Let Xm,n be the Segre-Veronese variety P m ×P embedded by the morphism given by O(1, 2). In this paper, we provide two functions s(m, n) ≤ s(m, n) such that the s secant variety of Xm,n has the expected dimension if s ≤ s(m,n) or s(m, n) ≤ s. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as c...

We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show that the singular locus of the secant varieties σr(Seg(P × P × P)) has codime...

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computation...

A simple secant-based fast gradient method is developed for problems whose objective function is convex and well-defined. The proposed algorithm extends the classical Nesterov gradient method by updating the estimate-sequence parameter with secant information whenever possible. This is achieved by imposing a secant condition on the choice of search point. Furthermore, the proposed algorithm emb...

in this paper, an effective technique is proposed to determine thenumerical solution of nonlinear volterra-fredholm integralequations (vfies) which is based on interpolation by the hybrid ofradial basis functions (rbfs) including both inverse multiquadrics(imqs), hyperbolic secant (sechs) and strictly positive definitefunctions. zeros of the shifted legendre polynomial are used asthe collocatio...

Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; this proof might be generalisable to cover...

Let O be a conic in the classical projective plane PG(2, q), where q is an odd prime power. With respect to O, the lines of PG(2, q) are classified as passant, tangent, and secant lines, and the points of PG(2, q) are classified as internal, absolute and external points. The incidence matrices between the secant/passant lines and the external/internal points were used in [5] to produce several ...

In this talk we present several representation theorems and averaging theorems for members of the difference class of secant updates introduced by Broadlie, Gourlay , and Greenstadt in 1973. The BFGS update is a well-known member of this difference class. We begin with the notion of an update kernel and use it to construct a new parametrization of the difference class of secant updates. Next we...

In this paper, an effective technique is proposed to determine thenumerical solution of nonlinear Volterra-Fredholm integralequations (VFIEs) which is based on interpolation by the hybrid ofradial basis functions (RBFs) including both inverse multiquadrics(IMQs), hyperbolic secant (Sechs) and strictly positive definitefunctions. Zeros of the shifted Legendre polynomial are used asthe collocatio...