The Median Formula and Related Inequalities in n—simplex

5l.TheMedianFormulaAseveryoneknows:in6ABC,letal=BC,a2=CA,a3=AB,;)11,)n2,n13aremedian1inesseparatelyonthethreeedgesBC,CA,AB,tI1entheequalityistrueasfollowsTheequality(l)iscalledmedianformulaofQABC,andwecanobtainfrom(l)Inthispaper,wesha1lextendtheequality(l)and(2)to,l-simplexin)l-EuclideanspaceH.Itisobtainedthemedianformulaof)I-simplex,andusingthisformula.wegetsomein-equalities.Themedianformulaofn-simp1exisobtainedasfollows:TheoremlFork=1,2,--',It 1andl相似文献   

Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability

Received July 16, 1998; accepted August 7, 1999     

Determinants and Symmetries in ‘Yetter—Drinfeld’ Categories

A Yetter—Drinfeld category over a Hopf algebra H with a bijective antipode, is equipped with a braiding which may be symmetric for some of its subcategories (e.g. when H is a triangular Hopf algebra). We prove that under an additional condition (which we term the u-condition) such symmetric subcategories completely resemble the category of vector spaces over a field k, with the ordinary flip map. Consequently, when Char k=0, one can define well behaving exterior algebras and non-commutative determinant functions.     

Conley pairs in geometry—Lusternik–Schnirelmann theory and more

Firstly, we wish to motivate that Conley pairs, realized via Salamon’s definition (Salamon, 1990), are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces (Weber, 2017). From this fell off quite naturally, firstly, an alternative proof (Weber, 2016) of the cell attachment theorem in Morse theory (Milnor, 1963) and, secondly, some ideas (Majer and Weber, 2015) how to try to organize the closures of the unstable manifolds of a Morse–Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934) proposed in Bott’s survey (Bott, 1982).Secondly, we shall use this opportunity to provide an exposition of Lusternik–Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934), cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.     

Modified Landweber Iteration in Banach Spaces—Convergence and Convergence Rates

We introduce and discuss an iterative method of modified Landweber type for regularization of nonlinear operator equations in Banach spaces. Under smoothness and convexity assumptions on the solution space we present convergence and stability results. Furthermore, we will show that under the so-called approximate source conditions convergence rates may be achieved by a proper a-priori choice of the parameter of the presented algorithm. We will illustrate these theoretical results with a numerical example.     

Arcs and k-sets with large nucleus — set in Hall planes

In this paper we give k-sets (k>-q+1) of the Hall planes with q-q nuclei. This shows that the results of BICHARA — KORCHMÁROS [1] and WETTL [16] cannot be extended to non-desarguesian planes. We construct some inherited arcs in the Hall planes as well. For example complete (q–1)-arcs of the Hall planes of odd order will be constructed.     

Binding of atoms and stability of molecules in hartree and thomas—fermi type theories

This paper is the sequel of a previous work where we showed a general necessary and sufficient condition for the stability of an arbitrary molecular system (possibly ionized) in the framework of Hartree or Thomas-Fermi type theories. This condition, roughly speaking, meant that certain particular subsystems have to be bound. We show here in particular that this condition reduces for general molecular system with nonnegative excess charge to the binding of all subsystems with the same property. For neutral inolecular systems, this reduces to the binding of all neutral subsystems. In both cases, all other subsystems can be bound. We also show that, for the Hartree-Fock and Hartree models, this condition involves only “physical” sulxystems We use these reduced conditions to conclude allout the stability or the binding in some particular cases. This work 1s also the second of a series devoted to these equations and we shall come back on the binding of neutral systems in Part 3.     

Invariant Mean—Value Property and M—Harmonicity in the Unit Ball of R^n

In 1993,Ahern,Flores and Rudin showed that,if f is integrable over the unit ball BC^n of C^n and satisfies∫BC^nfoφdv=f（φ（0)) for every φ∈Aut(BC^n),then f is M-harmonic if and only if n≤11.The present paper is about an analogous question in the context of the unit ball Bn of R^n as well as in the weighted setting.     

