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We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)\chi(G)\xi(G), \end{equation} where $\chi$ is a Dirichlet character…

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets…

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds, which is called the "sum…

We consider the convex optimization problem P: min {f(x): x in K} where "f" is convex continuously differentiable, and K is a compact convex set in Rn with representation {x: g_j(x) >=0, j=1,;;,m} for some continuously differentiable…

The problem of optimal estimation of linear functional ${{A}_{N}}\xi =\sum\limits_{k=0}^{N}{a(k)\xi (k)}\,$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary $n$-th increments from observations of the…

Let $(X_i,i\geq 1)$ be a sequence of i.i.d. random variables with values in $[0,1]$, and $f$ be a function such that $`E(f(X_1)^2)<+\infty$. We show a functional central limit theorem for the process $t\mapsto \sum_{i=1}^n f(X_i)1_{X_i\leq…

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…

Algorithms for min-max optimization and variational inequalities are often studied under monotonicity assumptions. Motivated by non-monotone machine learning applications, we follow the line of works [Diakonikolas et al., 2021, Lee and Kim,…

We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness…

The problem of minimizing the least squares functional with a Fr\'echet differentiable, lower semi-continuous, convex penalizer $J$ is considered to be solved. The penalizer maps the functions of Banach space $\mathcal{V}$ into…

The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the…

The Salem problem to verify whether Fourier-Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we…

In this paper, we study the zero sets of the confluent hypergeometric function $_{1}F_{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!(\gamma)_{n}}z^{n}$, where $\alpha, \gamma, \gamma-\alpha\not\in \mathbb{Z}_{\leq 0}$, and…

We consider the problem of finding the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized. While the set of permutations maximizing this value are easily determined,…

We propose an improvement of the splitting functions at small x which overcomes the apparent problems encountered by the BFKL approach. We obtain a stable expansion for the x-evolution function chi(M) near M=0 by including in it a sequence…

In this paper, we establish coincidence-like results in the case when the values of the correspondences are not convex. In order to do this, we define a new type of correspondences, namely properly quasi-convex-like. Further, we apply the…

Calculations using the (exact) fermionic functional renormalization group are usually truncated at the second order of the corresponding hierarchy of coupled ordinary differential equations. We present a method for the systematic…

Let $\mathbb{F}_{p^{n}}$ be the finite field with $p^n$ elements and $\operatorname{Tr}(\cdot)$ be the trace function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$, where $p$ is a prime and $n$ is an integer. Inspired by the works of…

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=\lambda \mathbf{x}…

We consider the eigenvalue problem $K x = \lambda x$. Our analysis focuses on the convergence rates of eigenvalue and spectral subspace approximations for compact linear integral operator $K$ with Green's kernels. By employing orthogonal…