A focus on focal surfaces

We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal surface (degree, class, class of its hyperplane section, sectional genus and degrees of the nodal and cuspidal curve). We study in particular the congruences of chords to a smooth curve and the congruences of bitangents or flexes to a smooth surface. We find that they possess unexpected components in their focal surface, and conjecture that they are the only ones with this property.Comment: Plain TeX, 33 pages with no figure

Numerical Regge pole analysis of resonance structures in elastic, inelastic and reactive state-to-state integral cross sections

We present a detailed description of a FORTRAN code for evaluation of the resonance contribution a Regge trajectory makes to the integral state-to-state cross section (ICS) within a specified range of energies. The contribution is evaluated with the help of the Mulholland formula (Macek et al., 2004) and its variants (Sokolovski et al., 2007; Sokolovski and Akhmatskaya, 2011). Regge pole positions and residues are obtained by analytically continuing S-matrix element, evaluated numerically for the physical values of the total angular momentum, into the complex angular momentum plane using the PADE-II program (Sokolovski et al., 2011). The code decomposes an elastic, inelastic, or reactive ICS into a structured, resonance, and a smooth, 'direct', components, and attributes observed resonance structure to resonance Regge trajectories. The package has been successfully tested on various models, as well as the F+H 2‚ÜíHF+H benchmark reaction. Several detailed examples are given in the text

Complex angular momentum theory of state-to-state integral cross sections: Resonance effects in the F+HD→HF(v′=3)+DF + HD \to HF(v' = 3) + D reaction

State-to-state reactive integral cross sections (ICSs) are often affected by quantum mechanical resonances, especially near a reactive threshold. An ICS is usually obtained by summing partial waves at a given value of energy. For this reason, the knowledge of pole positions and residues in the complex energy plane is not sufficient for a quantitative description of the patterns produced by resonance. Such description is available in terms of the poles of an S-matrix element in the complex plane of the total angular momentum. The approach was recently implemented in a computer code, available in the public domain [Comput. Phys. Commun., 2014, 185, 2127]. In this paper, we employ the package to analyse in detail, for the first time, the resonance patterns predicted for integral cross sections (ICSs) of the benchmark F + HD → HF(v′ = 3) + D reaction. The v = 0, j = 0, Ω = 0 → v′ = 3, j′ = 0, 1, 2, and Ω′ = 0, 1, 2 transitions are studied for collision energies from 58.54 to 197.54 meV. For these energies, we find several resonances, whose contributions to the ICS vary from symmetric and asymmetric Fano shapes to smooth sinusoidal Regge oscillations. Complex energies of metastable states and Regge pole positions and residues are found by Padé reconstruction of the scattering matrix elements. The accuracy of the code, relation between complex energies and Regge poles, various types of Regge trajectories, and the origin of the J-shifting approximation are also discussed

Rational parametrization of conchoids to algebraic curves

We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components

An Algebraic Analysis of Conchoids to Algebraic Curves

We study the conchoid to an algebraic affine plane curve C from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside C, the notion of conchoid involves a point A in the affine plane (the focus) and a nonzero field element d (the distance).We introduce the formal definition of conchoid by means of incidence diagrams.We prove that the conchoid is a 1-dimensional algebraic set having atmost two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to C, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve

Rational conchoid and offset constructions: algorithms and implementation

This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages

Vector bundles on fano 3-folds without intermediate cohomology

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate cohomology if and only if it decomposes as a direct sum of line bundles. It is also known that only on projective spaces and quadrics there is, up to a twist by a line bundle, a finite number of indecomposable vector bundles with no intermediate cohomology [see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182 (1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. In the paper under review the authors deal with vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5 and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of rank two vector bundles without intermediate cohomology on such 3-folds is given. In fact the authors prove that, up to a twist, there are only three indecomposable vector bundles without intermediate cohomology. Vector bundles of rank greater than two are also considered. Under an additional technical condition, the authors characterize the possible Chern classes of such vector bundles without intermediate cohomology

The Binding of Aβ42 Peptide Monomers to Sphingomyelin/Cholesterol/Ganglioside Bilayers Assayed by Density Gradient Ultracentrifugation

The binding of Aβ42 peptide monomers to sphingomyelin/cholesterol (1:1 mol ratio) bilayers containing 5 mol% gangliosides (either GM1, or GT1b, or a mixture of brain gangliosides) has been assayed by density gradient ultracentrifugation. This procedure provides a direct method for measuring vesicle-bound peptides after non-bound fraction separation. This centrifugation technique has rarely been used in this context previously. The results show that gangliosides increase by about two-fold the amount of Aβ42 bound to sphingomyelin/cholesterol vesicles. Complementary studies of the same systems using thioflavin T fluorescence, Langmuir monolayers or infrared spectroscopy confirm the ganglioside-dependent increased binding. Furthermore these studies reveal that gangliosides facilitate the aggregation of Aβ42 giving rise to more extended β-sheets. Thus, gangliosides have both a quantitative and a qualitative effect on the binding of Aβ42 to sphingomyelin/cholesterol bilayers.This work was supported in part by grants from the Spanish Ministry of Economy (grant FEDER MINECO PGC2018-099857-B-I00) and the Basque Government (grants No. IT1264-19 and IT1270-19)