Normalizers of maximal tori and real forms of Lie groups

For a complex reductive Lie group $G$ Tits defined an extension $W_G^T$ of the corresponding Weyl group $W_G$. The extended group is supplied with an embedding into the normalizer $N_G(H)$ of the maximal torus $H\subset G$ such that $W_G^T$ together with $H$ generate $N_G(H)$. We give an interpretation of the Tits classical construction in terms of the maximal split real form $G(\mathbb{R})\subset G(\mathbb{C})$, leading to a simple topological description of $W^T_G$. We also propose a different extension $W_G^U$ of the Weyl group $W_G$ associated with the compact real form $U\subset G(\mathbb{C})$. This results into a presentation of the normalizer of maximal torus of the group extension $U\ltimes {\rm Gal}(\mathbb{C}/\mathbb{R})$ by the Galois group ${\rm Gal}(\mathbb{C}/\mathbb{R})$. We also describe explicitly the adjoint action of $W_G^T$ and $W^U_G$ on the Lie algebra of $G$.Comment: 17 page

On Using the Decision Trees to Identify the Local Extrema in Parallel Global Optimization Algorithm

In the present work, the solving of the multidimensional global optimization problems using decision tree to reveal the attractor regions of the local minima is considered. The objective function of the problem is defined as a “black box”, may be non-differentiable, multi-extremal and computational costly. We assume that the function satisfies the Lipschitz condition with a priory unknown constant. Global search algorithm is applied for the search of global minimum in the problems of such type. It is well known that the solution complexity essentially depends on the presence of multiple local extrema. Within the framework of the global search algorithm, we propose a method for selecting the vicinity of local extrema of the objective function based on analysis of accumulated search information. Conducting such an analysis using machine learning techniques allows making a decision to run a local method, which can speed up the convergence of the algorithm. This suggestion was confirmed by the results of numerical experiments demonstrating the speedup when solving a series of test problems.In the present work, the solving of the multidimensional global optimization problems using decision tree to reveal the attractor regions of the local minima is considered. The objective function of the problem is defined as a “black box”, may be non-differentiable, multi-extremal and computational costly. We assume that the function satisfies the Lipschitz condition with a priory unknown constant. Global search algorithm is applied for the search of global minimum in the problems of such type. It is well known that the solution complexity essentially depends on the presence of multiple local extrema. Within the framework of the global search algorithm, we propose a method for selecting the vicinity of local extrema of the objective function based on analysis of accumulated search information. Conducting such an analysis using machine learning techniques allows making a decision to run a local method, which can speed up the convergence of the algorithm. This suggestion was confirmed by the results of numerical experiments demonstrating the speedup when solving a series of test problems

Dynamic Path Planning for a 7-DOF Robot Arm

Klanke S, Lebedev DV, Haschke R, Steil JJ, Ritter H. Dynamic Path Planning for a 7-DOF Robot Arm. In: Int. Conf. Intelligent Robots and Systems. IEEE; 2006: 3879-3884

Acyclic Preference Systems in P2P Networks

In this work we study preference systems natural for the Peer-to-Peer paradigm. Most of them fall in three categories: global, symmetric and complementary. All these systems share an acyclicity property. As a consequence, they admit a stable (or Pareto efficient) configuration, where no participant can collaborate with better partners than their current ones. We analyze the representation of the such preference systems and show that any acyclic system can be represented with a symmetric mark matrix. This gives a method to merge acyclic preference systems and retain the acyclicity. We also consider such properties of the corresponding collaboration graph, as clustering coefficient and diameter. In particular, studying the example of preferences based on real latency measurements, we observe that its stable configuration is a small-world graph

Serological diagnostics of myocardium diseases based on multivariate analysis of cardiotrophic autoantibodies' profiles

ABSTRACT We analyzed profiles of IgG autoantibodies to 16 cardiac specific proteins and their main immunogenic region B-epitopes, in the groups of already verified cardiac pathology: acute and chronic lymphocytic myocarditis, ST elevation myocardial infarction, postinfarction remodeling of myocardium, dilated cardiomyopathy and in healthy controls along with patients, suffered from gastritis (to evaluate immune response against cross-reactive B-epitopes). AAB specific patterns allowed us to distinguish cases among themselves by means of multiparametrical canonical discriminant analysis in approximately 95% of cases. Positive predictive value in the group of MYO reached 95%, in the STEMI-89%, in the PIR-99%, in the DCM-99%, in the group of gastritis-88%. Principal component analysis of mentioned cardiac pathologies extended current clinical knowledge of their immunopathogenesis. Obtained data markedly proved a usability of serum AAB profiling for non invasive screening, differential diagnostics and working hypothesis composition