Missouri Journal of Mathematical Sciences Articles (Project Euclid)

Editorial

Wed, 14 Sep 2011 16:37 EDT

Terry Goodman

Source: Missouri J. Math. Sci., Volume 23, Number 1, 1--2.

Separation Axioms and Lattice Equivalence

Wed, 14 Sep 2011 16:37 EDT

Sami Lazaar

Source: Missouri J. Math. Sci., Volume 23, Number 1, 3--11.

Abstract:
>This paper deals with the relation between lattice-equivalence and some separation axioms. We are concerned with two questions: The first one is to characterize topological spaces $X$ such that $X$ and $\mathbf{F}(X)$ are lattice equivalent for some covariant functors $\mathbf{F}$ from $\mathbf{TOP}$ to itself. In the second question, it is proved that $T_{(0,2)}, T_{(S,D)}, T_{(S,1)}$ and $T_{(0,3\frac{1}{2})}$ are lattice-invariant properties but $S$, $T_{(0,1)}$, $T_{(0,S)}$, $T_{(1,2)}$, $T_{(1,S)}$, $T_{(1,3\frac{1}{2})}$, and $T_{(0,D)}$ are not.

A Continuous Bijection from $\ell^{2}$ Onto a Subset of $\ell^{2}$ Whose Inverse is Everywhere Unboundedly Discontinuous, with an Application to Packing of Balls in $\ell^2$

Wed, 14 Sep 2011 16:37 EDT

Sam H. Creswell

Source: Missouri J. Math. Sci., Volume 23, Number 1, 12--18.

Abstract:
There is a continuous bijection from $ \ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere unboundedly discontinuous. If $B$ is a ball in $\ell^2$, then the continuous bijection defined on $\ell^2$ maps countably many mutually disjoint balls of $\ell^2$ into countably many mutually disjoint balls in $B$, making those images mutually disjoint.

Blind Men and Hypercubes

Wed, 14 Sep 2011 16:37 EDT

M. Bencze, O. Bagdasar, J. L. Díaz-Barrero

Source: Missouri J. Math. Sci., Volume 23, Number 1, 19--26.

Abstract:
In this paper an elementary probability question is solved and the procedure used is generalized to higher dimensions.

Generalized-designs

Wed, 14 Sep 2011 16:37 EDT

F. Ayatollah Zadeh Shirazi, M. Bagherian

Source: Missouri J. Math. Sci., Volume 23, Number 1, 27--47.

Abstract:
The generalization of designs can be considered as infinite designs. A finite set can be a part of an infinite set, a well-ordered set, a partial ordered set, or even a (discrete) topological space; in this paper designs are generalized due to this point of view. In this way for a mathematical structure ${\mathcal M}$, ${\mathcal M}$-designs are developed, especially some results in poset-designs are obtained.

Some Applications of the Fundamental Theorem of Hermitian K-Theory

Wed, 14 Sep 2011 16:37 EDT

Naoufel Battikh

Source: Missouri J. Math. Sci., Volume 23, Number 1, 48--64.

Abstract:
In this work we show how to use the Karoubi's fundamental theorem of Hermitian K-theory [6] to prove some results in L-Theory using these same results in algebraic K-Theory.

On $\gamma$-P-Regular Spaces

Wed, 14 Sep 2011 16:37 EDT

Sabir Hussain, Bashir Ahmad

Source: Missouri J. Math. Sci., Volume 23, Number 1, 65--74.

Abstract:
We define and explore $\gamma$-P-regularity which is a generalization of P-regularity [16] and $\gamma$-regularity [3]. We also define and discuss strongly $\gamma$-semi-continuous functions.

Limiting availability of a one-unit system backed by a spare under repair or preventive maintenance

Wed, 14 Sep 2011 16:37 EDT

Bruno Bieth, Liang Hong, Jyotirmooy Sarkar

Source: Missouri J. Math. Sci., Volume 23, Number 1, 75--87.

Abstract:
We consider a one-unit system under continuous monitoring, aided by an identical spare unit and serviced by a facility that performs repair on a failed unit or preventive maintenance on a recalled unit making it as good as new. We assume instantaneous commencement of service and installation to operation. We find the distribution of the system up time and down time when life-, recall- and service-times have arbitrary probability density functions. Hence, we obtain the limiting availability of the system. Also, we compute the servicing cost per unit time to determine whether preventive maintenance is preferable over a repair only model.

