Abstract:
The Plant, Costus afer Ker Gawl. belongs to the family of Costaceae and has various uses where they exist. Their use in folk medicine and phytomedicine is in the treatment and management of variety of human ailment, like diabetes mellitus, abdominal problems etc. The search for new antidiabetic therapies has become increasingly urgent due to the development of adverse effects and resistance by the chemically synthesized drugs on one hand and effectiveness with low cost of the plant materials on the other hand. The investigations carried out is to determine the long term effects of Costus afer leaf methanol extract, snail slime and the combined Costus afer and snail slime extracts on blood glucose levels of alloxan induced diabetic Swiss albino rats treated orally for 21 days on graded dose of (100 mg/kg and 300 mg/kg). From the determination, the snail slime showed positive effect on blood glucose lowering level but less effective when compared with similar dose of the Costus afer leaf methanol extract. The investigation indicated that there was 103 mg/dL and 87 mg/dL blood glucose reduction for the low dose of Costus afer and Snail slime respectively while the standard hypoglycemic drug (Glibenclamide, 5 mg/kg) used for comparison yielded a blood glucose level reduction of 103 mg/dL. Similarly, the high dose used in the study gave a blood glucose reduction of 99 mg/dL and 95 mg/dL for Costus afer leaf methanol extract and Snail slime respectively. The results obtained when alloxan induced rats was treated with C. afer leaf methanol extract, Snail slime extract, and combined C. afer and snail slime extracts was analysed using Statistix 8.0 American version. The result showed a dose dependent fashion and the difference obtained from the compared results was statistically significant at p < 0.05. This result supports the views of other researchers that some herbal anti-diabetic remedies which reduce blood glucose levels were similar to those of synthetic oral hypoglycemic drugs like metformin and sulfonylurea etc [1]. Still to that, medicinal and pharmacological activities of medicinal plants are often attributed to the presence of the so called secondary plant metabolites. Hence this regenerative capacity of snail slime and the fact that diabetes is characterized by damage of the pancreatic beta cells, may give credit to the hypoglycaemic effect observed in Costus afer methanol leaf extract and snail slime for possible drug formulation for

Abstract:
A simple kinematic model for the trajectories of Listeria monocytogenes is generalized to a dynamical system rich enough to exhibit the resonant Hopf bifurcation structure of excitable media and simple enough to be studied geometrically. It is shown how L. monocytogenes trajectories and meandering spiral waves are organized by the same type of attracting set.

Abstract:
The Kashiwara $B(\infty)$ crystal pertains to a Verma module for a Kac- Moody Lie algebra. Ostensibly it provides only a parametrisation of the global/canonical basis for the latter. Yet it is much more having a rich combinatorial structure from which one may read of a parametrisation of the corresponding basis for any integrable highest weight module, describe the decomposition of the tensor products of highest weight modules, the Demazure submodules of integrable highest weight modules and Demazure flags for translates of Demazure modules. $B(\infty)$ has in general infinitely many presentations as subsets of countably many copies of the natural numbers each given by successive reduced decompositions of Weyl group elements. In each presentation there is an action of Kashiwara operators determined by Kashiwara functions. These functions are linear in the entries. Thus a natural question is to show that in each presentation the subset $B(\infty)$ is polyhedral. Here a new approach to this question is initiated based on constructing dual Kashiwara functions and in this it is enough to show that the latter are also linear in the entries. The present work resolves one of the two very difficult obstacles in a step-wise construction, namely that the resulting functions must satisfy a sum, or simply S, condition. It depends very subtly on inequalities between the coefficients occurring in functions obtained from the previous step. The only remaining obstacle, that sufficiently many functions are obtained, can at least be verified in many families of cases, though this is to be postponed to a subsequent paper. This theory has some intriguing numerology which involves the Catalan numbers in two different ways.

Abstract:
A Lie theoretic interpretation is given to a pattern with five-fold symmetry occurring in aperiodic Penrose tiling based on isosceles triangles with length ratios equal to the Golden Section. Specifically a $B(\infty)$ crystal based on that of Kashiwara is constructed exhibiting this five-fold symmetry. It is shown that it can be represented as a Kashiwara $B(\infty)$ crystal in type $A_4$. Similar crystals with $(2n+1)$-fold symmetry are represented as Kashiwara crystals in type $A_{2n}$. The weight diagrams of the latter inspire higher aperiodic tiling. In another approach alcove packing is seen to give aperiodic tiling in type $A_4$. Finally $2m$-fold symmetry is related to type $B_m$.

