Related papers: Approximations of generating functions and a few c…
Program synthesis is a class of regression problems where one seeks a solution, in the form of a source-code program, mapping the inputs to their corresponding outputs exactly. Due to its precise and combinatorial nature, program synthesis…
This is a collection of examples showing how the GAP system can be used to compute information about the probabilistic generation of finite almost simple groups. It includes all examples that were needed for the computational results in the…
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum…
Improving upon previous work on the subject, we use Wright's Circle Method to derive an asymptotic formula for the number of parts in all partitions of an integer that are in any given arithmetic progression.
Approximate integer programming is the following: For a convex body $K \subseteq \mathbb{R}^n$, either determine whether $K \cap \mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity…
This is a conspectus of definite integrals, products and series. These formulae involve special functions in the integrand and summand functions and closed form solutions. Some of the special cases are stated in terms of fundamental…
We describe the design of Comlex Syntax, a computational lexicon providing detailed syntactic information for approximately 38,000 English headwords. We consider the types of errors which arise in creating such a lexicon, and how such…
Probabilistic programs encode stochastic models as ordinary-looking programs with primitives for sampling numbers from predefined distributions and conditioning. Their applications include, among many others, machine learning and modeling…
In 1930s Paul Erdos conjectured that for any positive integer $C$ in any infinite $\pm 1$ sequence $(x_n)$ there exists a subsequence $x_d, x_{2d}, x_{3d},\dots, x_{kd}$, for some positive integers $k$ and $d$, such that $\mid \sum_{i=1}^k…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
We classify all of the groups with twelve or fewer subgroups. This paper is the proof of the entries in a submission to the Online Encyclopedia of Integer Sequences.
In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for noncommutative symmetric functions which enables one to count words and permutations with restrictions on the lengths of their increasing runs. We generalize Gessel's theorem…
We present a software package that guesses formulae for sequences of, for example, rational numbers or rational functions, given the first few terms. We implement an algorithm due to Bernhard Beckermann and George Labahn, together with some…
In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
In this article, dedicated with admiration and gratitude to guru Neil Sloane on his 75-th birthday, we observe that the generating functions for multi-set permutations that do not contain an increasing subsequence of length d, and where…
Sequence-to-sequence models have lead to significant progress in keyphrase generation, but it remains unknown whether they are reliable enough to be beneficial for document retrieval. This study provides empirical evidence that such models…
In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all…
We present an algorithm for synthesizing program loops satisfying a given polynomial loop invariant. The class of loops we consider can be modeled by a system of algebraic recurrence equations with constant coefficients. We turn the task of…
With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation -- bringing two replicas of the same data structure as close as possible…