Turing machines can be efficiently simulated by the General Purpose Analog Computer

The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reductions. However, the situation is less clear in what concerns models of computation using real numbers, and no analog of the Church-Turing thesis exists for this case. Recently it was shown that some models of computation with real numbers were equivalent from a computability perspective. In particular it was shown that Shannon's General Purpose Analog Computer (GPAC) is equivalent to Computable Analysis. However, little is known about what happens at a computational complexity level. In this paper we shed some light on the connections between this two models, from a computational complexity level, by showing that, modulo polynomial reductions, computations of Turing machines can be simulated by GPACs, without the need of using more (space) resources than those used in the original Turing computation, as long as we are talking about bounded computations. In other words, computations done by the GPAC are as space-efficient as computations done in the context of Computable Analysis

Long-lived oscillons from asymmetric bubbles

The possibility that extremely long-lived, time-dependent, and localized field configurations (``oscillons'') arise during the collapse of asymmetrical bubbles in 2+1 dimensional phi^4 models is investigated. It is found that oscillons can develop from a large spectrum of elliptically deformed bubbles. Moreover, we provide numerical evidence that such oscillons are: a) circularly symmetric; and b) linearly stable against small arbitrary radial and angular perturbations. The latter is based on a dynamical approach designed to investigate the stability of nonintegrable time-dependent configurations that is capable of probing slowly-growing instabilities not seen through the usual ``spectral'' method.Comment: RevTeX 4, 9 pages, 11 figures. Revised version with a new approach to stability. Accepted to Phys. Rev.

From the Big Bang Theory to the Theory of a Stationary Universe

We consider chaotic inflation in the theories with the effective potentials phi^n and e^{\alpha\phi}. In such theories inflationary domains containing sufficiently large and homogeneous scalar field \phi permanently produce new inflationary domains of a similar type. We show that under certain conditions this process of the self-reproduction of the Universe can be described by a stationary distribution of probability, which means that the fraction of the physical volume of the Universe in a state with given properties (with given values of fields, with a given density of matter, etc.) does not depend on time, both at the stage of inflation and after it. This represents a strong deviation of inflationary cosmology from the standard Big Bang paradigm. We compare our approach with other approaches to quantum cosmology, and illustrate some of the general conclusions mentioned above with the results of a computer simulation of stochastic processes in the inflationary Universe.Comment: No changes to the file, but original figures are included. They substantially help to understand this paper, as well as eternal inflation in general, and what is now called the "multiverse" and the "string theory landscape." High quality figures can be found at http://www.stanford.edu/~alinde/LLMbigfigs

Cosmology, Particle Physics and Superfluid 3He

Many direct parallels connect superfluid 3He with the field theories describing the physical vacuum, gauge fields and elementary fermions. Superfluid $^3$He exhibits a variety of topological defects which can be detected with single-defect sensitivity. Modern scenarios of defect-mediated baryogenesis can be simulated by the interaction of the 3He vortices and domain walls with fermionic quasiparticles. Formation of defects in a symmetry-breaking phase transition in the early Universe, which could be responsible for large-scale structure formation and for microwave-background anisotropy, also may be modelled in the laboratory. This is supported by the recent observation of vortex formation in neutron-irradiated 3He-B where the "primordial fireball" is formed in an exothermic nuclear reaction.Comment: Invited talk at LT-21 Conference, 20 pages, 3 figures available at request, compressed ps file of the camera-ready format with 3 figures is at ftp://boojum.hut.fi/pub/publications/lowtemp/LTL-96006.ps.g