Partition Identities and the Coin Exchange Problem

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.Comment: 6 page

A matrix model for the latitude Wilson loop in ABJM theory

In ABJ(M) theory, we propose a matrix model for the exact evaluation of BPS Wilson loops on a latitude circular contour, so providing a new weak-strong interpolation tool. Intriguingly, the matrix model turns out to be a particular case of that computing torus knot invariants in $U(N_1|N_2)$ Chern-Simons theory. At weak coupling we check our proposal against a three-loop computation, performed for generic framing, winding number and representation. The matrix model is amenable of a Fermi gas formulation, which we use to systematically compute the strong coupling and genus expansions. For the fermionic Wilson loop the leading planar behavior agrees with a previous string theory prediction. For the bosonic operator our result provides a clue for finding the corresponding string dual configuration. Our matrix model is consistent with recent proposals for computing Bremsstrahlung functions exactly in terms of latitude Wilson loops. As a by-product, we extend the conjecture for the exact $B^{\theta}_{1/6}$ Bremsstrahlung function to generic representations and test it with a four-loop perturbative computation. Finally, we propose an exact prediction for $B_{1/2}$ at unequal gauge group ranks.Comment: 73 pages; v2: several improvements, JHEP published versio

Regenerative Composition Structures

A new class of random composition structures (the ordered analog of Kingman's partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the L{\'e}vy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of $[0,1]$ generated by excursions of a standard Bessel bridge of dimension $2 - 2 \alpha$ for some $\alpha \in [0,1]$

The Combinatorialization of Linear Recurrences

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings

An assessment of blockchain consensus protocols for the Internet of Things

In a few short years the Internet of Things has become an intrinsic part of everyday life, with connected devices included in products created for homes, cars and even medical equipment. But its rapid growth has created several security problems, with respect to the transmission and storage of vast amounts of customers data, across an insecure heterogeneous collection of networks. The Internet of Things is therefore creating a unique set of risk and problems that will affect most households. From breaches in confidentiality, which could allow users to be snooped on, through to failures in integrity, which could lead to consumer data being compromised; devices are presenting many security challenges to which consumers are ill equipped to protect themselves from. Moreover, when this is coupled with the heterogeneous nature of the industry, and the interoperable and scalability problems it becomes apparent that the Internet of Things has created an increased attack surface from which security vulnerabilities may be easily exploited. However, it has been conjectured that blockchain may provide a solution to the Internet of Things security and scalability problems. Because of blockchain’s immutability, integrity and scalability, it is possible that its architecture could be used for the storage and transfer of Internet of Things data. Within this paper a cross section of blockchain consensus protocols have been assessed against a requirement framework, to establish each consensus protocols strengths and weaknesses with respect to their potential implementation in an Internet of Things blockchain environment