A Rational-Function Identity Related to the Murnaghan–Nakayama Formula for the Characters of S n

The Murnaghan–Nakayama formula for the characters of S _n is derived from Young's seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Young's seminormal representations.     

The Néron–Severi group of a proper seminormal complex variety

We prove a Lefschetz (1,1)-Theorem for proper seminormal varieties over the complex numbers.     

Epimorphisms, dominions and seminormal identities

We find some sufficient conditions on semigroup identities in which both sides contain repeated variables to be preserved under epis in conjunction with any seminormal permutation identity.     

局部左正则纯正密群的一个等式

本文给出局部左正则纯正密群的一个等式.证明了一个完全正则半群是左半正规密群当且仅当它是局部左正则纯正密群.     

具有逆断面的左半正规纯正半群

左半正规纯正半群是幂等元集形成左半正规带的纯正半群．本文讨论了具有逆断面的左半正规纯正半群上的一些性质；给出该类半群的一个构造定理。     

Homotopy Spheres in Formal Language

The set of nonempty proper subwords of a word is either contractible or a homotopy n-sphere. There is a simple algorithm which computes n. The existence of spherical words is investigated, and the words which yield spheres are determined. A language which is closed under subwords has a finite number of components, and each component has a finitely generated fundamental group. For each n greater than 1, there is a language on two letters which has the homotopy type of an infinite cluster of n-sphere. There is a language on two letters which has nontrivial homology in each dimension greater than 1. If a language is closed under subwords and has bounded period, then it has the homotopy type of a finite polyhedron.     

半群上的模糊群同余

设 S是一个半群 ,ρ是 S上的一个模糊同余。引进半群的模糊半正规子半群的概念 ,证明ρ是 S上的一个模糊群同余当且仅当它的模糊核 K(ρ)是 S的模糊半正规子半群 ;而且对每个给定的模糊半正规子半群 μ可以构造一个模糊同余 ρμ 使得它的模糊核 K(ρμ) =μ.     

Finite groups with a seminormal Hall subgroup

Tests for π-solvability of a finite group with seminormal Hall π-subgroup are established and the nilpotency of the third commutator subgroup of any group with seminormal noncyclic Sylow subgroups is proved.     

Some distribution properties of 0,1-sequences

In this paper several distribution concepts of 0,1-sequences suggested by some work of E. Hlawka and D. E. Knuth are investigated in detail. In the first instance the number of occurrences of specified (closed) subblocks in the initial part of a sequence is considered, whereas in the second case we enumerate the occurrences of (not necessarily closed) subwords. In any of the two instances we study the case of fixed length of subblocks (resp.subwords) as well as the case where this length increases slowly.     

On the Extension of Seminormal Functors to the Category of Tychonoff Spaces

Chigogidze proposed a construction of extending a normal functor from the category Comp to the category Tych. We can apply his scheme to seminormal functors and study the properties of the original functor which are preserved under extension. We introduce the concept of functor having an invariant extension from Comp to Tych because the very existence of this invariance plays a key role in the preservation of the properties of a seminormal functor in its extension. It is proved that the superextension functor λ has an invariant extension. We check that if a seminormal functor has an invariant extension then its extension preserves a point, the empty set, intersection and is a monomorphic functor. If this functor has finite degree then its extension is continuous and hence a seminormal functor in Tych. If the functor is of infinite degree then continuity may be lost. Namely, we show that the extension of λ for Tych is not continuous.     

A note on maximal subgroups of free idempotent generated semigroups over bands

We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.     

On Rational Varieties Smooth Except at a Seminormal Singular Point

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ? ^m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal.     

The Katětov property for finite degree seminormal functors

A seminormal functor kF enjoys the Katěetov property (K-property) if for every compact set X the hereditary normality of kF(X) implies the metrizability of X. We prove that every seminormal functor of finite degree n>3 enjoys the K-property. On assuming the continuum hypothesis (CH) we characterize the weight preserving seminormal functors with the K-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming CH is a universal counterexample for the K-property in the class of weight preserving seminormal functors.     

Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part

Numerical Algorithms - For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of...     

Separating words by occurrences of subwords

We obtain some lower bounds for the complexity of word separation by occurrences of subwords. In the case of length 1 subwords, we show that the bound is exact up to a constant factor. Connection with the problem of separating words by automata is considered.     

Cluster Algebras and Binary Subwords

This paper establishes a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion w can be associated to a piecewise-linear poset P and a snake graph G. We construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti. We then present bijections from the subwords of w to the antichains of P and to the perfect matchings of G.

     

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.     

Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

A generalized Goulden–Jackson cluster method and lattice path enumeration

The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths – both with height bounded and unbounded – by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulas.