Stable pairs on nodal K3 fibrations

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.Comment: Published versio

The Solution of the N=2 Supersymmetric f-Toda Chain with Fixed Ends

The integrability of the recently introduced N=2 supersymmetric f-Toda chain, under appropriate boundary conditions, is proven. The recurrent formulae for its general solutions are derived. As an example, the solution for the simplest case of boundary conditions is presented in explicit form.Comment: 15 pages, latex, no figure

Thermodynamic interpretation of the uniformity of the phase space probability measure

Uniformity of the probability measure of phase space is considered in the framework of classical equilibrium thermodynamics. For the canonical and the grand canonical ensembles, relations are given between the phase space uniformities and thermodynamic potentials, their fluctuations and correlations. For the binary system in the vicinity of the critical point the uniformity is interpreted in terms of temperature dependent rates of phases of well defined uniformities. Examples of a liquid-gas system and the mass spectrum of nuclear fragments are presented.Comment: 11 pages, 2 figure

Reductions of the Volterra and Toda chains

The Volterra and Toda chains equations are considered. A class of special reductions for these equations are derived.Comment: LaTeX, 6 page

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

Archaeological evidence for historical navigation on the Mureş (Maros) river. Enquiries based on a medieval boat imprint from Bizere abbey (Romania)

The boat imprint unearthed at the site of the Benedictine abbey from Bizere (Frumuşeni, Romania) is a unique discovery for two reasons: its preservation as a negative imprint, due to its reuse for preparing mortar, and its dating back to the 12th century, based on the context of its discovery. It has been identified as a logboat, due to the absence of any technical details specific for plank boats, and now stands as the only vessel of this type with known dating for the territory of Romania. The article also enquires into the wider historical context of the discovery, thus bringing forth the archival data available with regard to medieval inland navigation

Imaging of Cherenkov and Transition Radiation from Thin Films and Particles

Cherenkov radiation and transition radiation, which are generated by high energy electrons with constant velocity, can be detected in a transmission electron microscope using a cathodoluminescence (CL) detection system. The characteristic peaks due to interference were observed in the emission spectra from thin films of mica, silicon and silver, and their dependence on sample thickness and accelerating voltage was studied. Particles of BaTiO3 and MgO also showed characteristic feature in the spectra which changed with their size. A recently developed imaging system revealed the two-dimensional intensity distribution of these radiations; for example, oscillating contrast, such as equal thickness contour appears in silicon, and hole edges in a silver thin film show bright fringe contrast due to radiative surface plasmon

The Toda lattice is super-integrable

We prove that the classical, non-periodic Toda lattice is super-integrable. In other words, we show that it possesses 2N-1 independent constants of motion, where N is the number of degrees of freedom. The main ingredient of the proof is the use of some special action--angle coordinates introduced by Moser to solve the equations of motion.Comment: 8 page

Integrable Discretizations of Chiral Models

A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and B\"acklund transformations for the class of models considered in this work.Comment: 14 pages, Late