Greg Markowsky
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View article: Courtship choreography is stabilised among genetically isolated populations
Courtship choreography is stabilised among genetically isolated populations Open
Sexual selection has sculpted diverse and intricate courtship displays throughout the animal kingdom, where failure to achieve the choreographic standards of a potential partner can be highly costly for reproductive success. Yet this raise…
View article: A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion, II
A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion, II Open
This paper is the sequel to another with the same name (Buttigieg et al., Comput. Methods Funct. Theory, 2023), and is concerned with results of the same type. We deduce a result on the moments of the exit time of Brownian motion from doma…
View article: An improved bound for strongly regular graphs with smallest eigenvalue $-m$
An improved bound for strongly regular graphs with smallest eigenvalue $-m$ Open
In 1979, Neumaier gave a bound on $λ$ in terms of $m$ and $μ$, where $-m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular graphs…
View article: A method of solution for the inverse problem for $h$-functions of planar Brownian motion
A method of solution for the inverse problem for $h$-functions of planar Brownian motion Open
Given a planar domain $D$, the harmonic measure distribution function $h_D(r)$, with base point $z$, is the harmonic measure with pole at $z$ of the parts of the boundary which are within a distance $r$ of $z$. Equivalently it is the proba…
View article: Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift
Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift Open
In this note, by an elementary use of Girsanov's transform we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In t…
View article: Banknote Life in India: A Survival Analysis Approach
Banknote Life in India: A Survival Analysis Approach Open
India’s currency-to-GDP ratio indicates a strong and persistent demand for cash. Despite the withdrawal of two high-value banknotes in 2016 and the recent economic contraction due to the COVID-19 pandemic, cash continues to rule in India. …
View article: On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes
On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes Open
We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results …
View article: Non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$
Non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$ Open
In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs w…
View article: Comments on the infinitely divisibility of the Conway--Maxwell--Poisson distribution
Comments on the infinitely divisibility of the Conway--Maxwell--Poisson distribution Open
In an elegant recent paper \cite{geng2022conway}, Geng and Xia settled the question of the infinite divisibility of the Conway--Maxwell--Poisson distribution, using in large part several results from complex analysis. In this note we show …
View article: A Collection of Results Relating the Geometry of Plane Domains and the Exit Time of Planar Brownian Motion
A Collection of Results Relating the Geometry of Plane Domains and the Exit Time of Planar Brownian Motion Open
We prove a number of results relating exit times of planar Brownian motion with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; …
View article: On the duration of stays of Brownian motion in domains in Euclidean space
On the duration of stays of Brownian motion in domains in Euclidean space Open
Let TD denote the first exit time of a Brownian motion from a domain D in Rn. Given domains U,W⊆Rn containing the origin, we investigate the cases in which we are more likely to have fast exits from U than W, meaning P(TUP(TWt)>P(TW>t) for…
View article: The transformation of edge-regular and pseudo strongly regular graphs under graph operations
The transformation of edge-regular and pseudo strongly regular graphs under graph operations Open
The graph $G$ is said to be strongly regular with parameters $(n,k,λ,μ)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $λ$ common neighbours; (3) any two non-adjacent vertic…
View article: The transformation of edge-regular and pseudo strongly regular graphs\n under graph operations
The transformation of edge-regular and pseudo strongly regular graphs\n under graph operations Open
The graph $G$ is said to be strongly regular with parameters\n$(n,k,\\lambda,\\mu)$ if the following conditions hold:\n (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have\n$\\lambda$ common neighbours; (3) any tw…
View article: A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion
A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion Open
We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similar…
View article: Binary sequences with a Cesàro limit
Binary sequences with a Cesàro limit Open
The Cesàro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Cesàro limits, this paper consider…
View article: Spectrum of Strongly Regular Graphs under Graph Operators
Spectrum of Strongly Regular Graphs under Graph Operators Open
In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well …
View article: On the finiteness of moments of the exit time of planar Brownian motion from comb domains
On the finiteness of moments of the exit time of planar Brownian motion from comb domains Open
A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from…
View article: On the duration of stays of Brownian motion in domains in Euclidean space
On the duration of stays of Brownian motion in domains in Euclidean space Open
Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${\mathbb R}^n$. Given domains $U,W \subseteq {\mathbb R}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from…
View article: Several consequences of adding or removing an edge from an electric network
Several consequences of adding or removing an edge from an electric network Open
In certain instances an electric network transforms in natural ways by the addition or removal of an edge. This can have interesting consequences for random walks, in light of the known relationships between electric resistance and random …
View article: Several consequences of adding or removing an edge from an electric\n network
Several consequences of adding or removing an edge from an electric\n network Open
In certain instances an electric network transforms in natural ways by the\naddition or removal of an edge. This can have interesting consequences for\nrandom walks, in light of the known relationships between electric resistance\nand rand…
View article: A theory of integration for Cesàro limits
A theory of integration for Cesàro limits Open
The Cesàro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and analysis. Surprisingly, spaces of sequences with Cesàro limits have not previously been studi…
View article: Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion
Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion Open
In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of\nself-intersection local time of fractional Brownian motion were defined, and\nexistence over certain regions of the Hurst parameter $H$ was proved. Utilizing\nt…
View article: Development of larvae of the Australian blowfly,<i>Calliphora augur</i>(Diptera: Calliphoridae), at constant temperatures
Development of larvae of the Australian blowfly,<i>Calliphora augur</i>(Diptera: Calliphoridae), at constant temperatures Open
Calliphora augur (Diptera: Calliphoridae) is a common carrion-breeding blowfly of forensic, medical and agricultural importance in eastern Australia. Despite this, detailed information on the developmental biology of C. augur is lacking. H…
View article: A VARIANT OF CAUCHY’S ARGUMENT PRINCIPLE FOR ANALYTIC FUNCTIONS WHICH APPLIES TO CURVES CONTAINING ZEROS
A VARIANT OF CAUCHY’S ARGUMENT PRINCIPLE FOR ANALYTIC FUNCTIONS WHICH APPLIES TO CURVES CONTAINING ZEROS Open
The standard version of Cauchy’s argument principle, applied to a holomorphic function f , requires that f has no zeros on the curve of integration. In this note, we give a generalisation of such a principle which covers the case when f ha…
View article: Planar Brownian Motion and Complex Analysis
Planar Brownian Motion and Complex Analysis Open
A graduate level textbook on the interplay between complex analysis and Brownian motion.