Fixed parameter tractable algorithms in combinatorial topology

To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an exponential-time enumeration. However, asymptotically most graphs do not result in any 3-manifold triangulation, which leads to significant "wasted time" in topological enumeration algorithms. Here we give a new algorithm to determine whether a given face pairing graph supports any 3-manifold triangulation, and show this to be fixed parameter tractable in the treewidth of the graph. We extend this result to a "meta-theorem" by defining a broad class of properties of triangulations, each with a corresponding fixed parameter tractable existence algorithm. We explicitly implement this algorithm in the most generic setting, and we identify heuristics that in practice are seen to mitigate the large constants that so often occur in parameterised complexity, highlighting the practicality of our techniques.Comment: 16 pages, 9 figure

Simple Mechanism of Softening Structure Functions at Low Transverse Momentum Region

The relevance of the dipole configurations of quarks in forming nucleus structure functions is discussed. It is shown that a radiation generated by dipole configurations while moving relativistically along their axises is described by distributions which are finite and infrared stable in low transverse momentum region. It is argued that there is an exponential transition to the perturbative regime of large transverse momenta and its power is defined by the distance between the dipole charges in its rest frame $m^{-1}_\pi$.Comment: 8 page

Orientational mobility and relaxation spectra of dendrimers : theory and computer simulation

The developed theory of the orientational mobility of individual segments of a perfectly branched dendrimer is used to calculate the relaxation spectrum of a dendrimer. Frequency dependences of NMR relaxation 1/T1 and of the nuclear Overhauser effect have been theoretically calculated from the Brownian dynamics simulation data. The dendrimer segmental orientational mobility is governed by three main relaxation processes: (i) the rotation of the dendrimer as a whole, (ii) the rotation of the dendrimer's branch originated from a given segment, and (iii) the local reorientation of the segment. The internal orientational mobility of an individual dendrimer segment depends only on the topological distance between this segment and the terminal shell of the dendrimer. Characteristic relaxation times of all processes and their contributions to the segmental mobility have been calculated. The influence of the number of generations and the number of the generation shell on the relaxation times has been studied. The correlation between the characteristic times and the calculated relaxation spectrum of the dendrimer has been established

Weak Charge Quantization on Superconducting Islands

We consider the Coulomb blockade on a superconductive quantum dot strongly coupled to a lead through a tunnelling barrier and/or normal diffusive metal. Andreev transport of the correlated pairs leads to quantum fluctuations of the charge on the dot. These fluctuations result in exponential renormalization of the effective charging energy. We employ two complimentary ways to approach the problem, leading to the coinciding results: the instanton and the functional RG treatment of the non-linear sigma model. We also derive the charging energy renormalization in terms of arbitrary transmission matrix of the multi-channel interface.Comment: 21 pages, 4 eps figures, RevTe

A TQFT associated to the LMO invariant of three-dimensional manifolds

We construct a Topological Quantum Field Theory (in the sense of Atiyah) associated to the universal finite-type invariant of 3-dimensional manifolds, as a functor from the category of 3-dimensional manifolds with parametrized boundary, satisfying some additional conditions, to an algebraic-combinatorial category. It is built together with its truncations with respect to a natural grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The TQFT(s) induce(s) a (series of) representation(s) of a subgroup ${\cal L}_g$ of the Mapping Class Group that contains the Torelli group. The N=1 truncation produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been considerably shorten, and section 3 has been slightly shorten, since they will constitute a separate paper. Section 4, which contained only announce of results, has been suprimated; it will appear in detail elsewhere. Consequently some statements have been re-numbered. No mathematical changes have been made.

Foliations of Isonergy Surfaces and Singularities of Curves

It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level. We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure

PEDAGOGIC SCHOOL OF THEORY OF WORKFLOW AND TESTING OF GAS TURBINE ENGINES OF SAMARA UNIVERSITY

Brief history ofpedagogic school of Theory of workflow and testing of gas turbine engines of Samara University is provided. The features and benefits of the courses of Theory of gas turbine engines, Theory and calculation of blade machines. Workflow theory of combustion chambers and Testing aircraft engines are considered

Casimir energy in multiply connected static hyperbolic Universes

We generalize a previously obtained result, for the case of a few other static hyperbolic universes with manifolds of nontrivial topology as spatial sections.Comment: accepted for publicatio

Coulomb blockade in metallic grains at large conductance

We study Coulomb blockade effects in the thermodynamic quantities of a weakly disordered metallic grain coupled to a metallic lead by a tunneling contact with a large conductance $g_T$. We consider the case of broken time-reversal symmetry and obtain expressions for both the {\em ensemble averaged} amplitude of the Coulomb blockade oscillations of the thermodynamic potential and the correlator of its {\em mesoscopic fluctuations} for a finite mean level spacing $\delta$ in the grain. We develop a novel method which allows for an exact evaluation of the functional integral arising from disorder averaging. The results and the method are applicable in the temperature range $\delta \ll T \ll E_C$.Comment: 18 pages, 3 figures (revised version

From nonassociativity to solutions of the KP hierarchy

A recently observed relation between 'weakly nonassociative' algebras A (for which the associator (A,A^2,A) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus A' of A) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with A' a matrix algebra, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.Comment: 7 pages, 4 figures, International Colloquium 'Integrable Systems and Quantum Symmetries', Prague, 15-17 June 200