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

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

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