Rocky Mountain Journal of Mathematics Articles (Project Euclid)
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The latest articles from Rocky Mountain Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 02 May 2011 10:11 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Rings Over which All Modules are Strongly Gorenstein Projective
http://projecteuclid.org/euclid.rmjm/1277385512
<strong>Driss Bennis</strong>, <strong>Najib Mahdoua</strong>, <strong>Khalid Ouarghi</strong><p><strong>Source: </strong>Rocky Mountain J. Math., Volume 40, Number 3, 749--759.</p>projecteuclid.org/euclid.rmjm/1277385512_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTInterpolation mixing hyperbolic functions and polynomialshttps://projecteuclid.org/euclid.rmjm/1528077625<strong>J.M. Carnicer</strong>, <strong>E. Mainar</strong>, <strong>J.M. Peña</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 443--461.</p><p><strong>Abstract:</strong><br/>
Exponential polynomials as solutions of differential equations with constant coefficients are widely used for approximation purposes. Recently, mixed spaces containing algebraic, trigonometric and exponential functions have been extensively considered for design purposes. The analysis of these spaces leads to constructions that can be reduced to Hermite interpolation problems. In this paper, we focus on spaces generated by algebraic polynomials, hyperbolic sine and hyperbolic cosine. We present classical interpolation formulae, such as Newton and Aitken-Neville formulae and a suggestion of implementation. We explore another technique, expressing the Hermite interpolant in terms of polynomial interpolants and derive practical error bounds for the hyperbolic interpolant.
</p>projecteuclid.org/euclid.rmjm/1528077625_20180603220043Sun, 03 Jun 2018 22:00 EDTAlmost compatible functions and infinite length gameshttps://projecteuclid.org/euclid.rmjm/1528077627<strong>Steven Clontz</strong>, <strong>Alan Dow</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 463--483.</p><p><strong>Abstract:</strong><br/>
${\mathcal{A}}'(\kappa)$ asserts the existence of pairwise almost compatible finite-to-one functions $A\to \omega$ for each countable subset $A$ of $\kappa$. The existence of winning $2$-Markov strategies in several infinite-length games, including the Menger game on the one-point Lindelofication $\kappa^\dagger$ of $\kappa$, are guaranteed by ${\mathcal{A}}'(\kappa)$. ${\mathcal{A}}'(\kappa)$ is implied by the existence of cofinal Kurepa families of size $\kappa$, and thus, holds for all cardinals less than $\aleph _\omega$. It is consistent that ${\mathcal{A}}'({\aleph _\omega })$ fails; however, there must always be a winning $2$-Markov strategy for the second player in the Menger game on $\omega_\omega^\dagger$.
</p>projecteuclid.org/euclid.rmjm/1528077627_20180603220043Sun, 03 Jun 2018 22:00 EDTSymmetry and monotonicity of solutions for equations involving the fractional Laplacian of higher orderhttps://projecteuclid.org/euclid.rmjm/1528077628<strong>Xuewei Cui</strong>, <strong>Weijie Song</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 485--499.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish symmetry and monotonicity of solutions to the equation involving fractional Laplacians of higher order. For this purpose, we first reduce the equation into a system via the composition of lower fractional Laplacians and then obtain symmetry and monotonicity of solutions to the system by applying the method of moving planes.
</p>projecteuclid.org/euclid.rmjm/1528077628_20180603220043Sun, 03 Jun 2018 22:00 EDTOn the algebra of WCE operatorshttps://projecteuclid.org/euclid.rmjm/1528077630<strong>Yousef Estaremi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 501--517.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the algebra of WCE operators on $L^p$-spaces, and we investigate some al\-ge\-braic properties of it. For instance, we show that the set of normal WCE operators is a unital finite Von Neumann algebra, and we obtain the spectral measure of a normal WCE operator on $L^2(\mathcal {F})$. Then, we specify the form of projections in the Von Neumann algebra of normal WCE operators, and we obtain that, if the underlying measure space is purely atomic, then all projections are minimal. In the non-atomic case, there is no minimal projection. Also, we give a non-commutative operator algebra on which the spectral map is subadditive and submultiplicative. As a consequence, we obtain that the set of quasinilpotents is an ideal, and we get a relation between quasinilpotents and commutators. Moreover, we give some sufficient conditions for an algebra of WCE operators to be triangularizable, and consequently, that its quotient space over its quasinilpotents is commutative.
</p>projecteuclid.org/euclid.rmjm/1528077630_20180603220043Sun, 03 Jun 2018 22:00 EDTSolutions for second order nonlocal BVPs via the generalized Miranda theoremhttps://projecteuclid.org/euclid.rmjm/1528077631<strong>Mateusz Krukowski</strong>, <strong>Katarzyna Szymańska-Debowska</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 519--528.</p><p><strong>Abstract:</strong><br/>
In this paper, the generalized Miranda theorem is applied for second-order systems of differential equations with one boundary condition given by Riemann-Stieltjes integral \[ x'' = f(t,x,x'), \quad x(0) = 0, \ x'(1) = \int _0^1 x(s) \, dg(s),\] where $f : [0,1]\times \mathbb{R} ^k\times \mathbb{R} ^k \to \mathbb{R} ^k$ is continuous and $g : [0,1] \to \mathbb{R} ^k$ has bounded variation. Under suitable assumptions upon $f$ and $g$ we prove the existence of solutions to such posed problem.
</p>projecteuclid.org/euclid.rmjm/1528077631_20180603220043Sun, 03 Jun 2018 22:00 EDTCharacterization of a two-parameter matrix valued BMO by commutator with the Hilbert transformhttps://projecteuclid.org/euclid.rmjm/1528077632<strong>Dario Mena</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 529--550.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove that the space of two parameter matrix-valued BMO functions can be char\-ac\-terized by considering iterated commutators with the Hilbert transform. Specifically, we prove that $$ \| B \|_{BMO} \!\lesssim \! \| [[M_B, H_1],H_2] \|_{L^2(\mathbb{R} ^2;\mathbb{C} ^d) \rightarrow L^2(\mathbb{R} ^2;\mathbb{C} ^d)} \!\lesssim \! \| B \|_{BMO}. $$ The upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.
</p>projecteuclid.org/euclid.rmjm/1528077632_20180603220043Sun, 03 Jun 2018 22:00 EDTAsymptotic behavior of integral closures, quintasymptotic primes and ideal topologieshttps://projecteuclid.org/euclid.rmjm/1528077633<strong>Reza Naghipour</strong>, <strong>Peter Schenzel</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 551--572.</p><p><strong>Abstract:</strong><br/>
Let $R$ be a Noetherian ring, $N$ a finitely generated $R$-module and $I$ an ideal of $R$. It is shown that the sequences $Ass _R R/(I^n)_a^{(N)}$, $Ass _R (I^n)_a^{(N)}/ (I^{n+1})^{(N)}_a$ and $Ass _R (I^n)_a^{(N)}/ (I^n)_a$, $n= 1,2, \ldots $, of associated prime ideals, are increasing and ultimately constant for large $n$. Moreover, it is shown that, if $S$ is a multiplicatively closed subset of $R$, then the topologies defined by $(I^n)_a^{(N)}$ and $S((I^n)_a^{(N)})$, $n\geq 1$, are equivalent if and only if $S$ is disjoint from the quintasymptotic primes of $I$. By using this, we also show that, if $(R, \mathfrak {m})$ is local and $N$ is quasi-unmixed, then the local cohomology module $H^{\dim N}_I(N)$ vanishes if and only if there exists a multiplicatively closed subset $S$ of $R$ such that $\mathfrak {m} \cap S \neq \emptyset $ and the topologies induced by $(I^n)_a^{(N)}$ and $S((I^n)_a^{(N)})$, $n\geq 1$, are equivalent.
</p>projecteuclid.org/euclid.rmjm/1528077633_20180603220043Sun, 03 Jun 2018 22:00 EDTCounting all self-avoiding walks on a finite lattice strip of width one and twohttps://projecteuclid.org/euclid.rmjm/1528077634<strong>M.A. Nyblom</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 573--605.</p><p><strong>Abstract:</strong><br/>
In this paper, a closed-form expression for counting all SAWs, irrespective of length, but restricted to the finite lattice strip $\{ -a,\ldots ,0,\ldots ,b\}\times \{0,1\}$, shall be obtained in terms of the non-negative integer parameters $a$ and $b$. In addition, the argument used to prove this result will be extended to establish an enumerating formula for counting all SAWs, irrespective of length, but restricted to the half-finite lattice strip of width two $\{ 0,1,\ldots ,n\}\times \{ 0,1,2\}$, in terms of $n$.
</p>projecteuclid.org/euclid.rmjm/1528077634_20180603220043Sun, 03 Jun 2018 22:00 EDTNew real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz typehttps://projecteuclid.org/euclid.rmjm/1528077635<strong>Chunyan Qi</strong>, <strong>Hui Zhang</strong>, <strong>Baode Li</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 607--637.</p><p><strong>Abstract:</strong><br/>
A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda $ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.
</p>projecteuclid.org/euclid.rmjm/1528077635_20180603220043Sun, 03 Jun 2018 22:00 EDTUniformly non-square points and representation of functionals of Orlicz-Bochner sequence spaceshttps://projecteuclid.org/euclid.rmjm/1528077636<strong>Zhongrui Shi</strong>, <strong>Yu Wang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 639--660.</p><p><strong>Abstract:</strong><br/>
In this work, a representation of functionals and a necessary and sufficient condition for uniformly non-square points of Orlicz-Bochner sequence spaces endowed with the Orlicz norm are given.
</p>projecteuclid.org/euclid.rmjm/1528077636_20180603220043Sun, 03 Jun 2018 22:00 EDTOn generalized weaving frames in Hilbert spaceshttps://projecteuclid.org/euclid.rmjm/1528077637<strong>Lalit K. Vashisht</strong>, <strong>Saakshi Garg</strong>, <strong> Deepshikha</strong>, <strong>P.K. Das</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 661--685.</p><p><strong>Abstract:</strong><br/>
Generalized frames (in short, $g$-frames) are a natural generalization of standard frames in separable Hilbert spaces. Motivated by the concept of weaving frames in separable Hilbert spaces by Bemrose, Casazza, Grochenig, Lammers and Lynch in the context of distributed signal processing, we study weaving properties of $g$-frames. Firstly, we present necessary and sufficient con\-ditions for weaving $g$-frames in Hilbert spaces. We extend some results of \cite Bemrose, Casazza, Grochenig, Lammers and Lynch, and Casazza and Lynch regarding conversion of standard weaving frames to $g$-weaving frames. Some Paley-Wiener type perturbation results for weaving $g$-frames are obtained. Finally, we give necessary and sufficient conditions for weaving $g$-Riesz bases.
</p>projecteuclid.org/euclid.rmjm/1528077637_20180603220043Sun, 03 Jun 2018 22:00 EDTHomological properties of the algebra of compact operators on a Banach spacehttps://projecteuclid.org/euclid.rmjm/1528077638<strong>G.A. Willis</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 2, 687--701.</p><p><strong>Abstract:</strong><br/>
The conditions on a Banach space $E$ under which the algebra $\mathcal {K}(E)$ of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra, and, it is shown that, for $\mathcal {K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces $E$ such that $\mathcal {K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.
</p>projecteuclid.org/euclid.rmjm/1528077638_20180603220043Sun, 03 Jun 2018 22:00 EDTTensor products and endomorphism rings of finite valuated groupshttps://projecteuclid.org/euclid.rmjm/1533230821<strong>Ulrich Albrecht</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 703--727.</p><p><strong>Abstract:</strong><br/>
This paper discusses homological properties of a finite valuated $p$-group $A$. A category equivalence between full subcategories of the category of valuated $p$-groups and the category of right modules over the endomorphism ring of $A$ is developed to study $A$-presented and $A$-valuated valuated $p$-groups. In particular, we show that these classes do not coincide if $|A/pA| \gt p$. Examples are given throughout the paper.
</p>projecteuclid.org/euclid.rmjm/1533230821_20180802132720Thu, 02 Aug 2018 13:27 EDTReduction graph and its application on algebraic graphshttps://projecteuclid.org/euclid.rmjm/1533230822<strong>Saeid Bagheri</strong>, <strong>Fatemeh Nabaei</strong>, <strong>Rashid Rezaeii</strong>, <strong>Karim Samei</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 729--751.</p><p><strong>Abstract:</strong><br/>
Evans, et al. (J. Graph Theory 18 (1994), 801--815), defined an equivalence relation $\sim $ on the set of vertices of a simple graph $G$ by taking $a\sim b$ if and only if their open neighborhoods are the same. They introduced a new graph $G_{red }={G}/{\sim } $, reduction graph of $G$, as follows. The vertices are $V (G_{red })=\{[a]: a \in V (G) \}$, and two equivalence classes $[a]$ and $[b]$ are adjacent if and only if $a$ and $b$ are adjacent in $G$. Recently, Anderson and LaGrange defined some equivalence relations on the set of vertices of the zero-divisor graph of a commutative ring, one of which yields the reduction graph of the zero-divisor graph. In this paper, we state some basic graph theoretic properties of $G_{red }$ and study the relations between some properties of graph $G$ and its subgraph, $G_{red }$, such as the chromatic number, clique number, girth and diameter. Moreover, we study the reduction graph of some algebraic graphs, such as the comaximal graph, zero-divisor graph and Cayley graph of a commutative ring. Among other results, we show that, for every commutative ring $R$, $\Gamma _2(R)_{red } \simeq \Gamma _1(\mathbb {Z}_2^n)$, where $\Gamma _1(\mathbb {Z}_2^n)$ is the zero-divisor graph of the Boolean ring $\mathbb {Z}_2^n$, $\Gamma _2(R)$ is the comaximal graph of $R$ and $n=|Max (R)|$.
</p>projecteuclid.org/euclid.rmjm/1533230822_20180802132720Thu, 02 Aug 2018 13:27 EDTPeriodic solution for second order damped differential equations with attractive-repulsive singularitieshttps://projecteuclid.org/euclid.rmjm/1533230823<strong>Zhibo Cheng</strong>, <strong>Jingli Ren</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 753--768.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate a kind of second-order nonlinear differential equation with attractive-repulsive singularities. By applications of Green's function and Schauder's fixed point theorem, we establish the existence of a positive periodic solution for this equation.
</p>projecteuclid.org/euclid.rmjm/1533230823_20180802132720Thu, 02 Aug 2018 13:27 EDTLefschetz properties of balanced 3-polytopeshttps://projecteuclid.org/euclid.rmjm/1533230824<strong>David Cook II</strong>, <strong>Martina Juhnke-Kubitzke</strong>, <strong>Satoshi Murai</strong>, <strong>Eran Nevo</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 769--790.</p><p><strong>Abstract:</strong><br/>
In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.
</p>projecteuclid.org/euclid.rmjm/1533230824_20180802132720Thu, 02 Aug 2018 13:27 EDTSemigroup asymptotics, the Funk-Hecke identity and the Gegenbauer coefficients associated with the spherical Laplacianhttps://projecteuclid.org/euclid.rmjm/1533230825<strong>Stuart Day</strong>, <strong>Ali Taheri</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 791--817.</p><p><strong>Abstract:</strong><br/>
A trace formulation of the Maclaurin spectral coefficients of the Schwartzian kernel of functions of the spherical Laplacian is given. A class of polynomials $\mathscr {P}^\nu _l(X)$ $(l \ge 0$, $\nu \gt -1/2)$ linking to the classical Gegenbauer polynomials through a differential-spectral identity is introduced, and its connection to the above spectral coefficients and their asymptotics analyzed. The paper discusses some applications of these ideas combined with the Funk-Hecke identity and semigroup techniques to geometric and variational-energy inequalities on the sphere and presents some examples.
</p>projecteuclid.org/euclid.rmjm/1533230825_20180802132720Thu, 02 Aug 2018 13:27 EDTA note on sums of rootshttps://projecteuclid.org/euclid.rmjm/1533230826<strong>Timothy Ferdinands</strong>, <strong>Annette Pilkington</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 819--829.</p><p><strong>Abstract:</strong><br/>
In this paper, we look at properties of roots which can be written as sums of roots in crystallographic root systems. We derive properties of the poset associated to such a sum consisting of the subsums which are themselves roots.
</p>projecteuclid.org/euclid.rmjm/1533230826_20180802132720Thu, 02 Aug 2018 13:27 EDTExistence of solutions for a nonlocal fractional boundary value problemhttps://projecteuclid.org/euclid.rmjm/1533230827<strong>Zhen Gao</strong>, <strong>Min Wang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 831--843.</p><p><strong>Abstract:</strong><br/>
We study a nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed as a series of functions by the perturbation approach. Criteria for the existence of solutions are obtained based upon it.
</p>projecteuclid.org/euclid.rmjm/1533230827_20180802132720Thu, 02 Aug 2018 13:27 EDTOn the periodic solutions of some systems of higher order difference equationshttps://projecteuclid.org/euclid.rmjm/1533230828<strong>Melih Gocen</strong>, <strong>Adem Cebeci</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 845--858.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain the general form of the periodic solutions of some higher order difference equations system \[x_{n+1}=\frac {\pm x_{n-k}y_{n-(2k+1)}}{y_{n-(2k+1)}\mp y_{n-k}},\] \[y_{n+1}=\frac {\pm y_{n-k}x_{n-(2k+1)}}{x_{n-(2k+1)}\mp x_{n-k}},\] $n,k\in \mathbb {N}_{0}$, where the initial values are arbitrary real numbers such that the denominator is always nonzero. Moreover, some numerical examples are presented to verify our theoretical results.
</p>projecteuclid.org/euclid.rmjm/1533230828_20180802132720Thu, 02 Aug 2018 13:27 EDTNonlinear spectral radius preservers between certain non-unital Banach function algebrashttps://projecteuclid.org/euclid.rmjm/1533230829<strong>Maliheh Hosseini</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 859--884.</p><p><strong>Abstract:</strong><br/>
Let $\alpha _0\in \mathbb {C} \setminus \{0\}$, $A$ and $B$ be Banach function algebras. Also, let $\rho _1:\Omega _1 \rightarrow A$, $\rho _2:\Omega _2 \rightarrow A$, $\tau _1: \Omega _1 \rightarrow B$ and $\tau _2:\Omega _2 \rightarrow B$ be surjections such that $\|\rho _1(\omega _1)\rho _2(\omega _2)+\alpha _0\|_\infty =\|\tau _1(\omega _1)\tau _2(\omega _2)+\alpha _0\|_\infty $ for all $\omega _1\in \Omega _1, \omega _2\in \Omega _2$, where $\Omega _1$, $\Omega _2$ are two non-empty sets. Motivated by recent investigations on such maps between unital Banach function algebras, in this paper we characterize these maps for certain non-unital Banach function algebras including pointed Lipschitz algebras and abstract Segal algebras of the Talamanca-Herz algebras when the underlying groups are first countable. Moreover, sufficient conditions are given to guarantee such maps induce weighted composition operators.
</p>projecteuclid.org/euclid.rmjm/1533230829_20180802132720Thu, 02 Aug 2018 13:27 EDTSome results on a class of mixed van der Waerden numbershttps://projecteuclid.org/euclid.rmjm/1533230830<strong>Kaushik Maran</strong>, <strong>Sai Praneeth Reddy</strong>, <strong>Dravyansh Sharma</strong>, <strong>Amitabha Tripathi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 885--904.</p><p><strong>Abstract:</strong><br/>
The mixed van der Waerden number ${w}(k_1, \ldots ,k_r;r)$ is the least positive integer $n$ such that every $r$-coloring of $[1,n]$ admits a monochromatic arithmetic progression of length $k_i$, for at least one $i$. We denote by ${w}_2(k;r)$ the case in which $k_1=\cdots =k_{r-1}=2$ and $k_r=k$. For $k \le r$, we give upper and lower bounds for ${w}_2(k;r)$, also indicating cases when these bounds are achieved. We determine exact values in the cases where $(k,r) \in \{(p,p),(p,p+1),(p+1,p+1)\}$ and give bounds in the cases where $(k,r) \in \{(p,p+2),(p+2,p+2)\}$, for primes $p$. We provide a table of values for the cases $k \le r$ with $3 \le k \le 10$ and for several values of $r$, correcting some known values.
</p>projecteuclid.org/euclid.rmjm/1533230830_20180802132720Thu, 02 Aug 2018 13:27 EDTInvariant means and property $T$ of crossed productshttps://projecteuclid.org/euclid.rmjm/1533230831<strong>Qing Meng</strong>, <strong>Chi-Keung Ng</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 905--912.</p><p><strong>Abstract:</strong><br/>
Let $\Gamma $ be a discrete group that acts on a semi-finite measure space $(\Omega , \mu )$ such that there is no $\Gamma $-invariant function in $L^1(\Omega , \mu )$. We show that the absence of the $\Gamma $-invariant mean on $L^\infty (\Omega ,\mu )$ is equivalent to the property $T$ of the reduced $C^*$-crossed product of $L^\infty (\Omega ,\mu )$ by $\Gamma $. In particular, if $\Lambda $ is a countable group acting ergodically on an infinite $\sigma $-finite measure space $(\Omega , \mu )$, then there exists a $\Lambda $-invariant mean on $L^\infty (\Omega , \mu )$ if and only if the corresponding crossed product does not have property $T$. Moreover, if $\Gamma $ is an ICC group, then $\Gamma $ is inner amenable if and only if $\ell ^\infty (\Gamma \setminus \{e\})\rtimes _{\mathbf {i},r} \Gamma $ does not have property $T$, where $\mathbf {i}$ is the conjugate action. On the other hand, a non-compact locally compact group $G$ is amenable if and only if $L^\infty (G)\rtimes _{\mathbf {lt}, r} G_\mathrm {d}$ does not have property $T$, where $G_\mathrm {d}$ is the group $G$ equipped with the discrete topology and $\mathbf {lt}$ is the left translation.
