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Michael Th. Rassias

Publications and source records attributed to Michael Th. Rassias.

At least 19 recordsLinked to original sources
arXiv2025

Stochastic forward-backward-half forward splitting algorithm with variance reduction

In this paper, we present a stochastic forward-backward-half forward splitting algorithm with variance reduction for solving the structured monotone inclusion problem composed of a maximally monotone operator, a maximally monotone operator and a cocoercive operator in a separable real Hilbert space. By deffining a Lyapunov function, we establish the weak almost sure convergence of the proposed algorithm, and obtain the linear convergence when one of the maximally monotone operators is strongly monotone. Numerical examples are provided to show the performance of the proposed algorithm.

math.OC↗
arXiv2025

Contractive difference-of-convex algorithms

The difference-of-convex algorithm (DCA) and its variants are the most popular methods to solve the difference-of-convex optimization problem. Each iteration of them is reduced to a convex optimization problem, which generally needs to be solved by iterative methods such as proximal gradient algorithm. However, these algorithms essentially belong to some iterative methods of fixed point problems of averaged mappings, and their convergence speed is generally slow. Furthermore, there is seldom research on the termination rule of these iterative algorithms solving the subproblem of DCA. To overcome these defects, we ffrstly show that the subproblem of the linearized proximal method (LPM) in each iteration is equal to the ffxed point problem of a contraction. Secondly, by using Picard iteration to approximately solve the subproblem of LPM in each iteration, we propose a contractive difference-ofconvex algorithm (cDCA) where an adaptive termination rule is presented. Both global subsequential convergence and global convergence of the whole sequence of cDCA are established. Finally, preliminary results from numerical experiments are promising.

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arXiv2024

Recent results on large gaps between primes

One of the themes of this paper is recent results on large gaps between primes. The first of these results has been achieved in the paper [12] by Ford, Green, Konyagin and Tao. It was later improved in the joint paper [13] of these four authors with Maynard. One of the main ingredients of these results are old methods due to Erdős and Rankin. Other ingredients are important breakthrough results due to Goldston, Pintz and Yildirim [15, 16, 17], and their extension by Maynard on small gaps between primes. All these previous results are discussed shortly. The results on the appearance of $k$-th powers of primes contained in those large gaps, obtained by the author in joint work with Maier [23, 24, 25] are based on a combination of the results just described with the matrix method of Maier.

math.NT↗
arXiv2021

The ternary Goldbach problem with a prime with a missing digit and primes of special types

Let $$γ^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ γ^*<\frac{1}{c_0}\leq 1\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace one of the Piatetski-Shapiro primes $p_2$ and $p_3$ by primes of the type $$p=x^2+y^2+1\:.$$

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arXiv2021

The ternary Goldbach problem with two Piatetski-Shapiro primes and a prime with a missing digit

Let $$γ^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results of A. Balog and J. Friedlander on Piatetski-Shapiro primes and the Hardy-Littlewood circle method in two variables. This is the first result on the ternary Goldbach problem with primes of mixed type which involves primes with missing digits.

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arXiv2018

Explicit estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound - essentially the number of terms - by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

math.CA↗
arXiv2017

From a cotangent sum to a generalized totient function

In this paper we investigate a certain category of cotangent sums and more specifically the sum $$\sum_{m=1}^{b-1}\cot\left(\frac{πm}{b}\right)\sin^{3}\left(2πm\frac{a}{b}\right)\:$$ and associate the distribution of its values to a generalized totient function $ϕ(n,A,B)$, where $$ϕ(n,A,B):=\sum_{\substack{A\leq k \leq B \\ (n,k)=1}}1\:.$$ One of the methods used consists in the exploitation of relations between trigonometric sums and the fractional part of a real number.

math.CA↗
arXiv2017

A criterion related to the Riemann Hypothesis

A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2π}\int_{-\infty}^\infty\left|1-ζA_N\left(\frac{1}{2}+it\right)\right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where the infimum is over all Dirichlet polynomials $$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we investigate $d_N^2$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line. Thus we obtain a criterion for the non validity of the Riemann Hypothesis.

math.CA↗
arXiv2017

Estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the Möbius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the Möbius function. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

math.CA↗