Missouri Journal of Mathematical Sciences Articles (Project Euclid)
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The latest articles from Missouri Journal of Mathematical Sciences on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2011 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Wed, 14 Sep 2011 16:37 EDTWed, 14 Sep 2011 16:37 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Editorial
http://projecteuclid.org/euclid.mjms/1312233177
<strong>Terry Goodman</strong><p><strong>Source: </strong>Missouri J. Math. Sci., Volume 23, Number 1, 1--2.</p>projecteuclid.org/euclid.mjms/1312233177_Wed, 14 Sep 2011 16:37 EDTWed, 14 Sep 2011 16:37 EDTAn Interesting Infinite Series and Its Implications to Operator Theoryhttp://projecteuclid.org/euclid.mjms/1474295355<strong>Melanie Henthorn-Baker</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 28, Number 1, 49--61.</p><p><strong>Abstract:</strong><br/>
The following is a discussion regarding a specific class of operators acting on
the space of entire functions, denoted $H(\mathbb{C})$. A diagonal operator $D$ on $H(\mathbb{C})$
is defined to be a continuous linear map, sending $H(\mathbb{C})$ into $H(\mathbb{C})$, that has the
monomials $z^n$ as its eigenvectors and $\{\lambda_n\}$ as the corresponding
eigenvalues. A closed subspace $M$ is invariant for $D$ if $Df\in M$ for all
$f\in M$. The study of invariant subspaces is a popular topic in modern operator
theory. We observe that the closed linear span of the orbit, which we write
$\overline{\mbox{span}}\{D^kf:k\geq0\}=\overline{\mbox{span}}\{\sum^{\infty}_{n=0}a_n\lambda_n^kz^n:k\geq0\}$,
is the smallest closed invariant subspace for $D$ containing $f$. If every
invariant subspace for a diagonal operator $D$ on $H(\mathbb{C})$ can be expressed as a
closed linear span of some subset of the eigenvectors of $D$, we say that $D$
admits spectral synthesis on $H(\mathbb{C})$. Until recently, it was not known whether or
not every diagonal operator on $H(\mathbb{C})$ admitted spectral synthesis. This article
focuses on using techniques from calculus and linear algebra to construct a
class of operators which fail spectral synthesis on $H(\mathbb{C})$. If the reader is not
familiar with the operator theory definitions provided in the background, he or
she can still appreciate the construction of an interesting infinite series
relying on properties of logarithms, various convergence tests, and Cramer's
Rule.
</p>projecteuclid.org/euclid.mjms/1474295355_20160919102916Mon, 19 Sep 2016 10:29 EDTCompact Weighted Composition Operators Between Generalized Fock Spaceshttp://projecteuclid.org/euclid.mjms/1474295356<strong>Waleed Al-Rawashdeh</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 28, Number 1, 62--75.</p><p><strong>Abstract:</strong><br/>
Let $\psi$ be an entire self-map of the $n$-dimensional Euclidean complex space
$\mathbb{C}^n$ and $u$ be an entire function on $\mathbb{C}^n$. A weighted
composition operator induced by $\psi$ with weight $u$ is given by
$(uC_{\psi}f)(z)= u(z)f(\psi(z))$, for $z \in \mathbb{C}^n$ and $f$ is the
entire function on $\mathbb{C}^n$. In this paper, we study weighted composition
operators acting between generalized Fock-types spaces. We characterize the
boundedness and compactness of these operators act between
$\mathcal{F}_{\phi}^{p}(\mathbb{C}^n)$ and
$\mathcal{F}_{\phi}^{q}(\mathbb{C}^n)$ for $0\lt p, q\leq\infty$. Moreover, we
give estimates for the Fock-norm of $uC_{\psi}:
\mathcal{F}_{\phi}^{p}\rightarrow \mathcal{F}_{\phi}^{q}$ when $0\lt p, q\lt
\infty$, and also when $p=\infty$ and $0\lt q\lt \infty$.
</p>projecteuclid.org/euclid.mjms/1474295356_20160919102916Mon, 19 Sep 2016 10:29 EDTIdentifying Outlying Observations in Regression Treeshttp://projecteuclid.org/euclid.mjms/1474295357<strong>Nicholas Granered</strong>, <strong>Samantha C. Bates Prins</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 28, Number 1, 76--87.</p><p><strong>Abstract:</strong><br/>
Regression trees are an alternative to classical linear regression models that
seek to fit a piecewise linear model to data. The structure of regression trees
makes them well-suited to the modeling of data containing outliers. We propose
an algorithm that takes advantage of this feature in order to automatically
detect outliers. This new algorithm performs well on the four test datasets
[7] that are considered to be necessary for a valid outlier
detection algorithm in a linear regression context, even though regression trees
lack the global linearity assumption. We also show the practical use of this
approach in detecting outliers in an ecological dataset collected in the
Shenandoah Valley.
</p>projecteuclid.org/euclid.mjms/1474295357_20160919102916Mon, 19 Sep 2016 10:29 EDTAn Alternate Cayley-Dickson Producthttp://projecteuclid.org/euclid.mjms/1474295358<strong>John W. Bales</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 28, Number 1, 88--96.</p><p><strong>Abstract:</strong><br/>
Although the Cayley-Dickson algebras are twisted group algebras, little
attention has been paid to the nature of the Cayley-Dickson twist. One reason is
that the twist appears to be highly chaotic and there are other interesting
things about the algebras to focus attention upon. However, if one uses a
doubling product for the algebras different from yet equivalent to the ones
commonly used, and if one uses a numbering of the basis vectors different from
the standard basis a quite beautiful and highly periodic twist emerges. This
leads easily to a simple closed form equation for the product of any two basis
vectors of a Cayley-Dickson algebra.