</p>projecteuclid.org/euclid.rmjm/1533230831_20180802132720Thu, 02 Aug 2018 13:27 EDTOn the planarity of cyclic graphshttps://projecteuclid.org/euclid.rmjm/1533230832<strong>Mohammad Naghshinehfard</strong>, <strong>Foroud Parvaneh</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 913--926.</p><p><strong>Abstract:</strong><br/>
We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases).
</p>projecteuclid.org/euclid.rmjm/1533230832_20180802132720Thu, 02 Aug 2018 13:27 EDTThe fibering map approach to a $p(x)$-Laplacian equation with singular nonlinearities and nonlinear Neumann boundary conditionshttps://projecteuclid.org/euclid.rmjm/1533230833<strong>Kamel Saoudi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 927--946.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to study the singular Neumann problem involving the p$(x)$-Laplace operator: \begin{equation} (P_\lambda)\qquad\begin{cases} - \Delta_{p(x)} u +|u|^{p(x)-2}u = \frac{\lambda a(x)}{u^{\delta(x)}} &\mbox{in }\Omega, \\ u\gt0 &\mbox{in } \Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu} = b(x) u^{q(x)-2}u &\mbox{on } \partial\Omega, \end{cases} \end{equation} where $\Omega\subset\mathbb{R}^N$, $N\geq 2$, is a bounded domain with $C^2$ boundary, $\lambda$ is a positive parameter, $a, b\in C(\overline{\Omega})$ are non-negative weight functions with compact support in $\Omega$ and $\delta(x),$ $p(x),$ $q(x) \in C(\overline{\Omega})$ are assumed to satisfy the assumptions (A0)--(A1) in Section 1. We employ the Nehari manifold approach and some variational techniques in order to show the multiplicity of positive solutions for the $p(x)$-Laplacian singular problems.
</p>projecteuclid.org/euclid.rmjm/1533230833_20180802132720Thu, 02 Aug 2018 13:27 EDTOn the structure of $S_2$-ifications of complete local ringshttps://projecteuclid.org/euclid.rmjm/1533230834<strong>Sean Sather-Wagstaff</strong>, <strong>Sandra Spiroff</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 947--965.</p><p><strong>Abstract:</strong><br/>
Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $Spec (R)-V(J)$, and the (finite simple) graph $\Gamma _R$ with vertex set $Min (R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of connected components of $\Gamma _R$ is equal to the maximum number of connected components of $Spec (R)-V(J)$ for all $J$ of height $2$. We further investigate this graph by exhibiting a technique for showing that certain graphs $G$ can be realized in the form $\Gamma _R$.
</p>projecteuclid.org/euclid.rmjm/1533230834_20180802132720Thu, 02 Aug 2018 13:27 EDTOn the parametric representation of univalent functions on the polydischttps://projecteuclid.org/euclid.rmjm/1533230835<strong>Sebastian Schleißinger</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 981--1001.</p><p><strong>Abstract:</strong><br/>
We consider support points of the class $S^0(\mathbb{D} ^n)$ of normalized univalent mappings on the polydisc $\mathbb{D} ^n$ with parametric representation, and we prove sharp estimates for coefficients of degree 2.
</p>projecteuclid.org/euclid.rmjm/1533230835_20180802132720Thu, 02 Aug 2018 13:27 EDTA class of Frobenius-type Eulerian polynomialshttps://projecteuclid.org/euclid.rmjm/1533230836<strong>H.M. Srivastava</strong>, <strong>M.A. Boutiche</strong>, <strong>M. Rahmani</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 1003--1013.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to prove several explicit formulas associated with the Frobenius-type Eulerian polynomials in terms of the weighted Stirling numbers of the second kind. As a consequence, we derive an explicit formula for the tangent numbers of higher order. We also give a recursive method for the calculation of the Frobenius-type Eulerian numbers and polynomials.
</p>projecteuclid.org/euclid.rmjm/1533230836_20180802132720Thu, 02 Aug 2018 13:27 EDTCyclicity of the left regular representation of a locally compact grouphttps://projecteuclid.org/euclid.rmjm/1533230837<strong>Zsolt Tanko</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 1015--1018.</p><p><strong>Abstract:</strong><br/>
We combine harmonic analysis and operator algebraic techniques to give a concise argument that the left regular representation of a locally compact group is cyclic if and only if the group is first countable, a result first proved by Greenleaf and Moskowitz.
</p>projecteuclid.org/euclid.rmjm/1533230837_20180802132720Thu, 02 Aug 2018 13:27 EDTOrthogonal rational functions on the extended real line and analytic on the upper half planehttps://projecteuclid.org/euclid.rmjm/1533230838<strong>Xu Xu</strong>, <strong>Laiyi Zhu</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 1019--1030.</p><p><strong>Abstract:</strong><br/>
Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.
</p>projecteuclid.org/euclid.rmjm/1533230838_20180802132720Thu, 02 Aug 2018 13:27 EDTThe log-convexity of $r$-derangement numbershttps://projecteuclid.org/euclid.rmjm/1533230839<strong>Feng-Zhen Zhao</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 1031--1042.</p><p><strong>Abstract:</strong><br/>
This paper focuses on the log-convexity of the sequence $\{D_r(n)\}_{n\ge r}$ of $r$-derangement numbers, where $r\ge 2$ is a positive integer. We mainly prove that $\{D_2(n)\}_{n\ge 2}$ and $\{D_3(n)\}_{n\ge 7}$ are log-convex. In addition, we also show that $\{\sqrt {D_2(n)}\}_{n\ge 2}$ and $\{\sqrt [3]{D_3(n)}\}_{n\ge 7}$ are log-balanced.
</p>projecteuclid.org/euclid.rmjm/1533230839_20180802132720Thu, 02 Aug 2018 13:27 EDTCrossing changes, Delta moves and sharp moves on welded knotshttps://projecteuclid.org/euclid.rmjm/1533668963<strong>Shin Satoh</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 3, 967--879.</p><p><strong>Abstract:</strong><br/>
We prove that the crossing changes, Delta moves and sharp moves are unknotting operations on welded knots.
</p>projecteuclid.org/euclid.rmjm/1533668963_20180807150943Tue, 07 Aug 2018 15:09 EDTNonlocal initial value problems for Hadamard-type fractional differential equations and inclusionshttps://projecteuclid.org/euclid.rmjm/1538272822<strong>Bashir Ahmad</strong>, <strong>Sotiris K. Ntouyas</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1043--1068.</p><p><strong>Abstract:</strong><br/>
In this paper, we study initial value problems of fractional differential equations and inclusions of Hadamard type, supplemented with nonlocal conditions. Some new existence and uniqueness results are obtained by using fixed point theorems for single valued and multi-valued maps. Examples illustrating the main results are also presented.
