Topological Phenomena in Normal Metals

This paper is devoted to topological phenomena in normal metals with rather complicated Fermi surface. The results of the article are based on the deep topological theorems concerning the geometry of non-compact plane sections of level surfaces of periodic function in 3-dimensional Euclidean space which are the quasi-classical electron orbits in the presence of homogeneous magnetic field. The main result is that the observation of electrical conductivity in strong magnetic fields can reveal such nontrivial topological characteristics of Fermi surface as integral planes, connected with conductivity tensor and locally stable under small rotations of magnetic field. This planes are connected with generic non-closed orbits on the Fermi surface. Some non-generic situations are also discussed.Comment: 21 pages, 9 Encapsulated Postscript figure

Dynamical Systems, Topology and Conductivity in Normal Metals

New observable integer-valued numbers of the topological origin were revealed by the present authors studying the conductivity theory of single crystal 3D normal metals in the reasonably strong magnetic field ($B \leq 10^{3} Tl$). Our investigation is based on the study of dynamical systems on Fermi surfaces for the motion of semi-classical electron in magnetic field. All possible asymptotic regimes are also found for $B \to \infty$ based on the topological classification of trajectories.Comment: Latex, 51 pages, 14 eps figure

Open level lines of a superposition of periodic potentials on a plane

We consider here open level lines of potentials resulting from the superposition of two different periodic potentials on the plane. This problem can be considered as a particular case of the Novikov problem on the behavior of open level lines of quasi-periodic potentials on the plane with four quasi-periods. At the same time, the formulation of this problem may have many additional features that arise in important physical systems related to it. Here we will try to give a general description of the emerging picture both in the most general case and in the presence of additional restrictions. The main approach to describing the possible behavior of the open level lines will be based on their division into topologically regular and chaotic level lines.Comment: 8 pages, 5 figures, revte

Geometry of quasiperiodic functions on the plane

The present article proposes a review of the most recent results obtained in the study of Novikov's problem on the description of the geometry of the level lines of quasi-periodic functions in the plane. Most of the paper is devoted to the results obtained for functions with three quasi-periods, which play a very important role in the theory of transport phenomena in metals. In this part, along with previously known results, a number of new results are presented that significantly refine the general description of the picture that arises in this case. New statements are also presented for the case of functions with more than three quasi-periods, which open up approaches to the further study of Novikov's problem in the most general formulation. The role of Novikov's problem in various fields of mathematical and theoretical physics is also discussed.Comment: 24 pages, 17 figures, late

Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics

Retrieving refractive index of single spheres using the phase spectrum of light-scattering pattern

We analyzed the behavior of the complex Fourier spectrum of the angle-resolved light scattering pattern (LSP) of a sphere in the framework of the Wentzel-Kramers-Brillouin (WKB) approximation. Specifically, we showed that the phase value at the main peak of the amplitude spectrum almost quadratically depends on the particle refractive index, which was confirmed by numerical simulations using both the WKB approximation and the rigorous Lorenz-Mie theory. Based on these results, we constructed a method for characterizing polystyrene beads using the main peak position and the phase value at this point. We tested the method both on noisy synthetic LSPs and on the real data measured with the scanning flow cytometer. In both cases, the spectral method was consistent with the reference non-linear regression one. The former method leads to comparable errors in retrieved particle characteristics but is 300 times faster than the latter one. The only drawback of the spectral method is a limited operational range of particle characteristics that need to be set a priori due to phase wrapping. Thus, its main application niche is fast and precise characterization of spheres with small variation range of characteristics.Comment: 16 pages, 9 figures, 2 table

Why nonlocal recursion operators produce local symmetries: new results and applications

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all known today recursion operators and are much easier to verify than those found in earlier work. We also give explicit formulas for the nonlocal parts of higher recursion operators, Poisson and symplectic structures of integrable systems in (1+1) dimensions. Using these two results we prove, under some natural assumptions, the Maltsev--Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.Comment: 10 pages, LaTeX 2e, final versio