Preliminary Analysis on the Probabilistic Feature of a Dark Cave of Digits

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Infinite-Dimensional Quadrature and Approximation of Distributions

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.      

Probabilistic Extensions of the Erdős–Ko–Rado Property

The classical Erdős–Ko–Rado (EKR) Theorem states that if we choose a family of subsets, each of size k, from a fixed set of size , then the largest possible pairwise intersecting family has size . We consider the probability that a randomly selected family of size t=t _n has the EKR property (pairwise nonempty intersection) as n and k=k _n tend to infinity, the latter at a specific rate. As t gets large, the EKR property is less likely to occur, while as t gets smaller, the EKR property is satisfied with high probability. We derive the threshold value for t using Janson’s inequality. Using the Stein–Chen method we show that the distribution of X ₀, defined as the number of disjoint pairs of subsets in our family, can be approximated by a Poisson distribution. We extend our results to yield similar conclusions for X _i , the number of pairs of subsets that overlap in exactly i elements. Finally, we show that the joint distribution (X ₀, X ₁, ..., X _b ) can be approximated by a multidimensional Poisson vector with independent components.      

A variant of Newton’s method based on Simpson’s three-eighths rule for nonlinear equations

We propose a new variant of Newton’s method based on Simpson’s three-eighth rule. It can be shown that the new method is cubically convergent.     

Analysis of astronomical records of King Wu’s Conquest

All related astronomical records of King Wu ’ s Conquest have been searched and analysed comprehensively. Constrained by the newest conclusions of archeology, philology and history in the Xia-ShangZhou Chronology Project and based mainly on dates inWucheng, Jupiter’s position inGuoyu and information on the season, our first choice of the date of King Wu’s Conquest is Jun. 20, BC104.6. This conclusion explains properly most relevant literature.     

Beppo Levi’s Analysis of the Paradoxes

This paper presents and comments the content of a note by Beppo Levi on logical paradoxes. Though the existence of this contribution is known, very little analysis of it is available in the literature. I put the emphasis on Levi’s usage of “elementation procedures” for solving the set-theoretical paradoxes, which is the most original part of Levi’s approach to the topic.     

Homological properties of Podle′s quantum spheres

The standard Podle′s quantum sphere is Artin-Schelter regular as showed by Kra¨hmer(2012).The non-standard Podle′s quantum spheres are proved to be Auslander-regular,Cohen-Macaulay and Artin-Schelter regular in this paper.     

Quadrature and Schatz’s Pointwise Estimates for Finite Element Methods

We investigate numerical integration effects on weighted pointwise estimates. We prove that local weighted pointwise estimates will hold, modulo a higher order term and a negative-order norm, as long as we use an appropriate quadrature rule. To complete the analysis in an application, we also prove optimal negative-order norm estimates for a corner problem taking into account quadrature. Finally, we present an example to show that our result is sharp. AMS subject classification (2000) 65N15     

On a Functional Equation Associated with Simpson’s Rule

In this paper, we determine the general solution of the functional equation $$f(x)-g(y)=(x-y)\lbrack h(x+y)+\psi (x)+\phi (y)\rbrack$$ for all real numbers x and y. This equation arises in connection with Simpson’s Rule for the numerical evaluation of definite integrals. The solution of this functional equation is achieved through the functional equation $$g(x)-g(y)=(x-y)f(x+y)+(x+y)f(x-y).$$      

Analysis of the hyper-chaos generated from Chen’s system

This letter does some further researches on a newly proposed hyper-chaos found in [Gao TG, Chen ZQ, Chen G. A hyper-chaos Generated from Chen’s system, Int J Mod Phys C 2006;17(4):471–8], and gets some new results with some parameters of the system are definite and the other one varies. The basic dynamical behavior of the system under various conditions is analyzed, and the new findings is not only demonstrated by computer simulations but also verified with bifurcation analysis.     

Retraction Notice to ＂A Mathematical Proof a Probabilistic Model of Hardy＇s Inequality＂

The Editor-in-Chief, after being alerted that this article is essentially the same as [On an Argument of K6rner and Hardy＇s Inequality, M. Sababheh, Analysis Mathematica, 34（2008）, 51-57], requested the retraction of the paper.     

Probabilistic Treatment of Schrodinger Equation With Infinite Gauge

Consider Dirichlet problem of Schrodinger equation     

On Non-additive Probabilistic Inequalities of H?lder-type

Non-additive measure is a generalization of additive probability measure. Integral inequalities play important roles in classical probability and measure theory. Some well-known inequalities such as the Minkowski inequality and the H?lder inequality play important roles not only in the theoretical area but also in application. Non-additive integrals are useful tools in several theoretical and applied statistics which have been built on non-additive measure. For instance, in decision theory and applied statistics, the use of the non-additive integrals can be envisaged from two points of view: decision under uncertainty and multi-criteria decision-making. In fact, the non-additive integrals provide useful tools in many problems in engineering and social choice where the aggregation of data is required. In this paper, H?lder and Minkowski type inequalities for semi(co)normed non-additive integrals are discussed. The main results of this paper generalize some previous results obtained by the authors.     

Analysis of stability boundaries of satellite’s equilibrium attitude in a circular orbit

An asymmetric satellite equipped with control momentum gyroscopes (CMGs) with the center of mass of the system moving uniformly in a circular orbit was considered. The stability of a relative equilibrium attitude of the satellite was analyzed using Lyapunov’s direct method. The Lyapunov function V is a positive definite integral of the total energy of the perturbed motion of the system. The asymptotic stability analysis of the stationary motion of the conservative system was based on the Barbashin–Krasovskii theorem on the nonexistence of integer trajectories of the set \(\dot V\), which was obtained using the differential equations of motion of the satellite with CMGs. By analyzing the sign definiteness of the quadratic part of V, it was found earlier by V.V. Sazonov that the stability region is described by four strict inequalities. The asymptotic stability at the stability boundary was analyzed by sequentially turning these inequalities into equalities with terms of orders higher than the second taken into account in V. The sign definiteness analysis of the inhomogeneous function V at the stability boundary involved a huge amount of computations related to the multiplication, expansion, substitution, and factorization of symbolic expressions. The computations were performed by applying a computer algebra system on a personal computer.     

Analysis of a generalized Lebesgue identity in Ramanujan’s Lost Notebook

We analyze a two-parameter q-series identity in Ramanujan’s Lost Notebook that generalizes the product part of the fundamental one-parameter Lebesgue identity. From reformulations of this two-parameter identity, we deduce new partition theorems including variants of the Gauss triangular number identity and Euler’s pentagonal number theorem. We discuss connections with a partial theta identity of Ramanujan and with several classical results such as those of Sylvester and Göllnitz–Gordon.     

Qualitative Analysis of Gradient-Type Systems with Oscillatory Nonlinearities on the Sierpi′nski Gasket

Under an appropriate oscillating behavior either at zero or at infinity of the nonlinear data, the existence of a sequence of weak solutions for parametric quasilinear systems of the gradient-type on the Sierpi′nski gasket is proved. Moreover, by adopting the same hypotheses on the potential and in presence of suitable small perturbations, the same conclusion is achieved. The approach is based on variational methods and on certain analytic and geometrical properties of the Sierpi′nski fractal as, for instance, a compact embedding result due to Fukushima and Shima.     

Probabilistic and fractal aspects of Lévy trees

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted -trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.