</p>projecteuclid.org/euclid.rmjm/1538272822_20180929220041Sat, 29 Sep 2018 22:00 EDTA generalization of the Bocher-Grace theoremhttps://projecteuclid.org/euclid.rmjm/1538272823<strong>John Clifford</strong>, <strong>Michael Lachance</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1069--1076.</p><p><strong>Abstract:</strong><br/>
The Bocher-Grace theorem can be stated as follows: let $p$ be a third degree complex polynomial. Then, there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of $p$, and the foci of the ellipse are the critical points of $p$. Here, we prove the following generalization: let $p$ be an $n$th degree complex polynomial, and let its critical points take the form \[ \alpha +\beta \cos k\pi /n,\quad k=1,\ldots ,n-1,\ \beta \ne 0. \] Then, there is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of $p$, and the foci of this ellipse are the two most extreme critical points of $p$: $\alpha \pm \beta \cos \pi /n$.
</p>projecteuclid.org/euclid.rmjm/1538272823_20180929220041Sat, 29 Sep 2018 22:00 EDTEigenvalues for systems of fractional $p$-Laplacianshttps://projecteuclid.org/euclid.rmjm/1538272824<strong>Leandro M. Del Pezzo</strong>, <strong>Julio D. Rossi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1077--1104.</p><p><strong>Abstract:</strong><br/>
We study the eigenvalue problem for a system of fractional $p$-Laplacians, that is, \begin{equation} \begin{cases} (-\Delta _p)^r u = \lambda \tfrac {\alpha }p|u|^{\alpha -2}u|v|^{\beta } &\mbox{in } \Omega ,\\ (-\Delta _p)^s v = \lambda \tfrac {\beta }p|u|^{\alpha }|v|^{\beta -2}v &\mbox{in } \Omega ,\\ u=v=0 &\mbox{in } \Omega ^c=\mathbb{R} ^N\setminus \Omega. \end{cases} \end{equation} We show that there is a first (smallest) eigenvalue that is simple and has associated eigenpairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $\lambda _n$ such that $\lambda _n\to \infty $ as $n\to \infty $.
In addition, we study the limit as $p\to \infty $ of the first eigenvalue, $\lambda _{1,p}$, and we obtain $[\lambda _{1,p}]^{{1}/{p}}\to \Lambda _{1,\infty }$ as $p\to \infty ,$ where $$\Lambda _{1,\infty }= \inf _{(u,v)} \bigg \{\frac {\max \{ [u]_{r,\infty } ; [v]_{s,\infty } \} }{ \| |u|^{\Gamma } |v|^{1-\Gamma } \|_{L^\infty (\Omega )}}\bigg \} = \bigg [ \frac {1}{R(\Omega )} \bigg ]^{ (1-\Gamma ) s + \Gamma r }. $$ Here, $$R(\Omega ):=\max _{x\in \Omega }dist (x,\partial \Omega )\mbox { and } [w]_{t,\infty } := \!\sup _{(x,y)\in \overline {\Omega }} \frac {| w(y) - w(x)|}{|x-y|^{t}}.$$
Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigenpairs.
</p>projecteuclid.org/euclid.rmjm/1538272824_20180929220041Sat, 29 Sep 2018 22:00 EDTWelschinger invariants of blow-ups of symplectic 4-manifoldshttps://projecteuclid.org/euclid.rmjm/1538272825<strong>Yanqiao Ding</strong>, <strong>Jianxun Hu</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1105--1144.</p><p><strong>Abstract:</strong><br/>
Using the degeneration technique, we study the behavior of Welschinger invariants under the blow-up and obtain some blow-up formulae of Welschinger invariants. To analyze the variation of Welschinger invariants when replacing a pair of real points in the real configuration by a pair of conjugated points, Welschinger introduced the $\theta $-invariant. In this paper, we also verify that the $\theta $-invariant is the Welschinger invariant of the blow-up of the symplectic $4$-manifold.
</p>projecteuclid.org/euclid.rmjm/1538272825_20180929220041Sat, 29 Sep 2018 22:00 EDTNontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operatorhttps://projecteuclid.org/euclid.rmjm/1538272826<strong>Abdeljabbar Ghanmi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1145--1158.</p><p><strong>Abstract:</strong><br/>
In this article, we study the existence of nontrivial solutions for the following $p(x)$ Kirchhoff-type problem \begin{equation} \begin{cases}\smash {\!-M\big (\textstyle \int _{\Omega }A(x,\nabla u)\,dx\big ){div}(a(x,\nabla u))} =\lambda h(x)\frac{\partial F}{\partial u} (x,u), \quad \mbox {in } \Omega \\ u=0, \quad \mbox {on } \partial \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb {R}^{n}$, $n\geq 3$, is a smooth bounded domain, $\lambda >0$, $h\in C(\Omega )$, $F:\overline {\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is continuously differentiable and $a, A:\Omega \times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ are continuous. The proof is based on variational arguments and the theory of variable exponent Sobolev spaces.
</p>projecteuclid.org/euclid.rmjm/1538272826_20180929220041Sat, 29 Sep 2018 22:00 EDTNumerical ranges of normal weighted composition operators on $\ell ^2(\mathbb N)$https://projecteuclid.org/euclid.rmjm/1538272827<strong>Mitu Gupta</strong>, <strong>B.S. Komal</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1159--1169.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain numerical ranges of normal weighted composition operators on $\ell ^{2}(\mathbb {N})$.
</p>projecteuclid.org/euclid.rmjm/1538272827_20180929220041Sat, 29 Sep 2018 22:00 EDTIntroverted subspaces of the duals of measure algebrashttps://projecteuclid.org/euclid.rmjm/1538272828<strong>Hossein Javanshiri</strong>, <strong>Rasoul Nasr-Isfahani</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1171--1189.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{G} $ be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalized functions, here we study the C$^*$-subalgebra $GL_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G} )^*$. In the case where $\mathcal{G} $ is non-compact, we show that any topological left invariant mean on $GL(\mathcal{G} )$ lies in $GL_0(\mathcal{G} )^\perp $. We then endow $GL_0(\mathcal{G} )^*$ with an Arens-type product, which contains $M(\mathcal{G} )$ as a closed subalgebra and $M_a(\mathcal{G} )$ as a closed ideal, which is a solid set with respect to absolute continuity in $GL_0(\mathcal{G} )^*$. Among other things, we prove that $\mathcal{G} $ is compact if and only if $GL_0(\mathcal{G} )^*$ has a non-zero left (weakly) completely continuous element.
</p>projecteuclid.org/euclid.rmjm/1538272828_20180929220041Sat, 29 Sep 2018 22:00 EDTOn the greatest common divisor of $n$ and the $n$th Fibonacci numberhttps://projecteuclid.org/euclid.rmjm/1538272829<strong>Paolo Leonetti</strong>, <strong>Carlo Sanna</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1191--1199.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#(\mathcal {A} \cap [1, x])\gg x / \log x$ for all $x \geq 2$ and that $\mathcal {A}$ has zero asymptotic density. Our proofs rely upon a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.
