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

Prime power indices in factorised groups

[EN] Let the group G = AB be the product of the subgroups A and B. We determine some structural properties of G when the p-elements in A. B have prime power indices in G, for some prime p. More generally, we also consider the case that all prime power order elements in A. B have prime power indices in G. In particular, when G = A = B, we obtain as a consequence some known results.The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain), and the second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain). The results in this paper are part of the third author's Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain).Felipe Román, MJ.; Martínez-Pastor, A.; Ortiz-Sotomayor, VM. (2017). Prime power indices in factorised groups. Mediterranean Journal of Mathematics. 14(6):1-15. https://doi.org/10.1007/s00009-017-1023-6S115146Amberg, B., Franciosi, S., de Giovanni, F.: Products of Groups. Oxford University Press Inc., New York (1992)Baer, R.: Group elements of prime power index. Trans. Am. Math. Soc. 75, 20–47 (1953)Ballester-Bolinches, A., Cossey, J., Li, Y.: Mutually permutable products and conjugacy classes. Monatsh. Math. 170, 305–310 (2013)Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of finite groups, vol. 53 of de Gruyter Expositions in Mathematics, Berlin (2010)Berkovich, Y., Kazarin, L.S.: Indices of elements and normal structure of finite groups. J. Algebra 283, 564–583 (2005)Camina, A.R., Camina, R.D.: Implications of conjugacy class size. J. Group Theory 1(3), 257–269 (1998)Camina, A.R., Shumyatsky, P., Sica, C.: On elements of prime-power index in finite groups. J. Algebra 323, 522–525 (2010)Chillag, D., Herzog, M.: On the length of the conjugacy classes of finite groups. J. Algebra 131, 110–125 (1990)Doerk, K., Hawkes, T.: Finite Soluble Groups, vol. 4 of de Gruyter Expositions in Mathematics, Berlin (1992)Felipe, M.J., Martínez-Pastor, A., Ortiz-Sotomayor, V.M.: On finite groups with square-free conjugacy class sizes. Int. J. Group Theory (to appear)Kurzweil, H., Stellmacher, B.: The theory of finite groups: an introduction. Springer, New York (2004)Liu, X., Wang, Y., Wei, H.: Notes on the length of conjugacy classes of finite groups. J. Pure Appl. Algebra 196, 111–117 (2005

A Map of the Nanoworld: Sizing up the Science, Politics, and Business of the Infinitesimal

Mapping out the eight main nodes of nanotechnology discourse that have emerged in the past decade, we explore how various scientific, social, and ethical islands of discussion have developed, been recognized, and are being continually renegotiated. We do so by (1) identifying the ways in which scientists, policy makers, entrepreneurs, educators, and environmental groups have drawn boundaries on issues relating to nanotechnology; (2) describing concisely the perspectives from which these boundaries are drawn; and (3) exploring how boundaries on nanotechnology are marked and negotiated by various nodes of nanotechnology discourse.Comment: 25 page

Virtual money, practices and moral orders in Second Life

Virtual monies present a limit case in debates about money's moral and political entanglements between sociologists, anthropologists, and economists. Digitized virtual monies seem ephemeral, almost ideal typical examples of money as a pure medium of exchange. This paper begins with the premise that virtual monies are as value-laden and morally entangled as any other form of money. This assertion is demonstrated by exploring how one type of virtual money, the Linden dollar (L$), and some of its associated practices are bound up with research participants' moral categories and judgments in the virtual world of Second Life (SL). Participants' accounts of virtual money practices are linked to moral attributes, sometimes in stark ‘good’ or ‘bad’ dichotomies, but also in more nuanced terms. These framings reproduce classifications of people and practices along a continuum with virtuousness at one end and maliciousness or harm at the other, passing through various states of possible moral dubiousness. For respondents, these two judgments go together; people are what they do with money. As a result, respondents decide what ‘people like that’ deserve. Evaluating someone's money practices means assessing the person. Participants' accounts of Linden dollar practices overlap with explanations of what SL is and how residents should live there. In SL, money is a form of material culture through which appropriate ways of being in the world are debated and reproduced

Locally finite groups containing a 2 -element with Chernikov centralizer

Suppose that a locally finite group G has a 2-element g with Chernikov centralizer. It is proved that if the involution in ⟨g⟩ has nilpotent centralizer, then G has a soluble subgroup of finite index

The Temporal Singularity: time-accelerated simulated civilizations and their implications

Provided significant future progress in artificial intelligence and computing, it may ultimately be possible to create multiple Artificial General Intelligences (AGIs), and possibly entire societies living within simulated environments. In that case, it should be possible to improve the problem solving capabilities of the system by increasing the speed of the simulation. If a minimal simulation with sufficient capabilities is created, it might manage to increase its own speed by accelerating progress in science and technology, in a way similar to the Technological Singularity. This may ultimately lead to large simulated civilizations unfolding at extreme temporal speedups, achieving what from the outside would look like a Temporal Singularity. Here we discuss the feasibility of the minimal simulation and the potential advantages, dangers, and connection to the Fermi paradox of the Temporal Singularity. The medium-term importance of the topic derives from the amount of computational power required to start the process, which could be available within the next decades, making the Temporal Singularity theoretically possible before the end of the century.Comment: To appear in the conference proceedings of the AGI-18 conference (published in the Springer's Lecture Notes in AI series

Scalable Massively Parallel Artificial Neural Networks

There is renewed interest in computational intelligence, due to advances in algorithms, neuroscience, and computer hardware. In addition there is enormous interest in autonomous vehicles (air, ground, and sea) and robotics, which need significant onboard intelligence. Work in this area could not only lead to better understanding of the human brain but also very useful engineering applications. The functioning of the human brain is not well understood, but enormous progress has been made in understanding it and, in particular, the neocortex. There are many reasons to develop models of the brain. Artificial Neural Networks (ANN), one type of model, can be very effective for pattern recognition, function approximation, scientific classification, control, and the analysis of time series data. ANNs often use the back-propagation algorithm for training, and can require large training times especially for large networks, but there are many other types of ANNs. Once the network is trained for a particular problem, however, it can produce results in a very short time. Parallelization of ANNs could drastically reduce the training time. An object-oriented, massively-parallel ANN (Artificial Neural Network) software package SPANN (Scalable Parallel Artificial Neural Network) has been developed and is described here. MPI was use

The Social and Ethical Acceptability of NBICs for Purposes of Human Enhancement: Why Does the Debate Remain Mired in Impasse?

The emergence and development of convergent technologies for the purpose of improving human performance, including nanotechnology, biotechnology, information sciences, and cognitive science (NBICs), open up new horizons in the debates and moral arguments that must be engaged by philosophers who hope to take seriously the question of the ethical and social acceptability of these technologies. This article advances an analysis of the factors that contribute to confusion and discord on the topic, in order to help in understanding why arguments that form a part of the debate between transhumanism and humanism result in a philosophical and ethical impasse: 1. The lack of clarity that emerges from the fact that any given argument deployed (arguments based on nature and human nature, dignity, the good life) can serve as the basis for both the positive and the negative evaluation of NBICs. 2. The impossibility of providing these arguments with foundations that will enable others to deem them acceptable. 3. The difficulty of applying these same arguments to a specific situation. 4. The ineffectiveness of moral argument in a democratic society. The present effort at communication about the difficulties of the argumentation process is intended as a necessary first step towards developing an interdisciplinary response to those difficulties