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We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(\varphi(x)-z)^2$, where $\varphi(x)\in…
Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…
In this paper, we use methods of exponential sums to derive a formula for estimating effective upper bounds of $|\zeta'(1/2+it)|$. Different effective upper bounds can be obtained by choosing different parameters.
In [On $IP^{\star}$sets and central sets, Combinatorica, 14 (1994) 269-277], N. Hindman and V.Bergelson proved additive $IP^{\star}$-sets contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto…
One of the most interesting formulas for multiple zeta values is the sum formula proved by Granville and Zagier independently in 1990s. Many variations and generalizations of it have been found since then. In this paper, we will provide a…
We explicitly describe an expansion of $e^{A+B}$ as an infinite sum of the products of $B$ multiplied by the exponential function of $A$. This is the explicit description of the Zassenhaus formula. We also express the…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
An algebraic framework in which to study infinite sums is proposed, complementing and augmenting the usual topological tools. The framework subsumes numerous examples in the literature. It is developed using many varied examples, with a…
A system of commutative complex numbers in 5 dimensions of the form u=x_0+h_1x_1+h_2x_2+h_3x_3+h_4x_4 is described in this paper, the variables x_0, x_1, x_2, x_3, x_4 being real numbers. The operations of addition and multiplication of the…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if…
We prove the enveloping property of the known divergent asymptotic expansions of the large real zeros of the cylinder and Airy functions, and thereby answering in the affirmative two conjectures posed by Elbert and Laforgia and by Fabijonas…
The evidence lower bound (ELBO) is one of the most central objectives for probabilistic unsupervised learning. For the ELBOs of several generative models and model classes, we here prove convergence to entropy sums. As one result, we…
We investigate inequalities for partial sums of complex numbers with bounded modulus and zero total sum, a topic referred to as "polygonal confinement". Starting from Steinitz's classical result, we provide detailed constructions yielding…
In this article, I introduce a group-theoretical method to prove positivity of certain linear combinations (with coefficients generally lying in $\mathbb{C}$) of exponential functions under a set of semidefinite linear constraints. The…
The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…
The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the…
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…