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In this paper we work on indivisibility of the class numbers of real quadratic function fields. We find an explicit expression for a lower bound of the density of real quadratic function fields (with constant field F ) whose class numbers are not divisible by a given prime ℓ . We point out that the explicit lower bound of such a density we found only depends on the prime ℓ , the degrees of the discriminants of real quadratic function fields, and the condition: either | F | ≡ 1 ( mod ℓ ) or not. [ABSTRACT FROM AUTHOR]

Abstract: We prove an extension of the Ocone–Karatzas integral representation, valid for all functions on the classical Wiener space. We also establish an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of composite random variables. [Copyright &y& Elsevier]

Abstract: The number , appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many partial results about the minimal polynomial of this algebraic number. Here we obtain the general formula and it is Möbius inversion for this minimal polynomial by means of the Dickson polynomials and the Möbius inversion theory. Moreover, we investigate the homogeneous cyclotomic, Chebychev and Dickson polynomials in two variables and we show that our main results in one variable case nicely extend to this situation. In this paper, the deep results concerning these polynomials are proved by elementary arguments. [Copyright &y& Elsevier]

Abstract: The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [Copyright &y& Elsevier]

Abstract: Vector space partitions of an -dimensional vector space over a finite field are considered in Bu (1980) , Heden (1984) , and more recently in a series of papers Blinco et al. (2008) , El-Zanati et al. (2008, 2009) . In this paper, we consider the generalization of a vector space partition which we call a -fold partition (or simply a -partition). In particular, for a given positive integer, , we define a -fold partition of to be a multiset of subspaces of such that every nonzero vector in is contained in exactly subspaces in the given multiset. A -fold spread as defined in Hirschfeld (1998) is one example of a -fold partition. After establishing some definitions in the introduction, we state some necessary conditions for a -fold partition of to exist, then introduce some general ways to construct such partitions. We also introduce the construction of a dual -partition as a way of generating -partitions from a given -partition. One application of this construction is that the dual of a vector space partition will, in general, be a -partition for some . In the last section, we discuss a connection between -partitions and some designs over finite fields. [Copyright &y& Elsevier]

Abstract: For any odd prime , we present a reflection theorem which for is the Scholz Reflection Theorem. We obtain some formulas for the -rank and the -rank(), where is a number field. With the help of our reflection theorem, we establish relations between the -divisibility of the order of and the -divisibility of the class number of some algebraic number fields . [Copyright &y& Elsevier]

Abstract: In this paper, we study potential convergence of modulus patterns. A modulus pattern is convergent if all complex matrices in (i.e. all matrices with modulus pattern ) are convergent. A modulus pattern is potentially (absolutely) convergent if there exists a (nonnegative) convergent matrix in . We also introduce types of potential convergence that correspond to diagonal and D-convergence, studied in [E. Kaszkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhauser, 2000]. Convergent modulus patterns have been completely characterized by Kaszkurewicz and Bhaya [E. Kaszkurewicz, A. Bhaya, Qualitative stability of discrete-time systems, Linear Algebra Appl. 117 (1989) 65–71]. This paper presents some techniques that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for modulus patterns is given. Complete characterizations of all introduced types of potential convergence for modulus patterns are also presented. [Copyright &y& Elsevier]

