Related papers: Cranks in Ramanujan's Lost Notebook
The aim of this note (as well as of the course itself) is to give a largely self-contained proof of two of the main results in the field of low-rank matrix recovery. This field aims for identification of low-rank matrices from only limited…
In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier…
A new transformation involving the error function $\textup{erf}(z)$, the imaginary error function $\textup{erfi}(z)$, and an integral analogue of a partial theta function is given along with its character analogues. Another complementary…
Some highlights of the recent researches in the field of relativistic jets are reviewed and critically analyzed. Given the extent of the available literature, this essay symbolically takes the baton from the outstanding and recent review by…
We examine computer experiments that can be performed to understand the dynamics of knots under self-repulsion. In the course of specific computer exploration we use the knot theory of rational knots and rational tangles to produce classes…
We respond to the criticism raised in the paper arXiv:1704.07831.
Bringmann, Lovejoy, and Osburn showed that the generating functions of the spt-overpartition functions spt(n), spt1(n), spt2(n), and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews,…
We describe briefly the recent advances in understanding the distributed nature of computations in the (neural) network structure of the brain. We discuss if such artificial networks will be able to perform mathematics and natural sciences.…
In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and…
History of Neutrino Physics is revied with special emphasis on anecdotal events often neglected.
While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origin and which of them are mere artifacts of the connection…
In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity.…
We generalize a beautiful method of Blasius and Ramakrishnan, that in order to exhibit particular instances of the Langlands functorial correspondence, it is enough to show that the correspondence holds in the semistable case, provided the…
We prove three variations of recent results due to Andrews on congruences for $NT(m,k,n)$, the total number of parts in the partitions of $n$ with rank congruent to $m$ modulo $k$. We also conjecture new congruences and relations for…
This is a comment on J. Schmittbuhl, A. Hansen, and G. G. Batrouni, Phys. Rev. Lett. 90, 045505 (2003). They offer a reply, in turn.
Studying corruption presents unique challenges. Recent work in the spirit of computational social science exploits newly available data and methods to give a fresh perspective on this important topic. In this chapter we highlight some of…
The Hardy-Ramanujan formula for the number of integer partitions of $n$ is one of the most popular results in partition theory. While the unabridged final formula has been celebrated as reflecting the genius of its authors, it has become…
We propose to study transformations on graphs, and more generally structures, by looking at how the cut-rank (as introduced by Oum) of subsets is affected when going from the input structure to the output structure. We consider…
In this essay dedicated to Yakov Eliashberg we survey the current state of the field of Lagrangian (un)knots, reviewing some constructions and obstructions along with a number of unsolved questions. The appendix by Georgios Dimitroglou…
This set of notes re-proves known results on weighted automata (over a field, also known as multiplicity automata). The text offers a unified view on theorems and proofs that have appeared in the literature over decades and were written in…