458 research outputs found
Mixed norm Bergman-Morrey-type spaces on the unit disc
We introduce and study the mixed-norm Bergman-Morrey space A (q;p,lambda) , mixednorm Bergman-Morrey space of local type A (loc) (q;p,lambda) , and mixed-norm Bergman-Morrey space of complementary type (C) A (q;p,lambda) on the unit disk D in the complex plane C. Themixed norm Lebesgue-Morrey space L (q;p,lambda) is defined by the requirement that the sequence of Morrey L (p,lambda)(I)-norms of the Fourier coefficients of a function f belongs to l (q) (I = (0, 1)). Then, A (q;p,lambda) is defined as the subspace of analytic functions in L (q;p,lambda) . Two other spaces A q;p,lambda loc and (C) A (q;p,lambda) are defined similarly by using the local Morrey L (loc) (p,lambda) (I)-norm and the complementary Morrey (C) L (p,lambda)(I)-norm respectively. The introduced spaces inherit features of both Bergman and Morrey spaces and, therefore, we call them Bergman-Morrey-type spaces. We prove the boundedness of the Bergman projection and reveal some facts on equivalent description of these spaces
Self-consistent theory of turbulence
A new approach to the stochastic theory of turbulence is suggested. The
coloured noise that is present in the stochastic Navier-Stokes equation is
generated from the delta-correlated noise allowing us to avoid the nonlocal
field theory as it is the case in the conventional theory. A feed-back
mechanism is introduced in order to control the noise intensity.Comment: submitted to J.Tech. Phys.Letters (St. Petersburg
Fractional Fokker-Planck Equation for Fractal Media
We consider the fractional generalizations of equation that defines the
medium mass. We prove that the fractional integrals can be used to describe the
media with noninteger mass dimensions. Using fractional integrals, we derive
the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski
equation). In this paper fractional Fokker-Planck equation for fractal media is
derived from the fractional Chapman-Kolmogorov equation. Using the Fourier
transform, we get the Fokker-Planck-Zaslavsky equations that have fractional
coordinate derivatives. The Fokker-Planck equation for the fractal media is an
equation with fractional derivatives in the dual space.Comment: 17 page
Distributed Order Calculus and Equations of Ultraslow Diffusion
We consider diffusion type equations with a distributed order derivative in
the time variable. This derivative is defined as the integral in of
the Caputo-Dzhrbashian fractional derivative of order with a
certain positive weight function. Such equations are used in physical
literature for modeling diffusion with a logarithmic growth of the mean square
displacement. In this work we develop a mathematical theory of such equations,
study the derivatives and integrals of distributed order.Comment: 39 pages. To appear in J. Math. Anal. App
Fractional Liouville and BBGKI Equations
We consider the fractional generalizations of Liouville equation. The
normalization condition, phase volume, and average values are generalized for
fractional case.The interpretation of fractional analog of phase space as a
space with fractal dimension and as a space with fractional measure are
discussed. The fractional analogs of the Hamiltonian systems are considered as
a special class of non-Hamiltonian systems. The fractional generalization of
the reduced distribution functions are suggested. The fractional analogs of the
BBGKI equations are derived from the fractional Liouville equation.Comment: 20 page
Transport Equations from Liouville Equations for Fractional Systems
We consider dynamical systems that are described by fractional power of
coordinates and momenta. The fractional powers can be considered as a
convenient way to describe systems in the fractional dimension space. For the
usual space the fractional systems are non-Hamiltonian. Generalized transport
equation is derived from Liouville and Bogoliubov equations for fractional
systems. Fractional generalization of average values and reduced distribution
functions are defined. Hydrodynamic equations for fractional systems are
derived from the generalized transport equation.Comment: 11 pages, LaTe
Levi-Civita cylinders with fractional angular deficit
The angular deficit factor in the Levi-Civita vacuum metric has been
parametrized using a Riemann-Liouville fractional integral. This introduces a
new parameter into the general relativistic cylinder description, the
fractional index {\alpha}. When the fractional index is continued into the
negative {\alpha} region, new behavior is found in the Gott-Hiscock cylinder
and in an Israel shell.Comment: 5 figure
Fractional Systems and Fractional Bogoliubov Hierarchy Equations
We consider the fractional generalizations of the phase volume, volume
element and Poisson brackets. These generalizations lead us to the fractional
analog of the phase space. We consider systems on this fractional phase space
and fractional analogs of the Hamilton equations. The fractional generalization
of the average value is suggested. The fractional analogs of the Bogoliubov
hierarchy equations are derived from the fractional Liouville equation. We
define the fractional reduced distribution functions. The fractional analog of
the Vlasov equation and the Debye radius are considered.Comment: 12 page
Langevin formulation for single-file diffusion
We introduce a stochastic equation for the microscopic motion of a tagged
particle in the single file model. This equation provides a compact
representation of several of the system's properties such as
Fluctuation-Dissipation and Linear Response relations, achieved by means of a
diffusion noise approach. Most important, the proposed Langevin Equation
reproduces quantitatively the \emph{three} temporal regimes and the
corresponding time scales: ballistic, diffusive and subdiffusive.Comment: 9 pages, 5 figures, 1 table, to appear in Physical Review
Polymer-mediated entropic forces between scale-free objects
The number of configurations of a polymer is reduced in the presence of a
barrier or an obstacle. The resulting loss of entropy adds a repulsive
component to other forces generated by interaction potentials. When the
obstructions are scale invariant shapes (such as cones, wedges, lines or
planes) the only relevant length scales are the polymer size R_0 and
characteristic separations, severely constraining the functional form of
entropic forces. Specifically, we consider a polymer (single strand or star)
attached to the tip of a cone, at a separation h from a surface (or another
cone). At close proximity, such that h<<R_0, separation is the only remaining
relevant scale and the entropic force must take the form F=AkT/h. The amplitude
A is universal, and can be related to exponents \eta governing the anomalous
scaling of polymer correlations in the presence of obstacles. We use
analytical, numerical and epsilon-expansion techniques to compute the exponent
\eta for a polymer attached to the tip of the cone (with or without an
additional plate or cone) for ideal and self-avoiding polymers. The entropic
force is of the order of 0.1 pN at 0.1 micron for a single polymer, and can be
increased for a star polymer.Comment: LaTeX, 15 pages, 4 eps figure
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