Related papers: Cranks in Ramanujan's Lost Notebook
A series of formula is presented that are all inspired by the Ramanujan Notebooks [6]. One of them appears in the notebooks II about Zeta(3). That formula inspired others that appeared in 1998, 2006 and 2009 on the author's website and…
Recurrent neural networks (RNNs) trained on low-dimensional tasks have been widely used to model functional biological networks. However, the solutions found by learning and the effect of initial connectivity are not well understood. Here,…
This note gives a very succinct conceptual proof of Brahmagupta's formula for cyclic quadrilaterals, and then a long discussion about how the proof is inspired by ideas from modern mathematics.
I review Chandra's achievements in the last two years in AGN research. I concentrate on some topics of my interest; some others are reviewed in an accompanying article by K. Nandra. I comment briefly on the need to allow for the discovery…
Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs…
This is a contribution to the review "50 Years of Quantum Chromdynamics" edited by F. Gross and E. Klempt [arXiv:2212.11107], to be published in EPJC. The contribution reviews the new baryon resonances derived from photoproduction…
Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We…
The ability to incorporate quantum phenomena in computing unlocks a host of new ways to make mistakes. This work surveys existing studies and approaches to debugging quantum programs. It then presents a set of examples that stem from…
Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including $spt$-crank, overpartition pairs, and $t$-core partitions. As corollaries, we obtain new proofs of…
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of…
Graph eigenvalues play a fundamental role in controlling structural properties, such as bisection bandwidth, diameter, and fault tolerance, which are critical considerations in the design of supercomputing interconnection networks. This…
The construction of the paths of all possible Brownian motions (in the sense of Knight) on a half line or a finite interval is reviewed.
The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.
Using quasimodular forms with respect to $\Gamma_0(4)$ we find exact relations between the M2-rank for partitions without repeated odd parts and three residual cranks. From these identities we are able to deduce various congruences mod 3…
We verify a recent conjecture of Sehrawat, Kumar and Ahlawat on the minimum bisection width (a notion that was rediscovered in 2020 by Acharya and Kureethara under the name of rna number) of powers of cycles.
Some aspects of analysis on disconnected open subsets of the plane with connected fractal boundary are discussed.
Let $N(\leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(\leq m,n)\leq…
Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. For example, we find the four tenth-order mock theta…
In this short note we use the umbral formalism to derive the Ramanujan Master Theorem and discuss its extension to more general cases.
Major breakthrough in quantum computation has recently been achieved using quantum annealing to develop analog quantum computers instead of gate based computers. After a short introduction to quantum computation, we retrace very briefly the…