arXiv subjects

Ofir David

Publications and source records attributed to Ofir David.

17 recordsLinked to original sources

arXiv2024

Unsupervised Discovery of Formulas for Mathematical Constants

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for $\pi$, $\ln(2)$, Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

cs.AI↗

arXiv2024

On the rate of convergence of continued fraction statistics of random rationals

We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator $q$, approaches the Gauss-Kuzmin statistics with polynomial rate in $q$. This improves on previous results giving the convergence without rate. As an application of this effective rate of convergence, we show that the statistics of a randomly chosen rational in the unit interval, with a fixed large denominator $q$ and prime numerator, also approaches the Gauss-Kuzmin statistics. Our results are obtained as applications of improved non-escape of mass and equidistribution statements for the geodesic flow on the space $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$.

math.DS↗

arXiv2023

Algorithm-assisted discovery of an intrinsic order among mathematical constants

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of $\zeta(3)$. Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

cs.AI↗

arXiv2023

On Euler polynomial continued fractions

In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing properties of simple continued fractions, polynomial continued fractions have many interesting patterns which can be exploited. Specifically, we explore the Euler continued fractions within this framework and present an algorithm for their identification

math.DS↗

arXiv2023

The conservative matrix field

We present a new structure called the "conservative matrix field", initially developed to elucidate and provide insight into the methodologies employed by Ap\'ery's in his proof of the irrationality of the Riemann zeta function at 3. This framework is also applicable to other well known mathematical constants, such as e, {\pi}, ln(2), and more, and can be used to study their properties. Moreover, the conservative matrix field exhibits inherent connections to various ideas and techniques in number theory, thereby indicating promising avenues for further applications and investigations.

math.GM↗

arXiv2022

Automated Search for Conjectures on Mathematical Constants using Analysis of Integer Sequences

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for $e, e^2, tan(1)$, and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

math.NT↗

arXiv2019

Shearing in the space of adelic lattices

In this notes we show how a problem regarding continued fractions of rational numbers, lead to several phenomena in number theory and dynamics, and eventually to the problem of shearing of divergent diagonal orbits in the space of adelic lattices. Finding these ideas quite interesting, the first half of these notes is about explaining theses ideas, the intuition and motivation behind them, and the second contains the details and proofs.

math.DS↗

arXiv2019

Graphs with large girth and free groups

We use Margulis' construction together with lattice counting arguments to build Cayley graphs on $\mathrm{SL}_{2}\left(\mathbb{F}_{p}\right),\;p\to\infty$ which are d-regular graphs with girth $\geq\frac{2}{3}\frac{\ln\left(n\right)}{\ln\left(d-1\right)+\ln\left(C\right)}$ for some absolute constant C.

math.GR↗

arXiv2019

Local to global property in free groups

The local to global property for an equation $\psi$ over a group G asks to show that $\psi$ is solvable in G if and only if it is solvable in every finite quotient of G. In this paper we focus that in order to prove this local to global property for free groups $G=F_k$, it is enough to prove for k less or equal the number of parameters in $\psi$. In particular we use it to show that the local to global property holds for m-powers in free groups.

math.GR↗

arXiv2017

Equidistribution of divergent orbits of the diagonal group in the space of lattices

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: The discriminant - an integer - and the type - an integer vector. We then study the question of the limit distributional behaviour of these orbits as the discriminant goes to infinity. Using entropy methods we prove that for divergent orbits of a specific type, virtually any sequence of orbits equidistribute as the discriminant goes to infinity. Using measure rigidity for higher rank diagonal actions we complement this result and show that in dimension 3 or higher only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.

math.DS↗

arXiv2017

Equidistribution of divergent orbits and continued fraction expansion of rationals

We establish an equidistribution result for push-forwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis we obtain new results regarding the asymptotic normality of the continued fraction expansions of most rationals with a high denominator as well as an estimate on the length of their continued fraction expansions. By similar methods we also establish a complementary result to Zaremba's conjecture. Namely, we show that given a bound M, for any large q, the number of rationals $p/q\in [0,1]$ for which the coefficients of the continued fraction expansion of p/q are bounded by M is $o(q^{1-\epsilon})$ for some $\epsilon>0$ which depends on M.

