How to find the holonomy algebra of a Lorentzian manifold

Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de~Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra $\mathfrak{g}\subset\mathfrak{so}(1,n-1)$ of a locally indecomposable Lorentzian manifold $(M,g)$ of dimension $n$ is different from $\mathfrak{so}(1,n-1)$, then it is contained in the similitude algebra $\mathfrak{sim}(n-2)$. There are 4 types of such holonomy algebras. Criterion how to find the type of $\mathfrak{g}$ are given, and special geometric structures corresponding to each type are described. To each $\mathfrak{g}$ there is a canonically associated subalgebra $\mathfrak{h}\subset\mathfrak{so}(n-2)$. An algorithm how to find $\mathfrak{h}$ is provided.Comment: 15 pages; the final versio

Structural transitions and nonmonotonic relaxation processes in liquid metals

Structural transitions in melts as well as their dynamics are considered. It is supposed that liquid represents the solution of relatively stable solid-like locally favored structures (LFS) in the surrounding of disordered normal-liquid structures. Within the framework of this approach the step changes of liquid Co viscosity are considered as liquid-liquid transitions. It is supposed that this sort of transitions represents the cooperative medium-range bond ordering, and corresponds to the transition of the "Newtonian fluid" to the "structured fluid". It is shown that relaxation processes with oscillating-like time behavior ($\omega \sim 10^{-2}$~$s^{-1}$) of viscosity are possibly close to this point

Next-to-Leading Order QCD Analysis of Polarized Deep Inelastic Scattering Data

We present a Next-to-Leading order perturbative QCD analysis of world data on the spin dependent structure functions $g_1^p, g_1^n$, and $g_1^d$, including the new experimental information on the $Q^2$ dependence of $g_1^n$. Careful attention is paid to the experimental and theoretical uncertainties. The data constrain the first moments of the polarized valence quark distributions, but only qualitatively constrain the polarized sea quark and gluon distributions. The NLO results are used to determine the $Q^2$ dependence of the ratio $g_1/F_1$ and evolve the experimental data to a constant $Q^2 = 5 GeV^2$. We determine the first moments of the polarized structure functions of the proton and neutron and find agreement with the Bjorken sum rule.Comment: 21 pages, 4 figures; final version to be published in Phys. Lett. B. References updated. Uses elsart.cls version 1996/04/22, 2e-1.4

Multiplicative updates for large margin classifiers

Abstract. Various problems in nonnegative quadratic programming arise in the training of large margin classifiers. We derive multiplicative updates for these problems that converge monotonically to the desired solutions for hard and soft margin classifiers. The updates differ strikingly in form from other multiplicative updates used in machine learning. In this paper, we provide complete proofs of convergence for these updates and extend previous work to incorporate sum and box constraints in addition to nonnegativity.