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We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ℙ m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal. [ABSTRACT FROM PUBLISHER]

This article presents a study regarding multivariate calculus, focusing on saddle points and inflection points. It discusses the similarities between saddle points on surfaces and inflection points on curves. It explores the direct connection between saddle points on surfaces and inflection points on curves. It presents the theorem which provides a simple constructive method to obtain continuous functions to connect the saddle points of functions of two variables with the inflection points of functions of one variable.

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

As an outgrowth of the study of algebraic geometry over groups, discriminating groups were introduced. Many important universal type groups such as Higman's universal group and Thompson's group F were shown to be discriminating. Squarelike groups were then introduced to better capture axiomatic properties of discrimination. In the present article squarelike groups are reinterpreted in terms of discrimination of quasi varieties, and the relationship with an older version of discrimination, termed varietal discrimination here, is studied. [ABSTRACT FROM AUTHOR]

The article traces the contributions of Pierre Hérigone in making mathematics algebraic rather than geometric. It discusses the algebraic concepts and notation of Hérigone which can be found in his volume on algebra. It introduces his universal symbolic language applicable to all branches of mathematics. It presents his algebraic procedures and subsections on identities, equations, geometric constructions and ambiguous equations. His influence on later mathematicians including Isaac Barrow and Gottfried Wilhelm von Leibniz is also discussed.

The article reviews several articles, including "The Hidden Math behind Alice in Wonderland," by Keith Devlin in the March 2010 issue of MAA Online, "Alice's Secrets in Wonderland," by Melanie Bayley in the December 2009 issue of "New Scientist," and "Algebra in Wonderland," by Melanie Bayley in the March 7, 2010 issue of "New York Times."