The simplest illustration of a steerable function is the first-order directional derivative of a two-dimensional Gaussian. Although we will only consider Gaussians, the principle of steerability may be extended to any differentiable function. For notational simplicity we will consider a unit-variant Gaussian and ignore the 1 p2 scaling constant: g(x; y) = e (x2+y2)=2: (1) Lets begin by consider...

Let EC denote the complexified adjoint Lie-algebra bundle associated to P . For the purposes of this summary, we will assume that E = EC; the more general case will be dealt with in section 7. Let dt : Ω (M ;EC)→Ω (M ;EC) denote the exterior derivative corresponding to At for each t. At t = 0, we wish to calculate the dimension of ker(D0), which gives the number of eigenvalues λα(t) of Dt passi...

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G. MSC 2000 : 20C15,17B99.

We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster’s operad for weak ω-categories.

Let M̃ be a (2m+ 1)-dimensional almost contact manifold with almost contact structure (φ,ξ,η), that is, a global vector field ξ, a (1,1) tensor field φ, and a 1-form η on M̃ such that φ2X =−X +η(X)ξ, η(ξ) = 1 for any vector field X on M̃. We consider a product manifold M̃×R, whereR denotes a real line. Then a vector field on M̃×R is given by (X , f (d/dt)), where X is a vector field tangent to M̃, t ...

A well known theorem of Duflo, the “annihilation theorem”, claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is centrally generated. For the Lie superalgebra osp(1, 2l), this result does not hold. In this article, we introduce a “correct” analogue of the centre for which the annihilation theorem does hold in the case osp(1, 2l). This sub...

Many familiar and useful spaces of continuous or differentiable functions are Hilbert or Banach spaces, with pleasant completeness properties, but many are not. Some are Fréchet spaces, thus still complete, but lacking some of the conveniences of Banach spaces. Some other important spaces are not Fréchet, either. Still, some of these important spaces are colimits of Fréchet spaces (or of Banach...

We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.

The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtain...

One can prove the following proposition (1) For all topological spaces S, T holds Ω[: S, T :] = [:ΩS , ΩT :]. Let X be a set and let Y be an empty set. Note that [:X, Y :] is empty. Let X be an empty set and let Y be a set. Observe that [:X, Y :] is empty. We now state the proposition (2) Let X, Y be non empty topological spaces and x be a point of X. Then Y 7−→ x is a continuous map from Y int...