1,870 research outputs found
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 (). 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 based on the topological
classification of trajectories.Comment: Latex, 51 pages, 14 eps figure
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
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
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