</p>projecteuclid.org/euclid.mjms/1474295358_20160919102916Mon, 19 Sep 2016 10:29 EDTOuterplanar Coarseness of Planar Graphshttp://projecteuclid.org/euclid.mjms/1474295359<strong>Paul C. Kainen</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 28, Number 1, 97--98.</p><p><strong>Abstract:</strong><br/>
The (outer) planar coarseness of a graph is the largest number of
pairwise-edge-disjoint non-(outer)planar subgraphs. It is shown that the maximum
outerplanar coarseness, over all $n$-vertex planar graphs, lies in the interval
$\;\big [\lfloor (n-2)/3 \rfloor, \lfloor (n-2)/2 \rfloor \big ]$.
</p>projecteuclid.org/euclid.mjms/1474295359_20160919102916Mon, 19 Sep 2016 10:29 EDTEditorialhttp://projecteuclid.org/euclid.mjms/1488423695<strong>Sam Creswell</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 1--1.</p>projecteuclid.org/euclid.mjms/1488423695_20170301220243Wed, 01 Mar 2017 22:02 ESTCenters and Generalized Centers of Nearrings Without Identityhttp://projecteuclid.org/euclid.mjms/1488423696<strong>G. Alan Cannon</strong>, <strong>Vincent Glorioso</strong>, <strong>Brad Bailey Hall</strong>, <strong>Taylor Triche</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 2--11.</p><p><strong>Abstract:</strong><br/>
The center of a nearring $N$, in general, is not a subnearring of $N$. The center,
however, is contained in a related structure, the generalized center, which is always a subnearring.
We give three constructions of nearrings without multiplicative identity and characterize their centers
and generalized centers. We find that the centers of these nearrings are always subnearrings.
</p>projecteuclid.org/euclid.mjms/1488423696_20170301220243Wed, 01 Mar 2017 22:02 ESTSolving Sudoku: Structures and Strategieshttp://projecteuclid.org/euclid.mjms/1488423697<strong>Hang Chen</strong>, <strong>Curtis Cooper</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 12--18.</p><p><strong>Abstract:</strong><br/>
We intend to solve Sudoku puzzles using various rules based on the structures and
properties of the puzzle. In this paper, we shall present several structures related to either one
potential solution or two potential solutions.
</p>projecteuclid.org/euclid.mjms/1488423697_20170301220243Wed, 01 Mar 2017 22:02 ESTReal Preimages of Duplication on Elliptic Curveshttp://projecteuclid.org/euclid.mjms/1488423698<strong>John Cullinan</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 19--26.</p><p><strong>Abstract:</strong><br/>
Let $E$ be an elliptic curve defined over the real numbers $R$ and let $P \in E(R)$. In
this note we give an elementary proof of necessary and sufficient conditions for the preimages
of $P$ under duplication to be real-valued.
</p>projecteuclid.org/euclid.mjms/1488423698_20170301220243Wed, 01 Mar 2017 22:02 ESTInteger Invariants of an Incidence Matrix Related to Rota's Basis Conjecturehttp://projecteuclid.org/euclid.mjms/1488423699<strong>Stephanie Bittner</strong>, <strong>Joshua Ducey</strong>, <strong>Xuyi Guo</strong>, <strong>Minah Oh</strong>, <strong>Adam Zweber</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 27--32.</p><p><strong>Abstract:</strong><br/>
We compute the spectrum and Smith normal form of the incidence matrix of disjoint transversals, a
combinatorial object closely related to the $n$-dimensional case of Rota's basis conjecture.
</p>projecteuclid.org/euclid.mjms/1488423699_20170301220243Wed, 01 Mar 2017 22:02 ESTLocal Separation Axioms Between Kolmogorov and Fr\'{e}chet Spaceshttp://projecteuclid.org/euclid.mjms/1488423700<strong>Raghu Gompa</strong>, <strong>Vijaya L. Gompa</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 33--42.</p><p><strong>Abstract:</strong><br/>
Several separation axioms on topological spaces are described between Kolmogorov and Fréchet
spaces as properties of the space at a particular point. After describing various equivalent descriptions,
implications are established. Various examples are studied in order to show that the implications are strict.
</p>projecteuclid.org/euclid.mjms/1488423700_20170301220243Wed, 01 Mar 2017 22:02 ESTStrongly Generalized Neighborhood Systemshttp://projecteuclid.org/euclid.mjms/1488423701<strong>Murad Arar</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 43--49.</p><p><strong>Abstract:</strong><br/>
In this paper we define strongly generalized neighborhood systems (in brief strongly $GNS$) and
study their properties. It's proved that every generalized topology $\mu$ on $X$ gives a unique strongly $GNS$
$\psi_{\mu}:X\rightarrow \exp{(\exp{X})}$. We prove that if a generalized topology $\mu$ is given, then
$\mu_{\psi_{\mu}}=\mu$; and if a strongly $GNS $ $\psi$ is given, then $\psi_{\mu_{\psi}}=\psi$.
