Related papers: Global and local behavior of zeros of nonpositive …

We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some…

We study the behaviour near s=1/2 of zeta functions of varieties over finite fields F_q with q a square. The main result is an Euler-characteristic formula for the square of the special value at s=1/2. The Euler-characteristic is…

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations.…

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots $\left\{ \rho_{n}\right\} _{n=1}^{\infty}$, as $n\to+\infty$ the moduli $\left|\rho_{n}\right|$ grow at least exponentially. Applications to entire…

Within the context of the functional renormalization group flow of gravity, we suggest that a generic f(R) ansatz (i.e. not truncated to any specific form, polynomial or not) for the effective action plays a role analogous to the local…

We review a result obtained with Andrew Ledoan and Marco Merkli. Consider a random analytic function $f(z) = \sum_{n=0}^{\infty} a_n X_n z^n$, where the $X_n$'s are i.i.d., complex valued random variables with mean zero and unit variance,…

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of…

We discuss the fractional Leibniz rule for periodic functions on the $d$-dimensional torus, including the endpoint cases. As an application, we present a product estimate, involving distributions of negative regularities.

The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= \lambda \rho(|x|) (1-u)^q &\mbox{in…

One-dimensional Yang-Mills Equations are considered from a point of view of a class of nonlinear Klein-Gordon-Fock models. The case of self-dual Nahm equations and non-self-dual models are discussed. A quasiclassical quantization of the…

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…

This paper treats two functional equations, the Kannppan-Van Vleck functional equation $$\mu(y)f(x\tau(y)z_0)\pm f(xyz_0) =2f(x)f(y), \;x,y\in S$$ and the following variant of it $$\mu(y)f(\tau(y)xz_0)\pm f(xyz_0) = 2f(x)f(y), \;x,y\in S,$$…

We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory.…

In the early 1980's, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in…

For \alpha in the interval [0,1], we consider the one-parameter family of \alpha-continued fraction maps, which include the Gauss map (\alpha=1) and the nearest integer (\alpha=1/2) and by-excess (\alpha=0) continued fraction maps. To each…

We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…

We study the question of positivity of the fundamental solution for fractional diffusion and wave equations of the form, which may be of fractional order both in space and time. We give a complete characterization for the positivity of the…

We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$ (-\Delta)^su=K(|x|)|u|^{2^*_s-2}u,\ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and…

Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized…

For the partial theta function $\theta (q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is…