Fragmentations with self-similar branching speeds

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

In this paper, we consider the sum Snξ = ξ1 + ... + ξn of possibly dependent and nonidentically distributed real-valued random variables ξ1, ... , ξn with consistently varying distributions. By assuming that collection {ξ1, ... , ξn} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((Snξ)α1(Snξ > x)) and E((Snξ – x)+)α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.

On the Operator ⊕k,m Related to the Wave Equation and Laplacian

In this article, we study the fundamental solution of the operator $\oplus _{m}^{k}$, iterated $k$-times and is defined by$$\oplus _{m}^{k} = \left[\left(\sum_{r=1}^{p} \frac{\partial^2} {\partial x_r^2}+m^{2}\right)^4 - \left( \sum_{j=p+1}^{p+q} \frac{\partial^2}{\partial x_{j}^2} \right)^4 \right ]^k,$$ where $m$ is a nonnegative real number, $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$,$x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n$, $k$ is a nonnegative integer. At first we study the fundamental solution of the operator $\oplus _{m}^{k}$ and after that, we apply such the fundamental solution to solve for the solution of the equation $\oplus _{m}^{k}u(x)= f(x)$, where $f(x)$ is generalized function and $u(x)$ is unknown function for $ x\in \mathbb{R}^{n}$.

Tails of the Moments for Sums with Dominatedly Varying Random Summands

The asymptotic behaviour of the tail expectation ?E(Snξ)α?{Snξ>x} is investigated, where exponent α is a nonnegative real number and Snξ=ξ1+…+ξn is a sum of dominatedly varying and not necessarily identically distributed random summands, following a specific dependence structure. It turns out that the tail expectation of such a sum can be asymptotically bounded from above and below by the sums of expectations ?Eξiα?{ξi>x} with correcting constants. The obtained results are extended to the case of randomly weighted sums, where collections of random weights and primary random variables are independent. For illustration of the results obtained, some particular examples are given, where dependence between random variables is modelled in copulas framework.

Minimum Variable Connectivity Index of Trees of a Fixed Order

The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a nonnegative real number, EG is the edge set of G, and dt denotes the degree of an arbitrary vertex t in G. Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors’ knowledge, mathematical properties of the variable connectivity index, for γ>0, have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order n, where n≥4.

On the Elementary Solution for the Partial Differential Operator $\circledcirc_c^{k}$ Related to the Wave Equation

In this article, we defined the operator $\diamondsuit _{m,c}^{k}$ which is iterated $k$-times and is defined by$$\diamondsuit _{m,c}^{k}=\left[\left(\frac{1}{c^2}\sum_{i=1}^{p}\frac{\partial ^{2}}{\partial x_{i}^{2}} +\frac{m^{2}}{2}\right)^{2} - \left(\sum_{j=p+1}^{p+q}\frac{\partial ^{2}}{\partial x_{j}^{2}} - \frac{m^{2}}{2}\right)^{2}\right]^{k},$$where $m$ is a nonnegative real number, $c$ is a positive real number and $p+q=n$ is the dimension of the $n$-dimensional Euclidean space $\mathbb{R}^{n}$, $x=(x_{1},\ldots x_{n})\in\mathbb{R}^{n}$ and $k$ is a nonnegative integer. We obtain a causal and anticausal solutionof the operator $\diamondsuit _{m,c}^{k}$, iterated $k$-times.

A NOTE ON THE FUNDAMENTAL THEOREM OF ALGEBRA

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

New ART2/ART2A Algorithm Apply to Entire Real Number Field

In this paper, two shortcomings of standard ART2/ART2A algorithm were revealed through theoretical analysis: (1)Standard ART2/ART2A algorithm is only suitable for the case in the nonnegative real number field because of a limit of pretreating process in F1layer; (2)Even through all input patterns are shifted to the nonnegative real number field through coordinate transformation, the standard ART2/ART2A algorithm can not correctly recognize those patterns which have same phase, but different amplitudes. As a result, the standard ART2/ART2A algorithm is not quite suitable for universal pattern recognition. So this paper presented a new nonlinear transforming function in F1layer and a new competitive learning formula in F2layer for traditional ART2/ART2A algorithm. The applicable scope of the new ART2/ART2A algorithm is expanded to entire real number field from nonnegative real number field. The result of typical calculation example shows that the presented algorithm is effective.

On the structure of Riemannian manifolds of almost nonnegative Ricci curvature

We study the structure of manifolds with almost nonnegative Ricci curvature. We prove a compact Riemannian manifold with bounded curvature, diameter bounded from above, and Ricci curvature bounded from below by an almost nonnegative real number such that the first Betti number havingcodimension two is an infranilmanifold or a finite cover is a sphere bundle over a torus. Furthermore, if we assume the Ricci curvature is bounded and volume is bounded from below, then the manifold must be an infranilmanifold.