Strongly $(\psi_{1},\psi_{2})$-continuity is defined. We prove that $f:X\rightarrow Y$ is
strongly $(\psi_{1},\psi_{2})$-continuous if and only if it is $(\mu_{\psi_{1}},\mu_{\psi_{2}})$-continuous.
</p>projecteuclid.org/euclid.mjms/1488423701_20170301220243Wed, 01 Mar 2017 22:02 ESTA Simple Pure Water Oscillatorhttp://projecteuclid.org/euclid.mjms/1488423702<strong>S. Lakshmivarahan</strong>, <strong>N. Trung</strong>, <strong>Barry Ruddick</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 50--79.</p><p><strong>Abstract:</strong><br/>
In this paper we analyze the properties of a pure water oscillator by
considering the pure water lake as a well mixed two layered system.
While there is heating of and evaporation from the shallow top layer,
the temperature of the deep bottom layer is assumed to be constant. By
exploiting the nonlinear dependence of the density of pure water on
temperature, we describe two complementary mathematical models to
capture the vertical instability resulting from the variation of the
density of the top layer with temperature.
</p>projecteuclid.org/euclid.mjms/1488423702_20170301220243Wed, 01 Mar 2017 22:02 ESTOn Nano Resolvable Spaceshttp://projecteuclid.org/euclid.mjms/1488423703<strong>M. Lellis Thivagar</strong>, <strong>J. Kavitha</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 80--91.</p><p><strong>Abstract:</strong><br/>
This paper introduces nano resolvable spaces and nano irresolvable spaces. Also, a new form
of nano subspace topology is established. Several new characterizations of nano strongly irresolvable
spaces are found and precise relationships are noted between nano strongly irresolvability and nano
irresolvable space. Some weaker forms related to nano irresolvable are discussed. Also, comparisons
between them are given.
</p>projecteuclid.org/euclid.mjms/1488423703_20170301220243Wed, 01 Mar 2017 22:02 ESTAtanassov's Intuitionistic Fuzzy Bi-Normed KU-Subalgebras of a KU-Algebrahttp://projecteuclid.org/euclid.mjms/1488423704<strong>Tapan Senapati</strong>, <strong>K. P. Shum</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 92--112.</p><p><strong>Abstract:</strong><br/>
In this paper, by using the $t$-norm $T$ and $t$-conorm $S$, we introduce the intuitionistic
fuzzy bi-normed $KU$-subalgebras of
a $KU$-algebra. Some properties of intuitionistic fuzzy bi-normed $KU$-subalgebras of a
$KU$-algebra under the homomorphism are discussed.
The direct product and the $(T,S)$-product of intuitionistic fuzzy bi-normed
$KU$-subalgebras are particularly investigated.
</p>projecteuclid.org/euclid.mjms/1488423704_20170301220243Wed, 01 Mar 2017 22:02 ESTAnnouncementshttp://projecteuclid.org/euclid.mjms/1488423705<p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 1, 113--114.</p>projecteuclid.org/euclid.mjms/1488423705_20170301220243Wed, 01 Mar 2017 22:02 ESTAn Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Masshttps://projecteuclid.org/euclid.mjms/1513306825<strong>Amir M. Rahimi</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 115--124.</p><p><strong>Abstract:</strong><br/>
This paper takes an interesting approach to conceptualize some power sum inequalities and uses them to develop limits on possible solutions to some Diophantine equations. In this work, we introduce how to apply center of mass of a $k$-mass-system to discuss a class of Diophantine equations (with fixed positive coefficients) and a class of equations related to Fermat's Last Theorem. By a constructive method, we find a lower bound for all positive integers that are not the solution for these type of equations. Also, we find an upper bound for any possible integral solution for these type of equations. We write an alternative expression of Fermat's Last Theorem for positive integers in terms of the product of the centers of masses of the systems of two fixed points (positive integers) with different masses. Finally, by assuming the validity of Beal's conjecture, we find an upper bound for any common divisor of $x$, $y$, and $z$ in the expression $ax^m+by^n = z^r$ in terms of $a, b, m({\rm or} ~n), r$, and the center of mass of the $k$-mass-system of $x$ and $y$.
</p>projecteuclid.org/euclid.mjms/1513306825_20171214220030Thu, 14 Dec 2017 22:00 ESTCubic Implicative Ideals of $BCK$-algebrashttps://projecteuclid.org/euclid.mjms/1513306826<strong>Tapan Senapati</strong>, <strong>K. P. Shum</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 125--138.</p><p><strong>Abstract:</strong><br/>
In this paper, we apply the concept of cubic sets to implicative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic implicative ideals, cubic subalgebras and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic implicative ideal. We define inverse images of cubic implicative ideals and establish how the inverse images of a cubic implicative ideal become a cubic implicative ideal. Finally we introduce products of cubic $BCK$-algebras.
</p>projecteuclid.org/euclid.mjms/1513306826_20171214220030Thu, 14 Dec 2017 22:00 EST$(\in,\in\vee q)$-bipolar Fuzzy $BCK/BCI$-algebrashttps://projecteuclid.org/euclid.mjms/1513306827<strong>Chiranjibe Jana</strong>, <strong>Madhumangal Pal</strong>, <strong>Arsham Borumand Saied</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 139--160.</p><p><strong>Abstract:</strong><br/>
In this paper, the concept of quasi-coincidence of a bipolar fuzzy point within a bipolar fuzzy set is introduced. The notion of $(\in,\in\vee q)$-bipolar fuzzy subalgebras and ideals of $BCK/BCI$-algebras are introduced and their related properties are investigated by some examples. We study bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-ideals by their level subalgebras and level ideals. We also provide the relationship between $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-subalgebras, and $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-ideals and bipolar fuzzy $BCK/BCI$-ideals by counter examples.
