Gopakumar-Vafa invariants via vanishing cycles

In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.Comment: 63 pages, many improvements of the exposition following referee comments, final version to appear in Inventione

Forcing vine regrowth to delay ripening and its association to changes in the hormonal balance

The quality and typicality of wines, strongly depends on the management techniques used for grapevine cultivation. Actually, the increment in the average world temperature due to climate change induces not only bigger irrigation necessities, but also earlier grape-ripening processes, which take place in warmer days and shorter nights. Thus, with the aim of delaying grape ripeness of at least two months, a technique has been proposed based on forcing vine regrowth. This technique consists on forcing vine regrowth from the formed latent buds after cutting the green shoots between the second and the third node; lateral shoots, leaves, and primary clusters are also removed. In this study, a forcing treatment was carried out at three different phenological stages (G, I and J). Depending on the phenological stage of vines during the forcing treatment, we wanted to determine the berry ripening delay and to explore how this mechanical pruning interacts with the hormonal balance to modulate bud growth just before shoot decapitation (Control) and later, within the following 7 and 14 days after cutting the green shoots. Forcing treatments carried out at stages G, I and J succeeded to delay ripening 18, 27 and 45 days respectively, as compared to unforced plants. Vine yield was significantly reduced in all treatments as compared to control plants, resulting in a high level of acidity in berries which might be associated with the loss of flowers, a reduction in the fruit set percentage or a combination of both. Endogenous cytokinin (CK) content in control latent buds decreased during the vine vegetative cycle. Contrarily, abscisic acid (ABA) and Jasmonic acid (JA) increased, while minor changes were found in the concentration of gibberellins (GAs), salicylic acid (SA) and the ethylene precursor 1-aminocyclopropane-1-carboxilic acid (ACC). Moreover, a clear modification of the hormonal balance was found in latent buds 7 and 14 days after forcing regrowth. CK content significantly increased while ABA rapidly decreased after pruning in all treatments. Thus, vine regrowth from the formed latent buds might have been upregulated by CK and promoted by the absence of ABA

Reflectionless analytic difference operators I. algebraic framework

We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schr\"odinger and Jacobi operators corresponding to KdV and Toda lattice solitons

Quantum Signatures of The Classical Disconnection Border

A quantum Heisenberg model with anisotropic coupling and all-to-all interaction has been analyzed using the Bose-Einstein statistics. In Ref.\cite{jsp} the existence of a classical energy disconnection border (EDB) in the same kind of models has been demonstrated. We address here the problem to find quantum signatures of the EDB. An independent definition of a quantum disconnection border, motivated by considerations strictly valid in the quantum regime is given. We also discuss the dynamical relevance of the quantum border with respect to quantum magnetic reversal. Contrary to the classical case the magnetization can flip even below the EDB through Macroscopic Quantum Tunneling. We evaluate the time scale for magnetic reversal from statistical and spectral properties, for a small number of particles and in the semiclassical limit.Comment: 5 pages, 5 figure

Point Symmetries of Generalized Toda Field Theories

A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point symmetries of these theories are found for an infinite, semi-infinite and finite number of fields. Special attention is accorded to conformal invariance and its breaking.Comment: 25 pages, no figures, Latex fil

Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background

We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class.Comment: 7 page

Vasa Nervorum in rat major pelvic ganglion are innervated by nitrergic nerve fibers

INTRODUCTION The vasa nervorum comprises a network of small diameter blood vessels that provide blood supply to nerves and ganglia. The cell bodies of autonomic nerves innervating the urogenital organs are housed in the major pelvic ganglia (MPG) in rats. The vasa nervorum of rat MPG have not been characterized previously, and it is not known whether these blood vessels are innervated by neuronal nitric oxide synthase (nNOS) containing nitrergic nerves. AIM To characterize the blood vessels in and around the rat MPG and to assess their nitrergic innervation. MAIN OUTCOME MEASURES Characterization of small blood vessels in and around the rat MPG and expression of nNOS in nerve fibers around those blood vessels. METHODS MPG were obtained from healthy Sprague Dawley rats, fixed in paraformaldehyde, frozen and sectioned using a cryostat. The blood vessels and their nitrergic innervation were assessed with immunohistochemistry using antibodies against alpha-smooth muscle actin (smooth muscle marker), CD31 (endothelial marker), collagen IV (basal membrane marker) and nNOS. The immunofluorescence was imaged using a laser scanning confocal microscope. RESULTS The neuronal cell bodies were contained within a capsule in the MPG. Blood vessels were observed within the capsule of the MPG as well as outside the capsule. The blood vessels inside the capsule were CD31-positive capillaries with no smooth muscle staining. Outside the capsule capillaries, arterioles and venules were observed. The extra-capsular arterioles and venules, but not the capillaries were innervated by nNOS-positive nerve fibers. CONCLUSIONS This study, to our knowledge, is the first to demonstrate the blood vessel distribution pattern and their nitrergic innervation in the rat MPG. While similar studies in human pelvic plexus are warranted, these results suggest that the blood flow in the MPG may be regulated by nitrergic nerve fibers and reveal a reciprocal relationship between nerves and blood vessels. Beetson KA, Smith SF, Muneer A, Cameron NE, Cotter MA, and Cellek S. Vasa nervorum in rat major pelvic ganglion are innervated by nitrergic nerve fibers. J Sex Med **;**:**-**

Evidence for moving breathers in a layered crystal insulator at 300K

We report the ejection of atoms at a crystal surface caused by energetic breathers which have travelled more than 10^7 unit cells in atomic chain directions. The breathers were created by bombardment of a crystal face with heavy ions. This effect was observed at 300K in the layered crystal muscovite, which has linear chains of atoms for which the surrounding lattice has C_2 symmetry. The experimental techniques described could be used to study breathers in other materials and configurations.Comment: 7 pages, 3 figure

Integrable dynamics of Toda-type on the square and triangular lattices

In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. In this paper we construct other examples of integrable dynamics of Toda-type on the square lattice, as well as on the triangular lattice, as nonlinear symmetries of the discrete Laplace equations on the square and triangular lattices. We also construct the $\tau$ - function formulations and the Darboux-B\"acklund transformations of these novel dynamics.Comment: 22 pages, 4 figure