</p>projecteuclid.org/euclid.rmjm/1538272829_20180929220041Sat, 29 Sep 2018 22:00 EDTStochastic predator-prey model with Leslie-Gower and Holling-type II schemes with regime switchinghttps://projecteuclid.org/euclid.rmjm/1538272830<strong>Jingliang Lv</strong>, <strong>Ke Wang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1201--1218.</p><p><strong>Abstract:</strong><br/>
A predator-prey model with Leslie-Gower and Holling-type II schemes with regime switching will be considered, that is, both white and color noises are taken into account. We firstly show that there exists a globally unique solution to the stochastic predator-prey model by use of the comparison theorem. Then, asymptotic properties of the system will be examined and the conditions under which the system is stochastically persistent will be given. Moreover, lastly, we analyze the optimal harvesting policy of the stochastic prey-predator model with Markovian switching.
</p>projecteuclid.org/euclid.rmjm/1538272830_20180929220041Sat, 29 Sep 2018 22:00 EDTMaps preserving quasi-isometries on Hilbert $C^*$-moduleshttps://projecteuclid.org/euclid.rmjm/1538272831<strong>Alireza Majidi</strong>, <strong>Maryam Amyari</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1219--1229.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal {K}(\mathcal {H})$ be the $C^*$-algebra of compact op\-erators on a Hilbert space $\mathcal {H}$. Let $E$ be a Hilbert $\mathcal {K}(\mathcal {H})$-mod\-ule and $\mathcal {L}(E)$ the $C^*$-algebra of all adjointable maps on $E$. In this paper, we prove that, if $\varphi :\mathcal {L}(E)\to \mathcal {L}(E)$ is a unital surjective bounded linear map, which preserves quasi-isometries in both directions, then there are unitary oper\-ators $U, V\in \mathcal {L}(E)$ such that \[ \varphi (T)=UTV\quad \mbox {or}\quad \varphi (T)=UT^{tr }V \] for all $T\in \mathcal {L}(E)$, where $T^{tr }$ is the transpose of $T$ with re\-spect to an arbitrary but fixed orthonormal basis of $E$.
</p>projecteuclid.org/euclid.rmjm/1538272831_20180929220041Sat, 29 Sep 2018 22:00 EDTAn analytic Novikov conjecture for semigroupshttps://projecteuclid.org/euclid.rmjm/1538272832<strong>Paul D. Mitchener</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1231--1253.</p><p><strong>Abstract:</strong><br/>
In this article, we formulate a version of the analytic Novikov conjecture for semigroups rather than groups and show that the descent argument from coarse geometry generalizes effectively to this new situation.
</p>projecteuclid.org/euclid.rmjm/1538272832_20180929220041Sat, 29 Sep 2018 22:00 EDTChern-Dirac bundles onnon-Kähler Hermitian manifoldshttps://projecteuclid.org/euclid.rmjm/1538272833<strong>Francesco Pediconi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1255--1290.</p><p><strong>Abstract:</strong><br/>
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with the Levi-Civita connection replaced by the Chern connection. We then show that the tensor product of the canonical and the anticanonical spinor bundles, called the $\mathcal{V} $-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic sections isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms among other types of harmonic sections of the $\mathcal{V} $-spinor bundle and twisted cohomology.
</p>projecteuclid.org/euclid.rmjm/1538272833_20180929220041Sat, 29 Sep 2018 22:00 EDTPeriodic solutions to nonlinear wave equations with $x$-dependent coefficients at resonancehttps://projecteuclid.org/euclid.rmjm/1538272834<strong>Wei Shi</strong>, <strong>Kai Liu</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1291--1306.</p><p><strong>Abstract:</strong><br/>
In this paper, the unique existence of generalized solutions to periodic boundary value problems for a class of systems of nonlinear equations with $x$-dependent coefficients is discussed under a resonance condition. The argument presented makes use of the global inverse theorem and Galerkin's method.
</p>projecteuclid.org/euclid.rmjm/1538272834_20180929220041Sat, 29 Sep 2018 22:00 EDTThe tracial Rokhlin property for actions of amenable groups on $C^*$-algebrashttps://projecteuclid.org/euclid.rmjm/1538272835<strong>Qingyun Wang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1307--1344.</p><p><strong>Abstract:</strong><br/>
In this paper, we present a definition of the tracial Rokhlin property for (cocyclic) actions of countable discrete amenable groups on simple $C^*$-algebras, which generalize Matui and Sato's definition. We show that generic examples, like Bernoulli shift on the tensor product of copies of the Jiang-Su algebra, has the weak tracial Rokhlin property, while it is shown in {Hirshberg 2014} that such an action does not have finite Rokhlin dimension. We further show that forming a crossed product from actions with the tracial Rokhlin property preserves the class of $C^*$-algebras with real rank $0$, stable rank $1$ and has strict comparison for pro\-jec\-tions, generalizing the structural results in {Osaka 2006}. We use the same idea of the proof with significant simplification. In another joint paper with Chris Phillips and Joav Orovitz, we shall show that pureness and $\mathcal{Z} $-stability could be preserved by crossed product of actions with the weak tracial Rokhlin property. The combination of these results yields an application to the classification program, which is discussed in the aforementioned paper. These results indicate that we have the correct definition of tracial Rokhlin property for actions of general countable discrete amenable groups.
</p>projecteuclid.org/euclid.rmjm/1538272835_20180929220041Sat, 29 Sep 2018 22:00 EDTOn a problem of Bharanedhar and Ponnusamy involving planar harmonic mappingshttps://projecteuclid.org/euclid.rmjm/1538272836<strong>Zhi-Gang Wang</strong>, <strong>Zhi-Hong Liu</strong>, <strong>Antti Rasila</strong>, <strong>Yong Sun</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1345--1358.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a negative answer to a problem presented by Bharanedhar and Ponnusamy concerning univalency of a class of harmonic mappings. More precisely, we show that for all values of the involved parameter, this class contains a non-univalent function. Moreover, several results on a new subclass of close-to-convex harmonic mappings, motivated by the work of Ponnusamy and Sairam Kaliraj, are obtained.
</p>projecteuclid.org/euclid.rmjm/1538272836_20180929220041Sat, 29 Sep 2018 22:00 EDTSmale spaces from self-similar graph actionshttps://projecteuclid.org/euclid.rmjm/1538272837<strong>Inhyeop Yi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 4, 1359--1384.</p><p><strong>Abstract:</strong><br/>
We show that, for contracting and regular self-similar graph actions, the shift maps on limit spaces are positively expansive local homeomorphisms. From this, we find that limit solenoids of contracting and regular self-similar graph actions are Smale spaces and that the unstable Ruelle algebras of the limit solenoids are strongly Morita equivalent to the Cuntz-Pimsner algebras by Exel and Pardo if self-similar graph actions satisfy the contracting, regular, pseudo free and $G$-transitive conditions.
