagraph.Weuse△(G)andδ(G)todenotethemaximum(vertex)degreeandtheminimum(vertex)degreeofG.Letw(G)＝min{d(u) d(v):uv∈E(G)}.Thegirthg(G)ofGistheminimumlengthofcycles.Thedensitymad(G)ofagraphGisthemaximumvalueof2|E(H)|/|V(H)|takenoverallsubgraphsHofG.Ifv6V(G),N(v)denotesthesetofvenicesadjacenttov,thedegreedG(v)isING(v)landNc(v)={aluEN(v)andd(u)=k}.Avertexofdegreekisc…     

The work of Lu Jia-xi

Lu Jia-xi was a high school teacher at the ninth middle school in Baot'ou China. The only work of his published in western journals were six brilliant papers published in the Journal of Combinatorial Theory-entitled on Lange Scts of Disjoint Triple Systems I,-VI, [LuI-VI]. This problem in one form or another dates back to the middle     

Solution of The Last Example

Solution Let E be the elevation of the platear(in meters).From the Slide Model for Subtraction.E-20=53.Solve this equation.     

Applying of The Pythagorean Theorem

~~Applying of The Pythagorean Theorem~~     

The Hausdorff Dimension of Sections

The notion of finite-type open set condition is defined to calculate the Hausdorff dimensions of the sections of some self-similar sets, such as the dimension of intersection of the Koch curve and the line x = a with a∈Q.     

The Stability of Generalized Solution of Integral Equation

When we seek the generalized solution ofthe second knid Fredholm integral equation,the integral domain is always perturbed.Is the generalized solution of the second kindFredholm integral equation stable,when the integral domain is perturbed?Before an-swering this question let us look at the following example.　 Example1　Consider the integral equationψ(x) -∫baex- yψ(y) dy =f(x) ,　　 x∈ [a,b] . (1 )Obviously,the solutionψ(x) of the equation(1 ) must satisfy thatψ(x) =f(x) + cex, (2 )wh…     

The spectrum of path factorization of bipartite multigraphs

LetλK_(m,n)be a bipartite multigraph with two partite sets having m and n vertices, respectively.A P_v-factorization ofλK_(m,n)is a set of edge-disjoint P_v-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).When v is an even number,Ushio,Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P_v-factorization ofλK_(m,n).When v is an odd number,we have proposed a conjecture.Very recently,we have proved that the conjecture is true when v=4k-1.In this paper we shall show that the conjecture is true when v = 4k 1,and then the conjecture is true.That is,we will prove that the necessary and sufficient conditions for the existence of a P_(4k 1)-factorization ofλK_(m,n)are(1)2km≤(2k 1)n,(2)2kn≤(2k 1)m,(3)m n≡0(mod 4k 1),(4)λ(4k 1)mn/[4k(m n)]is an integer.     

The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

The Solution of Scholz's Problem

Def. The natural n is called a satisfiable number of a(in first order logic)formula α if α has a model of cardinality n. Def.The set M is called a finite spectrum if M is the set of all the satis-fiable numbers of a certain formula of first order logic.     

The Popularization of the Weighting of Chebysher's Inequality

This paper presents the popularization of the weighting of Chebyshev's inequality and discusses the relation between this popularization and some famous inequalities.     

The Compressibility of Checkerboard Surfaces of Link Diagrams

Consider the checkerboard surfaces defined by some link diagrams. When they are not orientable, one considers the boundary surfaces of small regular neighborhoods of them. This article studies the compressibility problem of these kinds of surfaces in the link complements. The problem is solved by devising a normalization theory for the compressing discs, which brings up an algorithm to read out compressibility directly from the link diagrams. As an application of the algorithm, the compressibility changes under Reidermeister moves are studied. Diagrams from the knot tables are also studied, and surprisingly, some of them are shown to define completely compressible surfaces of this kind. Infinitely many examples of non-alternating knot diagrams with incompressible surfaces of this kind are also constructed.     

The structure of a class of Z-local rings

A local ring R is called Z-local if J(R) = Z(R) and J(R)2 = 0. In this paper the structure of a class of Z-local rings is determined.     

The Bandwidth of a Class of Line Graphs

本文给出了一类线图的带宽,并给出了它的最佳编号.