scholarly journals DYADIC TRIANGULAR HILBERT TRANSFORM OF TWO GENERAL FUNCTIONS AND ONE NOT TOO GENERAL FUNCTION

2015 ◽  
Vol 3 ◽  
Author(s):  
VJEKOSLAV KOVAČ ◽  
CHRISTOPH THIELE ◽  
PAVEL ZORIN-KRANICH
Keyword(s):  
Singular Integral ◽  
Hilbert Transform ◽  
Integral Form ◽  
General Function ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Bilinear Hilbert Transform ◽  
The One ◽  
Special Case ◽  
Dyadic Model

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

Uniform estimates for families of singular integrals and double Fourier series

1986 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Partial Sums ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Almost Everywhere ◽  
The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

1999 ◽  
Vol 51 (5) ◽  
pp. 915-935 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Christoph Leuenberger
Keyword(s):  
Unit Circle ◽  
Complex Flow ◽  
One Dimensional ◽  
Holomorphic Motions ◽  
Strictly Convex ◽  
Fixed Domain ◽  
Polynomial Hulls ◽  
Polynomial Hull ◽  
The One ◽  
Special Case

AbstractConsider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in C2 is studied in details.

Smooth solution to the one-dimensional inhomogeneous non-automorphic Landau–Lifshitz equation

2006 ◽  
Vol 462 (2072) ◽  
pp. 2397-2413 ◽  
Author(s):  
Junyu Lin ◽  
Shijin Ding
Keyword(s):  
Difference Method ◽  
Existence And Uniqueness ◽  
Smooth Solution ◽  
One Dimension ◽  
Initial Value ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Lifshitz Equation ◽  
Orthogonal Base ◽  
The One

Using the differential–difference method and viscosity vanishing approach, we obtain the existence and uniqueness of the global smooth solution to the periodic initial-value problem of the inhomogeneous, non-automorphic Landau–Lifshitz equation without Gilbert damping terms in one dimension. To establish the uniform estimates, we use some identities resulting from the fact and the fact that the vectors form an orthogonal base of the space .

scholarly journals Uniqueness of one-dimensional Néel wall profiles

2016 ◽  
Vol 472 (2187) ◽  
pp. 20150762 ◽  
Author(s):  
Cyrill B. Muratov ◽  
Xiaodong Yan
Keyword(s):  
Domain Wall ◽  
Energy Functional ◽  
Wall Structure ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Limiting Behaviour ◽  
Non Local ◽  
The One ◽  
Wall Profile ◽  
Domain Wall Structure

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin-film micromagnetic model, we analyse the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the Néel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behaviour at infinity. Thus, we establish uniqueness of the one-dimensional monotone Néel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.

Analytical Solution of 1-D Viscoelastic Consolidation of Soft Soils with Fractional Maxwell Model

2012 ◽  
Vol 157-158 ◽  
pp. 419-423
Author(s):  
Ya Peng Zhang ◽  
Feng Gao
Keyword(s):  
Analytical Solution ◽  
Soft Soil ◽  
Soil Layer ◽  
Viscoelastic Model ◽  
Integral Form ◽  
Maxwell Model ◽  
One Dimensional ◽  
The Laplace Transform ◽  
Fractional Maxwell Model ◽  
The One

Considering the rheological characteristics of soil, think the fractional maxwell with viscoelastic model can be described, the fractional maxwell model into integral form of saturated soft soil layer, the one dimensional compression, through the Laplace transform problems get instantaneous loading and single stage, the analytical solution of the loading conditions.

Unified geometric and analytical treatment of magneto– gasdynamic shocks. Part 2. The shock polars

1973 ◽  
Vol 10 (3) ◽  
pp. 397-423 ◽  
Author(s):  
Lee A. Bertram
Keyword(s):  
Particle Velocity ◽  
Two Dimensional ◽  
Perfect Gas ◽  
Analytical Treatment ◽  
Classification Schemes ◽  
One Dimensional ◽  
The One ◽  
Shock Solution ◽  
Special Case ◽  
Alfven Velocity

Previously derived shock solutions for a perfectly conducting perfect gas are used to compute shock polars for the one-dimensional unsteady and two- dimensional non-aligned shock representations. A new special-case shock solution, having a downstream particle velocity relative to the shock equal to upstream Alfvén velocity, is obtained, in addition to exhaustive analytical classification schemes for the shock polars. Eight classes of one-dimensional polars and twelve classes of two-dimensional polars are identified.

scholarly journals Variational bounds for a dyadic model of the bilinear Hilbert transform

2013 ◽  
Vol 57 (1) ◽  
pp. 105-119 ◽  
Author(s):  
Yen Do ◽  
Richard Oberlin ◽  
Eyvindur Ari Palsson
Keyword(s):  
Hilbert Transform ◽  
Bilinear Hilbert Transform ◽  
Variational Bounds ◽  
Dyadic Model

Exact damped sinusoidal electric field of nonlinear one-dimensional Vlasov-Maxwell equations

1984 ◽  
Vol 32 (2) ◽  
pp. 197-205 ◽  
Author(s):  
B. Abraham-Shrauner
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
Maxwell Equations ◽  
Angular Frequency ◽  
Wave Number ◽  
One Dimensional ◽  
Damping Decrement ◽  
Sinusoidal Electric Field ◽  
The One ◽  
Special Case

An exact solution for a temporally damped sinusoidal electric field which obeys the nonlinear, one-dimensional Vlasov-Maxwell equations is given. The electric field is a generalization of the O'Neil model electric field for Landau damping of plasma oscillations. The electric field is a special case of the form found from the invariance of the one-dimensional Vlasov equation under infinitesimal Lie group transformations. The time dependences of the damping decrement, of the wave-number and of the angular frequency are derived. Use of a time-dependent BGK one-particle distribution function is justified for weak damping where, in general, it is necessary to carry out a numerical calculation of the invariant of which the distribution function is a functional.

scholarly journals PHASE-DRIVEN INTERACTION OF WIDELY SEPARATED NONLINEAR SCHRÖDINGER SOLITONS

2012 ◽  
Vol 09 (03) ◽  
pp. 511-543 ◽  
Author(s):  
JUSTIN HOLMER ◽  
QUANHUI LIN
Keyword(s):  
Scattering Theory ◽  
General Framework ◽  
Nls Equation ◽  
One Dimensional ◽  
Initial Separation ◽  
The One ◽  
External Forces ◽  
Special Case ◽  
Equal Amplitude ◽  
Inverse Scattering Theory

We show that, for the one-dimensional cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for example, the dynamics of single solitons subject to external forces.

scholarly journals Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

1983 ◽  
Vol 24 (4) ◽  
pp. 392-416 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Nonlinear Elliptic ◽  
Special Cases ◽  
Exothermic Chemical Reaction ◽  
Dimensional Shape ◽  
The One ◽  
Special Case

AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

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