Stationary problems for equation of the KdV type and dynamical rr-matrices.

We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail.Comment: 25pp, LaTe

Physics and Mathematics of Calogero particles

We give a review of the mathematical and physical properties of the celebrated family of Calogero-like models and related spin chains.Comment: Version to appear in Special Issue of Journal of Physics A: Mathematical and Genera

Sigma-model approaches to exact solutions in higher-dimensional gravity and supergravity

Classical gravitating field theories reduced to three dimensions admit manifest gauge invariances and hidden symmetries, which together make up the invariance group G of the theory. If this group is large enough, the target space is a symmetric space G/H. New solutions may be generated by the action of invariance transformations on a seed solution. Another application is the construction of multicenter solutions from null geodesics of the target space. After a general introduction on this sigma-model approach, I will discuss the case of five-dimensional gravity, with invariace group SL(3,R), and minimal five-dimensional supergravity, with invariance group G_{2(2)}. I will also describe recent attempts at the generation of new charged rotating black rings.Comment: 28 pages, 1 figure, talk presented at the WE Heraeus Seminar on Models of Gravity in Higher Dimenions: From Theory to Experimental Search, Bremen, 25-29.8.200

Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media

In the present paper we investigate the mechanics of systems of affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry. Certain physical applications are possible in modelling of molecular crystals, granular media, and other physical objects. Particularly interesting are dynamical models invariant under the group underlying geometry of degrees of freedom. In contrary to the single body case there exist nontrivial potentials invariant under this group (left and right acting). The concept of relative (mutual) deformation tensors of pairs of affine bodies is discussed. Scalar invariants built of such tensors are constructed. There is an essential novelty in comparison to deformation scalars of single affine bodies, i.e., there exist affinely-invariant scalars of mutual deformations. Hence, the hierarchy of interaction models according to their invariance group, from Euclidean to affine ones, can be considered.Comment: 50 pages, 4 figure

Applications of Structural Balance in Signed Social Networks

We present measures, models and link prediction algorithms based on the structural balance in signed social networks. Certain social networks contain, in addition to the usual 'friend' links, 'enemy' links. These networks are called signed social networks. A classical and major concept for signed social networks is that of structural balance, i.e., the tendency of triangles to be 'balanced' towards including an even number of negative edges, such as friend-friend-friend and friend-enemy-enemy triangles. In this article, we introduce several new signed network analysis methods that exploit structural balance for measuring partial balance, for finding communities of people based on balance, for drawing signed social networks, and for solving the problem of link prediction. Notably, the introduced methods are based on the signed graph Laplacian and on the concept of signed resistance distances. We evaluate our methods on a collection of four signed social network datasets.Comment: 37 page

Fast Approximation of EEG Forward Problem and Application to Tissue Conductivity Estimation

Bioelectric source analysis in the human brain from scalp electroencephalography (EEG) signals is sensitive to the conductivity of the different head tissues. Conductivity values are subject dependent, so non-invasive methods for conductivity estimation are necessary to fine tune the EEG models. To do so, the EEG forward problem solution (so-called lead field matrix) must be computed for a large number of conductivity configurations. Computing one lead field requires a matrix inversion which is computationally intensive for realistic head models. Thus, the required time for computing a large number of lead fields can become impractical. In this work, we propose to approximate the lead field matrix for a set of conductivity configurations, using the exact solution only for a small set of basis points in the conductivity space. Our approach accelerates the computing time, while controlling the approximation error. Our method is tested for brain and skull conductivity estimation , with simulated and measured EEG data, corresponding to evoked somato-sensory potentials. This test demonstrates that the used approximation does not introduce any bias and runs significantly faster than if exact lead field were to be computed.Comment: Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]