Geometric investigations of a vorticity model equation

This article consists of a detailed geometric study of the one-dimensional vorticity model equation $$\omega_{t} + u\omega_{x} + 2\omega u_{x} = 0, \qquad \omega = H u_{x}, \qquad t\in\mathbb{R},\; x\in S^{1}\,,$$ which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on $\operatorname{Diff}(S^{1})$ when the latter is endowed with the right-invariant homogeneous $\dot{H}^{1/2}$-metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-C\'ordoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to C\'ordoba-C\'ordoba.Comment: 30 pages; added references; corrected typo

Geodesic completeness of the H3/2H^{3/2} metric on Diff(S1)\mathrm{Diff}(S^{1})

Of concern is the study of the long-time existence of solutions to the Euler--Arnold equation of the right-invariant $H^{3/2}$-metric on the diffeomorphism group of the circle. In previous work by Escher and Kolev it has been shown that this equation admits long-time solutions if the order $s$ of the metric is greater than $3/2$, the behaviour for the critical Sobolev index $s=3/2$ has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. The behaviour of the $H^{3/2}$-metric is, however, still different from its higher order counter parts, as it does not induce a complete Riemannian metric on any group of Sobolev diffeomorphisms

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators

Lateral-torsional Buckling of Ferritic Stainless Steel Beams in Case of Fire

audience: researcherThis work presents a numerical study of the behaviour of ferritic stainless steel Ibeams subjected to lateral-torsional buckling and compares the obtained results with the beam design curves of Eurocode 3. New formulae, for the lateraltorsional buckling, that approximate better the real behaviour of ferritic stainless steel structural elements in case of fire are proposed. These new formulae were based on numerical simulations using the program SAFIR, which was modified to take into account the material properties of the stainless steel

Ionic and electronic transport in calcium-substituted LaAlO3 perovskites prepared via mechanochemical route

The present work explores mechanosynthesis of lanthanum aluminate-based perovskite ceramics and corresponding effects on ionic-electronic transport properties. La1-xCaxAlO3-δ (x = 0.05-0.20) nanopowders were prepared via one-step high-energy mechanochemical processing. Sintering at 1450°C yielded dense ceramics with submicron grains. As-prepared powders and sintered ceramics were characterized by XRPD, XPS and SEM. Electrochemical studies showed that partial oxygen-ionic conductivity in prepared La1-xCaxAlO3-δ increases with calcium content up to 10 at.% in the lanthanum sublattice and then levels off at ~6×10-3 S/cm at 900°C. La1-xCaxAlO3-δ ceramics are mixed conductors under oxidizing conditions and ionic conductors with negligible contribution of electronic transport in reducing atmospheres. Oxygen-ionic contribution to the total conductivity is 20-68% at 900°C in air and increases with Ca content, with temperature and with reducing p(O2). Impedance spectroscopy results showed however that electrical properties of mechanosynthesized La1-xCaxAlO3-δ ceramics below ~800°C are determined by prevailing grain boundary contribution to the total resistivity.This work was supported by the Slovak Research and Development Agency APVV (contracts SK-PT-18-0039 and 15-0438) and the Slovak Grand Agency (contract No. 2/0055/19). BIAS and AAY would like to acknowledge financial support by the FCT, Portugal (bilateral project Portugal-Slovakia 2019-2020, project CARBOSTEAM (POCI01-0145-FEDER-032295) and project CICECO-Aveiro Institute of Materials (FCT ref. UID/CTM/50011/2019), financed by national funds through the FCT/MCTES and when appropriate co-financed by FEDER under the PT2020 Partnership Agreement). HK thanks to SAIA, n.o. for financial support within National Scholarship Programme of the Slovak republic (NSP).in publicatio

Implication of Complement System and its Regulators in Alzheimer’s Disease

Alzheimer’s disease (AD) is an age-related neurodegenerative disease that affects approximately 24 million people worldwide. A number of different risk factors have been implicated in AD, however, neuritic (amyloid) plaques are considered as one of the defining risk factors and pathological hallmarks of the disease. Complement proteins are integral components of amyloid plaques and cerebral vascular amyloid in Alzheimer brains. They can be found at the earliest stages of amyloid deposition and their activation coincides with the clinical expression of Alzheimer's dementia. This review emphasizes on the dual key roles of complement system and complement regulators (CRegs) in disease pathology and progression. The particular focus of this review is on currently evolving strategies for design of complement inhibitors that might aid therapy by restoring the fine balance between activated components of complement system, thus improving the cognitive performance of patients. This review discusses these issues with a view to inspiring the development of new agents that could be useful for the treatment of AD

On a two-component π\pi-Camassa--Holm system

A novel $\pi$-Camassa--Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa--Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.Comment: 12 page

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