Rth Mean Research Articles

On Bose–Einstein Condensation in One-Dimensional Noninteracting Bose Gases in the Presence of Soft Poisson Obstacles

We study Bose–Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on \(\mathbb R\) with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With a probability arbitrarily close to one when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the rth mean, \(r \ge 1\). Furthermore, in Poisson random potentials of any fixed strength a probability arbitrarily close to one for type-I g-BEC to occur is also obtained, but our upper bound for the number of macroscopically occupied one-particle states may be large in this case.

Hypersurfaces of constant higher-order mean curvature in $$M\times {\mathbb{R}}$$

We consider hypersurfaces of products $$M\times {\mathbb{R}}$$ with constant rth mean curvature $${H_{r}} \ge 0$$ (to be called $${{H}_{r}}$$ -hypersurfaces), where M is an arbitrary Riemannian n-manifold. We develop a general method for constructing them and employ it to produce many examples for a variety of manifolds M, including all simply connected space forms and the hyperbolic spaces $${\mathbb{H}}_{{\mathbb{F}}}^{m}$$ (rank one symmetric spaces of noncompact type). We construct and classify complete rotational $${{H}_{r}}(\ge 0)$$ -hypersurfaces in $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ and in $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ as well. They include spheres, Delaunay-type annuli and, in the case of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}},$$ entire graphs. We also construct and classify complete $${{H}_{r}}(\ge 0)$$ -hypersurfaces of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and strictly convex $${{H}_{r}}$$ -hypersurface of $${\mathbb{H}}^{n}\times {\mathbb{R}}$$ or $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ $$(n\ge 3)$$ is a rotational embedded sphere. Other uniqueness results for complete $${{H}_{r}}$$ -hypersurfaces of these ambient spaces are obtained.

Convergence of series of moments on general exponential inequality

For an array of random variables and a sequence of positive numbers, sufficient conditions are given under which, for all , where denotes the positive part of x and , . Our statements are announced in a general setting allowing to conclude the previous convergence for well-known dependent structures. As an application, we study complete consistency and consistency in the rth mean of cumulative sum type estimators of the change in the mean of dependent observations.

A short note on a class of Weingarten hypersurfaces in $$\mathbb {R}^{n + 1}$$

We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying $$H_r = aH + b$$ , for constants $$a, b \in \mathbb {R}$$ , where $$H_r$$ stands for the rth mean curvature and H the mean curvature of the hypersurface. Besides we are able to characterize those hypersurfaces for which these bounds are attained by showing that they must be a cylinder of the type $$\mathbb {R}\times \mathbb {S}^{n - 1}(\rho )$$ . Moreover, with the additional assumption that the Gauss–Kronnecker curvature of the hypersurface does not change sign, we prove that it is the cylinder $$\mathbb {R}\times \mathbb {S}^{n - 1}(\rho )$$ . In order to proof our results, we will use the method introduced in Alias and Garca-Martinez (Geom Dedic 156:31–47, 2012), Alias and Melendez (Geom Dedic 182:117–131, 2016), which is based on the so called principal curvature theorem due to Smyth and Xavier (Invent Math 90:443–450, 1987).

Consistency and asymptotic normality of wavelet estimator in a nonparametric regression model

In this paper, the nonparametric regression model with repeated measurements based on ϕ-mixing errors is considered. The rth mean consistency, strong consistency, strong convergence rate, complete consistency and the asymptotic normality of the wavelet estimator are established under some mild conditions on moments and mixing coefficients.

On a condition for type-I Bose–Einstein condensation in random potentials in d dimensions

In this paper we discuss Bose–Einstein condensation (BEC) in systems of pairwise non-interacting bosons in random potentials in d dimensions. Working in a rather general framework, we provide a “gap condition” which is sufficient to conclude existence of type-I BEC in probability and in the rth mean. We illustrate our results in the context of the well-known (one-dimensional) Luttinger–Sy model. Here, whenever the particle density exceeds a critical value, we show in addition that only the ground state is macroscopically occupied.

The asymptotic properties of the estimators in a semiparametric regression model

In this paper, we investigate the parametric component and nonparametric component estimators in a semiparametric regression model based on $$\varphi $$ -mixing random variables. The rth mean consistency, complete consistency, uniform rth mean consistency and uniform complete consistency are established under some suitable conditions. In addition, a simulation to study the numerical performance of the consistency of the nearest neighbor weight function estimators is provided. The results obtained in the paper improve the conditions in the literature and generalize the existing results of independent random errors to the case of $$\varphi $$ -mixing random errors.

Some convergence theorems for RM algorithm

In this paper, we consider the multi-dimensional Kesten algorithm studied in Xu and Dai (2012). The convergence in rth mean for the algorithm is established under the weaker assumptions than Xu and Dai (2012). Moreover, we obtain also the almost sure convergence of the algorithm under the conditions of Xu and Dai (2012).

