Related papers: On Lower Bounds for $s$-multiplicities
We exhibit some strong positivity properties of a certain function which implies a key inequality that in turn implies the lower bound formula for the probability of correct selection in the Levin-Robbins-Leu family of sequential subset…
In recent years, maximizing G\'al sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of G\'al type, with respect to the $L^1$-norm. We also consider…
Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so--called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and…
We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts…
We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer…
Let $T = (T_1, \ldots, T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an…
Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the…
We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the…
We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary…
Starting from Wigner's theory of elementary systems and following a recent approach of Schroer we define certain subspaces of localized wave functions in the underlying Hilbert space with the help of the theory of modular von-Neumann…
In this paper, we introduce the notion of multiplier of a Hilbert algebra. The space of bounded multipliers is a semifinite von Neumann algebra isomorphic to the left von Neumann algebra of the Hilbert algebra, as expected. However, in the…
This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior…
In [8], G. David suggested a new type of global minimizer for the Mumford-Shah functional in $\R^3$, for which the singular sets belong to a three parameters family of sets ($0<\delta\_1,\delta\_2,\delta\_3<\pi$). We first derive necessary…
In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler--Lagrange equations. We show that weak solutions are locally bounded…
In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(\Delta)_{p}^{s} u + V(x)|u|^{p-2}u &= \lambda f(u),&x\in\Omega; u&=0,&x\in…
These lecture notes focus on the application of ideas of locality, in particular Lieb-Robinson bounds, to quantum many-body systems. We consider applications including correlation decay, topological order, a higher dimensional…
We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot),…
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at…
For $1<p<\infty$ and $0<s<1$, let $\mathcal{Q}^p_ s (\mathbb{T})$ be the space of those functions $f$ which belong to $ L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset…
We prove the local Lipschitz regularity of the minimizers of functionals of the form \[ \mathcal I(u)=\int_\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in\phi+W^{1,1}_0(\Omega) \] where $g$ is bounded and $\phi$ satisfies the Lower Bounded…