1,809 research outputs found
Variational electrodynamics of Atoms
We generalize Wheeler-Feynman electrodynamics by the minimization of a finite
action functional defined for variational trajectories that are required to
merge continuously into given past and future boundary segments. We prove that
the boundary-value problem is well-posed for two classes of boundary data and
show that the well-posed solution in general has velocity discontinuities,
henceforth broken extrema. Along regular segments, broken extrema satisfy the
Euler-Lagrange neutral differential delay equations with state-dependent
deviating arguments. At points where velocities are discontinuous, broken
extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta
are continuous. The electromagnetic fields of the variational trajectories are
derived quantities that can be extended only to a bounded region B of
space-time. For extrema with a finite number of velocity discontinuities,
extended fields are defined for all point in B with the exception of sets of
zero measure. The extended fields satisfy the integral laws of classical
electrodynamics for most surfaces and curves inside B. As an application, we
study globally bounded trajectories with vanishing far-fields for the
hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses
solutions of the neutral differential delay equations along regular segments
and a variational approximation for the collisional segments. Each hydrogenoid
model predicts a discrete set of finitely measured neighbourhoods of orbits
with vanishing far-fields at the correct atomic magnitude and in quantitative
and qualitative agreement with experiment and quantum mechanics, i.e., the
spacings between consecutive discrete angular momenta agree with Planck's
constant within thirty-percent, while orbital frequencies agree with a
corresponding spectroscopic line within a few percent.Comment: Full re-write using same equations and back to original title
(version 18 compiled with the wrong figure 5). A few commas introduced and
all paragraphs broken into smaller ones whenever possibl
Periodic solutions to a -Laplacian neutral Duffing equation with variable parameter
We study a type of -Laplacian neutral Duffing functional differential equation with variable parameter to establish new results on the existence of -periodic solutions. The proof is based on a famous continuation theorem for coincidence degree theory. Our research enriches the contents of neutral equations and generalizes known results. An example is given to illustrate the effectiveness of our results
Double-slit and electromagnetic models to complete quantum mechanics
We analyze a realistic microscopic model for electronic scattering with the
neutral differential delay equations of motion of point charges of the
Wheeler-Feynman electrodynamics. We propose a microscopic model according to
the electrodynamics of point charges, complex enough to describe the essential
physics. Our microscopic model reaches a simple qualitative agreement with the
experimental results as regards interference in double-slit scattering and in
electronic scattering by crystals. We discuss our model in the light of
existing experimental results, including a qualitative disagreement found for
the double-slit experiment. We discuss an approximation for the complex neutral
differential delay equations of our model using piecewise-defined
(discontinuous) velocities for all charges and piecewise-constant-velocities
for the scattered charge. Our approximation predicts the De Broglie wavelength
as an inverse function of the incoming velocity and in the correct order of
magnitude. We explain the scattering by crystals in the light of the same
simplified modeling with Einstein-local interactions. We include a discussion
of the qualitative properties of the neutral-delay-equations of electrodynamics
to stimulate future experimental tests on the possibility to complete quantum
mechanics with electromagnetic models.Comment: 4 figures, the same post-publication typos over the published version
of Journal of Computational and Theoretical Nanoscience, only that these
correction are not marked in red as in V7, this one is for a recollectio
Existence of periodic positive solutions to a nonlinear Lotka-Votlerra competition systems
We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka–Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967–2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature
On the stability, boundedness, and square integrability of solutions of third order neutral delay differential equations
In this paper, sufficient conditions are established for the stability, boundedness and square integrability of solutions for some non-linear neutral delay differential equations of third order. Lyapunov’s direct method is used to obtain the results
Oscillation and nonoscillation of third order functional differential equations
A qualitative approach is usually concerned with the behavior of solutions of a given differential equation and usually does not seek specific explicit solutions;This dissertation is the analysis of oscillation of third order linear homogeneous functional differential equations, and oscillation and nonoscillation of third order nonlinear nonhomogeneous functional differential equations. This is done mainly in Chapters II and III. Chapter IV deals with the analysis of solutions of neutral differential equations of third order and even order. In Chapter V we study the asymptotic nature of nth order delay differential equations;Oscillatory solution is the solution which has infinitely many zeros; otherwise, it is called nonoscillatory solution;The functional differential equations under consideration are:(UNFORMATTED TABLE OR EQUATION FOLLOWS) (b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + (q[subscript]1y)[superscript]\u27 + q[subscript]2y[superscript]\u27 = 0, &(b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + q[subscript]1y + q[subscript]2y(t - [tau]) = 0, &(b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + qF(y(g(t))) = f(t), &(y(t) + p(t)y(t - [tau]))[superscript]\u27\u27\u27 + f(t, y(t), y(t - [sigma])) = 0, &(y(t) + p(t)y(t - [tau]))[superscript](n) + f(t, y(t), y(t - [sigma])) = 0, and &y[superscript](n) + p(t)f(t, y[tau], y[subscript]sp[sigma][subscript]1\u27,..., y[subscript]sp[sigma][subscript]n[subscript]1(n-1)) = F(t). (TABLE/EQUATION ENDS);The first and the second equations are considered in Chapter II, where we find sufficient conditions for oscillation. We study the third equation in Chapter III and conditions have been found to ensure the required criteria. In Chapter IV, we study the oscillation behavior of the fourth and the fifth equations. Finally, the last equation has been studied in Chapter V from the point of view of asymptotic nature of its nonoscillatory solutions
On solvability of periodic boundary value problems for second order linear functional differential equations
The periodic boundary value problem for second order linear functional differential equations with pointwise restrictions (instead of integral ones) is considered. Sharp sufficient conditions for the solvability are obtained
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