Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence

We deal with homogeneous isotropic turbulence and use the two-point velocity correlation tensor field (parametrized by the time variable t) of the velocity fluctuations to equip an affine space K3 of the correlation vectors by a family of metrics. It was shown in Grebenev and Oberlack (J Nonlinear Math Phys 18:109–120, 2011) that a special form of this tensor field generates the so-called semi-reducible pseudo-Riemannian metrics ds2(t) in K3. This construction presents the template for embedding the couple (K3, ds2(t)) into the Euclidean space R3 with the standard metric. This allows to introduce into the consideration the function of length between the fluid particles, and the accompanying important problem to address is to find out which transformations leave the statistic of length to be invariant that presents a basic interest of the paper. Also we classify the geometry of the particles configuration at least locally for a positive Gaussian curvature of this configuration and comment the case of a negative Gaussian curvature.

Generalizations of Lie Algebras

The generalizations of Lie algebras appeared in the modern mathematics and mathematical physics. In this paper we consider recent developments and remaining open problems on the subject. Some of that developments have been influenced by lectures given by Professor Jaime Keller in his research seminar. The survey includes Lie superalgebras, color Lie algebras, Lie algebras in symmetric categories, free Lie tau-algebras, and some generalizations with non-associative enveloping algebras: tangent algebras to analytic loops, bialgebras and primitive elements, non-associative Hopf algebras.

COMMUTATIVE ALGEBRAS IN DRINFELD CATEGORIES OF ABELIAN LIE ALGEBRAS

We describe (braided-) commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with a finite number of simple local modules.

New structural brain imaging endophenotype in bipolar disorder

Neuroimaging studies suggest anterior-limbic structural brain abnormalities in patients with bipolar disorder (BD), but few studies have shown these abnormalities in unaffected but genetically liable family members. In this study, we report morphometric correlates of genetic risk for BD using voxel-based morphometry. In 35 BD type I (BD-I) patients, 20 unaffected first-degree relatives (UAR) of BD patients and 40 healthy control subjects underwent 3 T magnetic resonance scanner imaging. Preprocessing of images used DARTEL (diffeomorphic anatomical registration through exponentiated lie algebra) for voxel-based morphometry in SPM8 (Wellcome Department of Imaging Neuroscience, London, UK). The whole-brain analysis revealed that the gray matter (GM) volumes of the left anterior insula and right inferior frontal gyrus showed a significant main effect of diagnosis. Multiple comparison analysis showed that the BD-I patients and the UAR subjects had smaller left anterior insular GM volumes compared with the healthy subjects, the BD-I patients had smaller right inferior frontal gyrus compared with the healthy subjects. For white matter (WM) volumes, there was a significant main effect of diagnosis for medial frontal gyrus. The UAR subjects had smaller right medial frontal WM volumes compared with the healthy subjects. These findings suggest that morphometric brain abnormalities of the anterior-limbic neural substrate are associated with family history of BD, which may give insight into the pathophysiology of BD, and be a potential candidate as a morphological endophenotype of BD. Molecular Psychiatry (2012) 17, 412-420; doi: 10.1038/mp.2011.3; published online 15 February 2011

On simple Lie algebras over a field of characteristic 2

We prove that the simple Lie algebras constructed by G. Jurman (2004) in 121 are isomorphic to Hamiltonian algebras. As a corollary we answer all questions formulated in G. Jurman (2004) [2] about isomorphisms of these algebras. (C) 2012 Elsevier Inc. All rights reserved.

Normal Enveloping Algebras

A full characterization is given of ordinary and restricted enveloping algebras which are normal with respect to the principal involution.

Symmetry insights for design of supercomputer network topologies: roots and weights lattices

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

Effect of continuous and interval exercise training on the PETCO2 response during a graded exercise test in patients with coronary artery disease

OBJECTIVE: The purpose of this study was to evaluate the following: 1) the effects of continuous exercise training and interval exercise training on the end-tidal carbon dioxide pressure (PETCO2) response during a graded exercise test in patients with coronary artery disease; and 2) the effects of exercise training modalities on the association between PETCO2 at the ventilatory anaerobic threshold (VAT) and indicators of ventilatory efficiency and cardiorespiratory fitness in patients with coronary artery disease. METHODS: Thirty-seven patients (59.7 +/- 1.7 years) with coronary artery disease were randomly divided into two groups: continuous exercise training (n = 20) and interval exercise training (n = 17). All patients performed a graded exercise test with respiratory gas analysis before and after three months of the exercise training program to determine the VAT, respiratory compensation point (RCP) and peak oxygen consumption. RESULTS: After the interventions, both groups exhibited increased cardiorespiratory fitness. Indeed, the continuous exercise and interval exercise training groups demonstrated increases in both ventilatory efficiency and PETCO2 values at VAT, RCP, and peak of exercise. Significant associations were observed in both groups: 1) continuous exercise training (PETCO(2)VAT and cardiorespiratory fitness r = 0.49; PETCO(2)VAT and ventilatory efficiency r = -0.80) and 2) interval exercise training (PETCO(2)VAT and cardiorespiratory fitness r = 0.39; PETCO(2)VAT and ventilatory efficiency r = -0.45). CONCLUSIONS: Both exercise training modalities showed similar increases in PETCO2 levels during a graded exercise test in patients with coronary artery disease, which may be associated with an improvement in ventilatory efficiency and cardiorespiratory fitness.