Solving variational inequalities with Stochastic Mirror-Prox algorithm

In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasability problem and Eigenvalue minimization

Existence of Universal Entangler

A gate is called entangler if it transforms some (pure) product states to entangled states. A universal entangler is a gate which transforms all product states to entangled states. In practice, a universal entangler is a very powerful device for generating entanglements, and thus provides important physical resources for accomplishing many tasks in quantum computing and quantum information. This Letter demonstrates that a universal entangler always exists except for a degenerated case. Nevertheless, the problem how to find a universal entangler remains open.Comment: 4 page

Automorphic properties of low energy string amplitudes in various dimensions

This paper explores the moduli-dependent coefficients of higher derivative interactions that appear in the low-energy expansion of the four-graviton amplitude of maximally supersymmetric string theory compactified on a d-torus. These automorphic functions are determined for terms up to order D^6R^4 and various values of d by imposing a variety of consistency conditions. They satisfy Laplace eigenvalue equations with or without source terms, whose solutions are given in terms of Eisenstein series, or more general automorphic functions, for certain parabolic subgroups of the relevant U-duality groups. The ultraviolet divergences of the corresponding supergravity field theory limits are encoded in various logarithms, although the string theory expressions are finite. This analysis includes intriguing representations of SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and two-loop string and supergravity amplitudes.Comment: 80 pages. 1 figure. v2:Typos corrected, footnotes amended and small clarifications. v3: minor corrections. Version to appear in Phys Rev

Index and stable linear forms of Lie algebras

We characterize, in a purely algebraic manner, certain linear forms, called stable, on a Lie algebra. As an application, we determine the index of a Borel subalgebra of a semi-simple Lie algebra. Finally, we give an example of a parabolic subalgebra of a semi-simple Lie algebra which does not admit any stable linear form.Comment: This paper is written in Frenc

Commuting varieties of rr-tuples over Lie algebras

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let \g be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th Frobenius kernel of $G$ is homeomorphic to the commuting variety of $r$-tuples of elements in the nilpotent cone of \g [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{10} (1997), 693--728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties C_r(\mathfrak{gl}_2), C_r(\fraksl_2) and $C_r(\N)$ where $\N$ is the nilpotent cone of \fraksl_2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when \g=\fraksl_2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of \fraksl_3, we are able to verify the aforementioned properties for C_r(\fraku). Finally, applying our calculations on the commuting variety C_r(\overline{\calO_{\sub}}) where \overline{\calO_{\sub}} is the closure of the subregular orbit in \fraksl_3, we prove that the nilpotent commuting variety $C_r(\N)$ has singularities of codimension $\ge 2$.Comment: To appear in Journal of Pure and Applied Algebr

The integrability of the 2-Toda lattice on a simple Lie algebra

We define the 2-Toda lattice on every simple Lie algebra g, and we show its Liouville integrability. We show that this lattice is given by a pair of Hamiltonian vector fields, associated with a Poisson bracket which results from an R-matrix of the underlying Lie algebra. We construct a big family of constants of motion which we use to prove the Liouville integrability of the system. We achieve the proof of their integrability by using several results on simple Lie algebras, R-matrices, invariant functions and root systems.Comment: 30 page