Differentiability of fractal curves

While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for parabolic arcs

Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials

Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

A complete family of solutions for the one-dimensional reaction-diffusion equation $$u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t)$$ with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet boundary conditions are considered however the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.Comment: 8 pages, 1 figure. Minor updates to the tex

Magnetic Field Suppression of the Conducting Phase in Two Dimensions

The anomalous conducting phase that has been shown to exist in zero field in dilute two-dimensional electron systems in silicon MOSFETs is driven into a strongly insulating state by a magnetic field of about 20 kOe applied parallel to the plane. The data suggest that in the limit of T -> 0 the conducting phase is suppressed by an arbitrarily weak magnetic field. We call attention to striking similarities to magnetic field-induced superconductor-insulator transitions

Instability of the Two-Dimensional Metallic Phase to Parallel Magnetic Field

We report on magnetotransport studies of the unusual two-dimensional metallic phase in high mobility Si-MOS structures. We have observed that the magnetic field applied in the 2D plane suppresses the metallic state, causing the resistivity to increase dramatically by more than 30 times. Over the total existence range of the metallic state, we have found three distinct types of the magnetoresistance, related to the corresponding quantum corrections to the conductivity. Our data suggest that the unusual metallic state is a consequence of both spin- and Coulomb-interaction effects.Comment: 6 pages, Latex, 4 ps fig

Flow diagram of the metal-insulator transition in two dimensions

The discovery of the metal-insulator transition (MIT) in two-dimensional (2D) electron systems challenged the veracity of one of the most influential conjectures in the physics of disordered electrons, which states that `in two dimensions, there is no true metallic behaviour'; no matter how weak the disorder, electrons would be trapped and unable to conduct a current. However, that theory did not account for interactions between the electrons. Here we investigate the interplay between the electron-electron interactions and disorder near the MIT using simultaneous measurements of electrical resistivity and magnetoconductance. We show that both the resistance and interaction amplitude exhibit a fan-like spread as the MIT is crossed. From these data we construct a resistance-interaction flow diagram of the MIT that clearly reveals a quantum critical point, as predicted by the two-parameter scaling theory (Punnoose and Finkel'stein, Science 310, 289 (2005)). The metallic side of this diagram is accurately described by the renormalization group theory without any fitting parameters. In particular, the metallic temperature dependence of the resistance sets in when the interaction amplitude reaches gamma_2 = 0.45 - a value in remarkable agreement with the one predicted by the theory.Comment: as publishe

Test of scaling theory in two dimensions in the presence of valley splitting and intervalley scattering in Si-MOSFETs

We show that once the effects of valley splitting and intervalley scattering are incorporated, renormalization group theory consistently describes the metallic phase in silicon metal-oxide-semiconductor field-effect transistors down to the lowest accessible temperatures

Magnetoresistance of a two-dimensional electron gas in a parallel magnetic field

The conductivity of a two-dimensional electron gas in a parallel magnetic field is calculated. We take into account the magnetic field induced spin-splitting, which changes the density of states, the Fermi momentum and the screening behavior of the electron gas. For impurity scattering we predict a positive magnetoresistance for low electron density and a negative magnetoresistance for high electron density. The theory is in qualitative agreement with recent experimental results found for Si inversion layers and Si quantum wells.Comment: 4 pages, figures included, PDF onl

Fire Protection Of Wooden Storage Containers For Explosive And Pyrotechnic Products

Analysis of the emergency storage facilities for explosive and pyrotechnic products is conducted. It is established that one of the greatest risks is their flammability. Since the explosive and pyrotechnic products are stored in wooden containers, there is a need for their fire protection. To determine the efficiency of fire resistant containers for packaging explosive products it is designed operating range of testing method. This method is necessary to establish mass loss, measuring the growth temperature and response time of the squibs. The results of the efficiency of the fire retardant treatment of wood and organic coated coating showed that when exposed to high–temperature destruction of the construction detonation of the squibs didn\u27t happen.Tests to determine the quality of the fire retardant treatment of wood coatings showed that the temperature on the inner surfaces of the untreated sample was more than 760 ºC, samples with fire retardant coatings – no more than 128 °C. The conclusion of the feasibility of using fire–retardants is not based on inorganic and organic binders for the treatment of wooden structures.Method of determining the fire protection is used to assess the efficiency of the fire protection of wooden structures. Method comprises determining the ratio of the sample rate of burnout, the temperature increment and the ignition time of untreated and treated samples. As a result of the firing testing it is established a speed burnout reduction of samples of the container with treated coatings compared with untreated coatings is decreased by 2,4-4,4 times and respectively fire protection efficiency factor of treated samples of the container compared to untreated is increased by1.8-4.1 times