location:  Publications → journals
Search results

Search: All articles in the CJM digital archive with keyword rate

 Expand all        Collapse all Results 1 - 13 of 13

1. CJM Online first

Cahn, Jordan; Jones, Rafe; Spear, Jacob
 Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the only such functions are those of the form $cx^j(\psi(x))^m$ with $\psi \in K(x)$, and for $m \leq 4$ we show the only additional cases are certain LattÃ¨s maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n \geq 0 : \phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ is a union of finitely many arithmetic progressions, where $\phi^{n}$ denotes the $n$th iterate of $\phi$ and $\lambda \in K(x)$ is any map MÃ¶bius-conjugate over $K$ to $x^m$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell-Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^m = \phi^{n}(x)$. We describe all $\phi$ for which these curves have an irreducible component of genus at most 1, and show that such $\phi$ must have two distinct iterates that are equal in $K(x)^*/K(x)^{*m}$. Keywords:arithmetic dynamics, iteration of rational functions, special orbits of rational function, genus of variables-separated curve, LattÃ¨s mapCategories:37P05, 11G05, 37P15

2. CJM 2015 (vol 68 pp. 129)

Shiozawa, Yuichi
 Lower Escape Rate of Symmetric Jump-diffusion Processes We establish an integral test on the lower escape rate of symmetric jump-diffusion processes generated by regular Dirichlet forms. Using this test, we can find the speed of particles escaping to infinity. We apply this test to symmetric jump processes of variable order. We also derive the upper and lower escape rates of time changed processes by using those of underlying processes. Keywords:lower escape rate, Dirichlet form, Markov process, time changeCategories:60G17, 31C25, 60J25

3. CJM 2015 (vol 67 pp. 1358)

Garcia Trillos, Nicolas; Slepcev, Dejan
 On the Rate of Convergence of Empirical Measures in $\infty$-transportation Distance We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty$-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points. Keywords:rate, convergenceCategory:01B01

4. CJM 2015 (vol 67 pp. 1161)

Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
 Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$. Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transformCategories:42B30, 42B35, 35J70

5. CJM 2013 (vol 66 pp. 354)

Kellerhals, Ruth; Kolpakov, Alexander
 The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is as such unique. Our approach provides a different proof for the analog situation in ${\mathbb H^2}$ where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7). Keywords:hyperbolic Coxeter group, growth rate, Salem numberCategories:20F55, 22E40, 51F15

6. CJM 2013 (vol 65 pp. 1095)

Sambou, Diomba
 RÃ©sonances prÃ¨s de seuils d'opÃ©rateurs magnÃ©tiques de Pauli et de Dirac Nous considÃ©rons les perturbations $H := H_{0} + V$ et $D := D_{0} + V$ des Hamiltoniens libres $H_{0}$ de Pauli et $D_{0}$ de Dirac en dimension 3 avec champ magnÃ©tique non constant, $V$ Ã©tant un potentiel Ã©lectrique qui dÃ©croÃ®t super-exponentiellement dans la direction du champ magnÃ©tique. Nous montrons que dans des espaces de Banach appropriÃ©s, les rÃ©solvantes de $H$ et $D$ dÃ©finies sur le demi-plan supÃ©rieur admettent des prolongements mÃ©romorphes. Nous dÃ©finissons les rÃ©sonances de $H$ et $D$ comme Ã©tant les pÃ´les de ces extensions mÃ©romorphes. D'une part, nous Ã©tudions la rÃ©partition des rÃ©sonances de $H$ prÃ¨s de l'origine $0$ et d'autre part, celle des rÃ©sonances de $D$ prÃ¨s de $\pm m$ oÃ¹ $m$ est la masse d'une particule. Dans les deux cas, nous obtenons d'abord des majorations du nombre de rÃ©sonances dans de petits domaines au voisinage de $0$ et $\pm m$. Sous des hypothÃ¨ses supplÃ©mentaires, nous obtenons des dÃ©veloppements asymptotiques du nombre de rÃ©sonances qui entraÃ®nent leur accumulation prÃ¨s des seuils $0$ et $\pm m$. En particulier, pour une perturbation $V$ de signe dÃ©fini, nous obtenons des informations sur la rÃ©partition des valeurs propres de $H$ et $D$ prÃ¨s de $0$ et $\pm m$ respectivement. Keywords:opÃ©rateurs magnÃ©tiques de Pauli et de Dirac, rÃ©sonancesCategories:35B34, 35P25

7. CJM 2012 (vol 64 pp. 1395)

Rodney, Scott
 Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form \begin{align*} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta \\ u&=\varphi\text{ on }\partial \Theta. \end{align*} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in previous works. Sawyer and Wheeden give a regularity theory for a subset of the class of equations dealt with here. Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutionsCategories:35A01, 35A02, 35D30, 35J70, 35H20

8. CJM 2012 (vol 65 pp. 544)

Deitmar, Anton; Horozov, Ivan
 Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants. Keywords:higher order forms, iterated integralsCategories:14F35, 11F12, 55D35, 58A10

9. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
 Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition. Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identitiesCategories:28A80, , , , 35P20, 35J05, 43A05, 47A75

10. CJM 2010 (vol 62 pp. 737)

Ditzian, Z.; Prymak, A.
 Approximation by Dilated Averages and K-Functionals For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$ normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of $f( \mathbf{x})$ is given by $A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}-t\mathbf{u})d\mu(\mathbf{u}).$ It will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has the equivalence $\|A_tf-f\|_B\approx \inf \{ (\|f-g\|_B+t^2 \|P(D)g\|_B): P(D)g\in B\}\quad\text{for }t>0,$ where $\varphi(t)\approx \psi(t)$ means $c^{-1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions for which translations are continuous isometries and $P(D)$ is an elliptic differential operator induced by $\mu$. Many applications are given, notable among which is the averaging operator with $d\mu(\mathbf{u})= \frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of $S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate $K$-functional. Keywords:rate of approximation, K-functionals, strong converse inequalityCategories:41A27, 41A35, 41A63

11. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
 Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$. Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groupsCategories:35H30, 26D10, 22E25

12. CJM 2009 (vol 62 pp. 74)

Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre
 Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) in $L^{p}$ spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the NFDE as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems. Keywords:neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectorsCategories:34K05, 35K57, 47A56, 47H20

13. CJM 2004 (vol 56 pp. 209)

Schmuland, Byron; Sun, Wei
 A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations In \cite{P69}, Walter Philipp wrote that \dots the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.'' Many authors \cite{DW80}, \cite{I68}, \cite{N91}, \cite{OY71}, \cite{Y79} have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables. We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn \cite{Sp86} for certain Gibbs measures. These do not fall into the $\phi$-mixing or strong mixing cases established in the literature, but are needed for our investigations \cite{SS01} into diffusions on configuration space. The proofs are all obtained by patching together standard results from \cite{OY71}, \cite{Y79} while keeping a careful eye on the correlations. Keywords:law of the iterated logarithmCategories:60F99, 60G60
 top of page | contact us | privacy | site map |