Related papers: Identical representation functions of linear forms

In 1978, Allan.Wm.Johnson obtained an example of the representation of 1 by the sum of reciprocals of the product of two distinct prime numbers. His example has 48 terms [1], and we had no examples which have less than 48 terms until now.…

Natural logic offers a powerful relational conception of meaning that is a natural counterpart to distributed semantic representations, which have proven valuable in a wide range of sophisticated language tasks. However, it remains an open…

We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We…

New elements of the dual cone of the set of fermion N-representable 2-density operators are proposed. So far, the explicit form of the corresponding necessary conditions for N-representability is obtained for N=3. In this case the new…

We examine the lattice generated by two pairs of supplementary vector subspaces of a finite-dimensional vector space by intersection and sum, with the aim of applying the results to the study of representations admitting two pairs of…

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions…

We study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices…

Natural language semantics has recently sought to combine the complementary strengths of formal and distributional approaches to meaning. More specifically, proposals have been put forward to augment formal semantic machinery with…

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…

We call a set of positive integers closed under taking unitary divisors a unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial…

This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and…

We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in…

Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L over a finite alphabet A. For these systems, we obtain a…

We prove some cases of a conjecture by Farhi on the representation of every positive integer as the sum of three terms of the sequence $ \left\lfloor\frac{n^2}{a}\right\rfloor$. This is done by generalizing a method used by Farhi in his…

An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More…

Computational models are quantitative representations of systems. By analyzing and comparing the outputs of such models, it is possible to gain a better understanding of the system itself. Though as the complexity of model outputs…

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the…

Random features are a powerful technique for rewriting positive-definite kernels as linear products. They bring linear tools to bear in important nonlinear domains like KNNs and attention. Unfortunately, practical implementations require…

We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…