Related papers: A model problem for multiplicative chaos in number…
In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent…
Hiary [3] has presented an algorithm which allows to evaluate the truncated theta function $\sum_{k=0}^n \exp(2\pi \i (zk+\tau k^2))$ to within $\pm \epsilon$ in $O(\ln(\tfrac{n}{\epsilon})^{\kappa})$ arithmetic operations for any real $z$…
We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask, whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We…
Consider the Hitting Set problem where, for a given universe $\mathcal{X} = \left\{ 1, ... , n \right\}$ and a collection of subsets $\mathcal{S}_1, ... , \mathcal{S}_m$, one seeks to identify the smallest subset of $\mathcal{X}$ which has…
The probabilistic satisfiability of a logical expression is a fundamental concept known as the partition function in statistical physics and field theory, an evaluation of a related graph's Tutte polynomial in mathematics, and the…
Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting…
We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As…
A connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results…
The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e. the optimal matching…
This thesis is divided in two parts. The first presents an overview of known results in statistical mechanics of disordered systems and its approach to random combinatorial optimization problems. The second part is a discussion of two…
The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…
We review the occurrence of the patterns of the onset of chaos in low-dimensional nonlinear dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics associated…
We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined…
We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval. We investigate the possibility of having power-law tails in the invariant measure by approximate solution of the…
In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $(N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients.…
In the weakened 16th Hilbert's Problem one asks for a bound of the number of limit cycles which appear after a polynomial perturbation of a planar polynomial Hamiltonian vector field. It is known that this number is finite for an individual…
In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-It\^o chaos representation to a nondegenerate Gaussian measure on a separable Hilbert space. Our first main result shows that,…
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.
We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted…
In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. Scaling limits of similar models have been studied recently \cite{FSa,FSb} going back to…