Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar\'e equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.Comment: 40 pages. Corrected typos and improved redactio

The curvature of semidirect product groups associated with two-component Hunter-Saxton systems

In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its $\mu$-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group \Diff(\S) with a space of scalar functions on $\S$ we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2$\mu$HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [J. Escher, M. Kohlmann, and J. Lenells, J. Geom. Phys. 61 (2011), 436-452].Comment: 19 page

The energy functional on the Virasoro-Bott group with the L2L^2-metric has no local minima

The geodesic equation for the right invariant $L^2$-metric (which is a weak Riemannian metric) on each Virasoro-Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular solutions of KdV don't define locally length-minimizing paths.Comment: 12 pages, revised versio

High-precision molecular dynamics simulation of UO2-PuO2: superionic transition in uranium dioxide

Our series of articles is devoted to high-precision molecular dynamics simulation of mixed actinide-oxide (MOX) fuel in the rigid ions approximation using high-performance graphics processors (GPU). In this article we assess the 10 most relevant interatomic sets of pair potential (SPP) by reproduction of the Bredig superionic phase transition (anion sublattice premelting) in uranium dioxide. The measurements carried out in a wide temperature range from 300K up to melting point with 1K accuracy allowed reliable detection of this phase transition with each SPP. The {\lambda}-peaks obtained are smoother and wider than it was assumed previously. In addition, for the first time a pressure dependence of the {\lambda}-peak characteristics was measured, in a range from -5 GPa to 5 GPa its amplitudes had parabolic plot and temperatures had linear (that is similar to the Clausius-Clapeyron equation for melting temperature).Comment: 7 pages, 6 figures, 1 tabl

A bi-Hamiltonian supersymmetric geodesic equation

A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.Comment: 9 pages, no figure

Molecular mechanisms of glucocorticoids action: implications for treatment of rhinosinusitis and nasal polyposis

Intra-nasal glucocorticoids are the most effective drugs available for rhinosinusitis and nasal polyposis treatment. Their effectiveness depends on many factors and not all of them have been well recognized so far. The authors present the basic information on molecular mechanisms of glucocorticoid action, direct and indirect effects of glucocorticoids on transcription of genes encoding inflammatory mediators. They focus on recently proved nongenomic mechanisms which appear quickly, from several seconds to minutes after glucocorticoid administration and discuss clinical implications resulting from this knowledge. Discovery of nongenomic glucocorticoid actions allows for better use of these drugs in clinical practice

Fractional Sobolev Metrics on Spaces of Immersed Curves

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves Imm(S1 , R ) and on its Sobolev completions ℐ (S1 , R ). We prove local well-posedness of the geodesic equations both on the Banach manifold ℐ (S1 , R ) and on the Fr´echetmanifold Imm(S1 , R ) provided the order of the metric is greater or equal to one. In addition we show that the -metric induces a strong Riemannian metric on the Banach manifold ℐ (S1 , R ) of the same order , provided > 3 2 . These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group

Surveillance round for SBAO-40 weapon and ammunition system. Concept of design

W referacie przedstawiono przegląd rozwiązań istniejących konstrukcji 40 mm naboi granatnikowych z pociskami obserwacyjnymi oraz wyniki analiz mających na celu określenie podstawowych parametrów konstrukcyjnych dla potrzeb opracowania analogicznego pod względem funkcjonalnym rozwiązania w Polsce. W rozważaniach tych uwzględniono krajowe warunki produkcyjne oraz możliwości zaopatrzenia w podzespoły elektroniczne. Nabój, który zostanie opracowany według zaprezentowanych założeń, uzupełni rodzinę krajowej amunicji 40x46 mm do systemu broni i amunicji obezwładniającej SBAO-40, znacznie rozszerzając zakres zadań realizowanych za pomocą granatnika podwieszanego i samodzielnego tej rodziny. Ponadto konstrukcja ta wpisuje się w koncepcję przyszłościowego indywidualnego systemu wyposażenia żołnierza polskiego, zapewniając prowadzenie rozpoznania obrazowego na najniższym szczeblu taktycznym.This article presents a review of existing 40mm grenade rounds equipped with video system and the results of an analysis helping to decide about the basic construction parameters of a new, similar design in Poland. The limitations of Polish industry facilities and available supply sources of electronic components were taken into consideration. The round designed according to presented assumptions would fill the family of 40x46mm ammunition of a SBAO-40 weapon and ammunition system, allowing to increase the amount of tasks possible to achieve with that equipment. Moreover it will fulfill demands of future individual weapon and equipment system of Polish soldier