Minimal D=7D=7 Supergravity and the supersymmetry of Arnold-Beltrami Flux branes

In this paper we study some properties of the newly found Arnold-Beltrami flux-brane solutions to the minimal $D=7$ supergravity. To this end we first single out the appropriate Free Differential Algebra containing both a gauge $3$-form $\mathbf{B}^{[3]}$ and a gauge $2$-form $\mathbf{B}^{[2]}$: then we present the complete rheonomic parametrization of all the generalized curvatures. This allows us to identify two-brane configurations with Arnold-Beltrami fluxes in the transverse space with exact solutions of supergravity and to analyze the Killing spinor equation in their background. We find that there is no preserved supersymmetry if there are no additional translational Killing vectors. Guided by this principle we explicitly construct Arnold-Beltrami flux two-branes that preserve $0$, $1/8$ and $1/4$ of the original supersymmetry. Two-branes without fluxes are instead BPS states and preserve $1/2$ supersymmetry. For each two-brane solution we carefully study its discrete symmetry that is always given by some appropriate crystallographic group $\Gamma$. Such symmetry groups $\Gamma$ are transmitted to the $D=3$ gauge theories on the brane world--volume that occur in the gauge/gravity correspondence. Furthermore we illustrate the intriguing relation between gauge fluxes in two-brane solutions and hyperinstantons in $D=4$ topological sigma-models.Comment: 56 pages, LaTeX source, 8 jpg figures, typos correcte

Calculation of Heat-Kernel Coefficients and Usage of Computer Algebra

The calculation of heat-kernel coefficients with the classical DeWitt algorithm has been discussed. We present the explicit form of the coefficients up to $h_5$ in the general case and up to $h_7^{min}$ for the minimal parts. The results are compared with the expressions in other papers. A method to optimize the usage of memory for working with large expressions on universal computer algebra systems has been proposed.Comment: 12 pages, LaTeX, no figures. Extended version of contribution to AIHENP'95, Pisa, April 3-8, 199

Cosmology with minimal length uncertainty relations

We study the effects of the existence of a minimal observable length in the phase space of classical and quantum de Sitter (dS) and Anti de Sitter (AdS) cosmology. Since this length has been suggested in quantum gravity and string theory, its effects in the early universe might be expected. Adopting the existence of such a minimum length results in the Generalized Uncertainty Principle (GUP), which is a deformed Heisenberg algebra between minisuperspace variables and their momenta operators. We extend these deformed commutating relations to the corresponding deformed Poisson algebra in the classical limit. Using the resulting Poisson and Heisenberg relations, we then construct the classical and quantum cosmology of dS and Ads models in a canonical framework. We show that in classical dS cosmology this effect yields an inflationary universe in which the rate of expansion is larger than the usual dS universe. Also, for the AdS model it is shown that GUP might change the oscillatory nature of the corresponding cosmology. We also study the effects of GUP in quantized models through approximate analytical solutions of the Wheeler-DeWitt (WD) equation, in the limit of small scale factor for the universe, and compare the results with the ordinary quantum cosmology in each case.Comment: 11 pages, 4 figures, to appear in IJMP

A probabilistic algorithm to test local algebraic observability in polynomial time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation.Comment: 26 pages. A Maple implementation is availabl

Methods for suspensions of passive and active filaments

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid--structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation

Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies

I introduce an approximation scheme that allows to deduce differential equations for the renormalization group $\beta$-function from a Schwinger--Dyson equation for the propagator. This approximation is proven to give the dominant asymptotic behavior of the perturbative solution. In the supersymmetric Wess--Zumino model and a $\phi^3_6$ scalar model which do not have divergent vertex functions, this simple Schwinger--Dyson equation for the propagator captures the main quantum corrections.Comment: Clarification of the presentation of results. Equations and results unchanged. Match the published version. 12 page

Formation and Evolution of Singularities in Anisotropic Geometric Continua

Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces. Several examples are given. The motivating example is the directed self assembly of a large number of particles for technological purposes such as nano-science processes, in which the particle interactions are anisotropic. This application leads to the derivation and analysis of gradient flow equations on Lie algebras. The Riemann structure of these gradient flow equations is also discussed.Comment: 38 pages, 4 figures. Physica D, submitte