We study the algebraic and geometric structures that underly the space of vacua of N = 1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a po...

Implicit sampling is a Monte Carlo (MC) method that focuses the computational effort on the region of high probability by first locating this region via numerical optimization and then solving random algebraic equations to explore it. Implicit sampling has been shown to be efficient in online state estimation and filtering (data assimilation) problems; we use it here to estimate a diffusion coe...

This report contains a collection of notes on abstract additive and multiplicative Schwarz methods for self-adjoint positive linear operator equations. We examine closely one of the most elegant and useful modern convergence theories for these methods, following the recent work in the finite element multigrid and domain decomposition literature. Our motivation is to fully understand the structu...

We consider a new adaptive finite element (AFEM) algorithm for self-adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue met...

Let G be a connected reductive affine algebraic group defined over the complex numbers, and K ⊂ G be a maximal compact subgroup. Let X, Y be irreducible smooth complex projective varieties and f : X → Y an algebraic morphism, such that π1(Y ) is virtually nilpotent and the homomorphism f∗ : π1(X)→ π1(Y ) is surjective. Define R ( π1(X), G ) = {ρ ∈ Hom ( π1(X), G ) | A ◦ ρ factors through f∗} , ...

The aim of this note is to remark that the injectivity theorems of Kollár and EsnaultViehweg can be used to give a quick algebraic proof of a strengthening (by dropping the positivity hypothesis) of the Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves proved in [EL], and to extend this result to higher cohomology classes a...

We present a symbolic algorithm generating finite-element schemes with interpolating Hermite polynomials intended for solving the boundary-value problems with self-adjoint second-order differential equation and implemented in the Maple computer algebra system. Recurrence relations for the calculation in analytical form of the interpolating Hermite polynomials with nodes of arbitrary multiplicit...

Polynomials suffice as finite element basis functions for triangles, parallelograms, and some other elements of little practical importance. Rational basis functions extend the range of allowed elements to the much wider class of well-set algebraic elements, where well-set is a convexity type constraint. The extension field from R(x, y) to R(x, y, √ x2 + y2) removes this quadrilateral constrain...

Let G be a semisimple algebraic group with Lie algebra g. In 1979, J. Dixmier proved that any vector field annihilating all G-invariant polynomials on g lies in the k[g]module generated by the ”adjoint vector fields”, i.e., vector fields ς of the form ς(y)(x) = [x, y], x, y ∈ g. A substantial generalisation of Dixmier’s theorem was found by Levasseur and Stafford. They explicitly described the ...

This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution σ of the adjoint bundle A gives rise, given certain algebraic conditions, to a unique preferred affine connection ∇ with covariantly constant rho-tensor P, compatible with the algebraic bracket on A. These co...