RECENT RESULTS ON CARDINALITY ESTIMATION AND INFORMATION THEORETIC INEQUALITIES

I would like to dedicate this little exposition to Prof. Phan Dinh Dieu, one of the giants and pioneers of Mathematics in Computer Science in Vietnam. In the past 15 years or so, new and exciting connections between fundamental problems in database theory and information theory have emerged. There are several angles one can take to describe this connection. This paper takes one such angle, influenced by the author's own bias and research results. In particular, we will describe how the cardinality estimation problem -- a corner-stone problem for query optimizers -- is deeply connected to information theoretic inequalities. Furthermore, we explain how inequalities can also be used to derive a couple of classic geometric inequalities such as the Loomis-Whitney inequality. A purpose of the article is to introduce the reader to these new connections, where theory and practice meet in a wonderful way. Another objective is to point the reader to a research area with many new open questions.

On Golden Lorentzian Manifolds Equipped with Generalized Symmetric Metric Connection

This research deals with the generalized symmetric metric U-connection defined on golden Lorentzian manifolds. We also derive sharp geometric inequalities that involve generalized normalized δ-Casorati curvatures for submanifolds of golden Lorentzian manifolds equipped with generalized symmetric metric U-connection.

Geometric Inequalities for Warped Products in Riemannian Manifolds

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product N1×fN2 into any Riemannian manifold Rm(c) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H2 . Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.