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The Sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis. Even for other functions, the Sinc approximation works accurately when combined…
Synthetic data is a standard component in training large language models, yet systematic comparisons across design dimensions, including rephrasing strategy, generator model, and source data, remain absent. We conduct extensive controlled…
We target the problem of provably computing the equivalence between two complex expression trees. To this end, we formalize the problem of equivalence between two such programs as finding a set of semantics-preserving rewrite rules from one…
This collection of sums and integrals has been harvested from the mathematical and physical literature in unstructured ways. Its main use is backtracking the original sources whenever an integral of the reader's application resembles one of…
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements…
We propose several procedures for creating new families of integer sequences based on the method of Cantor diagonalization. Then we modify and generalize this method. The paper includes explicit formulas for most proposed families of…
Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…
Sequence rotation consists of a circular shift of the sequence's elements by a given number of positions. We present the four classic algorithms to rotate a sequence; the loop invariants underlying their correctness; detailed correctness…
Some aspects of Computer Algebra (notably Computation Group Theory and Computational Number Theory) have some good databases of examples, typically of the form "all the X up to size n". But most of the others, especially on the polynomial…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a…
A program is a finite piece of data that produces a (possibly infinite) sequence of primitive instructions. From scratch we develop a linear notation for sequential, imperative programs, using a familiar class of primitive instructions and…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
In this paper, we analyze the complexity of functional programs written in the interaction-net computation model, an asynchronous, parallel and confluent model that generalizes linear-logic proof nets. Employing user-defined sized and…
Integer programming is concerned with solving linear systems of equations over the non-negative integers. The basic question is to find a solution which minimizes a given linear objective function for a fixed right hand side. Here we also…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
There is a widespread need for techniques that can discover structure from time series data. Recently introduced techniques such as Automatic Bayesian Covariance Discovery (ABCD) provide a way to find structure within a single time series…
In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…
We survey key techniques and results from approximation theory in the context of uniform approximations to real functions such as e^{-x}, 1/x, and x^k. We then present a selection of results demonstrating how such approximations can be used…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…