</p>projecteuclid.org/euclid.rmjm/1538272837_20180929220041Sat, 29 Sep 2018 22:00 EDTElliptic problems involving natural growth in the gradient and general absorption termshttps://projecteuclid.org/euclid.rmjm/1539936028<strong>Haydar Abdelhamid</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1385--1397.</p><p><strong>Abstract:</strong><br/>
In this paper, we treat the existence of solutions for a class of general elliptic problems whose prototype is the following: \begin{equation} \begin{cases}-\Delta _{p}u+h(x)|u|^{q-1}u=\beta |\nabla u|^{p}+\lambda f(x) &\mbox {in } \Omega , \\ u=0 &\mbox {on } \partial \Omega ,\end{cases} \end{equation} where $\Omega $ is a bounded open subset of $\mathbb {R}^{N}$ with $N>1$, $1\lt p\lt N$, $q\geq 1$, $\lambda \in \mathbb {R}$, $\beta \in \mathbb {R}$, $h\in L^{1}(\Omega )$ with $h\geq 0$ and $f\in L^{1}(\Omega )$. Assuming that the source term $f$ satisfies $$\lambda _{1}(f)=\inf \bigg \{ \frac {\int _{\Omega }\vert \nabla w\vert ^{p}dx}{\int _{\Omega }|f|\vert w\vert ^{p}dx}:w\in W_{0}^{1,p} (\Omega )\setminus \{ 0\} \bigg \}>0,$$ we obtain the existence of a solution $u\in W_{0}^{1,p}(\Omega )$ when $|\lambda |$ is sufficiently small.
</p>projecteuclid.org/euclid.rmjm/1539936028_20181019040128Fri, 19 Oct 2018 04:01 EDTOn nonlocal fractional Laplacian problems with oscillating potentialshttps://projecteuclid.org/euclid.rmjm/1539936029<strong>Vincenzo Ambrosio</strong>, <strong>Luigi D'Onofrio</strong>, <strong>Giovanni Molica Bisci</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1399--1436.</p><p><strong>Abstract:</strong><br/>
In this paper, we deal with the following fractional nonlocal $p$-Laplacian problem: \begin{equation} \begin{cases}(-\Delta )^{s}_{p}u= \lambda \beta (x) u^q + f(u) &\mbox {in } \Omega ,\\ u\geq 0,\ u\not \equiv 0 &\mbox {in } \Omega ,\\ u=0 &\mbox {in } \mathbb{R} ^{N}\setminus \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb{R} ^{N}$ is a bounded domain with a smooth boun\-dary of $\mathbb{R} ^N$, $s\in (0,1)$, $p\in (1, \infty )$, $N> s p$, $\lambda $ is a real parameter, $\beta \in L^\infty (\Omega )$ is allowed to be indefinite in sign, $q>0$ and $f:[0,+\infty )\to \mathbb{R} $ is a continuous function oscillating near the origin or at infinity. By using variational and topological methods, we obtain the existence of infinitely many solutions for the problem under consideration. The main results obtained here represent some new interesting phenomena in the nonlocal setting.
</p>projecteuclid.org/euclid.rmjm/1539936029_20181019040128Fri, 19 Oct 2018 04:01 EDTModeling rogue waves with the Kadomtsev--Petviashvili (KP) equationhttps://projecteuclid.org/euclid.rmjm/1539936030<strong>Ion Bica</strong>, <strong>Randy Wanye K</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1437--1454.</p><p><strong>Abstract:</strong><br/>
In this paper, we derive a new class of solutions for the Kadomtsev-Petviashvili (KP) equation, and we discuss their possible relevance to rogue waves. The nonlinear interaction of these solutions is considered.
</p>projecteuclid.org/euclid.rmjm/1539936030_20181019040128Fri, 19 Oct 2018 04:01 EDTOn Riemannian surfaces with conical singularitieshttps://projecteuclid.org/euclid.rmjm/1539936031<strong>Charalampos Charitos</strong>, <strong>Ioannis Papadoperakis</strong>, <strong>Georgios Tsapogas</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1455--1474.</p><p><strong>Abstract:</strong><br/>
The geometry of closed surfaces of genus $g\geq 2$ equipped with a Riemannian metric of variable bounded curvature with finitely many conical points is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
</p>projecteuclid.org/euclid.rmjm/1539936031_20181019040128Fri, 19 Oct 2018 04:01 EDTDifferential subordination of a harmonic mean to a linear functionhttps://projecteuclid.org/euclid.rmjm/1539936032<strong>Oliwia Chojnacka</strong>, <strong>Adam Lecko</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1475--1484.</p><p><strong>Abstract:</strong><br/>
In this paper, we examine differential subordination related to the harmonic mean in the case where a dominant is a linear function. A result for the first order Euler differential subordination of the nonlinear type is also discussed.
</p>projecteuclid.org/euclid.rmjm/1539936032_20181019040128Fri, 19 Oct 2018 04:01 EDTK-theory and index pairings for C*-algebras generated by $q$-normal operatorshttps://projecteuclid.org/euclid.rmjm/1539936033<strong>Ismael Cohen</strong>, <strong>Elmar Wagner</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1485--1510.</p><p><strong>Abstract:</strong><br/>
The paper presents a detailed description of the K-theory and K-homology of C*-algebras generated by $q$-normal operators, including generators and index pairing. The C*-algebras generated by $q$-normal operators can be viewed as a $q$-deformation of the quantum complex plane. In this sense, we find deformations of the classical Bott projections describing complex line bundles over the 2-sphere, but there are also simpler generators for the $K_0$-groups, for instance, one dimensional Powers-Rieffel type projections and elementary projections belonging to the C*-algebra. The index pairing between these projections and generators of the even K-homology group is computed, and the result is used to express the $K_0$-classes of quantized line bundles of any winding number in terms of the other projections.
</p>projecteuclid.org/euclid.rmjm/1539936033_20181019040128Fri, 19 Oct 2018 04:01 EDTContinuous-trace $k$-graph $C^*$-algebrashttps://projecteuclid.org/euclid.rmjm/1539936034<strong>Danny Crytser</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1511--1535.</p><p><strong>Abstract:</strong><br/>
A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the Drinen-Tomforde desingularization. A characterization of continuous-trace AF $C^*$-algebras is obtained. Partial results are given to characterize higher-rank graphs that yield $C^*$-algebras with continuous trace.
</p>projecteuclid.org/euclid.rmjm/1539936034_20181019040128Fri, 19 Oct 2018 04:01 EDTInvariant curves and integrability of planar $\mathcal C^r$ differential systemshttps://projecteuclid.org/euclid.rmjm/1539936035<strong>Antoni Ferragut</strong>, <strong>Jaume Llibre</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1537--1550.</p><p><strong>Abstract:</strong><br/>
We improve the known expressions of the $\mathcal {C}^r$ differential systems in the plane having a given $\mathcal {C}^{r+1}$ invariant curve, or a given $\mathcal {C}^{r+1}$ first integral. Their application to polynomial differential systems having either an invariant algebraic curve, or a first integral, also improves the known results on such systems.
