## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (191.996 recursos)

Rocky Mountain Journal of Mathematics

2. #### Content formulas for power series and Krull domains

Yin, Huayu; Chen, Youhua; Zhu, Xiaosheng
Let $R$ be an integral domain with quotient field $K$, and let $X$ be an indeterminate over $R$. In this paper, we consider content formulae for power series in terms of $*$-operations for PVMDs, Krull domains and Dedekind domains, where $*$ is the star-operation, $d$, $w$, $t$, or $v$. We prove that $R$ is a Krull domain if and only if $c(f/g)_w=(c(f)c(g)^{-1})_w$ for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal if and only if $c(f/g)_t=(c(f)c(g)^{-1})_t$ for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal, and $R$ is a Dedekind domain if and only if for all $f,g\in R[[X]]^*$ with...

3. #### Arithmetic and geometry of rational elliptic surfaces

Let $\mathscr {E}$ be a rational elliptic surface over a number field~$k$. We study the interplay between a geometric property, the configuration of its singular fibers, and arithmetic features such as its Mordell-Weil rank over the base field and its possible minimal models over~$k$.

4. #### On Schauder basis properties of multiply generated Gabor systems

Nielsen, Morten
Let $\mathcal{A}$ be a finite subset of $L^2(\mathbb{R} )$ and $p,q\in \mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R} )$ of the Gabor system $G(1,p/q,\mathcal{A} )=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z} , g\in \mathcal{A} \},$ with a specific ordering on $\mathbb{Z} \times \mathbb{Z} \times \mathcal{A}$. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.

5. #### Strongly copure projective, injective and flat complexes

Ma, Xin; Liu, Zhongkui
In this paper, we extend the notions of strongly copure projective, injective and flat modules to that of complexes and characterize these complexes. We show that the strongly copure projective precover of any finitely presented complex exists over $n$-FC rings, and a strongly copure injective envelope exists over left Noetherian rings. We prove that strongly copure flat covers exist over arbitrary rings and that $(\mathcal {SCF},\mathcal {SCF}^\bot )$ is a perfect hereditary cotorsion theory where $\mathcal {SCF}$ is the class of strongly copure flat complexes.

6. #### A functional equation and degenerate principal series

Lee, Juhyung
A functional equation between the $\zeta$ distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.

7. #### Substitution Markov chains and Martin boundaries

Koslicki, David; Denker, Manfred
Substitution Markov chains have been introduced \cite {KoslickiThesis2012} as a new model to describe molecular evolution. In this note, we study the associated Martin boundaries from a probabilistic and topological viewpoint. An example is given that, although having a boundary homeomorphic to the well-known coin tossing process, has a metric description that differs significantly.

8. #### Smooth center manifolds for random dynamical systems

Guo, Peng; Shen, Jun
In this paper, we will prove the existence and H\"{o}lder continuity of smooth center-unstable and center-stable manifolds for random dynamical systems based on their Lyapunov exponents. Furthermore, we obtain the existence and H\"{o}lder continuity of smooth center manifolds.

9. #### An explicit formula for Bernoulli polynomials in terms of $\boldsymbol r$-Stirling numbers of the second kind

Guo, Bai-Ni; Mező, István; Qi, Feng
In this paper, the authors establish an explicit formula for computing Bernoulli polynomials at nonnegative integer points in terms of $r$-Stirling numbers of the second kind.

10. #### Torsion of rational elliptic curves over cubic fields

González-Jiménez, Enrique; Najman, Filip; Tornero, José M.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We study the relationship between the torsion subgroup $E(mathbb{Q} )_{tors}$ and the torsion subgroup $E(K)_{tors}$, where $K$ is a cubic number field. In particular, we study the number of cubic number fields $K$ such that $E(\mathbb{Q} )_{tors}\neq E(K)_{tors}$.

11. #### Singular links and Yang-Baxter state models

Caprau, Carmen; Okano, Tsutomu; Orton, Danny
We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Speci\-fically, we consider a Yang-Baxter state model for the {\rm sl($n$)} polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra and seek conditions for further extending the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the {\rm sl($n$)} polynomial for classical links.

12. #### Amicable pairs and aliquot cycles for elliptic curves over number fields

Brown, Jim; Heras, David; James, Kevin; Keaton, Rodney; Qian, Andrew
Let $E/\mathbb{Q}$ be an elliptic curve. Silverman and Stange define primes $p$ and $q$ to be an elliptic, amicable pair if $\#E(\mathbb{F} _p) = q$ and $\#E(\mathbb{F} _q) = p$. More generally, they define the notion of aliquot cycles for elliptic curves. Here, we study the same notion in the case that the elliptic curve is defined over a number field~$K$. We focus on proving the existence of an elliptic curve~$E/K$ with aliquot cycle $(\mathfrak{p} _1, \ldots , \mathfrak{p} _{n})$ where the $\mathfrak{p} _{i}$ are primes of~$K$ satisfying mild conditions.