Spacelike hypersurfaces with constant rth mean curvature in steady state type spacetimes

We deal with spacelike hypersurfaces immersed with some constant rth mean curvature in a steady state type spacetime, that is, a generalized Robertson–Walker spacetime of the type \({-\mathbb{R} \times_{e^t} M^n}\). In this setting, supposing that the fiber M n of the ambient space has nonnegative constant sectional curvature, we establish characterization results concerning domains of the spacelike slices \({\{t\} \times M^n}\). Afterwards, we apply such characterization results to study the uniqueness of complete spacelike hypersurfaces with one end in such a ambient space.

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant rth mean curvature S r . We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with S r + 1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.

The Newton transformation and new integral formulae for foliated manifolds

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.

Understanding Convergence Concepts: A Visual-Minded and Graphical Simulation-Based Approach

This article describes the difficult concepts of convergence in probability, convergence almost surely, convergence in law, and convergence in rth mean using a visual-minded and graphical simulation-based approach. For this purpose, each probability of events is approximated by a frequency. An R package that reproduces all of the experiments cited in this article is available in CRAN. See the online Supplement for details.

Hypersurfaces in hyperbolic space associated with the conformal scalar curvature equation [formula omitted

We consider a class of oriented hypersurfaces in hyperbolic space satisfying ∑ r = 0 n ( c − n + 2 r ) ( n r ) H r = 0 , where H r is the rth mean curvature and c is a real constant. We show how this class is characterized by a harmonic map derived from the two hyperbolic Gauss maps. By looking at hypersurfaces as orthogonal to a congruence of geodesics, we also show the relation of such hypersurfaces with solutions of the equation Δ u + k u n + 2 n − 2 = 0 , where k ∈ { − 1 , 0 , 1 } . Finally, we apply the relation mentioned above to obtain examples and geometrical results for the hypersurfaces.

A new variational characterization of the Wulff shape

Given a positive function F on S n which satisfies a convexity condition, we define the rth anisotropic mean curvature function M r for hypersurfaces in R n + 1 which is a generalization of the usual rth mean curvature function. Let X : M → R n + 1 be an n-dimensional closed hypersurface with M r + 1 M r = constant , for some r with 1 ⩽ r ⩽ n − 1 , which is a critical point for a variational problem. We show that X ( M ) is stable if and only if X ( M ) is the Wulff shape.

Notions of exponential robust stochastic stability, ISS and their Lyapunov characterizations

AbstractConcepts of exponential global robust stability for stochastic control systems are analysed in terms of Lyapunov functions. The main result of the paper constitutes a generalization of a converse stability theorem due to Khasminskii for stochastic differential equations and establishes that, under certain hypotheses, the origin is robustly exponentially stable in the rth mean, if and only if the system admits a Lyapunov function which is smooth except possibly at the origin. The main result concerning robust asymptotic stability enable us to derive a Lyapunov‐like characterization for the concept of stochastic input‐to‐state stability (ISS). Copyright © 2003 John Wiley & Sons, Ltd.

A Reilly inequality for some natural elliptic operators on hypersurfaces

Let M n be a compact hypersurface of a real space form and L r the linearized operator of the first variation of the (r+1)th mean curvature, r∈{0,…,n}. In this paper, we obtain an optimal upper bound for the first eigenvalue of those operators L r , in terms of the rth mean curvatures of M

SEMIPARAMETRIC REGRESSION MODELS WITH LOCALLY GENERALIZED GAUSSIAN ERROR'S STRUCTURE

This paper proposes parametric component and nonparametric component estimators in a semiparametric regression models based on least squares and weight function's method, their strong consistency and rth mean consistency are obtained under a locally generalized Gaussian error's structure. Finally, the author showes that the usual weight functions based on nearest neighbor method satisfy the designed assumptions imposed.

On convergence rates of averages of weakly dependent random variables

Let {; X n } ∞ n=1 be a sequence of random variabls on a probability space, r > 1 and the delayed sum S m,n = ∑ m+n k=m+1 X k , where m ⩾ 0 and n ⩾ 1. Further, let the function ϱ( n) = sup m ⩾ 0 ∥(1/ n) S m,n ∥ r r satisfy ∑ ∞ n=1 ϱ( n)/ n < ∞, where ∥ X∥ r denotes the rth mean of random variable X. Then we prove that for 2 k ⩽ n < 2 k+1 and for m ⩾ 0, ∥(1/ n) S m,n ∥ r r ⩽ c r >;ϱ(2 k ) + ∑ k j=1 1 2 ((1 + c r )/2 r ) j−1 ϱ(2 k−j )} ≡ v k = o( k −1) as k → ∞, where c r = 2 r−1 . We also prove that for m ⩾ 0, p ⩾ 1, and ε > 0, > P } p n ⩾ 2 p ((1/ n)| S m,n |) > ε} ⩽ (2/ε r )∑ ∞ k=p v k → 0 as p → ∞.

A theorem of Reilly for the linearized operator of rth mean curvature and applications to stability

A theorem of Reilly for the linearized operator of rth mean curvature and applications to stability

A limit theorem for population-size-dependent branching processes

An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.