scholarly journals $(s,t)$-Cores: a Weighted Version of Armstrong’s Conjecture

10.37236/6161 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Matthew Fayers
Keyword(s):  
Group Action ◽  
Weighted Average ◽  
Symmetric Groups ◽  
Weighted Version ◽  
Average Size ◽  
Positive Integers ◽  
Coprime Positive Integers

The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores — partitions which are both $s$- and $t$-cores %mdash; playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang.In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525—1539], studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core.

Classification of irreducible weak modules over Virasoro vertex operator algebras

2019 ◽  
Vol 30 (11) ◽  
pp. 1950054
Author(s):  
Guobo Chen ◽  
Dejia Cheng ◽  
Jianzhi Han ◽  
Yucai Su
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Positive Integers ◽  
Coprime Positive Integers

The classification of irreducible weak modules over the Virasoro vertex operator algebra [Formula: see text] is obtained in this paper. As one of the main results, we also classify all irreducible weak modules over the simple Virasoro vertex operator algebras [Formula: see text] for [Formula: see text] [Formula: see text], where [Formula: see text] are coprime positive integers.

A NEW CHARACTERIZATION OF ALMOST SPORADIC GROUPS

2002 ◽  
Vol 01 (03) ◽  
pp. 267-279 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Sporadic Groups ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Order Components ◽  
Positive Integers ◽  
Coprime Positive Integers

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut (J2) and Aut (McL), and the automorphism group of PSL(2, 2n) where n=2sare also uniquely determined by their order components. Also we discuss about the characterizability of Aut (PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

scholarly journals Modular Invariant Representations of the Superconformal Algebra

2018 ◽  
Vol 2019 (24) ◽  
pp. 7659-7690 ◽  
Author(s):  
Ryo Sato
Keyword(s):  
Vertex Operator ◽  
Central Charge ◽  
Superconformal Algebra ◽  
Transformation Formula ◽  
Modular Invariant ◽  
S Matrix ◽  
Invariant Representations ◽  
Positive Integers ◽  
Coprime Positive Integers ◽  
Vertex Operator Superalgebra

Abstract We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p^{\prime }}=3\left (1-\frac{2p^{\prime }}{p}\right ),$ where (p, p′) is a pair of coprime positive integers such that p ≥ 2. When p′ = 1, the formula coincides with that of the $\mathcal{N}=2$ unitary minimal series found by F. Ravanini and S.-K. Yang. In addition, we study the properties of the corresponding “modular S-matrix”, which is no longer a matrix if p′≥ 2.

TEM measurement of surface area of automotive catalysts

1994 ◽  
Vol 52 ◽  
pp. 776-777
Author(s):  
M. H. Yao ◽  
D. R. Liu ◽  
R. J. Baird ◽  
R. K. Usmen ◽  
R. W. McCabe
Keyword(s):  
Electron Microscopy ◽  
Surface Area ◽  
Average Particle Size ◽  
Weighted Average ◽  
Average Particle ◽  
Automotive Catalysts ◽  
Surface Areas ◽  
Measuring Surface ◽  
Major Factors ◽  
Average Size

The specific surface area of supported noble metal particles in an automotive catalyst is defined as the exposed surface area per unit mass of these particles. It is of great importance to know this parameter, since this is one of the major factors that determine the effectiveness of the catalyst. Commonly used methods for characterizing catalysts, such as X-ray diffraction and TEM, do not directly provide a measure of surface area, but, instead, provide a measure of the “average size” of supported particles. Moreover, the “average sizes” obtained from different experimental techniques are often not comparable. Furthermore, many previous electron microscopy catalyst studies measured only simple average particle size, and no detailed procedure for measuring area-weighted average size or surface area appear to have been reported.In the current study, a procedure for measuring surface area of supported particles by transmission electron microscopy(TEM) was developed, and applied to measure surface areas of various production three-way automotive catalysts.

