29,273 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

### 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

### 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
$p-$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

### Infinite-dimensional Hamilton-Jacobi theory and $L$-integrability

The classical Liouvile integrability means that there exist $n$ independent
first integrals in involution for $2n$-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
$L$-integrability. As examples, we prove that the string vibration equation and
the KdV equation are $L$-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

### Average values of functionals and concentration without measure

Although there doesn't exist the Lebesgue measure in the ball $M$ of $C[0,1]$
with $p-$norm, the average values (expectation) $EY$ and variance $DY$ of some
functionals $Y$ on $M$ 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 $M$ exist and are derived out
even though the density of points in $M$ 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 $DY$ 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. $DY=0$ 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

### 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
[email protected]

### Weighted Community Detection and Data Clustering Using Message Passing

Grouping objects into clusters based on similarities or weights between them
is one of the most important problems in science and engineering. In this work,
by extending message passing algorithms and spectral algorithms proposed for
unweighted community detection problem, we develop a non-parametric method
based on statistical physics, by mapping the problem to Potts model at the
critical temperature of spin glass transition and applying belief propagation
to solve the marginals corresponding to the Boltzmann distribution. Our
algorithm is robust to over-fitting and gives a principled way to determine
whether there are significant clusters in the data and how many clusters there
are. We apply our method to different clustering tasks and use extensive
numerical experiments to illustrate the advantage of our method over existing
algorithms. In the community detection problem in weighted and directed
networks, we show that our algorithm significantly outperforms existing
algorithms. In the clustering problem when the data was generated by mixture
models in the sparse regime we show that our method works to the theoretical
limit of detectability and gives accuracy very close to that of the optimal
Bayesian inference. In the semi-supervised clustering problem, our method only
needs several labels to work perfectly in classic datasets. Finally, we further
develop Thouless-Anderson-Palmer equations which reduce heavily the computation
complexity in dense-networks but gives almost the same performance as belief
propagation.Comment: 21 pages, 13 figures, to appear in Journal of Statistical Mechanics:
Theory and Experimen

### The renormalization method based on the Taylor expansion and applications for asymptotic analysis

Based on the Taylor expansion, we propose a renormalization method for
asymptotic analysis. The standard renormalization group (RG) method for
asymptotic analysis can be derived out from this new method, and hence the
mathematical essence of the RG method is also recovered. The biggest advantage
of the proposed method is that the secular terms in perturbation series are
automatically eliminated, but in usual perturbation theory, we need more
efforts and tricks to eliminate these terms. At the same time, the mathematical
foundation of the method is simple and the logic of the method is very clear,
therefore, it is very easy in practice. As application, we obtain the uniform
valid asymptotic solutions to some problems including vector field, boundary
layer and boundary value problems of nonlinear wave equations. Moreover, we
discuss the normal form theory and reduction equations of dynamical systems.
Furthermore, by combining the topological deformation and the RG method, a
modified method namely the homotopy renormalization method (for simplicity,
HTR) wasproposed to overcome the weaknesses of the standard RG method. In this
HTR method, since there is a freedom to choose the first order approximate
solution in perturbation expansion, we can improve the global solution. In
particular, for those equations including no a small parameter, the HTR method
can also be applied. Some concrete applications including multi-solutions
problems, the forced Duffing equation and the Blasius equation are given.Comment: 44 page

### The essence of the homotopy analysis method

The generalized Taylor expansion including a secret auxiliary parameter $h$
which can control and adjust the convergence region of the series is the
foundation of the homotopy analysis method proposed by Liao. The secret of $h$
can't be understood in the frame of the homotopy analysis method. This is a
serious shortcoming of Liao's method. We solve the problem. Through a detailed
study of a simple example, we show that the generalized Taylor expansion is
just the usual Taylor's expansion at different point $t_1$. We prove that there
is a relationship between $h$ and $t_1$, which reveals the meaning of $h$ and
the essence of the homotopy analysis method. As an important example, we study
the series solution of the Blasius equation. Using the series expansion method
at different points, we obtain the same result with liao's solution given by
the homotopy analysis method.Comment: 5 page

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