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 LL-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

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 $M$ 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

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]