On the one-dimensional approximations to the quantum mechanics of π -electrons in chain and ring systems

1969 ◽  
Vol 313 (1514) ◽  
pp. 403-419 ◽  
Keyword(s):  
Beltrami Operator ◽  
Rate Of Change ◽  
Transverse Motion ◽  
Critical Examination ◽  
Small Displacement ◽  
Free Electrons ◽  
Mechanical Motion ◽  
One Dimensional ◽  
The One ◽  
Special Case

A critical examination is made of the one-dimensional approximations for the quantum mechanical motion of almost-free electrons confined near a general space curve. A special set of coordinates which separate the longitudinal from the transverse directions for the general curve are introduced to specify the motion. The Laplace-Beltrami operator is found and Schrödinger’s equation is investigated when the transverse motion is limited to a small displacement. The special case where the curve has isolated sharp kinks is shown to be soluble in terms of certain boundary conditions at the points of large rate of change of curvature. When the curvature varies slowly the transition energies are shown to follow from the onedimensional longitudinal approximation to a certain order in the transverse displacements. However, the absolute energies are strongly dependent on the transverse motions.

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.

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.

CONDUCTION ELECTRON IN THE ANISOTROPIC MEDIUM

1993 ◽  
Vol 07 (22) ◽  
pp. 3899-3905
Author(s):  
VLADIMIR L. SAFONOV
Keyword(s):  
Free Electrons ◽  
One Dimensional ◽  
Long Wavelength ◽  
Magnetic Components ◽  
Fermi Statistics ◽  
Proposed Model ◽  
Crystalline Anisotropy ◽  
Long Wavelength Approximation ◽  
Wavelength Approximation ◽  
The One

A new model for describing free electrons and holes in crystals in the long-wavelength approximation is proposed. The crystalline anisotropy in the framework of this model is introduced by means of corresponding space-time geometry. The generalized Dirac’s equation is constructed and non-relativistic Hamiltonian containing energy terms of the order of c–2 is calculated. It is shown that the spin magnetic components depend on corresponding effective cyclotron masses. Applicability of the results of the proposed model to different experiments is discussed. For the one-dimensional case, a hypothesis of para-Fermi statistics is suggested which may appear to explain one more mechanism of high-T c superconductivity.

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.

scholarly journals A theory of spectral partitions of metric graphs

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
James B. Kennedy ◽  
Pavel Kurasov ◽  
Corentin Léna ◽  
Delio Mugnolo
Keyword(s):  
Spectral Gaps ◽  
Qualitative Properties ◽  
Metric Graphs ◽  
One Dimensional ◽  
Graph Partitions ◽  
Planar Domains ◽  
Open Questions ◽  
The One ◽  
Abstract Framework ◽  
Special Case

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

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.

scholarly journals PARRONDO GAMES WITH SPATIAL DEPENDENCE II

2012 ◽  
Vol 11 (04) ◽  
pp. 1250030 ◽  
Author(s):  
S. N. ETHIER ◽  
JIYEON LEE
Keyword(s):  
Spin System ◽  
Spatial Dependence ◽  
Integer Lattice ◽  
Periodic Pattern ◽  
One Dimensional ◽  
Random Mixture ◽  
The Mean ◽  
The One ◽  
Positive Integers ◽  
Special Case

Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N ≥ 3 and p0, p1, p2, p3 ∈ [0, 1], and let game A be the special case p0 = p1 = p2 = p3 = 1/2. In previous work we investigated μB and μ(1/2, 1/2), the mean profits per turn to the ensemble of N players always playing game B and always playing the randomly mixed game (1/2)(A + B). These means were computable for 3 ≤ N ≤ 19, at least, and appeared to converge as N → ∞, suggesting that the Parrondo region (i.e., the region in which μB ≤ 0 and μ(1/2, 1/2) > 0) has nonzero volume in the limit. The convergence was established under certain conditions, and the limits were expressed in terms of a parameterized spin system on the one-dimensional integer lattice. In this paper we replace the random mixture with the nonrandom periodic pattern Ar Bs, where r and s are positive integers. We show that μ[r, s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern Ar Bs, is computable for 3 ≤ N ≤ 18 and r + s ≤ 4, at least, and appears to converge as N → ∞, albeit more slowly than in the random-mixture case. Again this suggests that the Parrondo region (μB ≤ 0 and μ[r, s] > 0) has nonzero volume in the limit. Moreover, we can prove this convergence under certain conditions and identify the limits.

A Source Flow Model for Continuum Gas-Particle Flow

1967 ◽  
Vol 34 (4) ◽  
pp. 860-865 ◽  
Author(s):  
David Migdal ◽  
V. D. Agosta
Keyword(s):  
Differential Equations ◽  
Control Volume ◽  
Mass Conservation ◽  
One Dimensional ◽  
Complete Set ◽  
Thermodynamic Variables ◽  
Basic Equations ◽  
The One ◽  
Special Case ◽  
Source Flow

A system of differential equations is derived from basic laws of mass conservation dynamics and thermodynamics for a gas-particle reacting system. The concept of two superimposed continua occupying the same control volume is used throughout. A number of terms appearing in the basic equations, physically identified with the presence of particles, are defined as source terms for the gas motion. With the introduction of phenomenological relations for these terms and the thermodynamic variables, a complete set of differential equations is obtained. The equations governing the one-dimensional flow in an arbitrarily shaped duct are presented as a special case.

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