RESIDUAL SMALLNESS AND WEAK CENTRALITY

2003 ◽  
Vol 13 (01) ◽  
pp. 35-59 ◽  
Author(s):  
KEITH A. KEARNES ◽  
EMIL W. KISS
Keyword(s):  
Finite Number ◽  
Abelian Variety ◽  
Nilpotent Algebra ◽  
Finite Algebras ◽  
Small Variety ◽  
Special Cases ◽  
Strongly Nilpotent ◽  
Nilpotent Algebras ◽  
Minimal Algebra

We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable E-minimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an E-minimal, or by a finite strongly nilpotent algebra. This establishes the RS-conjecture for E-minimal algebras.

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

scholarly journals Exact distributions of order statistics from ln,p-symmetric sample distributions

2017 ◽  
Vol 5 (1) ◽  
pp. 221-245 ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Distribution Function ◽  
Finite Number ◽  
Order Statistics ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Symmetric Distributions ◽  
Sample Distribution ◽  
Special Cases ◽  
Exact Distributions ◽  
Gaussian Sample

Abstract We derive the exact distributions of order statistics from a finite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end,we first review the special cases of order statistics fromspherical aswell as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure.We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ {1, . . . , n − 1}.

scholarly journals Complete addition laws on abelian varieties

2012 ◽  
Vol 15 ◽  
pp. 308-316 ◽  
Author(s):  
Christophe Arene ◽  
David Kohel ◽  
Christophe Ritzenthaler
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Galois Group ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Rational Points ◽  
Absolute Galois Group ◽  
Projective Embedding ◽  
Complete Set

AbstractWe prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

Singularity confinement for a class of m -th order difference equations of combinatorics

2007 ◽  
Vol 366 (1867) ◽  
pp. 877-922 ◽  
Author(s):  
Mark Adler ◽  
Pierre van Moerbeke ◽  
Pol Vanhaecke
Keyword(s):  
Finite Number ◽  
Difference Equations ◽  
Random Permutations ◽  
Order Difference ◽  
Painlevé Property ◽  
Special Cases ◽  
Singularity Confinement ◽  
Time Differential ◽  
Free Parameters ◽  
The Difference

In a recent publication, it was shown that a large class of integrals over the unitary group U ( n ) satisfy nonlinear, non-autonomous difference equations over n , involving a finite number of steps; special cases are generating functions appearing in questions of the longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painlevé property ; roughly speaking, this means that after a finite number of steps the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again (‘ singularity confinement ’). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the Painlevé property for the discrete relations is inherited from the Painlevé property of the (continuous) Toeplitz lattice.

scholarly journals Unified computational approach to nilpotent algebra classification problems

2021 ◽  
Vol 29 (2) ◽  
pp. 215-226
Author(s):  
Shirali Kadyrov ◽  
Farukh Mashurov
Keyword(s):  
Computational Approach ◽  
Computational Power ◽  
Classification Problems ◽  
Nilpotent Algebra ◽  
Wolfram Mathematica ◽  
Finite Dimensional ◽  
Nilpotent Algebras

Abstract In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.

scholarly journals Angle Sums of Schläfli Orthoschemes

2021 ◽  
Author(s):  
Thomas Godland ◽  
Zakhar Kabluchko
Keyword(s):  
Finite Number ◽  
Random Walks ◽  
Stirling Numbers ◽  
Convex Hulls ◽  
Tangent Cones ◽  
Expected Number ◽  
Intrinsic Volumes ◽  
Special Cases ◽  
Minkowski Sums ◽  
Finite Products

AbstractWe consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

scholarly journals Some improvements to Turner's algorithm for bracket abstraction

1990 ◽  
Vol 55 (2) ◽  
pp. 656-669 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Finite Number ◽  
Translation Process ◽  
Special Cases ◽  
Infinite Set

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.There are several ways in which these problems can be alleviated:(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.(2) The use of an extra clause (c), appropriate to βη reduction.(3) The introduction of a finite number of extra combinators.The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

scholarly journals Large Deviations Principle for Occupancy Problems with Colored Balls

2007 ◽  
Vol 44 (01) ◽  
pp. 115-141
Author(s):  
Paul Dupuis ◽  
Carl Nuzman ◽  
Phil Whiting
Keyword(s):  
Large Deviations ◽  
Finite Number ◽  
Scale Parameter ◽  
Large Deviations Principle ◽  
Special Cases ◽  
Boltzmann Statistics ◽  
Occupancy Problems

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are r balls thrown into n urns with the probability of a ball entering a given urn being 1/n (i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter, n, tending to infinity and r increasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.

Multigranulation roughness based on soft relations

2021 ◽  
pp. 1-16
Author(s):  
Muhammad Shabir ◽  
Jamalud Din ◽  
Irfan Ahmad Ganie
Keyword(s):  
Finite Number ◽  
Rough Set ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Soft Sets ◽  
Upper Approximation ◽  
Special Cases ◽  
Multigranulation Rough Set ◽  
Empty Subset ◽  
Lower And Upper Approximations

The original rough set model, developed by Pawlak depends on a single equivalence relation. Qian et al, extended this model and defined multigranulation rough sets by using finite number of equivalence relations. This model provide new direction to the research. Recently, Shabir et al. proposed a rough set model which depends on a soft relation from an universe V to an universe W . In this paper we are present multigranulation roughness based on soft relations. Firstly we approximate a non-empty subset with respect to aftersets and foresets of finite number of soft binary relations. In this way we get two sets of soft sets called the lower approximation and upper approximation with respect to aftersets and with respect to foresets. Then we investigate some properties of lower and upper approximations of the new multigranulation rough set model. It can be found that the Pawlak rough set model, Qian et al. multigranulation rough set model, Shabir et al. rough set model are special cases of this new multigranulation rough set model. Finally, we added two examples to illustrate this multigranulation rough set model.

scholarly journals INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

2018 ◽  
Vol 19 (3) ◽  
pp. 869-890 ◽  
Author(s):  
Anna Cadoret ◽  
Ben Moonen
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Finite Type ◽  
Galois Representation ◽  
Shimura Variety ◽  
General Statement ◽  
Tate Conjecture ◽  
Tate Module ◽  
Special Cases ◽  
Image Theorem

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

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