Webs of Lagrangian Tori in Projective Symplectic Manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.Comment: 18 pages, minor latex problem fixe

Borel-Cantelli sequences

A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set $$\underset{k=1}{\overset{\infty}{\cap}} \underset{n=k}{\overset{\infty}{\cup}} B(x_n, a_n))=\{x\in[0,1)\mid |x_n-x|<a_n \text{for} \infty \text{many}n\geq1\}$$ has full Lebesgue measure. (To put it informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds). The notion of BC sequences is motivated by the Monotone Shrinking Target Property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC and not BC sequences are presented. The property of a sequence to be BC is a delicate diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC while the orbits of a "generic" IET are not. The notion of BC sequences is extended to more general spaces.Comment: 20 pages. Some proofs clarifie

Foundations for Relativistic Quantum Theory I: Feynman's Operator Calculus and the Dyson Conjectures

In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson's second conjecture for QED. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman's path integral, and to prove Dyson's first conjecture that the divergences are in part due to a violation of Heisenberg's uncertainly relations

Quantum memory effects on the dynamics of electrons in small gold clusters

Electron dynamics in metallic clusters are examined using a time-dependent density functional theory that includes a 'memory term', i.e. attempts to describe temporal non-local correlations. Using the Iwamoto, Gross and Kohn exchange-correlation (XC) kernel we construct a translationally invariant memory action from which an XC potential is derived that is translationally covariant and exerts zero net force on the electrons. An efficient and stable numerical method to solve the resulting Kohn-Sham equations is presented. Using this framework, we study memory effects on electron dynamics in spherical Jellium 'gold clusters'. We find memory significantly broadens the surface plasmon absorption line, yet considerably less than measured in real gold clusters, attributed to the inadequacy of the Jellium model. Two-dimensional pump-probe spectroscopy is used to study the temporal decay profile of the plasmon, finding a fast decay followed by slower tail. Finally, we examine memory effects on high harmonic generation, finding memory narrows emission lines

Users, Economics, Technology: Unavoidable Interdynamics

This paper briefly presents some conclusions of a brainstorming session on the way technology is evolving in ICT. Technology advances have overcome society ability to answer, both in economic and in human aspects. The current design paradigms, of agnostic technology development, need to be reconsidered, and the user needs to be repositioned at the center of future developments

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin-Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Schmidt's inhomogeneous (1964) version of Khintchine's Theorem (1924). For example, given any real sequence $\{y_i \}$, we build a divergent series of non-negative reals $\psi(n)$ such that for any $y\in\{y_i\}$, almost no real number is inhomogeneously $\psi$-approximable with inhomogeneous parameter $y$. Furthermore, given any second sequence $\{z_i\}$ not intersecting the rational span of $\{1,y_i\}$, we can ensure that almost every real number is inhomogeneously $\psi$-approximable with any inhomogeneous parameter $z\in\{z_i\}$. (This extension depends on a dynamical version of Erdos' Covering Systems Conjecture (1950).) Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin-Schaeffer Conjecture and Duffin-Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher's Zero-One Law (1961) for inhomogeneous approximation by reduced fractions

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010

A better life through information technology? The techno-theological eschatology of posthuman speculative science

This is the pre-peer reviewed version of the article, published in Zygon 41(2) pp.267-288, which has been published in final form at http://www3.interscience.wiley.com/journal/118588124/issueThe depiction of human identity in the pop-science futurology of engineer/inventor Ray Kurzweil, the speculative-robotics of Carnegie Mellon roboticist Hans Moravec and the physics of Tulane University mathematics professor Frank Tipler elevate technology, especially information technology, to a point of ultimate significance. For these three figures, information technology offers the potential means by which the problem of human and cosmic finitude can be rectified. Although Moravec’s vision of intelligent robots, Kurzweil’s hope for immanent human immorality, and Tipler’s description of human-like von Neumann probe colonising the very material fabric of the universe, may all appear to be nothing more than science fictional musings, they raise genuine questions as to the relationship between science, technology, and religion as regards issues of personal and cosmic eschatology. In an attempt to correct what I see as the ‘cybernetic-totalism’ inherent in these ‘techno-theologies’, I will argue for a theology of technology, which seeks to interpret technology hermeneutically and grounds human creativity in the broader context of divine creative activity

Robot rights? Towards a social-relational justification of moral consideration \ud

Should we grant rights to artificially intelligent robots? Most current and near-future robots do not meet the hard criteria set by deontological and utilitarian theory. Virtue ethics can avoid this problem with its indirect approach. However, both direct and indirect arguments for moral consideration rest on ontological features of entities, an approach which incurs several problems. In response to these difficulties, this paper taps into a different conceptual resource in order to be able to grant some degree of moral consideration to some intelligent social robots: it sketches a novel argument for moral consideration based on social relations. It is shown that to further develop this argument we need to revise our existing ontological and social-political frameworks. It is suggested that we need a social ecology, which may be developed by engaging with Western ecology and Eastern worldviews. Although this relational turn raises many difficult issues and requires more work, this paper provides a rough outline of an alternative approach to moral consideration that can assist us in shaping our relations to intelligent robots and, by extension, to all artificial and biological entities that appear to us as more than instruments for our human purpose

Upper mantle control on the W isotope record of shallow level plume and intraplate volcanic settings

Several studies have revealed small heterogeneities in the relative abundance of 182W, the radiogenic nuclide of short-lived 182Hf (t1/2 = ∼9 Ma), in terrestrial rocks. Whereas the majority of Archean rocks display 182W excesses relative to bulk silicate Earth, many young ocean island basalts show small 182W deficits, in particular if they are sourced from deep-rooted mantle plumes. The origin of this anomaly is still ambiguous, proposed models focus on core-mantle interaction or the presence of reservoirs in the lower mantle that have been isolated since the Hadean. In order to evaluate the role of upper mantle reservoirs, we report the first 182W data for intraplate basalts where a deep plume origin is still debated (Ascension Island, Massif Central, Siebengebirge and Eifel) and intraplate volcanic rocks associated with either plume or subduction zone environments (Italian Magmatic Provinces) and compare them to new data for basalts that have a deep mantle plume origin (La Réunion and Baffin Island). The proto-Iceland plume basalts from Baffin Island have uniform and modern mantle-like W of around 0 despite extremely high (3He/4He). In contrast, basalts from both volcanic edifices from La Réunion span a range from modern upper mantle values to deficits as low as W = −8.8 ppm, indicating a heterogeneous source reservoir. The W in all other intraplate volcanic provinces overlap the composition of modern upper mantle to within 3 ppm. The absence of resolvable 182W anomalies in these intraplate basalts, which partially tap the lithospheric mantle, suggests that primordial components are neither present in the central and southern European lithosphere nor in the European asthenospheric reservoir (EAR). The general absence of 182W anomalies in European plume-related basalts can either be explained by a shallow mantle source or by the absence of isotopically anomalous and isolated domains in the deep mantle beneath the northern hemisphere, as also suggested by geophysical evidence