39,277 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 $L$-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

### 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

### 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

### 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 $\{A_i\}_{i=0}^p$ ($p\ge 1$),
where a nonsingular matrix $W$ (often referred to as diagonalizer) needs to be
found such that the matrices $W^{H}A_iW$'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 $W=[x_1, x_2, \dots,
x_n]\Pi$, where $\Pi$ is a permutation matrix, $x_i$'s are eigenvectors of the
matrix polynomial $P(\lambda)=\sum_{i=0}^p\lambda^i A_i$, satisfying that
$[x_1, x_2, \dots, x_n]$ is nonsingular, and the geometric multiplicity of each
$\lambda_i$ corresponding with $x_i$ 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-$N_c$ 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

### Lifetimes of Doubly Charmed Baryons

The lifetimes of doubly charmed hadrons are analyzed within the framework of
the heavy quark expansion (HQE). Lifetime differences arise from spectator
effects such as $W$-exchange and Pauli interference. The $\Xi_{cc}^{++}$ baryon
is longest-lived in the doubly charmed baryon system owing to the destructive
Pauli interference absent in the $\Xi_{cc}^+$ and $\Omega_{cc}^+$. In the
presence of dimension-7 contributions, its lifetime is reduced from
$\sim5.2\times 10^{-13}s$ to $\sim3.0\times 10^{-13}s$. The $\Xi_{cc}^{+}$
baryon has the shortest lifetime of order $0.45\times 10^{-13}s$ due to a large
contribution from the $W$-exchange box diagram. It is difficult to make a
precise quantitative statement on the lifetime of $\Omega_{cc}^+$. Contrary to
$\Xi_{cc}$ baryons, $\tau(\Omega_{cc}^+)$ becomes longer in the presence of
dimension-7 effects and the Pauli interference $\Gamma^{\rm int}_+$ even
becomes negative. This implies that the subleading corrections are too large to
justify the validity of the HQE. Demanding the rate $\Gamma^{\rm int}_+$ to be
positive for a sensible HQE, we conjecture that the $\Omega_c^0$ lifetime lies
in the range of $(0.75\sim 1.80)\times 10^{-13}s$. The lifetime hierarchy
pattern is $\tau(\Xi_{cc}^{++})>\tau(\Omega_{cc}^+)>\tau(\Xi_{cc}^+)$ and the
lifetime ratio $\tau(\Xi_{cc}^{++})/\tau(\Xi_{cc}^+)$ is predicted to be of
order 6.7.Comment: 17 pages, 1 figure, version to appear in PRD. arXiv admin note: text
overlap with arXiv:1807.0091

### 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

### Nonlinear Mixed-effects Scalar-on-function Models and Variable Selection for Kinematic Upper Limb Movement Data

This paper arises from collaborative research the aim of which was to model
clinical assessments of upper limb function after stroke using 3D kinematic
data. We present a new nonlinear mixed-effects scalar-on-function regression
model with a Gaussian process prior focusing on variable selection from large
number of candidates including both scalar and function variables. A novel
variable selection algorithm has been developed, namely functional least angle
regression (fLARS). As they are essential for this algorithm, we studied the
representation of functional variables with different methods and the
correlation between a scalar and a group of mixed scalar and functional
variables. We also propose two new stopping rules for practical usage.
This algorithm is able to do variable selection when the number of variables
is larger than the sample size. It is efficient and accurate for both variable
selection and parameter estimation. Our comprehensive simulation study showed
that the method is superior to other existing variable selection methods. When
the algorithm was applied to the analysis of the 3D kinetic movement data the
use of the non linear random-effects model and the function variables
significantly improved the prediction accuracy for the clinical assessment.Comment: 31 pages, 9 figures. Submitted to Annuals of applied statistic

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