Solutions

Wed, 14 Sep 2011 16:37 EDT

Source: Missouri J. Math. Sci., Volume 23, Number 1, 88--95.

Editorial

Fri, 11 Nov 2011 15:59 EST

Shing So

Source: Missouri J. Math. Sci., Volume 23, Number 2, 103--104.

Extending Landau's Theorem on Dirichlet Series with Non-Negative Coefficients

Fri, 11 Nov 2011 15:59 EST

Brian N. Maurizi

Source: Missouri J. Math. Sci., Volume 23, Number 2, 105--122.

Abstract:
A classical theorem of Landau states that, if an ordinary Dirichlet series has non-negative coefficients, then it has a singularity on the real line at its abscissa of convergence. In this article, we relax the condition on the coefficients while still arriving at the same conclusion. Specifically, we write $a_n$ as $|a_n| e^{i \theta _n}$ and we consider the sequences $\{ |a_n| \}$ and $\{ \cos{\theta _n} \}$. Let $M \in \mathbb{N}$ be given. The condition on $\{ |a_n| \}$ is that, dividing the sequence sequentially into vectors of length $M$, each vector lies in a certain convex cone $B \subset [0,\infty)^M$. The condition on $\{ \cos{\theta _n} \}$ is (roughly) that, again dividing the sequence sequentially into vectors of length $M$, each vector lies in the negative of the polar cone of $B$. We demonstrate the additional freedom allowed in choosing the $\theta _n$, compared to Landau's Theorem. We also obtain sharpness results.

On Integrals and Sums Involving Special Functions

Fri, 11 Nov 2011 15:59 EST

Ahmad Al-Salman, Mohamed Ben Haj Rhouma, Ali A. Al-Jarrah

Source: Missouri J. Math. Sci., Volume 23, Number 2, 123--141.

Abstract:
Integrals and sums involving special functions are in constant demand in applied mathematics. Rather than refer to a handbook of integrals or to a computer algebra system, we present a do-it-yourself systematic approach that shows how the evaluation of such integrals and sums can be made as simple as possible. Illustrating our method, we present several examples of integrals of Poisson type, Fourier transform, as well as integrals involving product of Bessel functions. We also obtain a new identity involving the sums of $_{2}F_{1}$.

Win-Loss Sequences for Generalized Roundrobin Tournaments

Fri, 11 Nov 2011 15:59 EST

Arthur Holshouser, John W. Moon, Harold Reiter

Source: Missouri J. Math. Sci., Volume 23, Number 2, 142--150.

Abstract:
In a tournament, each of $n$ teams wins or loses against each of the other $n-1$ teams $c$ times. If team $i$ wins a total of $w_i$ games, then the sequence $( w_1, w_2, \ldots , w_n )$ is called the score sequence of the tournament. In this paper we give necessary and sufficient conditions on a sequence in order that it be a score sequence for a tournament.

The Shortest Distance in Data Analysis

Fri, 11 Nov 2011 15:59 EST

Alan M. Safer, Kagba Suaray, Saleem Watson

Source: Missouri J. Math. Sci., Volume 23, Number 2, 151--158.

Abstract:
By representing a set of $N$ data points as a vector $\mathbf{x}$ in $\mathbb{R}^{N}$, we show that certain data analysis concepts, in particular regression and quantile regression, can be interpreted as vectors that minimize the distance to the vector $\mathbf{x}$, with respect to an appropriate metric or quasimetric.

On a Degree of Primality

Fri, 11 Nov 2011 15:59 EST

Joseph Pe

Source: Missouri J. Math. Sci., Volume 23, Number 2, 159--172.

Abstract:
In this paper we introduce a degree of primality for natural numbers, and hence, a measure of primality for intervals of consecutive numbers. We characterize maximally prime intervals of length $\le 3$ and their primalities. Maximally prime intervals of length 2 are those that contain Mersenne or Fermat primes; maximally prime intervals of length 3 are, but for a few exceptions, those whose midpoints are Dan numbers. There are relatively few maxprimes for larger lengths. We present a heuristic argument for an asymptotic form describing the distribution of maximal primalities. Finally, we mention open problems and directions for further research.

