Related papers: On an approximation by Vaughan in restricted sets …
We propose here a number of approaches to implement constraint propagation for arithmetic constraints on integer intervals. To this end we introduce integer interval arithmetic. Each approach is explained using appropriate proof rules that…
In this paper, a sample-based procedure for obtaining simple and computable approximations of chance-constrained sets is proposed. The procedure allows to control the complexity of the approximating set, by defining families of…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
We prove some results concerning the distribution of primes on the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi} \sqrt{x} \log x,x]$ for all $x \geq 2$; this improves a…
In this paper we improve the estimate for the remainder term in the asymptotic formula concerning the circle problem in an arithmetic progression.
We consider the learning task consisting in predicting as well as the best function in a finite reference set G up to the smallest possible additive term. If R(g) denotes the generalization error of a prediction function g, under reasonable…
The Laplace approximation is a popular method for constructing a Gaussian approximation to the Bayesian posterior and thereby approximating the posterior mean and variance. But approximation quality is a concern. One might consider using…
We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution $\frac{1}{2} +…
In this paper, we make some conjectures on prime numbers that are sharper than those found in the current literature. First we describe our studies on Legendre's Conjecture which is still unsolved. Next, we show that Brocard's Conjecture…
We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We study rate of convergence of recursive estimation procedures for the general…
We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…
The paper considers the asymptotic of the ratio of the number of primes not exceeding the primorial and the number of residues in the reduced system of residues for the given primorial. We study the relationship between asymptotic lower…
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we…
It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…
Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $\pi(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that…
We prove novel convergence results for a stochastic proximal gradient algorithm suitable for solving a large class of convex optimization problems, where a convex objective function is given by the sum of a smooth and a possibly non-smooth…
We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the…
If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.
Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We…
Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the…