Related papers: Singularity confinement for a class of $m$-th orde…
We investigate the robustness of singularity avoidance mechanisms in nonrelativistic quantum mechanics on the discretised real line when lattice points are allowed to approach a singularity of the classical potential. We consider the…
We establish the existence theory of several commonly used finite element (FE) nonlinear fully discrete solutions, and the convergence theory of a linearized iteration. First, it is shown for standard FE, SUPG and edge-averaged method…
We construct a universally Bayes consistent learning rule that satisfies differential privacy (DP). We first handle the setting of binary classification and then extend our rule to the more general setting of density estimation (with…
In the case of monotone independence, the transparent understanding of the mechanism to validate the central limit theorem (CLT) has been lacking, in sharp contrast to commutative, free and Boolean cases. We have succeeded in clarifying it…
There exist many situations where an ordinary differential equation admits a movable critical singularity which the test of Kowalevski and Gambier fails to detect. Some possible reasons are: existence of negative Fuchs indices, insufficient…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group $U(N)$ with respect to Haar measure, for finite $N$ and including fixed external matrices. When depending only on the eigenvalues of the…
We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in…
Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \mathbb{H} $ has a unique solution $ m $ with…
Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d random variables chosen from a fixed probability distribution p of mean 0, variance 1 and finite higher moments. Previous work [BDJ,HM] showed that the…
After a brief introduction to the Painlev\'{e} property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlev\'{e} tests. The tests are…
We give a new semi-combinatorial proof for the equality of the number of ballot permutations of length $n$ and the number of odd order permutations of length $n$, which is due to Bernardi, Duplantier and Nadeau. Spiro conjectures that the…
Picard-Lefschetz theory is applied to solutions of the Helmholtz equation, formulated in terms of sums of integrals of a proper-time, or `einbein', wave function $\Psi(\Lambda) = \exp(i\mathbb S(\Lambda))$ along complex contours bounded by…
We consider the minimization problem for an integral functional $J$, possibly non-convex and non-coercive in $W^{1,1}_0(\Omega)$, where $\Omega\subset\R^n$ is a bounded smooth set. We prove sufficient conditions in order to guarantee that a…
We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a…
In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…
Consider the solution $y(t)$ for the ordinary differential equation $y' = f(t, y)$ with $t$ complex. Second-order nonlinear differential equations often exhibit patterns in their poles, branch points, and essential singularities, explored…
The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of DP1 are classified under criterion of their behavior while argument tends to infinity. The Isomonodromic Deformations Method yields asymptotic…
It is known that discrete Painlev\'e equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlev\'e equations in which the symmetries become clearly visible. We know how to obtain…
A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has…