Categories with Strong Monomorphic Strong Coimages

Fri, 11 Nov 2011 15:59 EST

M. A. Al Shumrani

Source: Missouri J. Math. Sci., Volume 23, Number 2, 173--181.

Abstract:
Let $SE(\mathcal{C})$ (respectively, $SM(\mathcal{C})$) be the subcategory of a category $\mathcal{C}$ with the same objects and whose morphisms are strong epimorphisms (respectively, strong monomorphisms) of $\mathcal{C}$. In this paper, we give conditions in some categories $\mathcal{C}$ for an object $X$ of pro-$\mathcal{C}$ to be isomorphic to an object of pro-$SE(\mathcal{C})$ (respectively, pro-$SM(\mathcal{C})$). As an application, we give conditions under which objects of pro-categories are stable.

Further Review of Panjer's Recursion for Evaluation of Compound Negative Binomial Distribution Using R

Fri, 11 Nov 2011 15:59 EST

Kumar Pial Das, Shamim Sarker, Norou Diawara

Source: Missouri J. Math. Sci., Volume 23, Number 2, 182--191.

Abstract:
Panjer's recursion formula is used for the evaluation of compound distributions. The use of this algorithm has become a widespread standard technique for life and general insurance problems. This study reviews and extends Panjer's recursion formula for evaluation of compound negative binomial distributions. For gamma and a mixture of gamma distributions, the theory has been developed so that the software R can be directly applicable. The accuracy of the method used in this study is better and the computation time is faster.

Some Inequalities in Ptolemaic Spaces

Fri, 11 Nov 2011 15:59 EST

V. Adiyasuren, TS. Batbold

Source: Missouri J. Math. Sci., Volume 23, Number 2, 192--196.

Abstract:
In this note, we establish some new interesting inequalities in Ptolemaic spaces.

Solutions

Fri, 11 Nov 2011 15:59 EST

Source: Missouri J. Math. Sci., Volume 23, Number 2, 197--201.

Announcements

Fri, 11 Nov 2011 15:59 EST

Source: Missouri J. Math. Sci., Volume 23, Number 2, 202--204.

$A^p$ is Not an Algebra for $1 \lt p \lt 2$

Fri, 25 May 2012 08:54 EDT

Ryan Mullen

Source: Missouri J. Math. Sci., Volume 24, Number 1, 1--6.

Abstract:
Let $A^p$ be the Banach space of all continuous functions on the torus ${\mathbb T} = \{ z \in {\mathbb C} \vert \vert z \vert = 1 \}$ whose Fourier coefficients are in $\ell ^p$. We show that $A^p$ is not an algebra for all $1 \lt p \lt 2$.

On Legendre Multiplier Sequences

Fri, 25 May 2012 08:54 EDT

Kelly Blakeman, Emily Davis, Tamás Forgács, Katherine Urabe

Source: Missouri J. Math. Sci., Volume 24, Number 1, 7--23.

Abstract:
In this paper we give a complete characterization of linear, quadratic, and geometric Legendre multiplier sequences. We also prove that all Legendre multiplier sequences must be Hermite multiplier sequences, and describe the relationship between the Legendre and generalized Laguerre multiplier sequences. We conclude with a list of open questions for further research.

On $(1,2)^*$-Semi-$T_{1/3}$ Bitopological Spaces

Fri, 25 May 2012 08:54 EDT

M. Lellis Thivagar, Nirmala Mariappan

Source: Missouri J. Math. Sci., Volume 24, Number 1, 24--36.

Abstract:
The aim of this paper is to introduce a separation axiom using $(1,2)^*$-$\psi$-closed sets.

On $u-\omega$-Open and $q-\omega$-Open Sets in Bitopological Spaces

Fri, 25 May 2012 08:54 EDT

S. Al Ghour, S. Issa

Source: Missouri J. Math. Sci., Volume 24, Number 1, 37--53.

Abstract:
$\omega$-open sets are used to introduce two new classes of sets in bitopological spaces, namely, $u-\omega$-open sets and $q-\omega$-open sets. Several properties of these classes are given. The new classes of sets are used to introduce several types of continuity. Several results related to two known Lindelöfness bitopological concepts are introduced.

