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Extremal Combinatorics is among the most active topics in Discrete Mathematics, dealing with problems that are often motivated by questions in other areas, including Theoretical Computer Science and Information Theory. This paper contains a…
In the last two decades new techniques emerged to construct valuations on an infinite division ring $D,$ given a normal subgroup $N\subseteq D$ of finite index. These techniques were based on the commuting graph of $D^{\times}/N$ in the…
We derive inequalities for sums of eigenvalues of Schr\"{o}dinger operators on finite intervals and tori. In the first of these cases, the inequalities converge to the classical trace formulae in the limit as the number of eigenvalues…
It is well known that the graph isomorphism problem is polynomial-time reducible to the graph automorphism problem (in fact these two problems are polynomial-time equivalent). We show that, analogously, the group isomorphism problem is…
We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning…
The aim of this paper is to obtain new inequalities for a large family of generalizations of the Wiener Index and to characterize the set of extremal graphs with respect to them. Our main results provide upper and lower bounds for these…
Some proofs of the problems of the basic statistics proposed for numeric symbolic data.
We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
We investigate a functional obtained by summing the squared differences of the integral of an Ito process over disjoint intervals. The limit of this sum is shown to converge in probability to two thirds the quadratic variation of the…
A thesis submitted for the degree of Doctor of Philosophy of The Australian National University. In this work we introduce several new optimisation methods for problems in machine learning. Our algorithms broadly fall into two categories:…
In this paper an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Some applications of this method to approximation of distribution…
We develop some aspects of a general theory of presentations of subshifts by labelled directed graphs, in particular by compact graphs. Also considered are synchronization properties of subshifts that lead to presentations by countable…
A generalization of the H\"older inequality is considered. Its relations with a previously obtained improvement of the Cauchy--Schwarz inequality are discussed.
A form of the Laplace transform is reviewed as a paradigm for an entire class of fractional functional transforms. Various of its properties are discussed. Such transformations should be useful in application to differential/integral…
We review some recent results on the minimization of the energy associated to the nonlinear Schr\"odinger Equation on non-compact graphs. Starting from seminal results given by the author together with C. Cacciapuoti, D. Finco, and D. Noja…
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the…
Complex unit gain graphs may exhibit various kinds of symmetry. In this work, we explore structural symmetry, spectral symmetry and sign-symmetry in such graphs, and their respective relations to one-another. Our main result is a…
This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of…
We present some addition theorems for spin-weighted spherical harmonics, generalizing previous results for scalar (spin-zero) spherical harmonics. These addition theorems involve sums over the azimuthal quantum number of products of two…