Related papers: Cranks in Ramanujan's Lost Notebook
We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan's lost notebook, MacMahon's partition functions, and reciprocal sums of parts in partitions, through the lens of the classical Fa\`{a}…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…
Biomechanics is a scientific discipline that studies the forces acting on a body and the effects they produce. In this paper, we bring together biomechanists and networking researchers to shed light into how research efforts in…
This article introduces the R package csranks for estimation and inference involving ranks. First, we review methods for the construction of confidence sets for ranks, namely marginal and simultaneous confidence sets as well as confidence…
This entry reviews Rudolf Carnap's philosophical views on the quantum mechanics of his time. It also offers some thoughts on how Carnap might have reacted to some recent developments in the foundations of quantum mechanics.
Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.
We provide a rigorous formulation of Entry 17(v) in Ramanujan's Notebooks and show how this relates to the first Stieltjes constant. A new proposition 4.5 is included to show the close relationship with some analysis presented by…
In recent years there has been some progress in understanding how one might model the interactions of branes in M-theory despite not having a fundamental perturbative description. The goal of this review is to describe different approaches…
This is an elementary explanation of a cubic composition formula due to Ramanujan.
Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.
In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences, whose existence is predicted by…
We can view Brownian sheet as a sequence of interacting Brownian motions or slices. Here we present a number of results about the slices of the sheet. A common feature of our results is that they exhibit phase transition. In addition, a…
In this book, we review various aspects of the Lorentz symmetry breaking, both classical and quantum ones, with the special interest to perturbative generation of Lorentz-breaking terms. We present impacts of Lorentz symmetry breaking in…
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo…
These notes expound the recent use of the signature transform and rough path theory in data science and machine learning. We develop the core theory of the signature from first principles and then survey some recent popular applications of…
Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistics, the mex, and…
This note is the written version of conversations with young colleagues on unofficial history, general ideas, unexpected facts and open problems concerning tilting theory.
The Ramanujan polynomials arise in three intertwined contexts. As remarked by BerndtEvans-Wilson, no combinatorial perspective seems to be alluded to in the original definition of Ramanujan. On a different stage, Dumont-Ramamonjisoa…
We give a brief survey of some known results on intrinsically linked or knotted graphs.
Recent results on the study of the electromagnetic structure of nucleon resonances, the spin structure of proton and neutrons at small and intermediate photon virtualities, and the search for exotic pentaquark baryons are presented.