Related papers: A Tale of Three Coauthors
We establish nontrivial bounds for bilinear sums involving the M\"obius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as M\"obius-function analogues of the ternary Goldbach problem.…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
In this article we introduce associative Look-Up Tables. With their help, pseudo sums are correctly determined. The set of limit distributions in a pseudo-summation scheme of i.i.d. random variables is described. Also, two special cases…
This paper generalizes work of Buzzard and Kilford to the case $p=3$, giving an explicit bound for the overconvergence of the quotient $E_\kappa / V(E_\kappa)$ and using this bound to prove that the eigencurve is a union of countably many…
Two Turing Machines may be able to answer questions about each other that they cannot answer about themselves.
New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising…
Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend…
Storytelling algorithms aim to 'connect the dots' between disparate documents by linking starting and ending documents through a series of intermediate documents. Existing storytelling algorithms are based on notions of coherence and…
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Statistical models that analyse (pairwise) relations between variables encompass assumptions about the underlying mechanism that generated the associations in the observed data. In the present paper we demonstrate that three Ising model…
We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new…
We prove that the sumset or the productset of any finite set of real numbers, $A,$ is at least $|A|^{4/3-\epsilon},$ improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, $E(A,A).$
We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.
We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.
Motivated by the works of Andrews-Merca and Guo-Zeng, we establish some truncated identities of Gauss by using some summation formulas from the works of Zhi-Guo Liu. These give three new expansions for partial sums of Gauss' triangular…
We study bounded and unbounded representations of the $*$-algebra $Q_{n,\lambda}(*)$ generated by $n$ idempotents whose sum equals $\lambda e$ ($\lambda\in{\mathbb C}$, $e$ is the identity).
A new finite form of de Finetti's representation theorem is established using elementary information-theoretic tools. The distribution of the first $k$ random variables in an exchangeable vector of $n\geq k$ random variables is close to a…
In this paper, we present several new bounds for the norm and numerical radius of sums of Hilbert space operators. The obtained bounds form a new collection that enriches our understanding of these bounds. We compare our bounds with the…
In this exposition the space of at most 3-element subsets of the circle, first identified by Borsuk and Bott, is used as a motivation to introduce the readers to a variety of methods in (algebraic) topology.
We improve the known upper bound for short exponential sums and increase the range on which a sharp upper bound is known.