</p>projecteuclid.org/euclid.mjms/1513306827_20171214220030Thu, 14 Dec 2017 22:00 ESTGeometry of Polynomials with Three Rootshttps://projecteuclid.org/euclid.mjms/1513306828<strong>Christopher Frayer</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 161--175.</p><p><strong>Abstract:</strong><br/>
Given a complex-valued polynomial of the form $p(z) = (z-1)^k(z-r_1)^m(z-r_2)^n$ with $|r_1|=|r_2|=1$; $k,m,n \in \mathbb{N}$ and $m \neq n$, where are the critical points? The Gauss-Lucas Theorem guarantees that the critical points of such a polynomial will lie within the unit disk. This paper further explores the location and structure of these critical points. Surprisingly, the unit disk contains two ‘desert’ regions in which critical points cannot occur, and each $c$ inside the unit disk and outside of the desert regions is the critical point of exactly two such polynomials.
</p>projecteuclid.org/euclid.mjms/1513306828_20171214220030Thu, 14 Dec 2017 22:00 ESTSieving for the Primes to Prove Their Infinitudehttps://projecteuclid.org/euclid.mjms/1513306829<strong>Hunde Eba</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 176--183.</p><p><strong>Abstract:</strong><br/>
We prove the infinitude of prime numbers by the principle of contradiction, that is different from Euclid's proof in a way that it uses an explicit property of prime numbers. A sieve method that applies the inclusion-exclusion principle is used to give the property of the prime numbers in terms of the prime counting function.
</p>projecteuclid.org/euclid.mjms/1513306829_20171214220030Thu, 14 Dec 2017 22:00 ESTWeakly JU Ringshttps://projecteuclid.org/euclid.mjms/1513306830<strong>Peter V. Danchev</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 184--196.</p><p><strong>Abstract:</strong><br/>
We define and completely explore the so-called WJU rings . This class properly encompasses the class of JU rings, introduced and studied by the present author in detail in Toyama Math. J. (2016).
</p>projecteuclid.org/euclid.mjms/1513306830_20171214220030Thu, 14 Dec 2017 22:00 ESTOn $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy Soft $BCI$-algebrashttps://projecteuclid.org/euclid.mjms/1513306831<strong>Chiranjibe Jana</strong>, <strong>Madhumangal Pal</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 197--215.</p><p><strong>Abstract:</strong><br/>
Molodtsov initiated soft set theory which has provided a general mathematical framework for handling uncertainties that occur in various real life problems. The aim of this paper is to provide fuzzy soft algebraic tools in considering many problems that contain uncertainties. In this article, the notion of $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy soft $BCI$-subalgebra of $BCI$-algebra is introduced. Some operational properties on $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy soft $BCI$-subalgebras are discussed as well as lattice structures of this kind of fuzzy soft set on $BCI$-subalgebras are derived.
</p>projecteuclid.org/euclid.mjms/1513306831_20171214220030Thu, 14 Dec 2017 22:00 ESTOn a Problem of Hararyhttps://projecteuclid.org/euclid.mjms/1513306832<strong>Paul C. Kainen</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 216--218.</p><p><strong>Abstract:</strong><br/>
An exercise in Harary [1, p. 100] states that the product of the vertex independence number and the vertex covering number is an upper bound on the number of edges in a bipartite graph. In this note, we extend the bound to triangle-free graphs, and show that equality holds if and only if the graph is complete bipartite.
</p>projecteuclid.org/euclid.mjms/1513306832_20171214220030Thu, 14 Dec 2017 22:00 ESTSoccer Balls, Golf Balls, and the Euler Identityhttps://projecteuclid.org/euclid.mjms/1513306833<strong>Linda Lesniak</strong>, <strong>Arthur T. White</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 219--222.</p><p><strong>Abstract:</strong><br/>
We show, with simple combinatorics, that if the dimples on a golf ball are all 5-sided and 6-sided polygons, with three dimples at each “vertex”, then no matter how many dimples there are and no matter the sizes and distribution of the dimples, there will always be exactly twelve 5-sided dimples. Of course, the same is true of a soccer ball and its faces.
</p>projecteuclid.org/euclid.mjms/1513306833_20171214220030Thu, 14 Dec 2017 22:00 ESTAnnouncementshttps://projecteuclid.org/euclid.mjms/1513306834<p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 29, Number 2, 223--224.</p>projecteuclid.org/euclid.mjms/1513306834_20171214220030Thu, 14 Dec 2017 22:00 ESTMunchausen Numbers Reduxhttps://projecteuclid.org/euclid.mjms/1534384947<strong>Devin Akman</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 1--4.</p><p><strong>Abstract:</strong><br/>
A Munchausen number is a mathematical curiosity: raise each digit to the power of itself, add them all up, and recover the original number. In the seminal paper on this topic, D. Van Berkel derived a bound on such numbers for any given radix, which means that they can be completely enumerated in principle. We present a simpler argument which yields a bound one half the size and show that a radically different approach would be required for further reductions.
