This will be a continuation of the overview from the last week. Some details will be briefly recalled from the last time, for continuity and the benefit of new audience if any.

The talk will outline the development of the PascGalois Project. Its origins are in an exercise using Pascal's Triangle and modular arithmetic. Colors are assigned to the numbers 0, 1, ..., n-1, and Pascal's Triangle modulo n is drawn. The patterns in the triangle are then related to the properties of the cyclic group Zn. The process of drawing the triangles is then generalized to non-cyclic and non-abelian groups and the new patterns are examined in light of the properties of these groups. The images can help develop visual and intuitive understanding of concepts such as subgroup closure and quotient groups. They can also be used to discuss the relationship between mathematical properties and visual aesthetics. Finally we view Pascal's Triangle as a one-dimensional cellular automata and generalize to more general initial conditions and two dimensional automata. Many of the investigations in this project have been undertaken with students in undergraduate research projects and one outgrowth of the project has been the development of a set of visualization exercises to supplement the standard undergraduate course in abstract algebra. The web site for the project is at www.pascgalois.org.

We will explain how to attach an L-function to a motive, what the critical points of this L-function are, and Deligne's conjectures on the values of the L-function at critical points.

In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work.

In this talk, we study the basics of defining ideal of the Rees algebra of Ideal I and what makes the ideal to be of linear type. Further, we prove that ideals generated by a regular sequences are of linear type.

For a finite abelian group G with |G|= n, the arithmetical invariant EA(G) is defined to be the least integer k such that any sequence S with length k of elements in G has a A weighted zero-sum subsequence of length n. When A={1}, it is the Erdos-Ginzburg-Ziv constant and is denoted by E(G). Similarly, the Davenport Constant DA(G) is defined to be the least integer k such that any sequence S with length k of elements in G has a non-empty A weighted zero-sum subsequence. For certain sets A, we already know some general bounds for these weighted constants corresponding to the cyclic group Z_n. We try to find out bounds for these combinatorial invariants for random A. We got few results in this connection. In this talk I would like to present those results and discuss about an extremal problem related to the cardinality of A

In current automotive market, reliability and durability are important hygiene factors to the customer and customer’s perception about these parameters is established through length of product warranty period. Therefore, to be competitive in market, all companies aim to provide optimal warranty on their product. To decide optimal warranty period for a product, forecasting of cost to the companies due to particular warranty policy is essential. Automotive product warranty has two dimensions, time and mileage (t,m). Hence, to forecast warranty cost one has to model a bivariate distribution for (t,m) and project expected number of warranty returns within specified period (t,m). In this paper we bypass the bivariate distribution by using usage distribution and use multiple renewal processes induced by Weibull distributions to arrive at cost structure under various warranty policies.

We introduce standard mixture models and standard mixture designs as are well-known in the literature. Some of the less known models are also introduced briefly. Next we mention about known applications of mixture experiments in agriculture, food processing and pharmaceutical studies. Then we describe the framework of exact and approximate [or, continuous] mixture designs. A broad class of research problems posed and discussed in the published literature is presented with appropriate references.

In this talk, we study the basics of defining ideal of the Rees algebra of Ideal I and what makes the ideal to be of linear type. Further, we prove that ideals generated by a regular sequences are of linear type.