arXiv subjects

Tomas Ekholm

Publications and source records attributed to Tomas Ekholm.

12 recordsLinked to original sources

arXiv2015

Modular Responsive Web Design using Element Queries

Responsive Web Design (RWD) enables web applications to adapt to the characteristics of different devices such as screen size which is important for mobile browsing. Today, the only W3C standard to support this adaptability is CSS media queries. However, using media queries it is impossible to create applications in a modular way, because responsive elements then always depend on the global context. Hence, responsive elements can only be reused if the global context is exactly the same, severely limiting their reusability. This makes it extremely challenging to develop large responsive applications, because the lack of true modularity makes certain requirement changes either impossible or expensive to realize. In this paper we extend RWD to also include responsive modules, i.e., modules that adapt their design based on their local context independently of the global context. We present the ELQ project which implements our approach. ELQ is a novel implementation of so-called element queries which generalize media queries. Importantly, our design conforms to existing web specifications, enabling adoption on a large scale. ELQ is designed to be heavily extensible using plugins. Experimental results show speed-ups of the core algorithms of up to 37x compared to previous approaches.

cs.SE↗

arXiv2015

Estimates for the Lowest Eigenvalue of Magnetic Laplacians

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.

math.SP↗

arXiv2014

Hardy inequalities for p-Laplacians with Robin boundary conditions

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.

math.AP↗

arXiv2014

Weak perturbations of the p-Laplacian

We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.

math.AP↗

arXiv2013

A Magnetic Contribution to the Hardy Inequality

We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.

math.SP↗

arXiv2009

Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions

This paper considers Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-γ} < C_γ\int_{R^d} V(x)_+^{γ+ d/4} dx for gamma \geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_γ> 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained.

math.SP↗

arXiv2008

Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior

Consider a regular $d$-dimensional metric tree $Γ$ with root $o$. Define the Schroedinger operator $-Δ- V$, where $V$ is a non-negative, symmetric potential, on $Γ$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-γ}$ at infinity, where $1 < γ\leq d \leq 2, γ\neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Δ- V$. In other words, we will describe the asymptotical behavior of $\inf σ(-Δ- αV)$ as $α\to 0+$

math.SP↗

arXiv2007

Remarks about Hardy inequalities on metric trees

We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.

math.SP↗

arXiv2007

Eigenvalue estimates for Schroedinger operators on metric trees

We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit.

math.SP↗

arXiv2006

Lieb-Thirring inequalities on the half-line with critical exponent

We consider a Schrödinger operator on the half-line with a Dirichlet boundary condition at the origin and show that moments of its negative eigenvalues can be estimated by the part of the potential that is larger than the critical Hardy weight. The estimate is valid for the critical value of the moment parameter.

math.SP↗

arXiv2004

Spectrum of the Magnetic Schrodinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann boundary condition (see Section 2, Eq. (2.2) for the definition of this boundary condition}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.

math-ph↗

arXiv2004

Stability of the magnetic Schrödinger operator in a waveguide

The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own.

math-ph↗