Abstract:
Sinai-Ruelle-Bowen measures are the only physically observable invariant measures for billiard dynamical systems under small perturbations. These measures are singular, but as it was observed, marginal distributions of spatial and angular coordinates are absolutely continuous. We generalize these facts and provide full mathematical proofs.

Abstract:
We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+\epsilon a(x_n,y_n,\epsilon), \quad y_{n+1} = T_\epsilon y_n, \end{align*} where $T_\epsilon$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $\epsilon\to0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = \epsilon a(x,y,\epsilon), \quad \dot y = g_\epsilon(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.

Abstract:
This is the English translation of the short note where the first nontrivial tetrahedron relation (solution of the Zamolodchikov tetrahedron equation) with variables on the edges was presented.

Abstract:
The construction of invariants of three-dimensional manifolds with a triangulated boundary, proposed earlier by the author for the case when the boundary consists of not more than one connected component, is generalized to any number of components. These invariants are based on the torsion of acyclic complexes of geometric origin. The relevant tool for studying our invariants turns out to be F.A. Berezin's calculus of anti-commuting variables; in particular, they are used in the formulation of the main theorem of the paper, concerning the composition of invariants under a gluing of manifolds. We show that the theory obeys a natural modification of M. Atiyah's axioms for anti-commuting variables.

Abstract:
Two novel fermionic - expressed in terms of Grassmann--Berezin calculus of anticommuting variables - solutions of pentagon equation are proposed, both being deformations of the known solution related to the affine group.

Abstract:
We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$. We show that despite the absence of a spectral gap for all $\alpha \in (0,1)$ and despite nonsummable correlations for $\alpha \geq 1/2$, the map $\alpha \mapsto \int \varphi \, d\nu_\alpha$ is continuously differentiable for $\varphi \in L^{q}[0,1]$ for $q$ sufficiently large.

Abstract:
A new relation in Grassmann algebra is presented, corresponding naturally to the four-dimensional Pachner move 3 -> 3. This relation is obtained by deforming a known relation associated with an exotic chain complex built for a triangulated four-manifold. The possible deformation terms turn out to correspond to second (middle) homologies of one more exotic complex. Both sides of the new relation are nonlinear in the deformation: they contain terms of degrees 0, 1 and 2.

Abstract:
A new approach is developed to find the source azimuth of the ultra low frequency (ULF) electromagnetic (EM) signals believed to be emanating from well defined seismic zone. The method is test applied on magnetic data procured from the seismoactive region of Koyna-Warna, known for prolonged reservoir triggered seismicity. Extremely low-noise, high-sensitivity LEMI-30 search coil magnetometers were used to measure simultaneously the vector magnetic field in the frequency range 0.001–32 Hz at two stations, the one located within and another ~100 km away from the seismic active zone. During the observation campaign extending from 15 March to 30 June 2006 two earthquakes (EQs) of magnitude (ML>4) occurred, which are searched for the presence of precursory EM signals. Comparison of polarization ellipses (PE) parameters formed by the magnetic field components at the measurement stations, in select frequency bands, allows discrimination of seismo-EM signals from the natural background ULF signals of magnetospheric/ionospheric origin. The magnetic field components corresponding to spectral bands dominated by seismo-EM fields define the PE plane which at any instant contains the source of the EM fields. Intersection lines of such defined PE planes for distant observation stations clutter in to the source region. Approximating the magnetic-dipole configuration for the source, the magnetic field components along the intersection lines suggest that azimuth of the EM source align in the NNW-SSE direction. This direction well coincides with the orientation of nodal plane of normal fault plane mechanism for the two largest EQs recorded during the campaign. More significantly the correspondence of this direction with the tectonic controlled trend in local seismicity, it has been surmised that high pressure fluid flow along the fault that facilitate EQs in the region may also be the source mechanism for EM fields by electrokinetic effect.

Abstract:
New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally – using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated boundary, and present example calculations confirming its nontriviality.

Abstract:
Pachner move 3 o 3 deals with triangulations of four-dimensional manifolds. We present an algebraic relation corresponding in a natural way to this move and based, a bit paradoxically, on three-dimensional geometry.