Marcus–Lushnikov processes,Smoluchowski’s and Flory’s models

The Marcus–Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses x$x$ and y$y$ merges into a single particle at a given rate K(x,y)$K(x,y)$. We consider a strongly gelling   kernel behaving as K(x,y)=x^αy+xy^α$K(x,y)=x^{\alpha }y+x{y}^{\alpha }$ for some α∈(0,1]$\alpha \in (0,1]$. In such a case, it is well-known that gelation occurs, that is, giant particles emerge. Then two possible models for hydrodynamic limits of the Marcus–Lushnikov process arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with finite ones.     

用矩阵的初等变换解矩阵方程Am×nXn×s=Bm×s

本文通过对一般的矩阵方程Am×nXn×s=Bm×s?B5木卣驛和B作初等行变换及初等列变换，给出了一般矩阵方程的求解方法。     

On Saint Venant's compatibility conditions and Poincaré's lemma

Saint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in $L^2$ by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in $H^{\mathrm{?1}}$. Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The N-heap Wythoffs game is a two-player impartial game with N piles of tokens of sizes Players take turns removing any number of tokens from a single pile, or removing (a₁,..., a_N) from all piles - a_i tokens from the i-th pile, providing that where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N₁ such that when . ii) there exist integers N₂ and _2 such that when , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoffs game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

Perverse equivalences and Brouéʼs conjecture

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué?s abelian defect group conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups elementary abelian of order 9. We also examine extensions to automorphism groups in a general setting.     

The decision to raise firm value through a sports-business exchange: How much are Real Madrid’s goals worth to its president’s company’s goals?

Global brands emerging from the world of sports are becoming commonplace, and firms invest in the realm of sports, usually through sponsorship initiatives, to get a link with these global brands. Over and above just a mere business link, what if a company makes a personal commitment to get into the core of a renowned, celebrated sports team? This article provides managers with a procedure to analyze, in a weekly basis, how valuable this type of decision is. A conceptual model shows that the personal involvement of a firm’s figurehead in a first-class sports club can impact positively on firm value if the person is doing well in the task s/he is entrusted with by the club. The empirical application to the soccer club Real Madrid, over 1,409 days and 215 matches, finds that the club’s performance on the field has a significant impact on the economic returns of its president’s company, with asymmetrical effects on firm value in a “loss aversion” pattern, that is, lost matches have a greater effect on firm value than games won.     

Explicit sentences distinguishing Mcduff’s II<Subscript>1</Subscript> factors

Recently, Boutonnet, Chifan, and Ioana proved that McDuff’s examples of continuum many pairwise non-isomorphic separable II₁ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht–Fra¨?sse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.     

On <Emphasis Type="Italic">Saaty</Emphasis>’s and <Emphasis Type="Italic">Koczkodaj</Emphasis>’s inconsistencies of pairwise comparison matrices

The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the case of 3 ×  3 PRM, a differentiable one-to-one correspondence is given between Saaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4  ×  4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n ×  n PRM is formulated, the elements a _ij (i < j) of which were randomly chosen from the ratio scale

Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy ⁻¹) =  2f(x) and f(xy) + f(y ⁻¹ x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GL_n(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S _n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S _n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.     

Insights into Dedekind and Weber’s edition of Riemann’s <Emphasis Type="Italic">Gesammelte Werke</Emphasis>

In this paper, I present and analyse Dedekind’s and Weber’s editorial work which led to the edition of Riemann’s Gesammelte Werke in 1876. With several examples, I suggest that this editorial work is to be understood as a mathematical activity in and of itself and provide evidence for it.     

On the relation of Voevodsky’s algebraic cobordism to Quillen’s <Emphasis Type="Italic">K</Emphasis>-theory

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P¹-spectrum MGL of Voevodsky is considered as a commutative P¹-ring spectrum. Setting we regard the bigraded theory MGL ^p,q as just a graded theory. There is a unique ring morphism which sends the class [X]_MGL of a smooth projective k-variety X to the Euler characteristic of the structure sheaf . Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories

Katriel’s Operators for Products of Conjugacy Classes of S<Subscript>n</Subscript>

We define a family of differential operators indexed with fixed point free partitions. When these differential operators act on normalized power sum symmetric functions q(x), the coefficients in the decomposition of this action in the basis q(x) are precisely those of the decomposition of products of corresponding conjugacy classes of the symmetric group S_n. The existence of such operators provides a rigorous definition of Katriels elementary operator representation of conjugacy classes and allows to prove the conjectures he made on their properties.Work partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.Work partially supported by ECs Research Training Network Algebraic Combinatorics in Europe (grant HPRN-CT-2001-00272).     

Rényi＇s Problem in Short Intervals

In this paper, we study Rényi＇s problem and other related problems about the additive functions in short intervals. As corollaries, we improve Ivic＇s results.     

A variant of Poincaré's inequality

We show that if $\text{Ω?}\text{R}{}^{\mathrm{N}}\text{,}\text{N?2}$, is a bounded Lipschitz domain and $\text{(ρ}{}_{\mathrm{n}}\text{)?L}{}^1\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ is a sequence of nonnegative radial functions weakly converging to δ₀ then there exist C>0 and n₀?1 such that

$\text{∫}\text{Ω}\text{f??}\text{∫}\text{Ω}\text{f}{}^{\mathrm{p}}\text{?C}\text{∫}\text{Ω}\text{∫}\text{Ω}\text{|f(x)?f(y)|}{}^{\mathrm{p}}\text{|x?y|}{}^{\mathrm{p}}\text{ρ}{}_{\mathrm{n}}\text{(|x?y|)}\text{d}\text{x}\text{d}\text{y}\text{?f∈L}{}^{\mathrm{p}}\text{(Ω)}\text{?n?n}{}_0\text{.}$

The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu (in: Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). As n→∞ in (1) we recover Poincaré's inequality. We also extend a compactness result of Bourgain, Brezis and Mironescu. To cite this article: A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A sharp version of Bauer–Fike’s theorem

In this paper, we present a sharp version of Bauer–Fike’s theorem. We replace the matrix norm with its spectral radius or sign-complex spectral radius for diagonalizable matrices; 1-norm and ∞$\infty$-norm for non-diagonalizable matrices. We also give the applications to the pole placement problem and the singular system.     

Banaschewski’s theorem for <Emphasis Type="Italic">GMV</Emphasis>-algebras

Banaschewski’s theorem concerns subdirect product decompositions of lattice ordered groups. In the present paper we deal with the analogous investigation for the case of generalized MV -algebras (GMV -algebras, in short); we apply this notion in the sense studied by Galatos and Tsinakis.     

Lah numbers and Lindström's lemma

We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a consequence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.     

Kőnig’s edge-colouring theorem for all graphs

We show that the maximum degree of a graph G$G$ is equal to the minimum number of ocm sets covering  G$G$, where an ocm set is the vertex-disjoint union of elementary odd cycles and one matching, and a collection of ocm sets covers   G$G$ if every edge is in the matching of an ocm set or in some odd cycle of at least two ocm sets.