The regular representation of 𝑈_{𝜐}(𝔤𝔩_{𝔪|𝔫})

In [ 2 ] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [ 6 ]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro- p groups. Let [ ] be the algebraic closure of [ ] p and U be the formal power series algebra [ ][lobrk ] T [robrk ], with group of units U × . Let Q be a finitely generated abelian pro- p group. We write ℤ p [lobrk ] Q [robrk ] for the completed group algebra of Q over ℤ p . Let T ( Q ) be the abelian group Hom( Q , U × ) of continuous homomorphisms from Q to U × . We write 1 for the trivial homomorphism. Each v ∈ T ( Q ) extends to a unique continuous algebra homomorphism vmacr; from ℤ p [lobrk ] Q [robrk ] to U .

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the k \mathfrak {k} -finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

Norming the Algebra of Formal Power Series in n Indeterminates

A representation of solution of stochastic differential equations

Potentials for some tensor algebras

Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for {\it arbitrary} values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.

In this paper, we study symplectic form on low dimensional real Lie algebra. A symplectic form is very important in classifying of Lie algebra types. Based on their dimension and certain conditions, there are two types of Lie algebras. A lie algebra with odd dimension endowed with one-form such that is called a contact Lie algebra, while a Lie algebra whose dimension is even and it is endowed with zero index is called a Frobenius Lie algebra. The research aimed to give explicit formula of a symplectic form of low dimensional contact Lie algebras and Frobenius Lie algebras. We established that a one-form associated to simplectic form determine a type of a Lie algebra whether a contact or a Frobenius Lie algebras.To clearer the main results, we give some examples of one-form and symplectic form of Frobenius and contact Lie algebras.

Partial geometric lattices, II: association schemes

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ePdy/dx = A(x,e)y with A holomorphic at (x0,0), suppose that x0 is formally not a turning point in the sense that no singularities appear at x0 during the standard formal solution procedure with formal fractional power series in e. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x0. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.

We study elastic wave propagation in a fluid-saturated porous medium. Low-frequency asymptotic analysis of Biot’s model of poroelasticity yields the wave velocity, attenuation, and wave number as power series with respect to a small dimensionless parameter. This parameter is the product of kinematic reservoir fluid mobility, imaginary unity, and the frequency of the signal. The simplified asymptotic relationships lead to explicit formulae for reflection and transmission coefficients for a planar wave crossing an interface between two permeable media. Theoretical calculations predict a peak of reflection from a thin highly-permeable lens in a low-frequency range. An explicit formula relates the resonance frequency to reservoir fluid mobility and reservoir thickness. The results have immediate practical implications for seismic modeling and attribute analysis.

Preliminary Remarks The operator or operational calculus, the method of treating differential operators as algebraic objects, was once thought to have originated with the English physicist and electrical engineer Oliver Heaviside (1850–1925). Indeed, Heaviside revived and brilliantly applied this method to problems in mathematical physics. But the basic ideas can actually be traced back to Leibniz and Lagrange who must be given credit as the founders of the operational method. With his notation for the differential and integral, Leibniz was able to regard some results on derivatives and integrals as analogs of algebraic results. The later insight of Lagrange was to extend this analogy to infinite series of differentials so that, in particular, he could write the Taylor expansion as an exponential function of a differential operator. In fact, this formal approach to infinite series appeared in the work of Newton himself. For Newton, infinite series in algebra served a purpose analogous to infinite decimals in arithmetic: They were necessary to carry out the algebraic operations to their completion. Newton's insightful algorithms using formal power series were of very wide applicability in analysis, algebra, and algebraic geometry; their power lay precisely in their formal nature. Thus, the algebraic analysis of the eighteenth century can trace its origins to Newton's genius. A branch of algebraic analysis focusing on the combinatorial aspects of power series was developed by the eighteenth-century German combinatorial school.

This paper is concerned with a progressive wave of general form in an infinitely long estuary of uniform cross-section when there is a permanent current independent of the wave. The only approximation is the neglect of friction. Explicit formulae in the form of infinite series are found for the surface-elevation and for the current.In the special case in which there is no permanent current and when the oscillation at the mouth of the estuary reduces to a single harmonic constituent, the first five harmonic shallow water constituents at any place up the estuary are evaluated.