</p>projecteuclid.org/euclid.mjms/1534384947_20180815220242Wed, 15 Aug 2018 22:02 EDTCubic Commutative Ideals of $BCK$-algebrashttps://projecteuclid.org/euclid.mjms/1534384948<strong>Tapan Senapati</strong>, <strong>K. P. Shum</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 5--19.</p><p><strong>Abstract:</strong><br/>
In this paper, we apply the concept of cubic set to commutative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic commutative ideals, cubic subalgebras, and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic commutative ideal. We define inverse images of cubic commutative ideals and establish how the inverse images of a cubic commutative ideal becomes a cubic commutative ideal. We introduce products of cubic $BCK$-algebras. Finally, we discuss the relationships between (cubic) commutative ideals, implicative ideals, and positive implicative ideals in $BCK/BCI$-algebras.
</p>projecteuclid.org/euclid.mjms/1534384948_20180815220242Wed, 15 Aug 2018 22:02 EDTStrong Forms of $\mu$-Lindelöfness with Respect to Hereditary Classeshttps://projecteuclid.org/euclid.mjms/1534384949<strong>Abdo Qahis</strong>, <strong>Heyam Hussain AlJarrah</strong>, <strong>Takashi Noiri</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 20--31.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to introduce and study strong forms of $\mu$-Lindelöfness in generalized topological spaces with a hereditary class, called $\mathcal{S} \mu\mathcal{H}$-Lindelöfness and $\mathbf{S}-\mathcal{S}\mu\mathcal{H}$-Lindelöfness. Interesting characterizations of these spaces are presented. Several effects of various types of functions on them are studied.
</p>projecteuclid.org/euclid.mjms/1534384949_20180815220242Wed, 15 Aug 2018 22:02 EDTSome Connections Between Bunke-Schick Differential K-theory and Topological $\mathbb{Z}/k\mathbb{Z}$ K-theoryhttps://projecteuclid.org/euclid.mjms/1534384951<strong>Adnane Elmrabty</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 32--44.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to prove some results in Bunke-Schick differential K-theory and topological $\mathbb{Z}/k\mathbb{Z}$ K-theory. The first one is an index theorem for the odd-dimensional geometric families of $\mathbb{Z}/k\mathbb{Z}$-manifolds. The second one is an alternative proof of the Freed-Melrose $\mathbb{Z}/k\mathbb{Z}$-index theorem in the framework of differential K-theory.
</p>projecteuclid.org/euclid.mjms/1534384951_20180815220242Wed, 15 Aug 2018 22:02 EDTNew Type of Simultaneous Remotal Sets in Certain Banach Spaceshttps://projecteuclid.org/euclid.mjms/1534384952<strong>Sh. Al-Sharif</strong>, <strong>A. Awad</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 45--53.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a new concept of simultaneous remotal sets and farthest points in Banach spaces and we present various characterizations of such points in certain Banach spaces.
</p>projecteuclid.org/euclid.mjms/1534384952_20180815220242Wed, 15 Aug 2018 22:02 EDTMagnifying Elements in a Semigroup of Transformations with Restricted Rangehttps://projecteuclid.org/euclid.mjms/1534384954<strong>Ronnason Chinram</strong>, <strong>Samruam Baupradist</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 54--58.</p><p><strong>Abstract:</strong><br/>
Let $Y$ be a nonempty subset of a set $X$ and let $T(X,Y)$ be the semigroup (under composition) of all functions $X\rightarrow X$ whose range is a subset of $Y$. We give necessary and sufficient conditions for elements in $T(X,Y)$ to be left and right magnifying.
</p>projecteuclid.org/euclid.mjms/1534384954_20180815220242Wed, 15 Aug 2018 22:02 EDTSome Operators in Ideal Topological Spaceshttps://projecteuclid.org/euclid.mjms/1534384955<strong>H. Al-Saadi</strong>, <strong>A. Al-Omari</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 59--71.</p><p><strong>Abstract:</strong><br/>
In this paper, we give an extensive study of ideal topological spaces and introduce some new types of sets with the help of a local function. Several characterizations of these sets will also be discussed through this paper. Moreover, we obtain characterizations of $\Psi_{\omega}$-operator and $\omega$-codense.
</p>projecteuclid.org/euclid.mjms/1534384955_20180815220242Wed, 15 Aug 2018 22:02 EDTArbitrarily High Hausdorff Dimensions of Continuahttps://projecteuclid.org/euclid.mjms/1534384956<strong>R. Patrick Vernon</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 72--76.</p><p><strong>Abstract:</strong><br/>
It is well-known that Hausdorff dimension is not a topological invariant; that is, that two homeomorphic continua can have different Hausdorff dimension, although their topological dimension will be equal. We show that it is possible to take any continuum embeddable in $\mathbb{R}^n$ and embed it in such a way that its Hausdorff dimension is $n$. In doing so, we can obtain an arbitrarily high Hausdorff dimension for any nondegenerate continuum. As an example, we will give different embeddings of an arc whose Hausdorff dimension is any real number between $1$ and $\infty$, including an arc of infinite Hausdorff dimension.
</p>projecteuclid.org/euclid.mjms/1534384956_20180815220242Wed, 15 Aug 2018 22:02 EDTOn Constructing Chaotic Maps with a Prescribed Probability Distributionhttps://projecteuclid.org/euclid.mjms/1534384957<strong>Peter M. Uhl</strong>, <strong>Hannah Bohn</strong>, <strong>Noah H. Rhee</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 77--84.</p><p><strong>Abstract:</strong><br/>
In this paper we discuss how to construct piecewise linear chaotic maps with a prescribed probability distribution on a finite number of open intervals of equal length that form a partition of the unit interval. The idea and method of how to find such a map are given in [3]. But a formal proof is not given. In this paper we provide a formal proof.
