Related papers: History of the formulas and algorithms for pi
Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind).…
Sometimes only some digits of a numerical product or some terms of a polynomial or series product are required. Frequently these constitute the most significant or least significant part of the value, for example when computing initial…
Pebbles (calculos in Latin) are the "bits" used in the Ancients' four function calculator / computer. The Ancient Computer's normal mode is to work with numbers in what we would call exponential notation. Decimal numbers can have up to 10…
Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…
In terms of the hypergeometric method, we give the extensions of two known series for $\pi$. Further, other twenty-nine summation formulas for $\pi$, $\pi^2$ and $1/\pi$ with free parameters are also derived in the same way.
Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the…
Within the framework of computable infinitary continuous logic, we develop a system of hyperarithmetic numerals. These numerals are infinitary sentences in a metric language $L$ that have the same truth value in every interpretation of $L$.…
Software and hardware architectures are prone to modifications. We demonstrate how a mathematically founded powerful refinement calculus for a class of architectures, namely pipe and filter architectures, can be used to modify a system in a…
This study investigates the progress made in LP and MILP solver performance during the last two decades by comparing the solver software from the beginning of the millennium with the codes available today. On average, we found out that for…
In order to verify programs or hybrid systems, one often needs to prove that certain formulas are unsatisfiable. In this paper, we consider conjunctions of polynomial inequalities over the reals. Classical algorithms for deciding these not…
In the last couple of decades, the world has seen several stunning instances of quantum algorithms that provably outperform the best classical algorithms. For most problems, however, it is currently unknown whether quantum algorithms can…
BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive directly, without doing any searches, explicit digit extraction BBP-type formulas in…
A permutiple is the product of a digit preserving multiplication, that is, a number which is an integer multiple of some permutation of its digits. Certain permutiple problems, particularly transposable, cyclic, and, more recently,…
Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum…
Over the recent years the importance of numerical experiments has gradually been more recognized. Nonetheless, sufficient documentation of how computational results have been obtained is often not available. Especially in the scientific…
Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced,…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
Over the past two decades, the field of high-dimensional statistics has experienced substantial progress, driven largely by technological advances that have dramatically reduced the cost and effort for data collection and storage across a…
This paper presents a logical approach to the translation of functional calculi into concurrent process calculi. The starting point is a type system for the {\pi}-calculus closely related to linear logic. Decompositions of intuitionistic…