10,577 research outputs found
Counting solutions of equations over two-element algebras
Solving equations is one of the most important problems in computer science. Apart from the problem of existence of solutions of equations we may consider the problem of a number of solutions of equations. Such a problem is much more difficult than the decision one. This paper presents a complete classification of the complexity of the problem of counting solutions of equations over any fixed two-element algebra. It is shown that the complexity of such problems depends only on the clone of term operations of the algebra and for any fixed two-element algebra such a problem is either in FP or #Pcomplete
V.M. Miklyukov: from dimension 8 to nonassociative algebras
In this short survey we give a background and explain some recent
developments in algebraic minimal cones and nonassociative algebras. A good
deal of this paper is recollections of my collaboration with my teacher, PhD
supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that
motivated the present research. This paper is dedicated to his memory.Comment: 19 page
Essential Constants for Spatially Homogeneous Ricci-flat manifolds of dimension 4+1
The present work considers (4+1)-dimensional spatially homogeneous vacuum
cosmological models. Exact solutions -- some already existing in the
literature, and others believed to be new -- are exhibited. Some of them are
the most general for the corresponding Lie group with which each homogeneous
slice is endowed, and some others are quite general. The characterization
``general'' is given based on the counting of the essential constants, the
line-element of each model must contain; indeed, this is the basic contribution
of the work. We give two different ways of calculating the number of essential
constants for the simply transitive spatially homogeneous (4+1)-dimensional
models. The first uses the initial value theorem; the second uses, through
Peano's theorem, the so-called time-dependent automorphism inducing
diffeomorphismsComment: 26 Pages, 2 Tables, latex2
Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators
Descent equations play an important role in the theory of characteristic
classes and find applications in theoretical physics, e.g. in the Chern-Simons
field theory and in the theory of anomalies. The second Chern class (the first
Pontrjagin class) is defined as where is the
curvature 2-form and is an invariant scalar
product on the corresponding Lie algebra . The descent for
gives rise to an element of
mixed degree. The 3-form part is the Chern-Simons form. The 2-form
part is known as the Wess-Zumino action in physics. The 1-form
component is related to the canonical central extension of the loop
group .
In this paper, we give a new interpretation of the low degree components
and . Our main tool is the universal differential calculus
on free Lie algebras due to Kontsevich. We establish a correspondence between
solutions of the first Kashiwara-Vergne equation in Lie theory and universal
solutions of the descent equation for the second Chern class . In more
detail, we define a 1-cocycle which maps automorphisms of the free Lie
algebra to one forms. A solution of the Kashiwara-Vergne equation is mapped
to . Furthermore, the component is related to the
associator corresponding to . It is surprising that while and
satisfy the highly non-linear twist and pentagon equations, the elements
and solve the linear descent equation
On polynomial solutions of differential equations
A general method of obtaining linear differential equations having polynomial
solutions is proposed. The method is based on an equivalence of the spectral
problem for an element of the universal enveloping algebra of some Lie algebra
in the "projectivized" representation possessing an invariant subspace and the
spectral problem for a certain linear differential operator with variable
coefficients. It is shown in general that polynomial solutions of partial
differential equations occur; in the case of Lie superalgebras there are
polynomial solutions of some matrix differential equations, quantum algebras
give rise to polynomial solutions of finite--difference equations.
Particularly, known classical orthogonal polynomials will appear when
considering acting on . As examples, some
polynomials connected to projectivized representations of ,
, and are briefly discussed.Comment: 12p
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