Teracluster LSSC-Ⅱ——Its designing principles and applications in large scale numerical simulations

The teracluster LSSC-II installed at the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences is one of the most powerful PC clusters in China. It has a peek performance of 2Tflops. With a Unpack performance of 1.04Tflops, it is ranked at the 43rd place in the 20th TOP500 List (November 2002), 51st place in the 21st TOP500 List (June 2003), and the 82nd place in the 22nd TOP500     

Mathematics teaching and values education—an intersection in need of research

This paper raises issues concerning the teaching of values in the context of mathematics education. It argues that a focus on education for democracy inevitably involves educating about values. It reviews the major relevant research and theoretical perspectives and argues for more research attention to be paid to this area. Although there has been relevant research done in the affective domain, both in general and in relation to mathematics, and in social and cultural issues, there is little direct research focus on either values or valuing. Teachers are rarely aware of teaching values either explicitly or implicitly, yet values teaching clearly does take place, mostly implicitly. If there are desires to change the directions of mathematics teaching to be more attuned to life in modern democratic societies then this aspect of mathematics education needs to be better understood in order that it can be better taught.     

Twofold fuzzy sets and rough sets—Some issues in knowledge representation

This paper deals with the representation of sets where the membership of some elements may be ill-known rather than just a matter of degree as in a fuzzy set. The notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete. A twofold fuzzy set is made of a nested pair of fuzzy sets: the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong. Twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by Pawlak. Set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined. Twofold fuzzy relations are also introduced. Finally, various applications of twofold fuzzy sets in knowledge representation are briefly discussed.     

The dynamic properties of polyethylene and an acrylonitrile — Butadiene — Styrene material in a broad range of temperature and strain-rate variation

Conclusions The mechanical properties of an acrylonitrile-butadiene-styrene material and polyethylenes of densities =0.92 and 0.96 g/cm³ are studied within the temperature range of from 100 to –196°C and at strain rates of from 10^–3 to 1.3 · 10³ sec^–1. It is observed that under low-temperature and high-strain-rate conditions, the laws governing the mechanical properties of the acrylonitrile-butadiene-styrene material and polyethylenes are similar. Regions of strain rates and temperatures are found for which the behavior of the test materials is in good agreement with the Ree-Eyring equation, and, consequently, conforms to a temperature-time analogy. No embrittlement of the polyethylene is observed during tests at strain rates to 1.3 · 10³ sec^–1 and temperatures down to –196°C. In approximating these values of and T, the mechanical properties of polyethylenes will cease to be dependent on initial density.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, 1027–1033, November–December, 1978.     

Ramsey-goodness—and otherwise

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number Δ, there is a constant r _Δ such that, for any connected n-vertex graph G with maximum degree Δ, the Ramsey number R(G,G) is at most r _Δ n, provided n is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take r _Δ = Δ. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r _Δ > 2 ^cΔ for some constant c>0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n)=o(n), then R(G,G)≤(2χ(G)+4)n≤(2Δ+6)n, i.e., r _Δ =2Δ+6 suffices. On the other hand, we show that Burr’s conjecture itself fails even for P _n ^k , the kth power of a path P _n . Brandt showed that for any c, if Δ is sufficiently large, there are connected n-vertex graphs G with Δ(G)≤Δ but R(G,K ₃) > cn. We show that, given Δ and H, there are β>0 and n ₀ such that, if G is a connected graph on n≥n ₀ vertices with maximum degree at most Δ and bandwidth at most β _n , then we have R(G,H)=(χ(H)?1)(n?1)+σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ?(H) log n=log logn.     

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In 1970, Farris introduced a procedure that can be used to transform a tree metric into an ultra metric. Since its discovery, Farris’ procedure has been used extensively within phylogenetics where it has become commonly known as the Farris transform. Remarkably, the Farris transform has not only been rediscovered several times within phylogenetics, but also in other fields. In this paper, we will review some of its various properties and uses. The paper is divided into four parts and, altogether, 12 sections. In the first part, we introduce a standardized scheme for classifying those dissimilarity mappings to which the Farris transform can be applied—a scheme that has evolved over the years, but has apparently not been spelled out before in sufficient detail. In the second part, we will discuss how a straightforward generalization of the Farris transform naturally arises in T-Theory. The third part describes how this generalized Farris transform can be used to approximate dissimilarities by tree metrics. And in the final part, we describe some further, “non-standard” applications of the Farris transform. Received November 24, 2004     

BCC—Algebras and BCK—Algebras

ＢＣＣ－ＡｌｇｅｂｒａｓａｎｄＢＣＫ－ＡｌｇｅｂｒａｓＺｈａｎｇＸｉａｏｈｏｎｇ（张小红）；ＷｅｉＺｏｎｇｌｉ（卫宗礼）（ＨａｎｚｈｏｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，７２３０００）（ＬｕｏｙａｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，４７１０２２）...