</p>projecteuclid.org/euclid.mjms/1534384957_20180815220242Wed, 15 Aug 2018 22:02 EDTThe Smallest Self-dual Embeddable Graphs in a Pseudosurfacehttps://projecteuclid.org/euclid.mjms/1534384958<strong>Ethan Rarity</strong>, <strong>Steven Schluchter</strong>, <strong>J. Z. Schroeder</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 85--92.</p><p><strong>Abstract:</strong><br/>
A proper embedding of a graph $G$ in a pseudosurface $P$ is an embedding in which the regions of the complement of $G$ in $P$ are homeomorphic to discs and a vertex of $G$ appears at each pinchpoint of $P$; we say that a proper embedding of $G$ in $P$ is self dual if there exists an isomorphism from $G$ to its topological dual. We determine five possible graphs with 7 vertices and 13 edges that could be self-dual embeddable in the pinched sphere, and we establish, by way of computer-powered methods, that such a self-embedding exists for exactly two of these five graphs.
</p>projecteuclid.org/euclid.mjms/1534384958_20180815220242Wed, 15 Aug 2018 22:02 EDTGenerating Stern-Brocot Type Rational Numbers with Mediantshttps://projecteuclid.org/euclid.mjms/1534384959<strong>Harold Reiter</strong>, <strong>Arthur Holshouser</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 93--104.</p><p><strong>Abstract:</strong><br/>
The Stern–Brocot tree is a method of generating or organizing all fractions in the interval \((0,1)\) by starting with the endpoints \(\frac{0}{1} \) and \(\frac{1}{1}\) and repeatedly applying the mediant operation: \(m\left( \frac{a}{b},\frac{c}{d} \right) =\frac{a+c}{b+d}\). A recent paper of Aiylam considers two generalizations: one is to apply the mediant operation starting with an arbitrary interval \(\left( \frac{a}{b},\frac{c}{d} \right)\) (the fractions must be non-negative), and the other is to allow arbitrary reduction of generated fractions to lower terms. In the present paper, we give simpler proofs of some of Aiylam's results, and we give a simpler method of generating just the portion of the tree that leads to a given fraction.
</p>projecteuclid.org/euclid.mjms/1534384959_20180815220242Wed, 15 Aug 2018 22:02 EDTAnnouncementshttps://projecteuclid.org/euclid.mjms/1534384960<p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 1, 105--106.</p>projecteuclid.org/euclid.mjms/1534384960_20180815220242Wed, 15 Aug 2018 22:02 EDTGuassian Amicable Pairshttps://projecteuclid.org/euclid.mjms/1544151688<strong>Patrick Costello</strong>, <strong>Ranthony A. C. Edmonds</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 107--116.</p><p><strong>Abstract:</strong><br/>
This article defines amicable pairs in the complex numbers and finds that some amicable pairs in the natural numbers are also amicable in the complex numbers. Unlike the case in the natural numbers, it is proved that no $(2,1)$ pairs made up of natural numbers where the common factor is a power of $2$ exist as Gaussian amicable pairs. Many pairs are found with complex parts using the DivisorSigma function in Mathematica . The factorizations into primes is given so that the type of pair might be determined.
</p>projecteuclid.org/euclid.mjms/1544151688_20181206220136Thu, 06 Dec 2018 22:01 ESTMozes' Game of Numbers on Directed Graphshttps://projecteuclid.org/euclid.mjms/1544151689<strong>Rohan Hemasinha</strong>, <strong>Avinash J. Dalal</strong>, <strong>Donald McGinn</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 117--131.</p><p><strong>Abstract:</strong><br/>
In 1986, the contestants of the $27$th International Mathematical Olympiad were given a game of numbers played on a pentagon. In 1987, Mozes generalized this game to an arbitrary undirected, weighted, connected graph. The convergence properties and total number of moves of any convergent game have been resolved by Mozes using Weyl groups. Eriksson provided an alternate proof using matrix theory and graph theory. In this paper, we briefly discuss the results of Mozes and Eriksson on undirected graphs. Then we generalize this game to arbitrary directed, strongly connected graphs and investigate the convergence properties of the game of numbers.
</p>projecteuclid.org/euclid.mjms/1544151689_20181206220136Thu, 06 Dec 2018 22:01 ESTHyper Dice Backgammon of Finite Sizehttps://projecteuclid.org/euclid.mjms/1544151690<strong>Amir M. Rahimi</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 132--139.</p><p><strong>Abstract:</strong><br/>
In this note, we extend the notion of standard backgammon to a more general setting and call it hyper dice backgammon (or HD-gammon for short) of size $n \geq 6$ (a positive even integer) by extending the regular die to a hyper die (i.e., hyper cube) with $n$ faces and the board from 24 pips to $4n$ pips, where $n = 2k \geq 6$ and there are $4k+3$ checkers for each player. The rules of the game are similar to the rules of standard backgammon when $n = 2k = 6$ and the number of the $n$-sided dice depends on $n$. Finally, we include a list of references related to some theoretical studies on standard backgammon.
