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B q for parabolic measures

Caroline Sweezy (1998)

Studia Mathematica

If Ω is a Lip(1,1/2) domain, μ a doubling measure on ∂ p Ω , ∂ / ∂ t - L i , i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures ω 0 , ω 1 have the property that ω 0 ∈ B q ( μ ) implies ω 1 is absolutely continuous with respect to ω 0 whenever a certain Carleson-type condition holds on the difference function of the coefficients of L 1 and L 0 . Also ω 0 ∈ B q ( μ ) implies ω 1 ∈ B q ( μ ) whenever both measures are center-doubling measures. This is B. Dahlberg’s result for elliptic measures extended...

Backpropagation generalized delta rule for the selective attention Sigma-if artificial neural network

Maciej Huk (2012)

International Journal of Applied Mathematics and Computer Science

In this paper the Sigma-if artificial neural network model is considered, which is a generalization of an MLP network with sigmoidal neurons. It was found to be a potentially universal tool for automatic creation of distributed classification and selective attention systems. To overcome the high nonlinearity of the aggregation function of Sigma-if neurons, the training process of the Sigma-if network combines an error backpropagation algorithm with the self-consistency paradigm widely used in physics....

Backward doubly stochastic differential equations with infinite time horizon

Bo Zhu, Baoyan Han (2012)

Applications of Mathematics

We give a sufficient condition on the coefficients of a class of infinite horizon backward doubly stochastic differential equations (BDSDES), under which the infinite horizon BDSDES have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations.

Backward solutions to nonlinear integro-differential systems

Yuzhen Bai (2010)

Open Mathematics

In this paper, we show the backward uniqueness in time of solutions to nonlinear integro-differential systems with Neumann or Dirichlet boundary conditions. We also discuss reasonable physical interpretations for our conclusions.

Banach algebras of pseudodifferential operators and their almost diagonalization

Karlheinz Gröchenig, Ziemowit Rzeszotnik (2008)

Annales de l’institut Fourier

We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra 𝒜 over a lattice Λ we associate a symbol class M ∞ , 𝒜 . Then every operator with a symbol in M ∞ , 𝒜 is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra...

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form u ( x , t ) = t – α f ( | x | t – β ) with suitable and β . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection...

Baroclinic Kelvin Waves in a Rotating Circular Basin

R. N. Ibragimov (2012)

Mathematical Modelling of Natural Phenomena

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover,...

Barriers for a class of geometric evolution problems

Giovanni Bellettini, Matteo Novaga (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form u t + F ⁢ t , x , ∇ ⁡ u , ∇ 2 ⁡ u = 0 . If F is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace F with F + , which is defined as the smallest degenerate elliptic function above F .

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