高阶Lagrange中值定理“中值点”的渐近性

借助Stirling数研究了高阶Lagrange微分中值定理在f(n+1)(a)=0或f(n+1)(a)不存在时的“中值点”的渐近性,并给出了渐近性估计式.     

On the Bol Loops

In the present paper, the Bol loops and related groups are studied. We suggest a universal way to construct a Bol loop and find criteria for simplicity and finiteness of such loops.     

Topologische Loops

Ohne Zusammenfassung     

Poset Loops

Given a ring and a locally finite poset, an incidence loop or poset loop is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative and associative properties for loop elements. Nilpotence of the ring and height restrictions on the poset force the loop to become associative, or even commutative. Constraints on the appearance of nilpotent groups of class 2 as poset loops are given. The main result shows that the incidence loop of a poset of finite height is nilpotent, of nilpotence class bounded in terms of the height of the poset.     

On the Embedding of Zero- Dimensional Double Loops in Locally Euclidean Double Loops

It is shown that a closed (hence locally compact) zero-dimensional sub-double-loop of a locally Euclidean double loop is always tamely embedded and that its complement is simply connected.     

On the Specht Property of Varieties of Commutative Alternative Algebras over a Field of Characteristic 3 and Commutative Moufang Loops

Siberian Mathematical Journal -     

Brownian Loops Topology

In the seminal work of Symanzik (1969), Poisson ensembles of Brownian loops were implicitly used. Since the work of Lawler and Werner (Prob. Th. Rel. Fields 128:565–588 2004) on “loop soups”, these ensembles have also been the object of many investigations. The purpose of the present work is to determine the distributions related to their topological properties, using trace formula and zeta regularization. These results have been announced in Le Jan (2016) and Le Jan (2017).

     

Abelian Extensions and Solvable Loops

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.     

Simple Bol Loops

In this article we investigate the Bol loops and connected with them groups. We prove an analog of the Doro's theorem for Moufang loops and find a criterion for simplicity of Bol loops. One of the main results obtained is the following: If the right multiplication group of a connected finite Bol loop S is a simple group, then S is a Moufang loop.     

Abelian-by-Cyclic Moufang Loops

We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of examples of finite abelian-by-cyclic Moufang loops. The previously known [10 Rajah , A. ( 2001 ). Moufang loops of odd order pq ³ . J. Algebra 235 ( 1 ): 66 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]] loops of this type of odd order 3q ³, with prime q ≡ 1 (mod 3), are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.     

Doppelebenen und Loops

Generalizing the concept of difference sets in groups conditions are given for a loop (P, +) (with equality of left and right inverses) and a subset D of P such that (i) the left translates of D and the right translates of -D (the set of inverses for elements of D) resp. are the lines of two projective planes with point set P, forming a double plane (i.e. the lines of one plane are ovals of the other), (ii) the loop operation has a certain geometric interpretation in the double plane.

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Walter Benz zum 60. Geburtstag     

Diffusion of Loops

We study a lattice model that is closely related to the Ising model and can be regarded as describing diffusion of loops in two dimensions. The time development is given by a transfer matrix for a random surface model on a three-dimensional lattice. The transfer matrix is indexed by loops and is invariant under a group of motions in the loop space. The eigenvalues of the transfer matrix are calculated in terms of the partition function and the correlation functions of the Ising model.     

Lagrange and the Solution of Numerical Equations

In 1798 J.-L. Lagrange published an extensive book on the solution of numerical equations. Lagrange had developed four versions of a general systematic algorithm for detecting, isolating, and approximating, with arbitrary precision, all real and complex roots of a polynomial equation with real coefficients. In contrast to Newton's method, Lagrange's algorithm is guaranteed to converge. Some of his powerful ideas and techniques foreshadowed methods developed much later in geometry and abstract algebra. For instance, in order to make a more efficient algorithm for isolating roots, Lagrange essentially worked in a quotient ring of a polynomial ring. And to accelerate both the convergence and calculation of his continued fraction expansions of the roots, he employed nonsimple continued fractions and Möbius transformations. Copyright 2001 Academic Press.En 1798 J.-L. Lagrange publié un important traité sur la résolution d'équations numériques. Dans ce traité, Lagrange développe quatre versions d'un algorithme général qui systématiquement détecte, isole, et approxime, avec toute la précision voulue, toutes les racines réelles et complexes d'une équation polynomiale à coefficients réels. Contrairement à la méthode de Newton, l'algorithme de Lagrange converge toujours. Certaines de ses idées et techniques les plus brillantes ont ouvert la voie à des méthodes en géométrie et en algèbre développées beaucoup plus tard. Par exemple, pour augmenter l'efficacité de son algorithme afin d'isoler les racines, Lagrange travaille essentiellement dans un anneau quotient d'un anneau de polynômes. De plus, dans le but d'accélérer à la fois la convergence et les calculs de l'expansion des racines via des fractions continues, il emploie des fractions continues non-simples ainsi que des transformations de Möbius. Copyright 2001 Academic Press.1798 veröffentlichte J.-L. Lagrange ein ausführliches Buch über die Lösung von numerischen Gleichungen. Lagrange entwickelte vier Versionen eines allgemeinen systematischen Algorithmus für die Identifizierung, Isolation, und beliebig genaue Approximation, aller reellen und imaginären Wurzeln einer Polynomgleichung mit reellen Koeffizienten. Im Gegensatz zu Newtons Methode konvergiert Lagranges Algorithmus immer. Einige seiner machtvollen Ideen und Techniken waren Vorläufer von Methoden in der Geometrie und Algebra die erst viel später entwickelt wurden. Zum Beispiel arbeitete Lagrange im Grunde in einem Quotientenring eines Polynomringes um seinen Algorithmus für die Isolation von Wurzeln effizienter zu machen. Und um sowohl die Konvergenz wie auch die Berechnung seiner Kettenbruchexpansion der Wurzeln zu beschleunigen benützte er nichteinfache Kettenbrüche und Möbiustransformationen. Copyright 2001 Academic Press.MSC subject classifications: 01A50; 65-03; 65H05; 65B66; 00A30.