</p>projecteuclid.org/euclid.rmjm/1539936035_20181019040128Fri, 19 Oct 2018 04:01 EDTThe upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applicationshttps://projecteuclid.org/euclid.rmjm/1539936036<strong>Carmelo A. Finocchiaro</strong>, <strong>Marco Fontana</strong>, <strong>Dario Spirito</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1551--1583.</p><p><strong>Abstract:</strong><br/>
Given an arbitrary spectral space $X$, we consider the set $\mathcal{X} (X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on $\mathcal{X} (X)$ and, after observing that it coincides the upper Vietoris topology, we prove that $\mathcal{X} (X)$ is itself a spectral space, that this construction is functorial, and that $\mathcal{X} (X)$ provides an extension of $X$ in a more ``complete'' spectral space. Among the applications, we show that, starting from an integral domain $D$, $\mathcal{X} (Spec (D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$.
</p>projecteuclid.org/euclid.rmjm/1539936036_20181019040128Fri, 19 Oct 2018 04:01 EDTThe local $S$-class group of an integral domainhttps://projecteuclid.org/euclid.rmjm/1539936037<strong>Ahmed Hamed</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1585--1605.</p><p><strong>Abstract:</strong><br/>
In this paper, we define the local $S$-class group of an integral domain $D$. A nonzero fractional ideal $I$ of $D$ is said to be $S$-invertible if there exist an $s\in S$ and a fractional ideal $J$ of $D$ such that $sD \subseteq I, J \subseteq D$. The local $S$-class group of $D$, denoted G$ (D)$, is the group of fractional $t$-invertible $t$-ideals of $D$ under $t$-multiplication modulo its subgroup of $S$-invertible $t$-invertible $t$-ideals of $D$. We study the case {G }$(D)=0$, and we generalize some known results developed for the classic contexts of Krull and P$\upsilon $MD domains. Moreover, we investigate the case of isomorphism {G }$(D) \simeq$ {G }$(D[[X]])$. In particular, we give with an additional condition an answer to the question of Bouvier, that is, when is G$ (D)$ isomorphic to G$ (D[[X]])?$
</p>projecteuclid.org/euclid.rmjm/1539936037_20181019040128Fri, 19 Oct 2018 04:01 EDTThe solution of a new Caputo-like fractional $h$-difference equationhttps://projecteuclid.org/euclid.rmjm/1539936038<strong>Baoguo Jia</strong>, <strong>Xiang Liu</strong>, <strong>Feifei Du</strong>, <strong>Mei Wang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1607--1630.</p><p><strong>Abstract:</strong><br/>
Consider the Caputo fractional $h$-difference equation \[ _a\Delta ^\nu _{h,*}x(t)=c(t)x(t+\nu ), \quad 0\lt \nu \lt 1,\ t\in (h\mathbb{N} )_{a+(1-\nu )h}, \] where $_a\Delta ^\nu _{h,*}x(t)$ denotes the Caputo-like delta fractional $h$-difference of $x(t)$ on sets $(h\mathbb{N} )_{a+(1-\nu )h}$. Our main results are found in Theorems A and B in Section 1. In Section 3, we show that the proof of a recent result in Baleanu, Wu, Bai and Chen is incorrect. Finally, four numerical examples are given to illustrate the main results.
</p>projecteuclid.org/euclid.rmjm/1539936038_20181019040128Fri, 19 Oct 2018 04:01 EDTNearly Krull domains and nearly Prufer $v$-multiplication domainshttps://projecteuclid.org/euclid.rmjm/1539936039<strong>Jung Wook Lim</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1631--1646.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce some new concepts of almost factoriality of integral domains. More precisely, we investigate nearly Krull domains, nearly Prufer $v$-multiplication domains and some related integral domains.
</p>projecteuclid.org/euclid.rmjm/1539936039_20181019040128Fri, 19 Oct 2018 04:01 EDTOn the existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentialshttps://projecteuclid.org/euclid.rmjm/1539936040<strong>Gongming Wei</strong>, <strong>Xueliang Duan</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1647--1683.</p><p><strong>Abstract:</strong><br/>
We are concerned with the nonlinear fractional Schrödinger system \begin{equation} \begin{cases}(-\Delta )^{s} u+V_{1}(x)u=f(x,u)+\Gamma (x)|u|^{q-2}u|v|^{q} &\mbox {in } \mathbb {R}^{N},\\ (-\Delta )^{s} v+V_{2}(x)v=g(x,v)+\Gamma (x)|v|^{q-2}v|u|^{q} &\mbox {in } \mathbb {R}^{N},\\ u,v\in H^{s}(\mathbb {R}^{N}), \end{cases} \end{equation} where $(-\Delta )^{s}$ is the fractional Laplacian operator, $s\in (0,1)$, $N>2s$, $4\leq 2q\lt p\lt 2^{\ast }$, $2^{\ast }={2N}/({N-2s})$. $V_{i}(x)=V^{i}_{per }(x)+V^{i}_{loc }(x)$ is closed-to-periodic for $i=1,2$, $f$ and $g$ have subcritical growths and $\Gamma (x)\geq 0$ vanishes at infinity. Using the Nehari manifold minimization technique, we first obtain a bounded minimizing sequence, and then we adopt the approach of Jeanjean-Tanaka (2005) to obtain a decomposition of the bounded Palais-Smale sequence. Finally, we prove the existence of ground state solutions for the nonlinear fractional Schrödinger system.
</p>projecteuclid.org/euclid.rmjm/1539936040_20181019040128Fri, 19 Oct 2018 04:01 EDTGlobal asymptotic stability of positive steady states of a solid avascular tumor growth model with time delayshttps://projecteuclid.org/euclid.rmjm/1539936041<strong>Shihe Xu</strong>, <strong>Fangwei Zhang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1685--1702.</p><p><strong>Abstract:</strong><br/>
In this work, global stability of a free boundary problem modeling solid avascular tumor growth is studied. The model is considered with time delays during the proliferation process. We prove that the unique positive constant steady state is globally asymptotically stable under some assumptions. The proof uses the comparison principle and the iteration method.
</p>projecteuclid.org/euclid.rmjm/1539936041_20181019040128Fri, 19 Oct 2018 04:01 EDTGlobal existence and decay rate of strong solution to incompressible Oldroyd type model equationshttps://projecteuclid.org/euclid.rmjm/1539936042<strong>Baoquan Yuan</strong>, <strong>Yun Liu</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1703--1720.</p><p><strong>Abstract:</strong><br/>
This paper investigates the global existence and the decay rate in time of a solution to the Cauchy problem for an incompressible Oldroyd model with a deformation tensor damping term. There are three major results. The first is the global existence of the solution for small initial data. Second, we derive the sharp time decay of the solution in $L^{2}$-norm. Finally, the sharp time decay of the solution of higher order Sobolev norms is obtained.
</p>projecteuclid.org/euclid.rmjm/1539936042_20181019040128Fri, 19 Oct 2018 04:01 EDTViscous limits for a Riemannian problem to a class of systems of conservation lawshttps://projecteuclid.org/euclid.rmjm/1539936043<strong>Yanyan Zhang</strong>, <strong>Yu Zhang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 48, Number 5, 1721--1741.</p><p><strong>Abstract:</strong><br/>
In this paper, by a vanishing viscosity approach, we investigate Riemannian solutions containing delta shock waves with Dirac delta functions in both state variables for a class of non-strictly hyperbolic systems of conservation laws. The existence and uniqueness of solutions for the viscous problem are shown.
</p>projecteuclid.org/euclid.rmjm/1539936043_20181019040128Fri, 19 Oct 2018 04:01 EDT