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In this paper, we focus on the topic Synchronization and consensus of Complex Networks and their relationships. It is revealed that two topics are closely relating to each other and all results given in \cite{Li1,Li2} and many other papers…
Using an original method, we find the algebra of generalized symmetries of a remarkable (1+2)-dimensional ultraparabolic Fokker-Planck equation, which is also called the Kolmogorov equation and is singled out within the entire class of…
In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from…
The standard formula for the multi-section of the general linear three-term recurrence relation is simplified in terms of Chebyshev S-polynomials.
We prove a generalisation of Vinogradov's theorem by finding for $m\geqslant 3$ and fixed positive integers $c_1, \dots ,c_m, r_1, \dots , r_m$ the asymptotics of the number of sequences $(n_1, \dots ,n_m) \in \mathbf{N}^{m}$ such that…
In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More…
Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen \& Cuny who showed convergence of a series along randomly perturbed times for…
We study the free central limit theorem for not necessarily identically distributed free random variables where the limiting distribution is the semicircle distribution. Starting from an estimate for the Kolmogorov distance between the…
We study an extension of Montgomery's pair-correlation conjecture and its relevance in some problems on the distribution of prime numbers.
We study the convergence of centered and normalized sums of i.i.d. random elements of the space $\mathcal{D}$ of c{{\'a}}dl{{\'a}}g functions endowed with Skorohod's $J\_1$ topology, to stable distributions in $\mathcal D$. Our results are…
In this dissertation, we show that the Central Limit Theorem and the Invariance Principle for Discrete Fourier Transforms discovered by Peligrad and Wu can be extended to the quenched setting. We show that the random normalization…
We introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying {Q,Q} =0). We develop the theory of connections on modules over Q-algebras…
Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011)…
In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the…
We derive concentration inequalities for sums of independent and identically distributed random variables that yield non-asymptotic generalizations of several strong laws of large numbers including some of those due to Kolmogorov [1930],…
We give improved bounds for the equidistribution of (multiparameter) nilsequences subject to any degree filtration. The bounds we obtain are single exponential in dimension, improving on double exponential bounds of Green and Tao. To obtain…
Weak convergence of various general functionals of partial sums of dependent random variables to stochastic integral now play a major role in the modern statistics theory. In this paper, we obtain the weak convergence of various general…
Using the approach of N. Etemadi for the Strong Law of Large Numbers (SLLN) from 1981 and the elaboration of this approach by S. Cs\"org\H{o}, K. Tandori and V. Totik from 1983, I give weak conditions under which the SLLN still holds for…
In this paper, we establish a residue theorem for Malcev-Neumann series that requires few constraints, and includes previously known combinatorial residue theorems as special cases. Our residue theorem identifies the residues of two formal…
In a recent work [1, 2] Sjoberg remarked that generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to…