LR-normal orthogroups

A new kind of regular orthogroups, namely, the LR-normal orthogroups is introduced and investigated. In particular, we will introduce the gearing techniques for fitting semigroup components on a semilattice in the construction of LR-normal orthogroups.     

Odd prime values of the Ramanujan tau function

We study the odd prime values of the Ramanujan tau function, which form a thin set of large primes. To this end, we define LR(p,n):=τ(p ^n?1) and we show that the odd prime values are of the form LR(p,q) where p,q are odd primes. Then we exhibit arithmetical properties and congruences of the LR numbers using more general results on Lucas sequences. Finally, we propose estimations and discuss numerical results on pairs (p,q) for which LR(p,q) is prime.     

The likelihood ratio tests for the dimensionality of regression coefficients

In this paper we give a unified derivation of the likelihood ratio (LR) statistics for testing the hypothesis on the dimensionality of regression coefficients under a usual MANOVA model. We also derive the LR statistics under a general MANOVA model and study their asymptotic null and nonnull distributions. Further it is shown that the test statistic used by Bartlett [4] for testing the hypothesis that the last p?k canonical correlations are all zero is the LR statistic.     

Characterizations of certain completely regular varieties

We characterize orthogroups, local orthogroups and (left,right) cryptogroups within completely regular semigroups by means of absence of certain kind of subsemigroups. For each of these varieties V, we determine the complete set of minimal non-V -varieties. For each of the latter varieties, we determine the lattice of its subvarieties. We then give a generating semigroup and a basis of its identities for every variety which occurs in this way. The subvariety lattices are illustrated by three diagrams.     

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R³ with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies.     

Large a quasi-Jacobi form for J-normal matrices and inverse eigenvalue problems

A quasi-Jacobi form for J-unitarily diagonalizable J-normal matrices is given, extending a result for normal matrices due to Malamud. The inverse eigenvalue problem for J-normal matrices satisfying certain prescribed spectral conditions is investigated. It is shown that there exists unicity in the case of pseudo-Jacobi matrices.     

On a shifted LR transformation derived from the discrete hungry Toda equation

The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors’ paper (Fukuda et al., in Phys Lett A 375, 303–308, 2010), we associate the dhToda equation with a sequence of LR transformations for a totally nonnegative (TN) matrix, and then, in another paper (Fukuda et al. in Annal Math Pura Appl, 2011), based on the dhToda equation, we design an algorithm for computing the eigenvalues of the TN matrix. In this paper, in order to accelerate the convergence speed, we first introduce the shift of origin into the LR transformations associated with the dhToda equation, then derive a recursion formula for generating the shifted LR transformations.We next present a shift strategy for avoiding the breakdown of the shifted LR transformations. We finally clarify the asymptotic convergence and show that the sequence of TN matrices generated by the shifted LR transformations converges to a lower triangular matrix, exposing the eigenvalues of the original TN matrix. The asymptotic convergence is also verified through some numerical examples.     

Maps and f-normal spaces

It has long been known that hyper-real maps preserve realcompactness. In this paper we show that hyper-real maps preserve nearly realcompactness as well. We will also introduce the concepts of ε-perfect maps and f-normal spaces and explore them in a way that mirrors Rayburn's 1978 study of δ-perfect maps and h-normal spaces.     

Maximal Normal Orthogroups in Rings Containing No Infinite Semilattices

Let R be a ring regarded as a multiplicative semigroup which contains no infinite subsemilattices. We investigate subsemigroups of R which are normal orthogroups, and present a construction from which all such maximal normal orthogroups can be obtained. In particular, we construct all maximal normal orthogroups of matrices over a field under matrix multiplication.

Communicated by D. Easdown.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

Global determinism of normal orthogroups

If S is a semigroup, the global (or the power semigroup) of S is the set \(\mathcal {P}(S)\) of all nonempty subsets of S equipped with the naturally defined multiplication. A class \(\mathcal {K}\) of semigroups is globally determined if any two semigroups of \({\mathcal {K}}\) with isomorphic globals are themselves isomorphic. We study properties of globals of normal orthogroups and show, in particular, that the class of normal orthogroups is globally determined.     

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.     

Self-similar flows

Large classes of self-similar (isospectral) flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms. In particular there exist families of flows associated with the QR, LR, and Cholesky eigenvalue algorithms. This paper uses Lie theory to develop a general theory of self-similar flows which includes the QR, LR, and Cholesky flows as special cases. Also included are new families of flows associated with the SR and HR eigenvalue algorithms. The basic theory produces analogues of unshifted, single-step eigenvalue algorithms, but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple-step eigenvalue algorithms.     

Transfinite Extension to Q m-normality Theory

The theory of Q _m-normal families, m ∈ ?, was developed by P. Montel for the cases m = 0 (normal families) [5] and m = 1 (quasinormal families) [4] and later generalized by C.T. Chuang [2] for any m ≥ 0. In this paper, we extend the definition to an arbitrary ordinal number α as follows. Given E ? D, define the α-th derived set $E^{(\alpha)}_D$ of E with respect to D by $(E^{(\alpha-1)}_D)^{(1)}_D$ if α has an immediate predecessor and by ${\mathop \bigcap\limits_{\beta<\alpha}} E^{(\beta)}_D$ if α is a limit ordinal. Then a family ${\cal F}$ of meromorphic functions on a plane domain D is Q_α-normal if each sequence S of functions in ${\cal F}$ has a subsequence which converges locally χ-uniformaly on the domain DE, where E = E(S) ? D satisfies $E^{(\alpha)}_{D}=\emptyset$ . Inparticular, a Q ₀ -normal family is a normal family, and a Q ₁ -normal family is a quasi- normal family. We also give analogues to some basic results in Q_m-normality theory and extend Zalcman’s Lemma to Q _α -normal families where α is an infinite countable (enumerable) ordinal number.     

On Shape-Preserving Additions of Fuzzy Intervals

We prove an open problem suggested by Mesiar (1997, Fuzzy Sets and Systems86, 73-78) concerning continuous t-norm-based additions preserving the LR-fuzzy intervals.     

New Braided Crossed Categories and Drinfel'd Quantum Double for Weak Hopf Group Coalgebras

A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups.     

Nonnormality of Stoneham constants

This paper examines “Stoneham constants,” namely real numbers of the form $\alpha_{b,c} = \sum_{n \geq1} 1/(c^{n} b^{c^{n}})$ , for coprime integers b≥2 and c≥2. These are of interest because, according to previous studies, α _b,c is known to be b-normal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b ^?m . So, for example, the constant $\alpha_{2,3} = \sum_{n \geq1} 1/(3^{n} 2^{3^{n}})$ is 2-normal. More recently it was established that α _b,c is not bc-normal, so, for example, α _2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α _b,c is not B-normal, where B=b ^p c ^q r, for integers b and c as above, p,q,r≥1, neither b nor c divide r, and the condition D=c ^q/p r ^1/p /b ^c?1<1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α _b,c is nonnormal. We also show that the sum of two B-nonnormal Stoneham constants as defined above, subject to some restrictions, is B-nonnormal.     

Characterizations of certain completely regular varieties

Abstract. We characterize orthogroups, local orthogroups and (left,right) cryptogroups within completely regular semigroups by means of absence of certain kind of subsemigroups. For each of these varieties V , we determine the complete set of minimal non-V -varieties. For each of the latter varieties, we determine the lattice of its subvarieties. We then give a generating semigroup and a basis of its identities for every variety which occurs in this way. The subvariety lattices are illustrated by three diagrams.     

Classifying smooth lattice polytopes via toric fibrations

We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.