Mathematics Commons^™

On The Zero-Divisor Graphs Of Finite Free Semilattices, Kemal Toker

Turkish Journal of Mathematics

Let $SL_{X}$ be the free semilattice on a finite nonempty set $X$. There exists an undirected graph $\Gamma(SL_{X})$ associated with $SL_{X}$ whose vertices are the proper subsets of $X$, except the empty set, and two distinct vertices $A$ and $B$ of $\Gamma(SL_{X})$ are adjacent if and only if $A\cup B=X$. In this paper, the diameter, radius, girth, degree of any vertex, domination number, independence number, clique number, chromatic number, and chromatic index of $\Gamma(SL_{X})$ have been established. Moreover, we have determined when $\Gamma(SL_{X})$ is a perfect graph and when the core of $\Gamma(SL_{X})$ is a Hamiltonian graph.

New Oscillation Tests And Some Refinements For First-Order Delay Dynamic Equations, Başak Karpuz, Özkan Öcalan

Turkish Journal of Mathematics

In this paper, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales. We also present some examples to which none of the previous results in the literature can apply.

Veronese Transform And Castelnuovo-Mumford Regularity Of Modules, Marcel Morales, Nguyen Thi Dung

Turkish Journal of Mathematics

Veronese rings, Segre embeddings, or more generally Segre--Veronese embeddings are very important rings in algebraic geometry. In this paper we present an original, elementary way to compute the Hilbert--Poincar\'e series of these rings; as a consequence we compute their Castelnuovo--Mumford regularity and also the highest graded Betti number. Moreover, using the Castelnuovo--Mumford regularity of a Cohen--Macaulay finitely generated graded module, we compute that of its Veronese transforms.

An Improved Singular Trudinger-Moser Inequality In Dimension Two, Anfeng Yuan, Zhiyong Huang

Turkish Journal of Mathematics

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain and $W_0^{1,2}(\Omega)$ be the usual Sobolev space. Let $\beta$, $0\leq\beta1$, $$\lambda_{p,\beta}(\Omega)=\inf_{u\in W_0^{1,2}(\Omega),\,u\not\equiv 0}{\ \nabla u\ _2^2}/{\ u\ _{p,\beta}^2},$$ where $\ \cdot\ _2$ denotes the standard $L^2$-norm in $\Omega$ and $\ u\ _{p,\beta}=({\int_{\Omega} x ^{-\beta} u ^pdx})^{1/p}$. Suppose that $\gamma$ satisfies $\f{\gamma}{4\pi}+\f{\beta}{2}=1$. Using a rearrangement argument, the author proves that $$\sup_{u\in W_0^{1,2}(\Omega), \ \nabla u\ _2\leq 1}\int_{\Omega} x ^{-\beta}e^{\gamma u^2 \le(1+\alpha\ u\ _{p,\beta}^2\ri) }dx$$ is finite for any $\alpha$, $0\leq\alpha

On $*$-Commuting Mappings And Derivations In Rings With Involution, Nadeem Ahmad Dar, Shakir Ali

Turkish Journal of Mathematics

Let $R$ be a ring with involution $*$. A mapping $f:R\rightarrow R$ is said to be $*$-commuting on $R$ if $[f(x),x^*]=0$ holds for all $x\in R$. The purpose of this paper is to describe the structure of a pair of additive mappings that are $*$-commuting on a semiprime ring with involution. Furthermore, we study the commutativity of prime rings with involution satisfying any one of the following conditions: (i) $[d(x),d(x^*)]=0,$ (ii) $d(x)\circ d(x^*)=0$, (iii) $d([x,x^*])\pm [x,x^*]=0$ (iv) $d(x\circ x^*)\pm (x\circ x^*)=0,$ (v) $d([x,x^*])\pm (x\circ x^*)=0$, (vi) $d(x\circ x^*)\pm [x,x^*]=0$, where $d$ is a nonzero derivation of $R$. Finally, an example …

Overall Approach To Mizoguchi--Takahashi Type Fixed Point Results, Gülhan Minak, İshak Altun

Turkish Journal of Mathematics

In this work, inspired by the recent technique of Jleli and Samet, we give a new generalization of the well-known Mizoguchi--Takahashi fixed point theorem, which is the closest answer to Reich's conjecture about the existence of fixed points of multivalued mappings on complete metric spaces. We also provide a nontrivial example showing that our result is a proper generalization of the Mizoguchi--Takahashi result.

On The Comaximal Ideal Graph Of A Commutative Ring, Mehrdad Azadi, Zeinab Jafari, Changiz Eslahchi

Turkish Journal of Mathematics

Let $R$ be a commutative ring with identity. We use $\Gamma ( R )$ to denote the comaximal ideal graph. The vertices of $\Gamma ( R )$ are proper ideals of R that are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with the planarity and perfection of $\Gamma ( R )$.

