47,728 research outputs found
Basic theory of a class of linear functional differential equations with multiplication delay
By introducing a kind of special functions namely exponent-like function,
cosine-like function and sine-like function, we obtain explicitly the basic
structures of solutions of initial value problem at the original point for this
kind of linear pantograph equations. In particular, we get the complete results
on the existence, uniqueness and non-uniqueness of the initial value problems
at a general point for the kind of linear pantograph equations.Comment: 44 pages, no figure. This is a revised version of the third version
of the paper. Some new results and proofs have been adde
Infinite-dimensional Hamilton-Jacobi theory and -integrability
The classical Liouvile integrability means that there exist independent
first integrals in involution for -dimensional phase space. However, in the
infinite-dimensional case, an infinite number of independent first integrals in
involution don't indicate that the system is solvable. How many first integrals
do we need in order to make the system solvable? To answer the question, we
obtain an infinite dimensional Hamilton-Jacobi theory, and prove an infinite
dimensional Liouville theorem. Based on the theorem, we give a modified
definition of the Liouville integrability in infinite dimension. We call it the
-integrability. As examples, we prove that the string vibration equation and
the KdV equation are -integrable. In general, we show that an infinite
number of integrals is complete if all action variables of a Hamilton system
can reconstructed by the set of first integrals.Comment: 13 page
The geometrical origins of some distributions and the complete concentration of measure phenomenon for mean-values of functionals
We derive out naturally some important distributions such as high order
normal distributions and high order exponent distributions and the Gamma
distribution from a geometrical way. Further, we obtain the exact mean-values
of integral form functionals in the balls of continuous functions space with
norm, and show the complete concentration of measure phenomenon which means
that a functional takes its average on a ball with probability 1, from which we
have nonlinear exchange formula of expectation.Comment: 8 page
Transient behavior of the solutions to the second order difference equations by the renormalization method based on Newton-Maclaurin expansion
The renormalization method based on the Newton-Maclaurin expansion is applied
to study the transient behavior of the solutions to the difference equations as
they tend to the steady-states. The key and also natural step is to make the
renormalization equations to be continuous such that the elementary functions
can be used to describe the transient behavior of the solutions to difference
equations. As the concrete examples, we deal with the important second order
nonlinear difference equations with a small parameter. The result shows that
the method is more natural than the multi-scale method.Comment: 12 page
The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds
The renormalization method based on the Taylor expansion for asymptotic
analysis of differential equations is generalized to difference equations. The
proposed renormalization method is based on the Newton-Maclaurin expansion.
Several basic theorems on the renormalization method are proven. Some
interesting applications are given, including asymptotic solutions of quantum
anharmonic oscillator and discrete boundary layer, the reductions and invariant
manifolds of some discrete dynamics systems. Furthermore, the homotopy
renormalization method based on the Newton-Maclaurin expansion is proposed and
applied to those difference equations including no a small parameter.Comment: 24 pages.arXiv admin note: text overlap with arXiv:1605.0288
Average values of functionals and concentration without measure
Although there doesn't exist the Lebesgue measure in the ball of
with norm, the average values (expectation) and variance of some
functionals on can still be defined through the procedure of limitation
from finite dimension to infinite dimension. In particular, the probability
densities of coordinates of points in the ball exist and are derived out
even though the density of points in doesn't exist. These densities include
high order normal distribution, high order exponent distribution. This also can
be considered as the geometrical origins of these probability distributions.
Further, the exact values (which is represented in terms of finite dimensional
integral) of a kind of infinite-dimensional functional integrals are obtained,
and specially the variance is proven to be zero, and then the nonlinear
exchange formulas of average values of functionals are also given. Instead of
measure, the variance is used to measure the deviation of functional from its
average value. means that a functional takes its average on a ball with
probability 1 by using the language of probability theory, and this is just the
concentration without measure. In addition, we prove that the average value
depends on the discretization.Comment: 32 page
Heat Superconductivity
Electrons/atoms can flow without dissipation at low temperature in
superconductors/superfluids. The phenomenon known as
superconductivity/superfluidity is one of the most important discoveries of
modern physics, and is not only fundamentally important, but also essential for
many real applications. An interesting question is: can we have a
superconductor for heat current, in which energy can flow without dissipation?
Here we show that heat superconductivity is indeed possible. We will show how
the possibility of the heat superconductivity emerges in theory, and how the
heat superconductor can be constructed using recently proposed time crystals.
The underlying simple physics is also illustrated. If the possibility could be
realized, it would not be difficult to speculate various potential
applications, from energy tele-transportation to cooling of information
devices.Comment: 12 pages, 2 figures. Correct an issue pointed out by Jing-ning Zhang.
Figures and text update
Solving General Joint Block Diagonalization Problem via Linearly Independent Eigenvectors of a Matrix Polynomial
In this paper, we consider the exact/approximate general joint block
diagonalization (GJBD) problem of a matrix set (),
where a nonsingular matrix (often referred to as diagonalizer) needs to be
found such that the matrices 's are all exactly/approximately block
diagonal matrices with as many diagonal blocks as possible. We show that the
diagonalizer of the exact GJBD problem can be given by , where is a permutation matrix, 's are eigenvectors of the
matrix polynomial , satisfying that
is nonsingular, and the geometric multiplicity of each
corresponding with equals one. And the equivalence of all
solutions to the exact GJBD problem is established. Moreover, theoretical proof
is given to show why the approximate GJBD problem can be solved similarly to
the exact GJBD problem. Based on the theoretical results, a three-stage method
is proposed and numerical results show the merits of the method
Multipole scattering amplitudes in the Color Glass Condensate formalism
We evaluate the octupole in the large- limit in the McLerran-Venugopalan
model, and derive a general expression of the 2n-point correlator, which can be
applied in analytical studies of the multi-particle production in the
scatterings between hard probes and dense targets
Dynamical properties of two electrons confined in a line shape three quantum dot molecules driven by an ac-field
Using the three-site Hubbard model and Floquet theorem, we investigate the
dynamical behaviors of two electrons which are confined in a line-shape three
quantum dot molecule driven by an AC electric field. Because the Hamiltonian
contains no spin-flip terms, the six- dimension singlet state and
nine-dimensional triplet state sub-spaces are decoupled and can be discussed
respectively. In particular, the nine-dimensional triplet state sub-spaces can
also be divided into 3 three-dimensional state sub-spaces which are fully
decoupled. The analysis shows that the Hamiltonian in each three-dimensional
triplet state sub-space, as well as the singlet state sub-space for the no
double-occupancy case, has the same form similar to that of the driven two
electrons in two-quantum-dot molecule. Through solving the time-dependent
Sch\"odinger equation, we investigate the dynamical properties in the singlet
state sub-space, and find that the two electrons can maintain their initial
localized state driven by an appropriately ac-field. Particularly, we find that
the electron interaction enhances the dynamical localization effect. The use of
both perturbation analytic and numerical approach to solve the Floquet function
leads to a detail understanding of this effect.Comment: 15 pages, 3 figures. Reviews are welcomed to
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