</p>projecteuclid.org/euclid.mjms/1544151690_20181206220136Thu, 06 Dec 2018 22:01 ESTTranslational Surfaceshttps://projecteuclid.org/euclid.mjms/1544151691<strong>Andrew Crutcher</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 140--149.</p><p><strong>Abstract:</strong><br/>
A translational surface is a rational surface generated from two rational space curves by translating one curve along the other curve. In this paper, we utilize matrices to represent translational surfaces, and give necessary and sufficient conditions for a real rational surface to be a translational surface.
</p>projecteuclid.org/euclid.mjms/1544151691_20181206220136Thu, 06 Dec 2018 22:01 ESTIsoperimetry in Surfaces of Revolution with Densityhttps://projecteuclid.org/euclid.mjms/1544151692<strong>Eliot Bongiovanni</strong>, <strong>Alejandro Diaz</strong>, <strong>Arjun Kakkar</strong>, <strong>Nat Sothanaphan</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 150--165.</p><p><strong>Abstract:</strong><br/>
The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on $\mathbb{R}^n$, the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two different densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric.
</p>projecteuclid.org/euclid.mjms/1544151692_20181206220136Thu, 06 Dec 2018 22:01 ESTDouble Bubbles on the Line with Log-Convex Density $f$ with $(\log f)'$ Boundedhttps://projecteuclid.org/euclid.mjms/1544151693<strong>Nat Sothanaphan</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 166--175.</p><p><strong>Abstract:</strong><br/>
We extend results of Bongiovanni et al. [1] on double bubbles on the line with log-convex density to the case where the derivative of the log of the density is bounded. We show that the tie function between the double interval and the triple interval still exists, but may blow up to infinity in finite time. For the first time, a density is presented for which the blowup time is positive and finite.
</p>projecteuclid.org/euclid.mjms/1544151693_20181206220136Thu, 06 Dec 2018 22:01 ESTIndicators of Pointed Hopf Algebras of Dimensions $pq$ Over Characteristic $p$https://projecteuclid.org/euclid.mjms/1544151694<strong>Si Chen</strong>, <strong>Tiantian Liu</strong>, <strong>Linhong Wang</strong>, <strong>Xingting Wang</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 176--184.</p><p><strong>Abstract:</strong><br/>
Let $p$, $q$ be two distinct primes. We consider pointed Hopf algebras of dimension $pq$ over an algebraically closed field of characteristic $p$. We compute higher Frobenius-Schur indicators of these Hopf algebras through the associated graded Hopf algebras with respect to their coradical filtrations. The resulting indicators are gauge invariants for the monoidal representation categories of these algebras.
</p>projecteuclid.org/euclid.mjms/1544151694_20181206220136Thu, 06 Dec 2018 22:01 ESTA Note on Class $Q(N)$ Operatorshttps://projecteuclid.org/euclid.mjms/1544151695<strong>Shqipe Lohaj</strong>, <strong>Valdete Rexhëbeqaj Hamiti</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 185--196.</p><p><strong>Abstract:</strong><br/>
Let $T$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. In this paper we introduce two new classes of operators: class $Q(N)$ and class $Q^*(N)$. An operator $T\in \mathcal{L}(\mathcal{H})$ is of class $Q(N)$ for a fixed real number $N\geq 1$, if $T$ satisfies $N\|Tx\|^{2} \leq \| T^2 x\|^{2}+ \| x\|^{2}$ for all $x\in \mathcal{H}$. And an operator $T\in \mathcal{L}(\mathcal{H})$ is of class $Q^*(N)$ for a fixed real number $N\geq 1$, if $T$ satisfies $N\|T^*x\|^{2} \leq \| T^2 x\|^{2}+ \| x\|^{2}$ for all $x\in \mathcal{H}$. We prove the basic properties of these classes of operators.
</p>projecteuclid.org/euclid.mjms/1544151695_20181206220136Thu, 06 Dec 2018 22:01 ESTPolynomials in Base $x$ and the Prime-Irreducible Affintyhttps://projecteuclid.org/euclid.mjms/1544151696<strong>Fusun Akman</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 197--217.</p><p><strong>Abstract:</strong><br/>
Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique “base-$x$” representation of such polynomials that makes the ring $\mathbb{Z}[x]$ into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes $p$ and certain infinite sets of irreducible polynomials $f(x)$ that attain the value $p$ at sufficiently large $x$, each generated in finitely many steps from the $p$th cyclotomic polynomial. The base-$x$ representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.
</p>projecteuclid.org/euclid.mjms/1544151696_20181206220136Thu, 06 Dec 2018 22:01 ESTAnnouncementshttps://projecteuclid.org/euclid.mjms/1544151697<p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 30, Number 2, 218--219.</p>projecteuclid.org/euclid.mjms/1544151697_20181206220136Thu, 06 Dec 2018 22:01 ESTAn Application of Infinite Sums and Products Relating to Spectral Synthesishttps://projecteuclid.org/euclid.mjms/1559181622<strong>Melanie Henthorn-Baker</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 1--13.</p><p><strong>Abstract:</strong><br/>
The following is a discussion regarding a specific set of operators acting on the space of functions analytic on the unit disk. A diagonal operator is said to admit spectral synthesis if all of its invariant subspaces can be expressed as a closed linear span of a subset of its eigenvectors. This article employs various techniques for verifying the convergence of infinite products and infinite sums as a means of demonstrating that a certain class of operators fail to admit spectral synthesis.