Abstract:
Let g be a complex simple Lie algebra and h a Cartan subalgebra. The Clifford algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of the simple 2m+1-dimensional module of the principal s-triple obtained from the Langlands dual of g. Bazlov settled this conjecture positively in type A. The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of the Harish-Chandra map for the enveloping algebra U(g) composed with evaluation at the half sum of the positive roots. In an earlier work we settled an analogue of the Kostant conjecture obtained by replacing the Harish-Chandra map by a "generalized Harish-Chandra" map whose image is described via Zhelobenko invariants. Here we show that there are analogue Zhelobenko invariants which describe the image of the Harish-Chandra map. Following this a similar proof goes through.

Abstract:
Let g be a complex simple Lie algebra and let V be a finite dimensional U(g) module. A relative Yangian is defined with respect to this pair. According to recent work of Khoroshkin and Nazarov the finite dimensional simple modules of the Yangians for g = sl(n) or the twisted Yangians for g = sp(2n); so(n) are described by the simple modules of relative Yangians for some V using the appropriate simple Lie algebra g. Here a classifi?cation of the simple modules of a relative Yangian is obtained simply and briefly as an advanced exercise in Frobenius reciprocity inspired by a Bernstein- Gelfand equivalence of categories. An unexpected fact is that the dimension of these modules are determined by the Kazhdan-Lusztig polynomials and conversely the latter are described in terms of dimensions of certain extension groups associated to finite dimensional modules of relative Yangians.

Abstract:
Let $\mathfrak a$ be an algebraic Lie algebra. An adapted pair for $\mathfrak a$ is pair $(h,\eta)$ consisting of an ad-semisimple element of $h \in \mathfrak a$ and a regular element of $\eta \in \mathfrak a^*$ satisfying $(ad \ h)\eta=-\eta$. An adapted pair $(h,\eta)$ is said to satisfy integrality if $ad \ h$ has integer eigenvalues on $\mathfrak a$. Integrality is shown to hold for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple Lie algebra; but may fail in general. Call $\mathfrak a$ regular if there are no proper semi-invariant polynomial functions on $\mathfrak a^*$ and if the subalgebra of invariant functions is polynomial. In this case there are no known counter-examples to integrality. It is shown that if $\mathfrak a$ is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra $\mathfrak g$ which is regular and admits an adapted pair $(h,\eta)$, then the eigenvalues of $ad \ h$ on $\mathfrak a$ lie in $\frac{1}{m}\mathbb Z$, where $m$ is a coefficient of a simple root in the highest root of $\mathfrak g$. Let $\mathfrak a$ be a regular Lie algebra admitting an adapted pair $(h,\eta)$. Let $\mathfrak a_\mathbb Z$ be the subalgebra spanned by the eigensubspaces of $ad \ h$ with integer eigenvalue. It is shown that the canonical truncation of $\mathfrak a_\mathbb Z$ is regular. Sufficient knowledge of the relation between the generators for the invariant polynomial functions on $\mathfrak a^*$ and on $\mathfrak a^*_\mathbb Z$ can then lead to establishing the integrality of $(h,\eta)$. This method is used to show the integrality of an adapted pair for a truncated parabolic subalgebra of a simple Lie algebra of type $C$.

Abstract:
Let g be a complex simple Lie algebra and h a Cartan subalgebra. The Clifford algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of the simple 2m+1-dimensional module of the principal s-triple obtained from the Langlands dual. Bazlov settled this conjecture positively in type A. The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of the Harish-Chandra map for the enveloping algebra U(g) composed with evaluation at the half sum of the positive roots. Here an analogue of the Kostant conjecture is settled by replacing the Harish-Chandra map by a "generalized Harish-Chandra" map which had been studied notably by Zhelobenko. The proof involves a symmetric algebra version of the Kostant conjecture (settled in works of Alekseev-Bazlov-Rohr), the Zhelobenko invariants in the adjoint case and surprisingly the Bernstein-Gelfand-Gelfand operators introduced in their study of the cohomology of the flag variety.

Abstract:
Let g be a semisimple Lie algebra with h a Cartan subalgebra. The orbit method attempts to assign representations of g to orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits which should lead to highest weight representations of g. It is known that all unitary highest weight representations can be obtained in this fashion. A hypersurface orbital variety is one which is of codimension 1 in the nilradical of a parabolic. Their classification for g = sl(n) obtains from general results. Recently Benlolo and Sanderson conjectured the form of the (non-linear) element describing such a variety. This paper proves that conjecture and further constructs a simple module with integral highest weight which ``quantizes'' the variety in the precise sense that the regular functions of its closure is given a g module structure compatible (up to shift by the highest weight) with its h module structure.