A note on almost sure uniform and complete convergences of a sequence of random variables

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.     

Characterizing optimal sampling of binary contingency tables via the configuration model

A binary contingency table is an m × n array of binary entries with row sums r = (r₁, …, r_m) and column sums c = (c₁, …, c_n). The configuration model generates a contingency table by considering r_i tokens of type 1 for each row i and c_j tokens of type 2 for each column j, and then taking a uniformly random pairing between type‐1 and type‐2 tokens. We give a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as \begin{align*}N=\sum_{i=1}^m r_i=\sum_{j=1}^n c_j\end{align*} goes to ∞. Our finding shows surprising differences from recent results for binary symmetric contingency tables. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Solutions of the Multivariate Inverse Frobenius–Perron Problem

We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ, find a deterministic map M such that iterations of M tend to ρ in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps.     

一类非自治Brinkman-Forchheimer方程的一致吸引子

This paper is concerned with the three-dimensional non-autonomous BrinkmanForchheimer equation.By Galerkin approximation method,we give the existence and uniqueness of weak solutions for non-autonomous Brinkman-Forchheimer equation.And we investigate the asymptotic behavior of the weak solution,the existence and structures of the(H,H)-uniform attractor and(H,V)-uniform attractor.Then we prove that an L 2-uniform attractor is actually an H 1-uniform attractor.     

An algorithm for the uniform sampling of iso-energy surfaces and for the calculation of microcanonical averages

In this article an algorithm is proposed to efficiently perform the uniform sampling of an iso-energy surface corresponding to a fixed potential energy U of a molecular system, and for calculating averages of certain quantities over microstates having this energy (microcanonical averages). The developed sampling technique is based upon the combination of a recently proposed method for performing constant potential energy molecular dynamics simulations [Rapallo, A. J Chem Phys 2004, 121, 4033] with well-established thermostatting techniques used in the framework of standard molecular dynamics simulations, such as the Andersen thermostat, and the Nose-Hoover chain thermostat. The proposed strategy leads to very accurate and drift-free potential energy conservation during the whole sampling process, and, very important, specially when dealing with high-dimensional or complicated potential functions, it does not require the calculation of the potential energy function hessian. The technique proved to be very reliable for sampling both low- and high-dimensional surfaces.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

What is the smallest possible constant in Céa’s lemma?

We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝ^d with d ∈ {1, 2, 3, ...} The constant C ⩾ 1 appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to as the discretization parameter h tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented. This research was supported by Shandong Province Young Scientists Foundation of China 2005BS01008, Institutional Research Plan AV02 101 90503, and by Grant No A 1019201 of the Academy of Sciences of the Czech Republic.     

拟柱面激光二极管阵列端面泵浦方式提高固体激光器性能

就高功率激光二极管阵列端面泵浦大口径放大器提出一种新的耦合方式:激光二极管阵列拟柱面排布,即所有激光二极管面阵成1维圆弧型排列,圆弧的圆心在增益介质的几何中心,其后紧接一个空心导管进行耦合传输。建立了3维光线追迹程序对这种新耦合方式的特性进行模拟。模拟计算结果表明:这种耦合方式中二极管阵列排布方式灵活,当二极管阵列面阵单元以1×12(圆弧方向)、2×6(圆弧方向)、3×4(圆弧方向)这3种排布方式排布时,在较大的圆半径变化范围内均能实现高的输出耦合效率和高的能量沉积效率;当增益介质紧贴导管输出放置时,3种方式排列均能在增益介质中实现均匀平顶分布;当快轴方向所排阵列个数与慢轴方向所排阵列个数之比接近慢轴发散角与快轴发散角之比时,能获得更好的耦合效果。     

亥姆霍兹线圈磁场的探究

定量计算了亥姆霍兹线圈空间磁场的分布,并用Mathmatic进行空间模拟,形象地描述了亥姆霍兹线圈的空间磁场分布,同时就其均匀性与三线圈和载荷旋转圆盘的磁场进行了对比讨论.