Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, a certain capacity called the tau invariant, and the total length of a metric graph. To put our formula in a broader context, we relate our work to the computation of heights attached to principally polarized abelian varieties.

One of the approaches to the Riemann Hypothesis is the Nyman–Beurling criterion. Cotangent sums play a significant role in this criterion. Here we investigate the values of these cotangent sums for various shifts of the argument.

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt−1 + Bt, t ∈ Z, where ((At, Bt))t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, p ∈ R. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

This paper discusses an experimental investigation of the behavior of sharp-notched circular tubes subjected to cyclic bending. The sharp-notched circular tubes of 304 stainless steel with three different diameter-to-thickness ratios (Do / t) were tested under symmetric curvature-controlled cyclic bending. It has been shown that the moment-curvature curves exhibited the loops with cyclic hardening and gradually becoming steady after a few cycles for all tested tubes. The ovalization-curvature curves revealed unsymmetric, ratcheting and increasing behavior with the number of cycles. In addition, five almost parallel lines corresponding to five different notch depths for each Do / t ratio were found from the experimental relationship between the cyclic controlled curvature and the number of cycles necessary to produce buckling on a log-log scale. Finally, an empirical relationship was proposed so it could be used for simulating the aforementioned relationship. By comparing with the experimental finding, the derived empirical relationship was in good agreement with the experimental data.

Recently, Wang and Xiao studied a four-dimensional competitive Lotka-Volterra systemwithin a deterministic environment in [11]. Withthe help of numerical example they showed the existence of a chaotic attractor through theperiod doubling route. In this paper, we are interested to study the dynamics of the samemodel in presence of environmental driving forces. To incorporate the environmentaldriving force into the deterministic system, we perturb the growth rates of each speciesby white noise terms. Then we prove that the unique positive global solution exists forthe noise added system and the general p-th order moment of it is boundedfor p ≥ 1, which ensures that the solution is stochasticallybounded. It is also shown that the solution of the stochastic system is stochasticallypermanent under some simple conditions. Finally , we demonstrate the noise inducedoscillation for the concerned model with the help of numerical example..

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

In a unified approach, this paper presents distributional properties of a Voronoi tessellation generated by a homogeneous Poisson point process in the Euclidean space of arbitrary dimension. Probability density functions and moments are given for characteristics of the ‘typical’ edge in lower-dimensional section hyperplanes (edge lengths, adjacent angles). We investigate relationships between edges and their neighbours, called Poisson points or centres; namely angular distributions, distances, and positions of neighbours relative to the edge. The approach is analytical, and the results are given partly explicitly and partly as integral expressions, which are suitable for the numerical calculations also presented.

The present paper considers methods for computing moments connected with subcritical multitype branching processes. By stressing the interplay between moment and cumulant structure, it is possible to prove simultaneously the existence of the moments. The kinds of moments considered are those arising in (1) times to extinction, (2) ultimate numbers of particles starting from a single particle, and (3) equilibrium numbers of particles for subcritical processes experiencing immigration.

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.