Hamiltonian-Connected Graphs with Large Neighborhoods and Degrees

Fri, 25 May 2012 08:54 EDT

Zhao Kewen, Hong-Jian Lai, Ju Zhou

Source: Missouri J. Math. Sci., Volume 24, Number 1, 54--66.

Abstract:
For a simple graph $G$, let $C(G) = \min \{ \vert N(u) \cup N(v) \vert : u,v \in V(g), \ uv \notin E(G) \}$. In this paper we prove that if $NC(G) + \delta (G) \ge \vert V(G) \vert$, then either $G$ is Hamiltonian-connected, or $G$ belongs to a well-characterized class of graphs. The former result by Dirac, Ore and Faudree et al. are extended.

Existence of Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Delay Difference Equations

Fri, 25 May 2012 08:54 EDT

Zhenyu Guo, Min Liu, Mingming Chen

Source: Missouri J. Math. Sci., Volume 24, Number 1, 67--75.

Abstract:
This paper studies the existence of nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation \begin{align*} \Delta \big(a_{kn} \cdots & \Delta (a_{2n} \triangle (a_{1n} \Delta (x_n + b_n x_{n-d}) ) ) \big) \\ &{} + f(n,x_{n-r_{1n}}, x_{n-r_{2n}}, \ldots , x_{n-r_{sn}} ) = 0, \ \ n \ge n_0, \end{align*} where $n_0 \ge 0$, $n \ge 0$, $d > 0$, $k > 0$, $s > 0$ are integers, $\{ a_{in} \} _{n \ge n_0}$ ($i = 1, 2, \ldots , k)$) and $\{ b_n \} _{n \ge n_0}$ are real sequences, $f \colon \{ n : n \ge n_0 \} \times {\mathbb R}^n \to {\mathbb R}$ is a mapping and $\bigcup _{j=1}^s \{ r_{jn} \} _{n \ge n_0} \subseteq {\mathbb Z}$. By applying Krasnoselskii's Fixed Point Theorem, some sufficient conditions for the existence of nonoscillatory solutions of this equation are established and indicated through five theorems according to the range of value of the sequence $b_n$.

An Annotated Bibliography on the Thickness, Outerthickness, and Arboricity of a Graph

Fri, 25 May 2012 08:54 EDT

Erkki Mäkinen, Timo Poranen

Source: Missouri J. Math. Sci., Volume 24, Number 1, 76--87.

Abstract:
The bibliography introduces literature on graph thickness, outerthickness, and arboricity. In addition to the pointers to the literature we also give some conjectures concerning known open problems on the field.

Unsteady Natural Convection Flow of Electrically Conducting Fluid in the Presence of Magnetic Field Past an Accelerated Vertical Heated Plate in a Thermally Stratified Fluid

Fri, 25 May 2012 08:54 EDT

U. S. Rajput, Surendra Kumar

Source: Missouri J. Math. Sci., Volume 24, Number 1, 88--95.

Abstract:
Unsteady natural convection flow of electrically conducting fluid in the presence of magnetic field past an accelerated vertical plate in a thermally stratified fluid is studied here. The governing equations involved in the present analysis are solved by the Laplace-transform technique. The velocity, skin friction, and Nusselt numbers are studied for different parameters like Prandtl numbers, thermal Grashof numbers, magnetic field parameters, stratification parameters, and time.

Announcements

Fri, 25 May 2012 08:54 EDT

Source: Missouri J. Math. Sci., Volume 24, Number 1, 96--97.

Isoperimetric Regions in the Hyperbolic Plane Between Parallel Horocycles

Mon, 05 Nov 2012 13:02 EST

Marcio Fabiano da Silva

Source: Missouri J. Math. Sci., Volume 24, Number 2, 102--115.

Abstract:
In this work the following problem in the hyperbolic plane is investigated. Find the perimeter-minimizing regions of prescribed area between two parallel horocycles. An explicit and detailed description of all such regions is given through isoperimetric inequalities.

Looking for Fibonacci Base-2 Pseudoprimes

Mon, 05 Nov 2012 13:02 EST

Daniel J. Monfre, Dominic Klyve

Source: Missouri J. Math. Sci., Volume 24, Number 2, 116--123.

