### ${\partial}_{\psi}$- difference calculus Bernoulli-Taylor formula

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Introduction: This article will present just one example of a general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution. It was the motivating example from two lectures on the BGG resolution given at the 19th Czech Winter School on Geometry and Physics held in Srní in January 1999. This article may be seen as a technical example to go with a more elementary introduction which will appear elsewhere [M. Eastwood, Notices Am. Math. Soc. 46, No. 11, 1368-1376 (1999)]. In fact, there...

Summary: The Ado theorem is a fundamental fact, which has a reputation of being a `strange theorem'. We give its natural proof.

An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\{,\cdots ,\}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$, i.e., $$\sum _{\sigma \in {S}_{2n-1}}(sign\sigma )\{\{{f}_{{\sigma}_{1}},\cdots ,{f}_{{\sigma}_{n}}\},{f}_{{\sigma}_{n+1}},\cdots ,{f}_{{\sigma}_{2n-1}}\}=0,$$${S}_{2n...}$

Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${\mathcal{D}}_{K}^{k}$ for any positive integer $k$. This is spanned over $K$ by ${d}_{0},...,{d}_{k}$, and has comultiplication $\Delta $ and counit $\epsilon $ defined by $\Delta \left({d}_{i}\right)={\sum}_{j=0}^{i}{d}_{j}\otimes {d}_{i-j}$ and $\epsilon \left({d}_{i}\right)={\delta}_{0,i}$ (Kronecker’s delta) for any $i$. This note presents a representation of the coalgebra ${\mathcal{D}}_{K}^{k}$ by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces.

Summary: In this paper the isometries of the dual space were investigated. The dual structural equations of a Killing tensor of order two were found. The general results are applied to the case of the flat space.

From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant’s result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package $LiE$ offers the data structures and corresponding procedures for computing...

This paper gives an exposition of algebraic K-theory, which studies functors ${K}_{n}:\text{Rings}\to \text{Abelian}\phantom{\rule{4.0pt}{0ex}}\text{Groups}$, $n$ an integer. Classically $n=0,1$ introduced by Bass in the mid 60’s (based on ideas of Grothendieck and others) and $n=2$ introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite,...

A certain family of homogeneous spaces is investigated. Basic invariant operators for each of these structures are presented and some analogies to Levi-Civita connections of Riemannian geometry are pointed out.

Summary: In order to get a mathematical understanding of the BRS-transformation and the Slavnov-Taylor identities, we treat them in a finite dimensional setting. We show that in this setting the BRS-transformation is a vector field on a certain supermanifold. The connection to the BRS-complex will be established. Finally we treat the generating functional and the Slavnov-Taylor identity in this setting.

[For the entire collection see Zbl 0742.00067.]We are interested in partial differential equations on domains in ${\mathcal{C}}^{n}$. One of the most natural questions is that of analytic continuation of solutions and domains of holomorphy. Our aim is to describe the domains of holomorphy for solutions of the complex Laplace and Dirac equations. We call them cells of harmonicity. We deduce their properties mostly by examining geometrical properties of the characteristic surface (which is the same for both equations),...

[For the entire collection see Zbl 0742.00067.]For the purpose of providing a comprehensive model for the physical world, the authors set up the notion of a Clifford manifold which, as mentioned below, admits the usual tensor structure and at the same time a spin structure. One considers the spin space generated by a Clifford algebra, namely, the vector space spanned by an orthonormal basis $\{{e}_{j}:j=1,\cdots ,n\}$ satisfying the condition $\{{e}_{i},{e}_{j}\}\equiv {e}_{i}{e}_{j}={e}_{j}{e}_{i}=2I{\eta}_{ij}$, where $I$ denotes the unit scalar of the algebra and (${\eta}_{ij}$) the nonsingular Minkowski...

This is an exposition of a general machinery developed by M. G. Eastwood, T. N. Bailey, C. R. Graham which analyses some real integral transforms using complex methods. The machinery deals with double fibrations $M\subset \Omega \stackrel{\eta}{\to}\leftarrow \tilde{\Omega}@>\tau >>X$$(\Omega $ complex manifold; $M$ totally real, real-analytic submanifold;...