A Contribution To The Analysis Of A Reduction Algorithm For Groups With An Extraspecial Normal Subgroup, Abdullah Çağman, Nurullah Ankaralioğlu

Turkish Journal of Mathematics

Reduction algorithms are an important tool for understanding structural properties of groups. They play an important role in algorithms designed to investigate matrix groups over a finite field. One such algorithm was designed by Brooksbank et al. for members of the class $C_6$ in Aschbacher's theorem, namely groups $N$ that are normalizers in $GL(d,q)$ of certain absolutely irreducible symplectic-type $r$-groups $R$, where $r$ is a prime and $d=r^n$ with $n>2$. However, the analysis of this algorithm has only been completed when $d=r^2$ and when $d=r^n$ and $n>2$, in the latter case under the condition that $G/RZ(G)\cong N/RZ(N)$. We …

Bell-Shaped Curve For Productivity Growth: An Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent analysis of the productivity growth data shows, somewhat surprisingly, that the dependence of the 20-century productivity growth on time can be reasonably well described by a Gaussian formula. In this paper, we provide a possible theoretical explanation for this observation.

A Generalization Of The Difference Of Slopes Test To Poisson Regression With Three-Way Interaction, Melinda Bierhals

Theses, Dissertations and Capstones

Linear regression models involving interaction can use the difference of slopes test to compare slopes for various situations. We will be generalizing this process to develop a procedure to compare rates in a Poisson regression model, allowing us to consider unbounded count data as opposed to continuous data. We will apply this process to an educational data set from a sample of students located in two different Los Angeles high schools. Our model will include a three-way interaction and address the following questions:

• Does language ability impact the relationship between math ability and attendance in the same way for …

Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, Sunil Chetty

Mathematics Faculty Publications

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than ℤ. We then study the growth of the p^∞- Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers k ⊂ K ⊂ F in which [F : K] is not a p-power extension.

Sphere Representations, Stacked Polytopes, And The Colin De Verdière Number Of A Graph, Lon Mitchell, Lynne Yengulalp

Mathematics Faculty Publications

We prove that a k-tree can be viewed as a subgraph of a special type of (k + 1)- tree that corresponds to a stacked polytope and that these “stacked” (k + 1)-trees admit representations by orthogonal spheres in R k+1. As a result, we derive lower bounds for Colin de Verdi`ere’s µ of complements of partial k-trees and prove that µ(G) + µ(G) > |G| − 2 for all chordal G.

Serre Weights And Wild Ramification In Two-Dimensional Galois Representations, Lassina Dembélé, Fred Diamond, David P. Roberts

Mathematics Publications

A generalization of Serre’s Conjecture asserts that if F is a totally real field, then certain characteristic p representations of Galois groups over F arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over p. This characterization of the weights, which is formulated using p-adic Hodge theory, is known under mild technical hypotheses if p > 2. In this paper we give, under the assumption that p is unramified in F, a conjectural alternative description for the set of weights. …

Generalized Eulerian Numbers And Multiplex Juggling Sequences, Esther M. Banaian

All College Thesis Program, 2016-2019

We consider generalizations of both juggling sequences and non-attacking rook placements. We demonstrate the important connection between these objects, and also propose a generalization of the Eulerian numbers. These generalizations give rise to several interesting counting problems, which we explore.

On Coprimely Structured Rings, Nesli̇han Ayşen Özki̇ri̇şci̇, Kürşat Hakan Oral, Ünsal Teki̇r

Turkish Journal of Mathematics

In this paper, we define coprimely structured rings, which are the generalization of strongly 0-dimensional rings. Furthermore, we investigate coprimely structured rings and give some relations between other rings such as Artinian rings, strongly 0-dimensional rings, and h-local domains.

Almost Co-K\"{A}Hler Manifolds Satisfying Some Symmetry Conditions, Yaning Wang

Turkish Journal of Mathematics

Let $M^{2n+1}$ be an almost co-K\"{a}hler manifold of dimension $>3$ with K\"{a}hlerian leaves. In this paper, we first prove that if $M^{2n+1}$ is locally symmetric, then either it is a co-K\"{a}hler manifold with locally symmetric K\"{a}hlerian leaves, or the Reeb vector field $\xi$ is harmonic and in this case $M^{2n+1}$ is non-co-K\"{a}hler. We also prove that any almost co-K\"{a}hler manifold of dimension $3$ is $\phi$-symmetric if and only if it is locally isometric to either a flat Euclidean space $\mathbb{R}^3$ or a Riemannian product $\mathbb{R}\times N^2(c)$, where $N^2(c)$ denotes a K\"{a}hler surface of constant curvature $c\neq0$.

An Exploration Of “The Enumeration Of Heterofullerenes”, Kayla Penkava

Conspectus Borealis

No abstract provided.

Visual Properties Of Generalized Kloosterman Sums, Paula Burkhardt '16, Alice Zhuo-Yu Chan '14, Gabriel Currier '16, Stephan Ramon Garcia, Florian Luca, Hong Suh '16

Pomona Faculty Publications and Research

For a positive integer m and a subgroup A of the unit group (Z/mZ)^x, the corresponding generalized Kloosterman sum is the function K(a, b, m, A) = Σ_uEA e(au+bu^-1/m). Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.

Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

Pomona Faculty Publications and Research

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of …

Bernstein’S Lethargy Theorem In Fréchet Spaces, Asuman Güven Aksoy, Grzegorz Lewicki

CMC Faculty Publications and Research

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {V_n} be a nested sequence of subspaces of X such that V_n ⊆ V_n+1 for any n ∈ N and X = S∞ _n=1 V_n. Let e_n be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, V_n)}, we prove that there exists x ∈ X and no ∈ N such that

e_n/3 ≤ dist(x, V …

Analysis Of Optimal Error Estimates And Superconvergence Of The Discontinuous Galerkin Method For Convection-Diffusion Problems In One Space Dimension, Mahboub Baccouch, Helmi Temimi

Mathematics Faculty Publications

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(h^p+1) and O(h^p) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h^2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG …

An Extension Of The Compression-Expansion Fixed Point Theorem Of Functional Type, Richard I. Avery, Douglas R. Anderson, Johnny Henderson

Research & Publications

In this article we use an interval of functional type as the underlying set in our compression-expansion fixed point theorem argument which can be used to exploit properties of the operator to improve conditions that will guarantee the existence of a fixed point in applications. An example is provided to demonstrate how intervals of functional type can improve conditions in applications to boundary value problems. We also show how one can use suitable k-contractive conditions to prove that a fixed point in a functionaltype interval is unique.

Adjoint Fuzzy Partition And Generalized Sampling Theorem, Irina Perfilieva, Michal Holčapek, Vladik Kreinovich

Departmental Technical Reports (CS)

A new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its F-transform components is analyzed. An analogy with the Nyquist-Shannon-Kotelnikov sampling theorem is discussed.

Bayesian Exponential Random Graph Models With Nodal Random Effects, Stephanie Thiemichen, Nial Friel, Alberto Caimo, Goeran Kauermann

Articles

We extend the well-known and widely used Exponential Random Graph Model (ERGM) by including nodal random effects to compensate for heterogeneity in the nodes of a network. The Bayesian framework for ERGMs proposed by Caimo and Friel (2011) yields the basis of our modelling algorithm. A central question in network models is the question of model selection and following the Bayesian paradigm we focus on estimating Bayes factors. To do so we develop an approximate but feasible calculation of the Bayes factor which allows one to pursue model selection. Two data examples and a small simulation study illustrate our mixed …

Pandey's Method Of Cube Root Extraction: Is It Better Than Aryabhata’S Method?, Deepak Basyal

Mathematics and Statistics

We compare two methods of cube root extraction: one proposed by the Nepali mathematician Gopal Pandey in the 19th century, which uses proportionality, and another one provided by the Indian mathematician and astronomer Aryabhata.

Robustness As A Criterion For Selecting A Probability Distribution Under Uncertainty, Songsak Sriboonchitta, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Often, we only have partial knowledge about a probability distribution, and we would like to select a single probability distribution $\rho(x)$ out of all probability distributions which are consistent with the available knowledge. One way to make this selection is to take into account that usually, the values $x$ of the corresponding quantity are also known only with some accuracy. It is therefore desirable to select a distribution which is the most robust -- in the sense the x-inaccuracy leads to the smallest possible inaccuracy in the resulting probabilities. In this paper, we describe the corresponding most robust probability distributions, …

Why Dependence Of Productivity On Group Size Is Log-Normal, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical analysis shows that, on average, the productivity of a group log-normally depends on its size. The current explanations for this empirical fact are based on reasonably complex assumptions about the human behavior. In this paper, we show that the same conclusion can be made in effect, from first principles, without making these complex assumptions.

Why Locating Local Optima Is Sometimes More Complicated Than Locating Global Ones, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most applications, practitioners are interested in locating global optima. In such applications, local optima that result from some optimization algorithms are an unnecessary side effect. In other words, in such applications, locating global optima is a much more computationally complex problem than locating local optima. In several practical applications, however, local optima themselves are of interest. Somewhat surprisingly, it turned out that in many such applications, locating all local optima is a much more computationally complex problem than locating all global optima. In this paper, we provide a theoretical explanation for this surprising empirical phenomenon.

Constrained Adaptive Sensing, Mark A. Davenport, Andrew K. Massimino, Deanna Needell, Tina Woolf

CMC Faculty Publications and Research

Suppose that we wish to estimate a vector x∈Cn from a small number of noisy linear measurements of the form y=Ax+z, where z represents measurement noise. When the vector x is sparse, meaning that it has only s nonzeros with s≪n, one can obtain a significantly more accurate estimate of x by adaptively selecting the rows of A based on the previous measurements provided that the signal-to-noise ratio (SNR) is sufficiently large. In this paper we consider the case where we wish to realize the potential of adaptivity but where the rows of A are subject to physical constraints. In …

Follower And Extender Sets In Symbolic Dynamics, Thomas Kelly French

Electronic Theses and Dissertations

Given a word w in the language of a one-dimensional shift space X, the follower set of w, denoted F_X(w), is the set of all right-infinite sequences which follow w in some point of X. Extender sets are a generalization of follower sets and are defined similarly. To a given shift space X, then, we may associate a follower set sequence {|F_X(n)|} which records the number of distinct follower sets in X corresponding to words of length n. Similarly, we may define an extender set sequence {|E …