We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie alge...

Fix a manifold M , and let V be an infinite dimensional simple Lie subalgebra of the Lie algebra VecM of vector fields on M . Assume that V contains a finite dimensional simple maximal subalgebra a(V). We define an a(V)-quantization of a V-module of differential operators on M to be a decomposition of the module into irreducible a(V)-modules. In this article we survey some recent results and op...

in this paper we will determine the multiple point manifolds of certain self-transverse immersions in euclidean spaces. following the triple points, these immersions have a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to dold manifold $v^5$ or a boundary. we will show there is an immersion of $s^7times p^2$ in $mathbb{r}^{13...

Algebra Situs is a branch of mathematics which has its roots in Jones’ construction of his polynomial invariant of links and Drinfeld’s work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and sta...

A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group-categories, in hopes of finding clues to a better u...

It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetric, and specializes in the projectively flat case to the one constructed previ...

Abstract It is known since the 1970s that no more than 23 multiplications are required for computing product of two 3 × -matrices. For non-commutative coefficient rings, it not whether can also be done with fewer multiplications. However, there several mutually inequivalent ways doing job In this article, we extend list considerably by providing 17,000 new and schemes multiplying -matrices usin...

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and sta...

The new generalization of the gauge interaction for the bosonic strings is found. We consider some quasi-equivariant maps from the space of metrics on the worldsheet to the space of n-tuples of oneand two-dimensional loops. The two-dimensional case is based on the cylinders interacted with a path space connection. The special 2-gauge string model is formulated using two 1-connections, nonabelia...

We generalize recent results by G.A. Freiman, M. Herzog and coauthors on the structure theory of product-sets from the context of linearly (i.e., strictly and totally) ordered groups to linearly ordered semigroups. In particular, we find that if S is a finite subset of a linearly ordered semigroup generating a nonabelian subsemigroup, then |S2| ≥ 3 |S| − 2. On the road to this goal, we also pro...