math.DS↗

arXiv2016

On statistical learning via the lens of compression

This work continues the study of the relationship between sample compression schemes and statistical learning, which has been mostly investigated within the framework of binary classification. The central theme of this work is establishing equivalences between learnability and compressibility, and utilizing these equivalences in the study of statistical learning theory. We begin with the setting of multiclass categorization (zero/one loss). We prove that in this case learnability is equivalent to compression of logarithmic sample size, and that uniform convergence implies compression of constant size. We then consider Vapnik's general learning setting: we show that in order to extend the compressibility-learnability equivalence to this case, it is necessary to consider an approximate variant of compression. Finally, we provide some applications of the compressibility-learnability equivalences: (i) Agnostic-case learnability and realizable-case learnability are equivalent in multiclass categorization problems (in terms of sample complexity). (ii) This equivalence between agnostic-case learnability and realizable-case learnability does not hold for general learning problems: There exists a learning problem whose loss function takes just three values, under which agnostic-case and realizable-case learnability are not equivalent. (iii) Uniform convergence implies compression of constant size in multiclass categorization problems. Part of the argument includes an analysis of the uniform convergence rate in terms of the graph dimension, in which we improve upon previous bounds. (iv) A dichotomy for sample compression in multiclass categorization problems: If a non-trivial compression exists then a compression of logarithmic size exists. (v) A compactness theorem for multiclass categorization problems.

cs.LG↗

arXiv2016

Shapes of unit lattices and escape of mass

We study the collection of points on the modular surface obtained from the logarithm embeddings of the groups of units in totally real cubic number fields. We conjecture that this set is dense and show that its closure contains countably many explicit curves and give a strategy to show that it has non-empty interior. The results are obtained by constructing explicit families of orders (generalizing the so called simplest cubic fields) and calculating their groups of units. We also address the question of escape of mass for the compact orbits of the diagonal group associated to these orders.

math.NT↗

arXiv2014

On group gradings on PI-algebras

We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \leq exp(W)^K$. A G-grading $W = \bigoplus_{g \in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \neq 0$ for any $r \geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.

math.RA↗

arXiv2014

The center of the generic G-crossed product

Let G be a finite group and let F be a field of characteristic zero. In this paper we construct a generic G-crossed product over F using generic graded matrices. The center of this generic G-crossed product, denoted by F(G), is then the invariant field of a suitable G action on a field of rational functions in several indeterminates. The main goal of this paper is to study the extensions F(G)/F given that F contains enough roots of unity and determine how close they are to being purely transcendental. In particular we show that F(G)/F is a stably rational extension for $G = C_2 \times C_{2n}$ where n is odd and for $G=<{\sigma},{\tau} | {\sigma}^n = {\tau}^{2m} = e, {\tau}{\sigma}{\tau}^{-1}={\sigma}^{-1}>$ where $gcd(n, 2m) = 1$. Furthermore, we prove that if H, K are groups of coprime orders, then $F(H \times K)$ is the fraction field of $F(H) \otimes F(K)$.

math.RA↗

arXiv2013

On regular G-grading

Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A= Σ_{g in G} A_g that satisfies the following two conditions: (1) for every integer n>=1 and every n-tuple (g_1,g_2,...,g_n) in G^n, there are elements, a_i in A_{g_i}, i=1,...,n, such that a_1*a_2*...*a_n != 0. (2) for every g,h in G and for every a_g in A_g,b_h in A_h, we have a_{g}b_{h}=theta(g,h)b_{h}a_{g}. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

math.RA↗

arXiv2012

Graded Embeddings of Finite Dimensional Simple Graded Algebras

Let A,B be finite dimensional G-graded algebras over an algebraically closed field K with char(K)=0, where G is an abelian group, and let Id_G(A) be the set of graded identities of A (res. Id_G(B)). We show that if A,B are G-simple then there is a graded embedding of A in B iff Id_G(B) is contained in Id_G(A). We also give a weaker generalization for the case where A is G-semisimple and B is arbitrary.

math.RA↗