A cube Slice that is not a Zonoid

Let $Q$ be the unit cube in $\mathbb{R}^n$ and $H$ a hyperplane thru the Origin. The intersection is called Cube slice and was investigated by Henesley, Vaaler, Ball and others. We give an example of a cube slice in $\mathbb{R}^4$ that is not a zonoid. This contrasts with a result in $\mathbb{R}^3$ that follows from a Theorem due to Herz and Lindenstrauss where every cube slice is a zonoid. The volume of this slice is computed and used to determine the likely known result, the value of the sinc integral $I_4$

Volume product

In this expository paper we discuss the volume product P(K) of convex bodies K in $R^n$; this is the product of volumes of K and its polar K*. The Blaschke- Santalo inequalities state that always $P(K) \le P(B_2)$ and $P(B_1)\le P(K)$ . Here the Closed unit Ball in norm of $l_2$ is denoted by $B_2$ and like wise for the $l_1$ unit ball. The upper bound is classical, due to Santalo in general and Blaschke and Mahler much earlier in 1930 s for dimensions 2 and 3. The lower bound is open for general K. However the result of Gordon, Meyer and Riesner says that the class of zonoids K attain the lower bound. There is Bianchi and Kelly' s proof of the upper bound in general,involving Paley-Wiener Theorem , as generalized to $R^n$ by Stein . For the lower bound, there is the result of of Kim and Zvavitch on its stability under perturbations of unconditional K . Further there are results on " functional" versions of Blashke -Santalo Inequality . In this context we discuss the newer concept of "Polar f* " for certain type of functions f and its relevance here and mention as a "functional" example a result by S.Artstein, B.Klartag and V.Milman . We mention a later one by Huang and Ai- Jun Li and discuss Ball's Inequality for unconditional bodies,another strengthening of the Blaschke-Santalo inequality .Comment: 22 pg

The Iron Pillar at Delhi

The massive Iron Pillar located in South Delhi has been an object of considerable interest to modern scientists and technologists for two main reasons viz., the amazing technology by which a metallic object weighing nearly seven tons could be fabricated over fifteen centuries ago and the phenomenal corrosion resistance displayed by this ancient monument despite exposure to sun, rain, wind and dust for so long. In this paper all available material on this metallurgical marvel is examined scientifically and systematically and an attempt made to answer such questions as are likely to arise in the minds of discern-ing visitors to this impressive monument. The following important conclusions are arrived at: (1) Date of Erec-tion : 370-375 A.D., (2) Date of the Inscription : 380-385 A.D., (3) Mode of Fabrication.: Hammer forging and welding ball of hot pasty iron in many steps, and (4) Reasons for Restlessness : Il4any viz., unusual chemical composition, protective oxide film, favourable Delhi climate and slag particles at grain boundaries

On proper Hamiltonian-connection number of graphs

A graph G is Hamiltonian-connected if every two vertices of G are connected by a Hamilton path. A bipartite graph H is Hamiltonian-laceable if any two vertices from different partite sets of H are connected by a Hamilton path. An edge-coloring (adjacent edges may receive the same color) of a Hamiltonian-connected (respectively, Hamiltonian-laceable) graph G (resp. H) is a proper Hamilton path coloring if every two vertices u and v of G (resp. H) are connected by a Hamilton path Puv such that no two adjacent edges of Pᵤᵥ are colored the same. The minimum number of colors in a proper Hamilton path coloring of G (resp. H) is the proper Hamiltonian-connection number of G (resp. H). In this paper, proper Hamiltonian-connection numbers are determined for some classes of Hamiltonian-connected graphs and that of Hamiltonian-laceable graphs.Publisher's Versio

Some open questions in "wave chaos"

The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent

Organizational factors and total quality management - an empirical study

The level of awareness of Total Quality Management (TQM) has increased considerably over the last few years. Different sets of organizational requirements are prescribed by quality management gurus and practitioners for the effective practice of TQM. These requirements do not seem to have been formulated on the basis of systematic empirical research. Many researchers point out that tacit factors, e.g. employee empowerment, open culture and executive commitment, and not TQM tools and techniques alone, could drive TQM success, and that organizations would need to acquire these factors to stay successful. Many TQM advocates have also suggested that a conducive organizational environment would be essential for an effective practice of TQM. However, they did not offer any empirical evidence. There appears to be no empirical study reported in the literature that could establish a relation between TQM and organizational factors. The objective of this paper is to describe an empirical research on TQM conducted in Indian business units carried out recently by considering some organizational factors, e.g. quality of work life, organizational climate and communication. The methodology and findings are discussed in detail