arXiv subjects
Publications and source records attributed to Yahel Manor.
We introduce the notion of $GL(n)$-dependence of matrices, which is a generalization of linear dependence taking into account the matrix structure. Then we prove a theorem, which generalizes, on the one hand, the fact that $n+1$ vectors in an $n$-dimensional vector space are linearly dependent and, on the other hand, the fact that the natural action of the group $GL(n,{\cal K})$ on ${\cal K}^n\setminus\{0\}$ is transitive.
Lifting theorems are theorems that bound the communication complexity of a composed function $f\circ g^{n}$ in terms of the query complexity of $f$ and the communication complexity of $g$. Such theorems constitute a powerful generalization of direct-sum theorems for $g$, and have seen numerous applications in recent years. We prove a new lifting theorem that works for every two functions $f,g$ such that the discrepancy of $g$ is at most inverse polynomial in the input length of $f$. Our result is a significant generalization of the known direct-sum theorem for discrepancy, and extends the range of inner functions $g$ for which lifting theorems hold.
Linear recursions with integer coefficients, such as the one generating the Fibonacci sequence, have been intensely studied over millennia and yet still hide new mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $ζ(3)$, later named the Apéry constant. Apéry's proof used a specific linear recursion containing integer polynomials forming a continued fraction; called polynomial continued fractions (PCFs). Similar polynomial recursions prove the irrationality of other mathematical constants such as $π$ and $e$. More generally, the sequences generated by PCFs form Diophantine approximations (DAs), which are ubiquitous in areas of math such as number theory. It is not known which polynomial recursions create useful DAs and whether they prove irrationality. Here, we present general conclusions and conjectures about DAs created from PCFs. Specifically, we generalize Apéry's work, going beyond his particular choice of PCF, finding the conditions under which a PCF proves irrationality or provides an efficient DA. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $π$, $e$, $ζ(3)$, and the Catalan constant G. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new DA conjectures based on PCFs. Our findings motivate future research on sequences created by any linear recursions with integer coefficients, to aid the development of systematic algorithms for finding DAs of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions, such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $ζ(5)$).
Fundamental mathematical constants like $e$ and $π$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and usually discovered sporadically. Here we propose a novel and systematic approach that leverages algorithms for deriving mathematical formulas for fundamental constants and help reveal their underlying structure. Our algorithms find dozens of well-known as well as previously unknown continued fraction representations of $π$, $e$, Catalan's constant, and values of the Riemann zeta function. Two example conjectures found by our algorithm and so far unproven are: \begin{equation*} \frac{24}{π^2} = 2 + 7\cdot 0\cdot 1+ \frac{8\cdot1^4}{2 + 7\cdot 1\cdot 2 + \frac{8\cdot2^4}{2 + 7\cdot 2\cdot 3 + \frac{8\cdot3^4}{2 + 7\cdot 3\cdot 4 + \frac{8\cdot4^4}{..}}}} \quad\quad,\quad\quad \frac{8}{7 ζ(3)} = 1\cdot 1 - \frac{1^6}{3\cdot 7 - \frac{2^6}{5\cdot 19 - \frac{3^6}{7\cdot 37 - \frac{4^6}{..}}}} \end{equation*} We present two algorithms that proved useful in finding conjectures: a Meet-In-The-Middle (MITM) algorithm and a Gradient Descent (GD) tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values and thus they conjecture formulas without providing proofs and without requiring prior knowledge on any underlying mathematical structure. This approach is especially attractive for constants for which no mathematical structure is known, as it reverses the conventional approach of sequential logic in formal proofs. Instead, our work supports a different approach for research: algorithms utilizing numerical data to unveil mathematical structures, thus trying to play the role of intuition of great mathematicians of the past, providing leads to new mathematical research.