Related papers: Unfocused notes on the Markoff equation and T-Sing…

We study the behavior of the Horrocks-Mumford bundle when restricted to a plane P^2 in P^4, looking for all possible minimal free resolutions for the restricted bundle. To each of the 6 resolutions (4 stable and 2 unstable) we find, we then…

In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our…

In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…

An $L^2(R^d)$-valued stochastic N-interacting particle systems is investigated. Existence and uniqueness of solutions for the degenerate nonlinear Fokker-Planck equation for probability measures that corresponds to the mean field limit…

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new…

Here, in \cite{KrumWicb} and in \cite{KrumWicc} we study the nature of an $n$-dimensional locally area minimising rectifiable current $T$ of codimension $\geq 2$ near its typical (i.e.\ ${\mathcal H}^{n-2}$ a.e.) singular points. Our…

Let $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, be the Gegenbauer weight function, and $\Vert\cdot\Vert$ denote the associated $L_2$-norm, i.e., $$ \Vert f\Vert:=\Big(\int_{-1}^{1}w_{\lambda}(t)\vert f(t)\vert^2\,dt\Big)^{1/2}.…

The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are…

For a linear differential equation with a mild condition on its singularities, we discuss generalized continued fractions converging to expressions in its solutions and their derivatives. In the case of an order two linear differential…

The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate on the solutions, the continuous dependence of the solutions on initial conditions as well as the existence and uniqueness of…

For a finite $E \subset \R^2$, $f:E \rightarrow \R$, and $p>2$, we produce a continuous $F:\R^2 \rightarrow \R$ depending linearly on $f$, taking the same values as $f$ on $E$, and with $L^{2,p}(\R^2)$ semi-norm minimal up to a factor…

We obtain solutions of the nonlinear degenerate parabolic equation \[ \frac{\partial \rho}{\partial t} = {div} \Big\{\rho \nabla c^\star [ \nabla (F^\prime(\rho)+V) ] \Big\} \] as a steepest descent of an energy with respect to a convex…

A time-discrete approach avoids the assumption of an 'integration sense'. New path increments (in a short time step) are complete in the order of that step, and not Gaussian distributed when the noise is multiplicative; this eliminates an…

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as…

We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of…

We compute three first steps of a minimal projective resolution for the trivial module over U_{F_2}(sl^+_3).

We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$ \int F(\nabla u, u, x) dx $$ under the assumption that a certain integral grows at most quadratically at infinity. As a…

We develop a new approach to build the eigenfunctions of a translationally shape-invariant potential. For this we show that their logarithmic derivatives can be expressed as terminating continued fractions in an appropriate variable. We…

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) =…

We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the…