Murphy bases of generalized Temperley-Lieb algebras

In this article we show how to use some results of G. E. Murphy on the so-called standard basis of Hecke-Algebras of Type A to derive a similar basis for generalized Temperley-Lieb algebras. This standard basis is compared to the usual diagrammatic basis of the original Temperley-Lieb algebra used in knot theory and statistical physics.     

Geometric bases and topological equivalence

There are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê-Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.     

Geometric interpretation of tight closure and test ideals

We study tight closure and test ideals in rings of characteristic using resolution of singularities. The notions of -rational and -regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal -Gorenstein ring of characteristic , the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings.

     

Geometric interpretation of some Cauchy related methods

This paper shows that a class of methods for solving linear equations, including the Cauchy?CBarzilai?CBorwein method, can be interpreted by means of a simple geometric object, the Bézier parabola. This curve is built from the current iterate using a transformation characterizing the system to be solved. The localization of the next iterates in the plane of the parabola sheds some light on the behavior of the methods and provides some new understanding of their relative efficiency.     

Perturbation of orthonormal bases with an application to diagonalization

Given an orthonormal basis {e _n } _n=1 in a Hilbert spaceH, and a dense linear manifoldDH, we show that there exists a unitary operatorV onH such thatI-V is a trace-class operator with arbitrarily small trace norm, andVe _jD for allj. This result can be used to simplify certain arguments of J. Xia concerning the simultaneous diagonalization of operators on a space of square integrable functions.     

Geometric construction of crystal bases for quantum generalized Kac-Moody algebras

We provide a geometric realization of the crystal B(∞) for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.     

Geometric interpretation of the optimality conditions in multifacility location and applications

Several corrections to Ref. 1 are pointed out.     

Geometric interpretation of the optimality conditions in multifacility location and applications

Geometrical optimality conditions are developed for the minisum multifacility location problem involving any norm. These conditions are then used to derive sufficient conditions for coincidence of facilities at optimality; an example is given to show that these coincidence conditions seem difficult to generalize.     

Geometric interpretation of curvature and torsion tensors in a generalized Finsler space

We give geometric interpretations of a torsion tensor and curvature tensors in a generalized Finsler space (with an asymmetric basic tensor).     

Geometric interpretation of Hamiltonian cycles problem via singularly perturbed Markov decision processes

We consider the Hamiltonian cycle problem (HCP) embedded in a singularly perturbed Markov decision process (MDP). More specifically, we consider the HCP as an optimization problem over the space of long-run state-action frequencies induced by the MDP's stationary policies. We also consider two quadratic functionals over the same space. We show that when the perturbation parameter, ? is sufficiently small the Hamiltonian cycles of the given directed graph are precisely the maximizers of one of these quadratic functionals over the frequency space intersected with an appropriate (single) contour of the second quadratic functional. In particular, all these maximizers have a known Euclidean distance of z _m (?) from the origin. Geometrically, this means that Hamiltonian cycles, if any, are the points in the frequency polytope where the circle of radius z _m (?) intersects a certain ellipsoid.     

Geometric properties, invariants, and the Toeplitz structure of minimal bases of rational vector spaces

The algebraic and geometric aspects of a minimal base of a rationalvector space are further developed by exploring the structureof an ordered minimal base (omb) and establishing the propertiesof its Toeplitz representation. The R(s)-prime modules are introducedas new invariants of , and each module is characterized by invariantreal spaces: the high, low, and prime spaces respectively. Usingthe Topelitz representation of ombs, the families of primitiveand composite spaces are introduced as new invariants of andtheir properties are established. The geometric results presentedhere have implications in the study of the dynamics of polynomialsystem models and in the computation of minimal bases.     

A new interpretation of Cauchy type singular integrals with an application to singular integral equations

Cauchy type integrals were given the interpretation of the principal value for points inside the integration interval. Here this interpretation is modified and generalized in a very simple manner. The new interpretation in general is not equivalent to the classical one. The relationship between the new interpretation and the classical one is investigated and various applications of the new interpretation (to the Plemelj formulas, the Riemann-Hilbert boundary value problem, singular integral equations, the inversion formula, quadrature rules and interface crack problems) are presented.     

Kazhdan-Lusztig Cells and the Murphy Basis

Let H be the Iwahori–Hecke algebra associated with S_n,the symmetric group on n symbols. This algebra has two importantbases: the Kazhdan–Lusztig basis and the Murphy basis.We establish a precise connection between the two bases, allowingus to give, for the first time, purely algebraic proofs fora number of fundamental properties of the Kazhdan–Lusztigbasis and Lusztig's results on the a-function. 2000 MathematicsSubject Classification 20C08.     

Geometric torsions and an Atiyah-style topological field theory

We generalize the construction of invariants of three-dimensional manifolds with a triangulated boundary that we previously proposed for the case where the boundary consists of not more than one connected component to any number of components. These invariants are based on the torsion of acyclic complexes of geometric origin. An adequate tool for studying such invariants turns out to be Berezin’s calculus of anticommuting variables; in particular, they are used to formulate our main theorem, concerning the composition of invariants under a gluing of manifolds. We show that the theory satisfies a natural modification of Atiyah’s axioms for anticommuting variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 405–418, March, 2009.     

Geometric interpretation of the Euler-Knopp summation method in the problem of inverting the Laplace transform

In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series

$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $

(1)

First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form

$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $

It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure

$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$

Under this approach, the original function is represented by the series

$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$

where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.

     

关于n维单形的几个几何不等式

建立了关于n维单形的几个几何不等式,并给出了它们的一些应用.     

Geometric meshes and their application to Volterra integro-differential equations with singularities

In this paper we introduce a new kind of mesh, a 'geometricmesh', and discuss the corresponding ß-polynomialcollocation method for Volterra integro-differential equationswith weakly singular kernels. It will be shown that superconvergenceproperties may be obtained by using appropriate collocationparameters and such meshes. The advantage of geometric meshesis that the cost of computing the ß-polynomial collocationapproximations can be decreased greatly.     

Height bases, level bases, and the equality of the height and the level characteristics of an M-matrix

Let A be an M-matrix. We introduce the concepts of height basis, level basis, and height-level basis for the generalized nullspace of A. We explore the properties of such bases and of induced matrices. We use these results to prove some new conditions for the equality of the (spectral) height (Weyr) characteristic and the (graph theoretic) level characteristic of A, and to simplify proofs of known conditions. We also prove the existence of a Jordan basis for the generalized nullspace with all chains of maximal length nonnegative.