The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space

It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the three dimensional upper half space is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.Comment: 9 page

Ground state energy of large polaron systems

The last unsolved problem about the many-polaron system, in the Pekar-Tomasevich approximation, is the case of bosons with the electron-electron Coulomb repulsion of strength exactly 1 (the 'neutral case'). We prove that the ground state energy, for large $N$, goes exactly as $-N^{7/5}$, and we give upper and lower bounds on the asymptotic coefficient that agree to within a factor of $2^{2/5}$.Comment: 16 page

Coherent optical implementations of the fast Fourier transform and their comparison to the optical implementation of the quantum Fourier transform

Optical structures to implement the discrete Fourier transform (DFT) and fast Fourier transform (FFT) algorithms for discretely sampled data sets are considered. In particular, the decomposition of the FFT algorithm into the basic Butterfly operations is described, as this allows the algorithm to be fully implemented by the successive coherent addition and subtraction of two wavefronts (the subtraction being performed after one has been appropriately phase shifted), so facilitating a simple and robust hardware implementation based on waveguided hybrid devices as employed in coherent optical detection modules. Further, a comparison is made to the optical structures proposed for the optical implementation of the quantum Fourier transform and they are shown to be very similar

The functionalist's body

Interview with professor Robert D Rupert

Cognitive systems and the supersized mind

In Supersizing the Mind: Embodiment, Action, and Cognitive Extension (Clark, 2008), Andy Clark bolsters his case for the extended mind thesis and casts a critical eye on some related views for which he has less enthusiasm. To these ends, the book canvasses a wide range of empirical results concerning the subtle manner in which the human organism and its environment interact in the production of intelligent behavior. This fascinating research notwithstanding, Supersizing does little to assuage my skepticism about the hypotheses of extended cognition and extended mind. In particular, Supersizing fails to make the case for the extended view as a revolutionary thesis in the theoretical foundations of cognitive science

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.Comment: 33 pages. arXiv admin note: text overlap with arXiv:0709.1103 by other author

Group Minds and Natural Kinds

The claim is frequently made that structured collections of individuals who are themselves subjects of mental and cognitive states – such collections as courts, countries, and corporations – can be, and often are, subjects of mental or cognitive states. And, to be clear, advocates for this so-called group-minds hypothesis intend their view to be interpreted literally, not metaphorically. The existing critical literature casts substantial doubt on this view, at least on the assumption that groups are claimed to instantiate the same species of mental and cognitive properties as individual humans. In this essay, I evaluate a defensive move made by some proponents of the group-oriented view: to concede that group states and individual states aren’t of the same specific natural kinds, while holding that groups instantiate different species of mental or cognitive states – perhaps a different species of cognition itself – from those instantiated by humans. In order to evaluate this defense of group cognition, I develop a view of natural kinds – or at least of the sort of evidence that supports inferences to the sameness of natural kind – a view I have previous dubbed the ‘tweak-and-extend’ theory. Guided by the tweak-and-extend approach, I arrive at a tentative conclusion: that what is common to models of individual cognitive processing and models of group processing does not suffice to establish sameness of cognitive (or mental) kinds, properties, or state-types, not even at a generic or overarching level

Production Controls.

2 p