Yang-Mills fields on CR manifolds

We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold.Comment: 52 page

Boundary regularity for the Poisson equation in reifenberg-flat domains

This paper is devoted to the investigation of the boundary regularity for the Poisson equation {{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $\alpha\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally H\"older continuous provided that $\Omega$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem

A Probabilistic proof of the breakdown of Besov regularity in LL-shaped domains

{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We also recover Grisvard's classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale

How self-organization can guide evolution

Self-organization and natural selection are fundamental forces that shape the natural world. Substantial progress in understanding how these forces interact has been made through the study of abstract models. Further progress may be made by identifying a model system in which the interaction between self-organization and selection can be investigated empirically. To this end, we investigate how the self-organizing thermoregulatory huddling behaviours displayed by many species of mammals might influence natural selection of the genetic components of metabolism. By applying a simple evolutionary algorithm to a wellestablished model of the interactions between environmental, morphological, physiological and behavioural components of thermoregulation, we arrive at a clear, but counterintuitive, prediction: rodents that are able to huddle together in cold environments should evolve a lower thermal conductance at a faster rate than animals reared in isolation. The model therefore explains how evolution can be accelerated as a consequence of relaxed selection, and it predicts how the effect may be exaggerated by an increase in the litter size, i.e. by an increase in the capacity to use huddling behaviours for thermoregulation. Confirmation of these predictions in future experiments with rodents would constitute strong evidence of a mechanism by which self-organization can guide natural selection

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

Layered control architectures in robots and vertebrates

We revieiv recent research in robotics, neuroscience, evolutionary neurobiology, and ethology with the aim of highlighting some points of agreement and convergence. Specifically, we com pare Brooks' (1986) subsumption architecture for robot control with research in neuroscience demonstrating layered control systems in vertebrate brains, and with research in ethology that emphasizes the decomposition of control into multiple, intertwined behavior systems. From this perspective we then describe interesting parallels between the subsumption architecture and the natural layered behavior system that determines defense reactions in the rat. We then consider the action selection problem for robots and vertebrates and argue that, in addition to subsumption- like conflict resolution mechanisms, the vertebrate nervous system employs specialized selection mechanisms located in a group of central brain structures termed the basal ganglia. We suggest that similar specialized switching mechanisms might be employed in layered robot control archi tectures to provide effective and flexible action selection

Cooperation and the evolution of intelligence

The high levels of intelligence seen in humans, other primates, certain cetaceans and birds remain a major puzzle for evolutionary biologists, anthropologists and psychologists. It has long been held that social interactions provide the selection pressures necessary for the evolution of advanced cognitive abilities (the ‘social intelligence hypothesis’), and in recent years decision-making in the context of cooperative social interactions has been conjectured to be of particular importance. Here we use an artificial neural network model to show that selection for efficient decision-making in cooperative dilemmas can give rise to selection pressures for greater cognitive abilities, and that intelligent strategies can themselves select for greater intelligence, leading to a Machiavellian arms race. Our results provide mechanistic support for the social intelligence hypothesis, highlight the potential importance of cooperative behaviour in the evolution of intelligence and may help us to explain the distribution of cooperation with intelligence across taxa

Slutsky Matrix Norms and Revealed Preference Tests of Consumer Behaviour

Given any observed finite sequence of prices, wealth and demand choices, we characterize the relation between its underlying Slutsky matrix norm (SMN) and some popular discrete revealed preference (RP) measures of departures from rationality, such as the Afriat index. We show that testing rationality in the SMN aproach with finite data is equivalent to testing it under the RP approach. We propose a way to "summarize" the departures from rationality in a systematic fashion in finite datasets. Finally, these ideas are extended to an observed demand with noise due to measurement error; we formulate an appropriate modification of the SMN approach in this case and derive closed-form asymptotic results under standard regularity conditions