Let X (n,m) (1,d) denote the Segre-Veronese embedding of Pn×Pm via the sections of the sheaf O(1, d). We study the dimensions of higher secant varieties of X (n,m) (1,d) and we prove that there is no defective sth secant variety, except possibly for n values of s. Moreover when (m+d d ) is a multiple of (m+ n+ 1), the sth secant variety of X (n,m) (1,d) has the expected dimension for every s.

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an ...

FIGURE 1 Two-dimensional representation of a spherical structure presenting a morphological particularity on a portion of its surface . AB, arc corresponding to the cross-section of the cap of the sphere presenting the characteristic ; OA = OB = R; 0, angle defining the cap ; v, vector defining the secant planes. S I and S2 are two-dimensional representation of secant planes which intersect (S2...

Many in the mathematical community believe that the secant method arose from Newton’s method using a finite difference approximation to the derivative, most likely because that is the way that it is taught in contemporary texts. However, we were able to trace the origin of the secant method all the way back to the Rule of Double False Position described in the 18th-century B.C. Egyptian Rhind P...

A well known theorem of Helly [l] asserts that if $ is a finite family of convex subsets of En in which each re + 1 members have a common point, then there is a point common to all members of g. In this connection, Vincensini [5; 6] has raised the following question: For «^2 and Ogrgn-1, is there an integer k = k(n, r) such that if § is an arbitrary finite family of convex subsets of En in whic...

This paper is an elementary introduction to the methods of Landsberg and Manivel, [3] for finding the ideals of secant varieties to Segre varieties. We cover only the most basic topics from [3], but hope that since this is a topic which is rarely made explicit, these notes will be of some use. We assume the reader is familiar with the basic operations of multilinear algebra: tensor, symmetric, ...

To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X̃ ⊂ P K be a variety of minimal degree and of codimension at le...

Solving the static equilibrium position is one of the most important parts of dynamic coefficients calculation and further coupled calculation of rotor system. The main contribution of this study is testing the superlinear iteration convergence method—twofold secant method, for the determination of the static equilibrium position of journal bearing with finite length. Essentially, the Reynolds ...

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations which are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model i...