DSpace Collection:
http://hdl.handle.net/10077/33583
Sun, 05 Feb 2023 14:00:53 GMT2023-02-05T14:00:53ZHilbert curves of quadric fibrations over smooth surfaces
http://hdl.handle.net/10077/34262
Title: Hilbert curves of quadric fibrations over smooth surfaces
Authors: Fania, Maria Lucia; Lanteri, Antonio
Abstract: Let (X,L) be a complex polarized n-fold with the structure of a geometric quadric fibration over a smooth projective surface. The Hilbert curve of (X,L) is a complex affine plane curve of degree n, containing n − 3 evenly spaced parallel lines. This paper is devoted to a detailed study of the cubic representing the residual component. Reducibility, existence of triple points, and properties of the irreducible components are analyzed in connection with the structure of (X,L).Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10077/342622022-01-01T00:00:00ZMonotonicity theorems and inequalities for certain sine sums
http://hdl.handle.net/10077/34261
Title: Monotonicity theorems and inequalities for certain sine sums
Authors: Alzer, Horst; Kwong, Man Kam
Abstract: Inspired by the work of Askey-Steinig, Szeg\"o, and Schweitzer, we provide several monotonicity theorems and inequalities for certain sine sums. Among others, we prove that for $n\geq 1$ and $x\in (0,\pi/2)$, we have
$$
\frac{d}{dx} \frac{C_n(x)}{1-\cos(x)}<0 \quad\mbox{and} \quad \frac{d}{dx} \left(1-\cos(x)\right)C_n(x)>0,
$$
where
$$
C_n(x)=\sum_{k=1}^n\frac{\sin((2k-1)x)}{2k-1}
$$
denotes Carslaw's sine polynomial. Another result states that the inequality
$$
\sum_{k=1}^n (n-k+a)(n-k+b) k \sin(kx)>0 \quad (a,b\in \mathbb{R})
$$
holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $a=b=1$.
Many corollaries and applications of these results are given. Among them, we present a two-parameter class of absolutely monotonic rational functions.Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10077/342612022-01-01T00:00:00ZOn the quadratic equations for odeco tensors
http://hdl.handle.net/10077/34260
Title: On the quadratic equations for odeco tensors
Authors: Biaggi, Benjamin; Draisma, Jan; Seynnaeve, Tim
Abstract: Elina Robeva discovered quadratic equations satisfied by orthogonally decomposable (“odeco”) tensors. Boralevi-Draisma-Emil Horobeț-Robeva then proved that, over the real numbers, these equations characterise odeco tensors. This raises the question to what extent they also characterise the Zariski-closure of the set of odeco tensors over the complex numbers. In the current paper we restrict ourselves to symmetric tensors of order three, i.e., of format n×n×n. By providing an explicit counterexample to one of Robeva’s conjectures, we show that for n ≥ 12, these equations do not suffice. Furthermore, in the open subset where the linear span of the slices of the tensor contains an invertible matrix, we show that Robeva’s equations cut out the limits of odeco tensors for dimension n ≤ 13, and not for n ≥ 14. To this end, we show that Robeva’s equations essentially capture the Gorenstein locus in the Hilbert scheme of n points and we use work by Casnati-Jelisiejew- Notari on the (ir)reducibility of this locus.Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10077/342602022-01-01T00:00:00ZPencils of singular quadrics of constant rank and their orbits
http://hdl.handle.net/10077/34102
Title: Pencils of singular quadrics of constant rank and their orbits
Authors: Boralevi, Ada; Mezzetti, Emilia
Abstract: We give a geometric description of singular pencils of quadrics of constant rank, relating them to the splitting type of some naturally associated vector bundles on P1. Then we study their orbits in the Grassmannian of lines, under the natural action of the general linear group.Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10077/341022022-01-01T00:00:00Z