39,277 research outputs found

    Basic theory of a class of linear functional differential equations with multiplication delay

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    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

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    The classical Liouvile integrability means that there exist nn independent first integrals in involution for 2n2n-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 LL-integrability. As examples, we prove that the string vibration equation and the KdV equation are LL-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

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    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 pβˆ’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

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    Although there doesn't exist the Lebesgue measure in the ball MM of C[0,1]C[0,1] with pβˆ’p-norm, the average values (expectation) EYEY and variance DYDY of some functionals YY on MM 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 MM exist and are derived out even though the density of points in MM 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 DYDY 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=0DY=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

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    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

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    In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set {Ai}i=0p\{A_i\}_{i=0}^p (pβ‰₯1p\ge 1), where a nonsingular matrix WW (often referred to as diagonalizer) needs to be found such that the matrices WHAiWW^{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=[x1,x2,…,xn]Ξ W=[x_1, x_2, \dots, x_n]\Pi, where Ξ \Pi is a permutation matrix, xix_i's are eigenvectors of the matrix polynomial P(Ξ»)=βˆ‘i=0pΞ»iAiP(\lambda)=\sum_{i=0}^p\lambda^i A_i, satisfying that [x1,x2,…,xn][x_1, x_2, \dots, x_n] is nonsingular, and the geometric multiplicity of each Ξ»i\lambda_i corresponding with xix_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

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    We evaluate the octupole in the large-NcN_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

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    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 WW-exchange and Pauli interference. The Ξcc++\Xi_{cc}^{++} baryon is longest-lived in the doubly charmed baryon system owing to the destructive Pauli interference absent in the Ξcc+\Xi_{cc}^+ and Ξ©cc+\Omega_{cc}^+. In the presence of dimension-7 contributions, its lifetime is reduced from ∼5.2Γ—10βˆ’13s\sim5.2\times 10^{-13}s to ∼3.0Γ—10βˆ’13s\sim3.0\times 10^{-13}s. The Ξcc+\Xi_{cc}^{+} baryon has the shortest lifetime of order 0.45Γ—10βˆ’13s0.45\times 10^{-13}s due to a large contribution from the WW-exchange box diagram. It is difficult to make a precise quantitative statement on the lifetime of Ξ©cc+\Omega_{cc}^+. Contrary to Ξcc\Xi_{cc} baryons, Ο„(Ξ©cc+)\tau(\Omega_{cc}^+) becomes longer in the presence of dimension-7 effects and the Pauli interference Ξ“+int\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 Ξ“+int\Gamma^{\rm int}_+ to be positive for a sensible HQE, we conjecture that the Ξ©c0\Omega_c^0 lifetime lies in the range of (0.75∼1.80)Γ—10βˆ’13s(0.75\sim 1.80)\times 10^{-13}s. The lifetime hierarchy pattern is Ο„(Ξcc++)>Ο„(Ξ©cc+)>Ο„(Ξcc+)\tau(\Xi_{cc}^{++})>\tau(\Omega_{cc}^+)>\tau(\Xi_{cc}^+) and the lifetime ratio Ο„(Ξcc++)/Ο„(Ξcc+)\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

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    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

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    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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