The Matsumoto–Yor Property and Its Converse on Symmetric Cones

The Matsumoto–Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As was observed in Letac and Weso?owski (Ann Probab 28:1371–1383, 2000), the natural framework for the multivariate MY property is symmetric cones; however, they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.     

On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of <Emphasis Type="Italic">p</Emphasis>-Adic Numbers

We prove the following theorem. Let X be a discrete field, and \(\xi \) and \(\eta \) be independent identically distributed random variables with values in X and distribution \(\mu \). The random variables \(S=\xi +\eta \) and \(D=(\xi -\eta )^2\) are independent if and only if \(\mu \) is an idempotent distribution. A similar result is also proved in the case when \(\xi \) and \(\eta \) are independent identically distributed random variables with values in the field of p-adic numbers \({\mathbf {Q}}_p\), where \(p>2\), assuming that the distribution \(\mu \) has a continuous density.     

Distance-regular Cayley graphs with least eigenvalue $$$$

We classify the distance-regular Cayley graphs with least eigenvalue \(-2\) and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.     

Linearly embeddable designs

We address the problem of determining when a plane algebraic cubic curve is complete as an (n, 3)-arc in \(\mathrm {PG}(2,q)\). Theoretical results are given for absolutely irreducible singular cubic curves, while computer based results are given for \(q\le 81\).     

Classification of partial difference sets in Abelian groups of order $$4p^2$$

In this article we provide a complete classification of regular partial difference sets in Abelian groups of order \(4p^2\), p an odd prime. It turns out that the known examples are the only examples. These are, up to complements, the trivial examples, the PCP examples, and a sporadic example in \(\mathbb {Z}_2^2\times \mathbb {Z}_3^2\).     

Characterizing Gaussian Flows Arising from Itô’s Stochastic Differential Equations

In order to identify which of the strong solutions of Itô’s stochastic differential equations (SDEs) are Gaussian, we introduce a class of diffusions which ‘depend deterministically on the initial condition’ and then characterize the class. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flows generated by the SDEs, for an extended class of initial conditions, are the unique solutions of the class of stochastic partial differential equations introduced in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162 2008), ‘Probabilistic Representations of Solutions of the Forward Equations’.     

A Notion of Fine Continuity for BV Functions on Metric Spaces

In the setting of a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with respect to the codimension 1 Hausdorff measure.     

In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r ^α for r∈(0,1] and any point x from metric space.     

Fast algorithm for singly linearly constrained quadratic programs with box-like constraints

This paper focuses on a singly linearly constrained class of convex quadratic programs with box-like constraints. We propose a new fast algorithm based on parametric approach and secant approximation method to solve this class of quadratic problems. We design efficient implementations for our proposed algorithm and compare its performance with two state-of-the-art standard solvers called Gurobi and Mosek. Numerical results on a variety of test problems demonstrate that our algorithm is able to efficiently solve the large-scale problems with the dimension up to fifty million and it substantially outperforms Gurobi and Mosek in terms of the running time.     

Optimal transport and a bilevel location-allocation problem

In this paper a two-stage optimization model is studied to find the optimal location of new facilities and the optimal partition of the consumers (location-allocation problem). The social planner minimizes the social costs, i.e. the fixed costs plus the waiting time costs, taking into account that the citizens are partitioned in the region according to minimizing the capacity costs plus the distribution costs in the service regions. By using optimal transport tools, existence results of solutions to the location-allocation problem are presented together with some examples.