807 research outputs found
A well-posed optimal spectral element approximation for the Stokes problem
A method is proposed for the spectral element simulation of incompressible flow. This method constitutes in a well-posed optimal approximation of the steady Stokes problem with no spurious modes in the pressure. The resulting method is analyzed, and numerical results are presented for a model problem
Spectral collocation methods
This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2
Four dimensional Lie symmetry algebras and fourth order ordinary differential equations
Realizations of four dimensional Lie algebras as vector fields in the plane
are explicitly constructed. Fourth order ordinary differential equations which
admit such Lie symmetry algebras are derived. The route to their integration is
described.Comment: 12 page
The rings of n-dimensional polytopes
Points of an orbit of a finite Coxeter group G, generated by n reflections
starting from a single seed point, are considered as vertices of a polytope
(G-polytope) centered at the origin of a real n-dimensional Euclidean space. A
general efficient method is recalled for the geometric description of G-
polytopes, their faces of all dimensions and their adjacencies. Products and
symmetrized powers of G-polytopes are introduced and their decomposition into
the sums of G-polytopes is described. Several invariants of G-polytopes are
found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers
and congruence classes of the polytopes. The definitions apply to
crystallographic and non-crystallographic Coxeter groups. Examples and
applications are shown.Comment: 24 page
On character generators for simple Lie algebras
We study character generating functions (character generators) of simple Lie
algebras. The expression due to Patera and Sharp, derived from the Weyl
character formula, is first reviewed. A new general formula is then found. It
makes clear the distinct roles of ``outside'' and ``inside'' elements of the
integrity basis, and helps determine their quadratic incompatibilities. We
review, analyze and extend the results obtained by Gaskell using the Demazure
character formulas. We find that the fundamental generalized-poset graphs
underlying the character generators can be deduced from such calculations.
These graphs, introduced by Baclawski and Towber, can be simplified for the
purposes of constructing the character generator. The generating functions can
be written easily using the simplified versions, and associated Demazure
expressions. The rank-two algebras are treated in detail, but we believe our
results are indicative of those for general simple Lie algebras.Comment: 50 pages, 11 figure
Optimally defined Racah-Casimir operators for su(n) and their eigenvalues for various classes of representations
This paper deals with the striking fact that there is an essentially
canonical path from the -th Lie algebra cohomology cocycle, ,
of a simple compact Lie algebra \g of rank to the definition of its
primitive Casimir operators of order . Thus one obtains a
complete set of Racah-Casimir operators for each \g and nothing
else. The paper then goes on to develop a general formula for the eigenvalue
of each valid for any representation of \g, and thereby
to relate to a suitably defined generalised Dynkin index. The form of
the formula for for is known sufficiently explicitly to make
clear some interesting and important features. For the purposes of
illustration, detailed results are displayed for some classes of representation
of , including all the fundamental ones and the adjoint representation.Comment: Latex, 16 page
Study of the risk-adjusted pricing methodology model with methods of Geometrical Analysis
Families of exact solutions are found to a nonlinear modification of the
Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM)
incorporates both transaction costs and the risk from a volatile portfolio.
Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM
equation. It gives us the possibility to describe an optimal system of
subalgebras and correspondingly the set of invariant solutions to the model. In
this way we can describe the complete set of possible reductions of the
nonlinear RAPM model. Reductions are given in the form of different second
order ordinary differential equations. In all cases we provide solutions to
these equations in an exact or parametric form. We discuss the properties of
these reductions and the corresponding invariant solutions.Comment: larger version with exact solutions, corrected typos, 13 pages,
Symposium on Optimal Stopping in Abo/Turku 200
Multimedia as a modernization direction in the course of teaching "History of Ukraine"
Використання мультимедійних презентацій в системі сучасної освіти займає все більше місце та стає певною повсякденністю. Мультимедія під час викладання дисципліни "Історія України" є важливим елементом освітнього процесу, яка покликана мотивувати студентів до навчання, поліпшити сприйняття інформації, зробити навчальний процес сучасним, цікавим та продуктивним. Мультимедійні презентації
створені викладачами та студентами постійно вдосконалюються та являються модернізаційним напрямком навчання та комунікації.The use of multimedia presentations in the system of modern education takes up an increasing number of places and becomes a "daily routine". Multimedia during the teaching of the discipline "History of Ukraine" is an important element of the educational process. During lectures and seminars, using the multimedia technologies is a topical issue today. Multimedia is designed to motivate students to study, improve perceptions of information, make the learning process interesting and productive. Multimedia presentations created by lecturers and students serve as a kind of communication. They are constantly improving and being a modernization training area
Vacuum Plane Waves in 4+1 D and Exact solutions to Einstein's Equations in 3+1 D
In this paper we derive homogeneous vacuum plane-wave solutions to Einstein's
field equations in 4+1 dimensions. The solutions come in five different types
of which three generalise the vacuum plane-wave solutions in 3+1 dimensions to
the 4+1 dimensional case. By doing a Kaluza-Klein reduction we obtain solutions
to the Einstein-Maxwell equations in 3+1 dimensions. The solutions generalise
the vacuum plane-wave spacetimes of Bianchi class B to the non-vacuum case and
describe spatially homogeneous spacetimes containing an extremely tilted fluid.
Also, using a similar reduction we obtain 3+1 dimensional solutions to the
Einstein equations with a scalar field.Comment: 16 pages, no figure
Einstein billiards and spatially homogeneous cosmological models
In this paper, we analyse the Einstein and Einstein-Maxwell billiards for all
spatially homogeneous cosmological models corresponding to 3 and 4 dimensional
real unimodular Lie algebras and provide the list of those models which are
chaotic in the Belinskii, Khalatnikov and Lifschitz (BKL) limit. Through the
billiard picture, we confirm that, in D=5 spacetime dimensions, chaos is
present if off-diagonal metric elements are kept: the finite volume billiards
can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody
algebras. The most generic cases bring in the same algebras as in the
inhomogeneous case, but other algebras appear through special initial
conditions.Comment: 27 pages, 10 figures, additional possibility analysed in section 4.3,
references added, typos correcte
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