Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in $L_2$ for small complex $L_\infty$ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for $k$-forms are well posed for small perturbations of block matrices.Comment: Some changes made in the introduction of the pape

Vertical versus conical square functions

We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator.Comment: 21 page

Riesz transform on manifolds and heat kernel regularity

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is $L^p$ bounded on such a manifold, for $p$ ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain $L^p$ estimate in the same interval of $p$'s.Comment: to appear in Annales de l'Ecole Normale Superieure de Pari

Carleson measures, trees, extrapolation, and T(b)T(b) theorems

The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation for Carleson measures, with two separate proofs, as well as a two-sided local dyadic $T(b)$ theorem which generalizes earlier $T(b)$ theorems of David, Journe, Semmes, and Christ.Comment: 50 pages, 3 figures, to appear, Publications Matematiques Barcelona. A new proof of the extrapolation lemma (due to John Garnett) is now include

Learning Aerial Image Segmentation from Online Maps

This study deals with semantic segmentation of high-resolution (aerial) images where a semantic class label is assigned to each pixel via supervised classification as a basis for automatic map generation. Recently, deep convolutional neural networks (CNNs) have shown impressive performance and have quickly become the de-facto standard for semantic segmentation, with the added benefit that task-specific feature design is no longer necessary. However, a major downside of deep learning methods is that they are extremely data-hungry, thus aggravating the perennial bottleneck of supervised classification, to obtain enough annotated training data. On the other hand, it has been observed that they are rather robust against noise in the training labels. This opens up the intriguing possibility to avoid annotating huge amounts of training data, and instead train the classifier from existing legacy data or crowd-sourced maps which can exhibit high levels of noise. The question addressed in this paper is: can training with large-scale, publicly available labels replace a substantial part of the manual labeling effort and still achieve sufficient performance? Such data will inevitably contain a significant portion of errors, but in return virtually unlimited quantities of it are available in larger parts of the world. We adapt a state-of-the-art CNN architecture for semantic segmentation of buildings and roads in aerial images, and compare its performance when using different training data sets, ranging from manually labeled, pixel-accurate ground truth of the same city to automatic training data derived from OpenStreetMap data from distant locations. We report our results that indicate that satisfying performance can be obtained with significantly less manual annotation effort, by exploiting noisy large-scale training data.Comment: Published in IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSIN

Analyticity of layer potentials and L2L^{2} solvability of boundary value problems for divergence form elliptic equations with complex L∞L^{\infty} coefficients

We consider divergence form elliptic operators of the form L=-\dv A(x)\nabla, defined in $R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic, complex and $t$-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on $L^2(\mathbb{R}^{n})=L^2(\partial\mathbb{R}_{+}^{n+1})$, is stable under complex, $L^{\infty}$ perturbations of the coefficient matrix. Using a variant of the $Tb$ Theorem, we also prove that the layer potentials are bounded and invertible on $L^2(\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (and thus, by our stability result, also when $A$ is complex, $\Vert A-A^0\Vert_{\infty}$ is small enough and $A^0$ is real, symmetric, $L^{\infty}$ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with $L^2$ (resp. $\dot{L}^2_1)$ data, for small complex perturbations of a real symmetric matrix. Previously, $L^2$ solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients $A_{j,n+1}=0=A_{n+1,j}$, $1\leq j\leq n$, which corresponds to the Kato square root problem

Linking Mine Action and SSR through Human Security

Security sector reform (SSR) and mine action share a strong common conceptual basis, which draws from a shared understanding of security. They both reflect a conceptualization of security that is not limited to the level of the state, but takes into account security threats and needs at societal and individual levels. This common basis provides opportunities for synergies between SSR and mine action. However, empirical evidence demonstrates that the strong conceptual basis is not fully reflected in concrete activities, and the linkages remain limited and underexplored. Despite this gap, there are positive examples showing the potential for synergies between SSR and mine action. Ultimately, this paper maintains that the concept of human security provides a comprehensive framework which can bridge the differences and open broader opportunities for cooperation, which in turn will increase the impact of interventions in SSR and mine action

Linking Mine Action and SSR through Human Security

Security sector reform (SSR) and mine action share a strong common conceptual basis, which draws from a shared understanding of security. They both reflect a conceptualization of security that is not limited to the level of the state, but takes into account security threats and needs at societal and individual levels. This common basis provides opportunities for synergies between SSR and mine action. However, empirical evidence demonstrates that the strong conceptual basis is not fully reflected in concrete activities, and the linkages remain limited and underexplored. Despite this gap, there are positive examples showing the potential for synergies between SSR and mine action. Ultimately, this paper maintains that the concept of human security provides a comprehensive framework which can bridge the differences and open broader opportunities for cooperation, which in turn will increase the impact of interventions in SSR and mine action

Trust in everyday life

Although trust plays a pivotal role in many aspects of life, very little is known about the manifestation of trust and distrust in everyday life. In this work, we integrated several prior approaches to trust and investigated the prevalence and key determinants of trust (vs. distrust) in people’s natural environments, using preregistered experience-sampling methodology. Across more than 4,500 social interactions from a heterogeneous sample of 427 participants, results showed high average levels of trust, but also considerable variability in trust across contexts. This variability was attributable to aspects of trustee perception, social distance, as well as three key dimensions of situational interdependence: conflict of interests, information (un)certainty, and power imbalance. At the dispositional level, average everyday trust was shaped by general trust, moral identity, and zero-sum beliefs. The social scope of most trust-related traits, however, was moderated by social distance: Whereas moral identity buffered against distrusting distant targets, high general distrust and low social value orientation amplified trust differences between close vs. distant others. Furthermore, a laboratory-based trust game predicted everyday trust only with regard to more distant but not close interaction partners. Finally, everyday trust was linked to self-disclosure and to cooperation, particularly in situations of high conflict between interaction partners’ interests. We conclude that trust can be conceptualized as a relational hub that interconnects the social perception of the trustee, the relational closeness between trustor and trustee, key structural features of situational interdependence, and behavioral response options such as self-disclosure

Absence of superconductivity in ultra-thin layers of FeSe synthesized on a topological insulator

The structural and electronic properties of FeSe ultra-thin layers on Bi$_{2}$Se$_{3}$ have been investigated with a combination of scanning tunneling microscopy and spectroscopy and angle-resolved photoemission spectroscopy. The FeSe multi-layers, which are predominantly 3-5 monolayers (ML) thick, exhibit a hole pocket-like electron band at \bar{\Gamma} and a dumbbell-like feature at \bar{M}, similar to multi-layers of FeSe on SrTiO$_{3}$. Moreover, the topological state of the Bi2Se3 is preserved beneath the FeSe layer, as indicated by a heavily \it{n}-doped Dirac cone. Low temperature STS does not exhibit a superconducting gap for any investigated thickness down to a temperature of 5 K