Abstract:
In this paper, we examine computationally the results of combining two well-known, simple, and imperfect tests for primality: the Fermat base-2 test, and the Fibonacci test. Although considerable attention has been paid to various properties of composite integers which pass the base-2 test ( base-2 pseudoprimes ), no comparable study of Fibonacci and base-2 Fibonacci tests exists in the literature. Our study tabulates various empirical properties of these numbers. Among other things, we conclude that there are no base-2 Fibonacci pseudoprimes less than $10^{15}$ which are congruent to 2 or 3 (modulo 5).

Generalized Groups that Distribute Over Stars

Mon, 05 Nov 2012 13:02 EST

Harold Reiter, Arthur Holshouser

Source: Missouri J. Math. Sci., Volume 24, Number 2, 124--155.

Abstract:
If $(S, \ast)$ is an arbitrary mathematical structure on a set $S$, three universal problems are to find all groups $(S, \cdot)$ on the same set that left-distribute or right-distribute or both left-distribute and right-distribute over $(S, \ast)$, if such a group exists. These concepts are defined in this paper. Also, we give a solution to the first two of these three problems for a naturally occurring example that involves what we call an $n$-star (which is structurally the same as $n$ lines in the plane intersecting in ${n \choose 2}$ district points).

Some New Characteristic Properties of the A-Pedal Hypersurfaces in $E^{n+1}$

Mon, 05 Nov 2012 13:02 EST

Ayhan Sarioğlugil, Sidika Tül, Nuri Kuruoğlu

Source: Missouri J. Math. Sci., Volume 24, Number 2, 156--166.

Abstract:
The primary purpose of this paper is to present the definition of the a-pedal hypersurface with respect to a point in the interior of a closed, convex and smooth hypersurface $M$. The secondary purpose of this paper is to give some new characteristic properties of the a-pedal hypersurfaces related to the support function, Gauss curvature, mean curvature, the first and second fundamental forms and their coefficients of $M$ (Section 3). Using the classical methods of the hypersurfaces in differential geometry we have established that the support function $h_{a}$ of the a-pedal hypersurface $M_{a}$ is equal to $\frac{h^{a+1}}{P_{a}}$ where $P_{a}^{2}=h^{2}+a^{2}\overset{III}{\nabla }(h,h)$.

Weak Approach to Planar Soap Bubble Clusters

Mon, 05 Nov 2012 13:02 EST

Wacharin Wichiramala

Source: Missouri J. Math. Sci., Volume 24, Number 2, 167--181.

Abstract:
The planar soap bubble problem seeks the least perimeter way to enclose and separate $m$ regions of $m$ given areas. We discuss a useful approach, especially for $m\le 8$.

A Compact Embedding for Sequence Spaces

Mon, 05 Nov 2012 13:02 EST

James Bisgard

Source: Missouri J. Math. Sci., Volume 24, Number 2, 182--189.

Abstract:
The Rellich-Kondrachov Theorem is a fundamental result in the theory of Sobolev spaces. We prove an analogue of this theorem in the context of sequence spaces, using elementary methods.

Lattice Properties of $T_1-L$ Topologies

Mon, 05 Nov 2012 13:02 EST

Raji George, T. P. Johnson

Source: Missouri J. Math. Sci., Volume 24, Number 2, 190--194.

Abstract:
We study the lattice structure of the set $ \Omega (X)$ of all $T_1$-$L$ topologies on a given set $X$. It is proved that $\Omega ( X ) $ has dual atoms (anti atoms) if and only if membership lattice $L$ has dual atoms (anti atoms). Some other properties of this lattice are also discussed.

Invariant Relations for the Derivatives of Two Arbitrary Polynomials

Mon, 05 Nov 2012 13:02 EST

Arthur Holshouser, Harold Reiter

Source: Missouri J. Math. Sci., Volume 24, Number 2, 195--201.

Abstract:
Suppose $P(x)$ and $Q(x)$ are two arbitrary polynomials. In this paper we use the theory of resultants of two polynomials including Sylvester's Matrix to specify a large number of polynomial relations involving $P(x)$, $Q(x)$ and their derivatives. We also include a research problem for the reader to consider.