Shock waves for the Burgers equation and curvatures of diffeomorphism groups

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This establishes an intrinsic connection between ideal Euler hydrodynamics (via Arnold's approach), shock formation in the multidimensional Burgers equation and the Wasserstein geometry of the space of densities.Comment: 11 pages, 2 figure

Euler equations on homogeneous spaces and Virasoro orbits

We show that the following three systems related to various hydrodynamical approximations: the Korteweg--de Vries equation, the Camassa--Holm equation, and the Hunter--Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces. We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits.Comment: 26 pages, 4 figures, LaTeX. Advances in Mathematics (to appear

Asymptotic directions, Monge-Ampere equations and the geometry of diffeomorphism groups

In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curvature. Finally, we exhibit the common origin of the Monge-Ampere equations in 2D fluid dynamics and mass transport.Comment: 10 pages, 1 fig., to appear in J. of Math. Fluid Mechanic

Geometry of the Madelung transform

The Madelung transform is known to relate Schr\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\"ahler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the problem of conservation of the Willmore energy in binormal flows.Comment: 27 pages, 2 figure

Geometric Hydrodynamics via Madelung Transform

We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schr\"odinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a K\"ahler map when the space of densities is equipped with the Fisher-Rao information metric. We describe several dynamical applications of these results.Comment: 17 pages, 2 figure

Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

We study an equation lying `mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped, as well as smooth, traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.Comment: 30 pages, 2 figure

Possibilities of surgical correction of vocal cord palsy after thyroid gland operations

  Wstęp: Powikłania po zabiegach chirurgicznych gruczołu tarczowego są stale najczęstszą przyczyną obustronnego porażenia fałdów głosowych, a zaburzenia oddychania są głównym problemem w tych sytuacjach. Istnieje wiele procedur operacyjnych poszerzenia szpary głośni w przypadku obustronnego porażenia fałdów głosowych. Celem pracy była prezentacja możliwości poszerzenia szpary głośni, ocena wykonywanych technik operacyjnych oraz wyników leczenia w przypadku porażeń fałdów głosowych po zabiegach operacyjnych tarczycy. Materiał i metody: W badaniu oceniono pięć technik operacyjnych: laserowa tylna chordektomia według Denisa i Kashimy, laserowa obustronna arytenoidektomia przyśrodkowa według Crumlaya, laserowa tylna wentrykulo-chordektomia według Pia, laserowa całkowita arytenoidektomia z chordektomią tylną według Ossoffa oraz laterofiksacja według Lichtenbergera. Subiektywnej oceny poprawy wentylacyjnej dokonywano przy użyciu skali VAS. Wyniki: W latach 1998–2014 autorzy zoperowali 270 pacjentów z obustronnym porażeniem fałdów głosowych, z czego 255 (94,4%) przypadków jatrogennych po operacjach tarczycy, a 15 (7,6%) z innych przyczyn. U znacznej większości pacjentów — 77,6%, zastosowano laserową całkowitą arytenoidektomię z chordektomią tylną, a u 13,7% laterofiksację metodą Lichtenbergera. Zabieg operacyjny według Ossoffa pozwala na osiągnięcie dobrych wyników wentylacyjnych: udana dekaniulacja (62,9% po pierwszym zabiegu, 97,6% jako ostateczny odsetek) oraz znaczna lub istotna stopnia subiektywna poprawa wydolności oddechowej u 96% pacjentów. Wnioski: Arytenoidektomia według Ossoffa jest bezpieczną procedurą dającą zadowalające wyniki wentylacyjne. Pacjenci relacjonują satysfakcjonującą jakość życia oraz możliwość powrotu do życia zawodowego. Zdaniem autorów pracy, laterofiksacja powinna pozostać jako alternatywa dla czasowej tracheotomii raczej niż zabieg zasadniczy. (Endokrynol Pol 2015; 66 (5): 412–416)    Introduction: Surgery of the thyroid gland remains the main cause of bilateral vocal cord palsy (VCP). Ventilation problem is the main problem in such situations. There are a couple of corrective surgical procedures in the case of VCP. The aim of our study was to show the possibility of widening of the glottis, and to evaluate the techniques and effects of surgical treatments due to bilateral VCP resulting from thyroid gland surgery. Material and methods: Five methods of surgical treatment were used: laser-assisted posterior cordectomy, according to Denis and Kashima; laser-assisted bilateral medial arytenoidectomy, as proposed by Crumley; laser-assisted posterior ventriculocordectomy, as described by Pia; laser-assisted total arytenoidectomy with posterior cordectomy, as presented by Ossoff; and laterofixation, according to Lichtenberger. The postoperative patient’s subjective improvement was assessed using visual analogue scale. Results: Between 1998 and 2014 we operated on 270 patients with bilateral VCP. Paresis occurred as the result of the iatrogenic effect of thyroid gland surgery in 255 patients (94.4%) vs. 15 (7.6%) from other causes. The majority of our patients (77.6%) had undergone laser arytenoidectomy with posterior partial cordectomy, and in 13.7% of them Lichtenberger laterofixation had been performed. Ossoff ’s surgery gives good ventilation results: successful decannulation (62.9% after first surgery; 97.6% final rate) and significant subjective ventilation improvement in 96% of patients. Conclusions: Ossoff ’s laser arytenoidectomy with posterior cordectomy is a safe procedure that gives acceptable ventilation improvement. Patients report satisfactory quality of life and the possibility of returning to active professional life. Laterofixation should be considered as an alternative for tracheotomy rather than permanent procedure. (Endokrynol Pol 2015; 66 (5): 412–416)

EFORMING THE ELECTORATE: PATTERNS OF TECHNOLOGY USE

This paper discusses research-in-progress on a longitudinal study of technology usage by incumbent members of the U.S. Congress on their official Congressional homepages to “routinely” communicate with the public and with their constituencies as well as technology usage on campaign homepages for the purpose of political campaigning. By examining the types of information technologies embedded on and linked to from both types of homepages, we propose to track technology usage over time in order to study the diffusion and adoption of information technologies within the political arena. Drawing on diffusion of innovation and institutional isomorphism as our theoretical framework, we propose that examining patterns of diffusion of three categories of information technology – Web 1.0, Web 2.0, and Social Networking technologies – will provide us with insights into the changing nature of political communication strategies enabled by information technology, as well as insights into factors underlying institutional technology diffusion