Abstract: For integers and with and , let denote the family of 2-edge-connected graphs such that for every bond with two or three edges, each component of has at least vertices. In this note we get: (1) If and , then is supereulerian if and only if cannot be contracted to some well classified special graphs. (2) If , and , then , the line graph of , is Hamilton-connected if and only if . Our results extend some earlier results in [P.A. Catlin, X.W. Li, Supereulerian graphs of minimum degree at least 4, J. Adv. Math. 28 (1999) 65–69], [H.J. Broersma, L.M. Xiong, A note on minimum degree conditions for supereulerian graphs, Discrete Appl. Math. 120 (2002) 35–43] and [D.X. Li, H.-J. Lai, M.Q. Zhan, Eulerian subgraphs and hamilton-connected line graphs, Discrete Appl. Math. 145 (2005) 422–428] by Catlin and Li, by Broersma and Xiong, and by Li, Lai and Zhan. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the status of some ‘unbelievable results’ presented in the paper ‘On Some Contradictory Computations in Multi-Dimensional Mathematics’ [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] published in Nonlinear Analysis, a journal indexed in the Science Citation Index. Among some of the unbelievable results ‘proved’ in the paper we can find statements like: (i) a rotation , , is inconsistent with arithmetic, (ii) complex number theory is inconsistent. Besides these ‘results’ of mathematical nature, [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] offers also a ‘proof’ that Special Relativity is inconsistent. Now, we are left with only two options (a) the results of [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] are correct and in this case we need a revolution in Mathematics (and also in Physics) or (b) the paper is a potpourri of nonsense. We show that option (b) is the correct one. All ‘proofs’ appearing in [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] are trivially wrong, being based on a poor knowledge of advanced calculus notions. There are many examples (some of them discussed in [A.L.T. Carvalho, W.A. Rodrigues Jr., The non sequitur mathematics and physics of the ‘new electrodynamics’ proposed by the AIAS group, Random Oper. Stoch. Equs. 9 (2001) 161–206. http://arxiv.org/abs/physics/0302016; E. Capelas de Oliveira, W.A. Rodrigues Jr., Dotted and undotted algebraic spinor fields in general relativity, Int. J. Mod. Phys. D 13 (2004) 1637–1659. http://arxiv.org/abs/math-ph/0407024; W.A. Rodrigues Jr., E. Capelas de Oliveira, Clifford valued differential forms and some issues on gravitation, electromagnetism and “unified” theories, Int. J. Mod. Phys. D 13 (2004) 1879–1915. http://arxiv.org/abs/math-ph/0407025; W.A. Rodrigues Jr., Q.A.G. Souza, An ambiguous statement called ‘tetrad postulate’ and the correct field equations satisfied by the tetrad fields, Int. J. Mod. Phys. D 14 (2005) 2095–2150. http://arxiv.org/abs/math-ph/0411085; W.A. Rodrigues Jr., A comment on Emergent Gravity, http://arxiv.org/abs/gr-qc/0602111]) of completely wrong papers using non sequitur Mathematics in the Physics literature. Taking into account also that a paper like [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] appeared in a Mathematics journal we think that it is time for editors and referees of scientific journals to become more careful in order to avoid the dissemination of nonsense. [Copyright &y& Elsevier]

Abstract: Nguyen [1996. An algorithmic approach to constructing supersaturated designs. Technometrics 38, 69–73] and Tang and Wu [1997. -optimality of supersaturated designs. Statist. Sinica 7, 929–939] independently derived a lower bound for the value of an N run, m factor supersaturated design (SSD). This bound can be achieved only if m is a multiple of when or if m is an even multiple of when . One important question is whether Nguyen–Tang–Wu bound can be achieved in all of these cases. In this paper, based on a construction method by , we present a theoretical method for finding as many positive integers t as possible such that there is an -optimal SSD achieving the Nguyen–Tang–Wu bound with N runs and factors when and factors when . This method is applied to the cases. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the n-order neutral delay differential equation with unstable typewhere n ⩾2 is an even integer, α ⩾1, τ >0, σ >0 and p, q ∈ C([t 0,+∞), R +). We first prove that this equation always has an unbounded positive solution, then some new bounded oscillation and nonoscillation criteria are obtained. [Copyright &y& Elsevier]

Abstract: Given a strictly increasing sequence s of non-negative integers, filtering a word by s consists in deleting the letters such that i is not in the set . By a natural generalization, denote by , where L is a language, the set of all words of L filtered by s. The filtering problem is to characterize the filters s such that, for every regular language L, is regular. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions. [Copyright &y& Elsevier]

Abstract: We study the action of the mapping class group of an oriented genus g surface with n punctures and a disc removed on a Poisson algebra which arises in the combinatorial description of Chern–Simons gauge theory when the gauge group is a semidirect product . We prove that the mapping class group acts on this algebra via Poisson isomorphisms and express the action of Dehn twists in terms of an infinitesimally generated G-action. We construct a mapping class group representation on the representation spaces of the associated quantum algebra and show that Dehn twists can be implemented via the ribbon element of the quantum double and the exchange of punctures via its universal R-matrix. [Copyright &y& Elsevier]