</p>projecteuclid.org/euclid.mjms/1559181622_20190529220033Wed, 29 May 2019 22:00 EDTComputation of Inverse 1-Center Location Problem on the Weighted Trapezoid Graphshttps://projecteuclid.org/euclid.mjms/1559181623<strong>Biswanath Jana</strong>, <strong>Sukumar Mondal</strong>, <strong>Madhumangal Pal</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 14--35.</p><p><strong>Abstract:</strong><br/>
Let $T_{TRP}$ be the tree corresponding to the weighted trapezoid graph $G=(V,E)$. The eccentricity $e(v)$ of the vertex $v$ is defined as the sum of the weights of the vertices from $v$ to the vertex farthest from $v \in T_{TRP}$. A vertex with minimum eccentricity in the tree $T_{TRP}$ is called the 1-center of that tree. In an inverse 1-center location problem, the parameter of the tree $T_{TRP}$ corresponding to the weighted trapezoid graph $G=(V,E)$, like vertex weights, have to be modified at minimum total cost such that a pre-specified vertex $s \in V$ becomes the 1-center of the trapezoid graph $G$. In this paper, we present an optimal algorithm to find an inverse 1-center location on the weighted tree $T_{TRP}$ corresponding to the weighted trapezoid graph $G=(V,E)$, where the vertex weights can be changed within certain bounds. The time complexity of our proposed algorithm is $O(n)$, where $n$ is the number of vertices of the trapezoid graph $G$.
</p>projecteuclid.org/euclid.mjms/1559181623_20190529220033Wed, 29 May 2019 22:00 EDTShift Up-Filters and Decompositions of Up-Filters in Up-Algebrashttps://projecteuclid.org/euclid.mjms/1559181624<strong>Young Bae Jun</strong>, <strong>Aiyared Iampan</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 36--45.</p><p><strong>Abstract:</strong><br/>
The decomposition of a UP-filter is first discussed, and then the notion of a shift UP-filter is introduced, and several properties are investigated. Relations between a UP-filter, a comparative UP-filter, and a shift UP-filter are considered. Conditions for a UP-filter to be a shift UP-filter, and for a comparative UP-filter to be a shift UP-filter are provided. Characterizations of a shift UP-filter are considered, and an extension property for a shift UP-filter is established.
</p>projecteuclid.org/euclid.mjms/1559181624_20190529220033Wed, 29 May 2019 22:00 EDTDirect Proofs of the Fundamental Theorem of Calculus for the Omega Integralhttps://projecteuclid.org/euclid.mjms/1559181625<strong>C. Bryan Dawson</strong>, <strong>Matthew Dawson</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 46--55.</p><p><strong>Abstract:</strong><br/>
When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow immediately, a much more satisfying approach would be to provide direct proofs not relying on the Riemann integral. This note provides those proofs.
</p>projecteuclid.org/euclid.mjms/1559181625_20190529220033Wed, 29 May 2019 22:00 EDTCommutative Ideals of BCK-Algebras Based on Uni-Hesitant Fuzzy Set Theoryhttps://projecteuclid.org/euclid.mjms/1559181626<strong>Shuaa Aldhafeeri</strong>, <strong>G. Muhiuddin</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 56--65.</p><p><strong>Abstract:</strong><br/>
The notion of uni-hesitant fuzzy commutative ideals in BCK-algebras is introduced, and related properties are investigated. Relations between uni-hesitant fuzzy commutative ideals and uni-hesitant fuzzy ideals are discussed. Characterizations of uni-hesitant fuzzy commutative ideals are considered. Conditions for a uni-hesitant fuzzy ideal to be a uni-hesitant fuzzy commutative ideal are provided. Extension property for a uni-hesitant fuzzy commutative ideal is established.
</p>projecteuclid.org/euclid.mjms/1559181626_20190529220033Wed, 29 May 2019 22:00 EDTSums of Powers of Integershttps://projecteuclid.org/euclid.mjms/1559181627<strong>Hunde Eba</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 66--78.</p><p><strong>Abstract:</strong><br/>
We present different methods to generalize sums of powers of positive integers in terms of recurrence relations using the Taylor series, and in closed form using a finite difference method and an integral method. The result gained through the integral method is similar to Bernoulli's sum formula, but it is expressed in terms of a certain recursive sequence $H_i$.
</p>projecteuclid.org/euclid.mjms/1559181627_20190529220033Wed, 29 May 2019 22:00 EDTOn the Kauffman-Jones Polynomial for Virtual Singular Linkshttps://projecteuclid.org/euclid.mjms/1559181628<strong>Carmen Caprau</strong>, <strong>Kelsey Friesen</strong>. <p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 79--104.</p><p><strong>Abstract:</strong><br/>
We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in $\mathbb{Z}[A^2, A^{-2}, h]$. The decomposition of the resulting polynomial into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]h$ yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]A^2$, where both components are invariants for virtual singular links.
</p>projecteuclid.org/euclid.mjms/1559181628_20190529220033Wed, 29 May 2019 22:00 EDTAnnouncementshttps://projecteuclid.org/euclid.mjms/1559181629<p><strong>Source: </strong>Missouri Journal of Mathematical Sciences, Volume 31, Number 1, 105--106.</p>projecteuclid.org/euclid.mjms/1559181629_20190529220033Wed, 29 May 2019 22:00 EDT