13. #### Symbol calculus of square-integrable operator-valued maps

Beltiţă, Ingrid; Beltiţă, Daniel; Măntoiu, Marius
We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method, we show that it behaves well with respect to infinite tensor products. This construction subsumes examples from the study of magnetic Weyl calculus, magnetic pseudo-differential Weyl calculus, metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of some special infinite-dimensional Lie groups, and square-integrability properties shared by arbitrary irreducible representations of nilpotent Lie groups.

14. #### Exponential dichotomies for impulsive equations via quadratic functions

Barreira, Luis; Valls, Claudia
We give a characterization of the existence of a nonuniform exponential dichotomy for a linear impulsive differential equation. The characterization uses quadratic functions and the symmetric matrices defining them. As an application, we give a simple proof of the robustness property of a nonuniform exponential dichotomy under sufficiently small linear perturbations.

15. #### On the divisibility graph for finite sets of positive integers

Let $X$ be a finite set of positive integers. The divisibility graph $\mathscr {D}\,(X)$ is a directed graph with vertex set $X\backslash \{1\}$ and an arc from $a$ to $b$ whenever $a$ divides $b$. Since the divisibility graph and its underlying graph have the same number of connected components, we consider the underlying graph of $\mathscr {D}\,(X)$, and we denote it by $\rm D (X)$. In this paper, we will find some graph theoretical parameters of $\rm D (X)$, some relations between the structure of $\rm D (X)$, and the structure of known graphs $\Gamma (X)$, $\Delta (X)$ and $B(X)$...

16. #### Resolutions and stability of $C$-Gorenstein flat modules

Zhao, Guoqiang; Yan, Xiaoguang
In this paper, we first investigate the relationship between $\mathcal{W}$-(co)resolutions and $\mathcal{X}$-(co)resolutions for two full subcategories $\mathcal{W}$ and $\mathcal{X}$ of an abelian category with $\mathcal{W} \subseteq \mathcal{X}$. Then some applications are given. In particular, we obtain the stability of the category of $C$-Gorenstein flat modules under the procedure used to define these entities, which is different from that established by Sather-Wagstaff, Sharif and White.

17. #### Affine-periodic solutions for nonlinear differential equations

Wang, Chuanbiao; Yang, Xue; Li, Yong
The existence of affine-periodic solutions is studied. These types of solutions may be periodic, harmonic or even quasi-periodic. Mainly, via the topological degree theory, a general existence theorem is proved, which asserts the existence of affine-periodic solutions, extending some classical results. The theorem is applied to establish the Lyapunov function type theorem and the invariant region principle relative to affine-periodic solutions.

18. #### Notes on $\log (\zeta (s))^\prime \prime$

Stopple, Jeffrey
Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann $\zeta$ function, in particular, the zeros of this function. Theorem~1.2 gives a zero-free region. Theorem~1.4 gives an asymptotic estimate for the number of nontrivial zeros to height $T$. Theorem~1.7 is a zero density estimate.

19. #### Existence of periodic solutions for 2$n$th-order nonlinear $p$-Laplacian difference equations

Shi, Haiping; Liu, Xia; Zhang, Yuanbiao
By using the critical point theory, the existence of periodic solutions for 2$n$th-order nonlinear $p$-Laplacian difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point theorem. The problem is to solve the existence of periodic solutions for 2$n$th-order $p$-Laplacian difference equations. The results obtained successfully generalize and complement the existing ones.

20. #### A certain class of approximations for the $q$-digamma function

Salem, Ahmed
In this paper, we derive a class of approximations of the $q$-digamma function $\psi _q(x)$. The infinite family $I_a(x;q)=\log [x+a]_q+\frac {q^x\log q}{1-q^x}-\bigg (\frac 12-a\bigg )H(q-1)\log q,$ $a\in [0,1]$; $q>0$, can be used as approximating functions for $\psi _q(x)$, where $[x]_q=(1-q^x)/(1-q)$ and $H(\cdot )$ is the Heaviside step function. We show that, for all $a\in [0,1]$, $I_a$ is asymptotically equivalent to $\psi _q(x)$ for $q>0$ and is a good pointwise approximation.

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