Related papers: Automated Search for Conjectures on Mathematical C…
We present a case study in {\it experimental} yet {\it rigorous} mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that {\it automatically conjectures}, and then {\it automatically proves}, closed-form…
In the words of the esteemed mathematician Paul Erd\"os, the mathematician's task is to \emph{prove and conjecture}. These two processes form the bedrock of all mathematical endeavours, and in the recent years, the mathematical community…
We present a comprehensive demonstration of how automated reasoning can assist mathematical research, both in the discovery of conjectures and in their verification. Our focus is a discrete geometry problem: What is $\mu_{5}(n)$, the…
Retrieving mathematical knowledge is a central task in both human-driven research, such as determining whether a result already exists, finding related results, and identifying historical origins, and in emerging AI systems for mathematics,…
Simple and robust algorithms are developed for compressible Euler equations with the stiffened gas equation of state (EOS), representing gaseous mixtures in thermal equilibrium and without chemical reactions. These algorithms use a fully…
Over the last few decades, many distinct lines of research aimed at automating mathematics have been developed, including computer algebra systems (CASs) for mathematical modelling, automated theorem provers for first-order logic, SAT/SMT…
Formula retrieval is an important topic in Mathematical Information Retrieval. We propose SSEmb, a novel embedding framework capable of capturing both structural and semantic features of mathematical formulas. Structurally, we employ Graph…
While contemporary deep learning malware detectors define a dominant defense paradigm, their sophistication also exposes them to novel structural evasion attacks, a limitation we attribute to their inherent inability to express epistemic…
Zaremba's Conjecture concerns the formation of continued fractions with partial quotients restricted to a given alphabet. In order to answer the numerous questions that arrive from this conjecture, it is best to consider a semi-group, often…
The Stochastic Approximation EM (SAEM) algorithm, a variant stochastic approximation of EM, is a versatile tool for inference in incomplete data models. In this paper, we review the fundamental EM algorithm and then focus especially on the…
Experimental mathematics is an experimental approach to mathematics in which programming and symbolic computation are used to investigate mathematical objects, identify properties and patterns, discover facts and formulas and even…
It is well-known that Abstract State Machines (ASMs) can simulate "step-by-step" any type of machines (Turing machines, RAMs, etc.). We aim to overcome two facts: 1) simulation is not identification, 2) the ASMs simulating machines of some…
Rowland and Zeilberger devised an approach to algorithmically determine the modulo $p^r$ reductions of values of combinatorial sequences representable as constant terms (building on work of Rowland and Yassawi). The resulting $p$-schemes…
There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln…
We propose a numerical test of fundamental physics based on the complexity measure of a general set of functions, which is directly related to the Kolmogorov (or algorithmic) complexity studied in mathematics and computer science. The…
In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we…
In this paper we explore how machine learning techniques can be applied to the discovery of efficient mathematical identities. We introduce an attribute grammar framework for representing symbolic expressions. Given a set of grammar rules…
This paper presents a new approach to statistical similarity assessment based on sequence alignment. The algorithm performs mutual matching of two random sequences by successively searching for common elements and by applying sequence…
Conjecturing and theorem proving are activities at the center of mathematical practice and are difficult to separate. In this paper, we propose a framework for completing incomplete conjectures and incomplete proofs. The framework can turn…
Brent and McMillan introduced in 1980 a new algorithm for the computation of Euler's constant $\gamma$, based on the use of the Bessel functions I\_0(x) and K\_0(x). It is the fastest known algorithm for the computation of $\gamma$. The…