On Certain Polynomials of Gaussian Type

1979 ◽  
Vol 31 (2) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel Reich
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Orbit Space ◽  
Partition Functions ◽  
Poincaré Polynomial ◽  
Complete Answer ◽  
Lie Group Action ◽  
Image Position ◽  
Positive Coefficients ◽  
Positive Integers

Introduction. We shall consider functions of the formwhere {ri} and {si} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {si} to be pairwise relatively prime, and asked the following two questions:(a) When is ƒ(t) a polynomial?(b) When does ƒ(t) have positive coefficients?These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {si}.

scholarly journals SOLUTIONS OF THE DIOPHANTINE EQUATION xy+yz+zx=n!

2008 ◽  
Vol 50 (2) ◽  
pp. 217-232
Author(s):  
MIHAI CIPU (BUCHAREST) ◽  
FLORIAN LUCA (MORELIA) ◽  
MAURICE MIGNOTTE (STRASBOURG)
Keyword(s):  
Diophantine Equation ◽  
Image Position ◽  
Positive Integers ◽  
Coprime Positive Integers

AbstractWe prove that the only solutions in coprime positive integers to the equation are (x, y, z)=(n!–2, 1, 1, n), n≥3.

scholarly journals On the laws of certain varieties of groups

1985 ◽  
Vol 31 (1) ◽  
pp. 145-154
Author(s):  
Robert B. Howlett ◽  
Richard Levingston
Keyword(s):  
Normal Subgroup ◽  
Positive Integers ◽  
Coprime Positive Integers

Let m and n be coprime positive integers. The variety (consisting of all groups G such that for some normal subgroup H of G, H is abelian of exponent dividing m and G/H is abelian of exponent dividing n) and the variety both satisfy the following three laws:all elements have order dividing mn;the commutator of two mth powers has order dividing m;the commutator of two nth powers has order dividing n.It is proved that any law which holds in both these varieties (notably that commutators commute) is a consequence of the above three laws.

scholarly journals Some results on quotients of triangle groups

1984 ◽  
Vol 30 (1) ◽  
pp. 73-90 ◽  
Author(s):  
Marston D.E. Conder
Keyword(s):  
Symmetric Groups ◽  
Computational Techniques ◽  
Triangle Group ◽  
Triangle Groups ◽  
Regular Maps ◽  
Rubik's Cube ◽  
Rubik’S Cube ◽  
Alternating And Symmetric Groups ◽  
Positive Integers

Given positive integers k, l, m, the (k, l, m) triangle group has presentation δ(k, l, m) = < X, Y, Z | Xk = Yl = Zm = XYZ = 1 >. This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

scholarly journals Improved Average Estimation in Seemingly Unrelated Regressions

Econometrics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 15
Author(s):  
Ali Mehrabani ◽  
Aman Ullah
Keyword(s):  
Returns To Scale ◽  
Weighted Average ◽  
Generalized Least Squares ◽  
Quadratic Loss ◽  
Moment Matrix ◽  
Mean Squared Errors ◽  
Cost System ◽  
Seemingly Unrelated Regressions ◽  
Average Size ◽  
Plausible Reason

In this paper, we propose an efficient weighted average estimator in Seemingly Unrelated Regressions. This average estimator shrinks a generalized least squares (GLS) estimator towards a restricted GLS estimator, where the restrictions represent possible parameter homogeneity specifications. The shrinkage weight is inversely proportional to a weighted quadratic loss function. The approximate bias and second moment matrix of the average estimator using the large-sample approximations are provided. We give the conditions under which the average estimator dominates the GLS estimator on the basis of their mean squared errors. We illustrate our estimator by applying it to a cost system for United States (U.S.) Commercial banks, over the period from 2000 to 2018. Our results indicate that on average most of the banks have been operating under increasing returns to scale. We find that over the recent years, scale economies are a plausible reason for the growth in average size of banks and the tendency toward increasing scale is likely to continue

scholarly journals Descents after maxima in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Generating Functions ◽  
Mellin Transforms ◽  
Asymptotic Expressions ◽  
International Audience ◽  
Average Size ◽  
Positive Integers

Combinatorics International audience We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ1+σ2+...+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functions and Mellin transforms, we obtain asymptotic expressions for the average size of these descents. Finally, we show with the use of a simple bijection between the compositions of n for n>1, that on average the descent after the last maximum is greater than the descent after the first.

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