Averaging algebras, Schr\"oder numbers, rooted trees and operads

In this paper, we study averaging operators from an algebraic and combinatorial point of view. We first construct free averaging algebras in terms of a class of bracketed words called averaging words. We next apply this construction to obtain one and two variable generating functions for subsets of averaging words when the averaging operator is taken to be idempotent. When the averaging algebra has an idempotent generator, the generating function in one variable is twice the generating function for large Schr\"oder numbers, leading us to give interpretations of large Schr\"oder numbers in terms of bracketed words and rooted trees, as well as a recursive formula for these numbers. We also give a representation of free averaging algebras by unreduced trees and apply it to give a combinatorial description of the operad of averaging algebras.Comment: 30 page

Quantum state transfer between three ring-connected atoms

A robust quantum state transfer scheme is discussed for three atoms that are trapped by separated cavities linked via optical fibers in ring-connection. It is shown that, under the effective three-atom Ising model, arbitrary quantum state can be transferred from one atom to another deterministically via an auxiliary atom with maximum unit fidelity. The only required operation for this scheme is replicating turning on/off the local laser fields applied to the atoms for two steps with time cost $\frac{\sqrt{2}\pi}{\Gamma_{0}}$. The scheme is insensitive to cavity leakage and atomic position due to the condition $\Delta \approx \kappa\gg g$. Another advantage of this scheme is that the cooperative influence of spontaneous emission and operating time error can reduce the time cost for maximum fidelity and thus speed up the implementation of quantum state transfer.Comment: 5 pages, 3 figures. arXiv admin note: text overlap with arXiv:0805.4493 by other author

Rota-Baxter operators on sl(2,C)sl(2,C) and solutions of the classical Yang-Baxter equation

We explicitly determine all Rota-Baxter operators (of weight zero) on $sl(2,C)$ under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in $sl(2,C)$, confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebra $sl(2,C) \ltimes_{{\rm ad}^{\ast}} sl(2,C)^{\ast}$. They also give rise to 3-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other 6-dimensional Lie algebras.Comment: 17 page

Splitting of operads and Rota-Baxter operators on operads

This paper establishes a uniform procedure to split the operations in any algebraic operad, generalizing previous known notions of splitting algebraic structures from the dendriform algebra of Loday that splits the associative operation to the successors that split any binary operad. Examples are provided for various $n$-associative algebras, $n$-Lie algebras, $A_\infty$ algebras and $L_\infty$ algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra context and then generalized to any binary operads, is generalized to arbitrary operads. The classical links from the Rota-Baxter associative algebra to the dendriform algebra and its numerous generalizations are further generalized and unified as the link from the Rota-Baxter operator on an operad to the splitting of the operad. Finally, the remarkable fact that any dendriform algebra can be recovered from a relative Rota-Baxter operator is generalized to the context of operads with the generalized notion of a relative Rota-Baxter operator for any operad.Comment: 30 page

Identification of the newly observed Σb(6097)±\Sigma_b(6097)^\pm baryons from their strong decays

Two bottom $\Sigma_b(6097)^\pm$ baryons were observed in the final states $\Lambda_b^0\pi^-$ and $\Lambda_b^0\pi^+$ in $pp$ collision by LHCb collaboration, whose masses and widths were measured. In a $^{3}P_{0}$ model, the strong decay widths of two ground $S$-wave and seven excited $P$-wave $\Sigma_b$ baryons have been systematically computed. Numerical results indicate that the newly observed $\Sigma_b(6097)^\pm$ are very possibly $\Sigma_{b2}^1({3\over 2}^-)$ with $J^P={3\over 2}^-$ or $\Sigma_{b2}^1({5\over 2}^-)$ with $J^P={5\over 2}^-$. The predicted decay widths of $\Sigma_b(6097)^\pm$ are consistent with experimental measurement from LHCb. In particular, it may be possible to distinguish these two assignments through ratios $\Gamma({\Sigma_b(6097)^\pm\to \Sigma_b^\pm\pi^0})/\Gamma({\Sigma_b(6097)^\pm\to \Sigma_b^{*\pm}\pi^0})$, which can be measured by experiments in the future. In the meantime, our results support the assignments that $\Sigma_b^\pm$ and $\Sigma_b^{*\pm}$ are the ground $S$-wave $\Sigma_b$ baryons with $J^P={1\over 2}^+$ and $J^P={3\over 2}^+$, respectively.Comment: 5 pages, 6 tables, RevTe

Strong Decays of observed Λc\Lambda_c Baryons in the 3P0^3P_0 Model

The strong decay widths and some important branching ratios of possible Okubo-Zweig-Iizuka(OZI)-allowed strong decay channels of $\Lambda_c(2595)^+$, $\Lambda_c(2625)^+$, $\Lambda_c(2765)^+$ ($\Sigma_c(2765)^+$), $\Lambda_c(2860)^+$, $\Lambda_c(2880)^+$ and $\Lambda_c(2940)^+$ are computed in a $^{3}P_{0}$ model, and possible assignments of these $\Lambda_c$ are given. (1), $\Lambda_c(2595)^+$ and $\Lambda_c(2625)^+$ are possibly the $1P$-wave charmed baryons $\Lambda_{c1}(\frac{1}{2}^-)$ and $\Lambda_{c1}(\frac{3}{2}^-)$, respectively. (2), $\Lambda_c(2765)^+$ ($\Sigma_c(2765)^+$) seems impossibly the $1P$-wave $\Lambda_{c}$, it could be the $2S$-wave or $1D$-wave charmed baryon. So far, the experimental information has not been sufficient for its identification. (3), $\Lambda_c(2860)^+$ seems impossibly $2S$-wave charmed baryon, it may be the $P$-wave $\tilde\Lambda_{c2}^{ }(\frac{3}{2}^-)$ or $\tilde\Lambda_{c2}^{ }(\frac{5}{2}^-)$, it could also be the $D$-wave $\check\Lambda_{c1}^{2}(\frac{1}{2}^+)$ or $\check\Lambda_{c1}^{2}(\frac{3}{2}^+)$. If the hypothesis that $\Lambda_c(2860)^+$ has $J^P={3\over 2}^+$ is true, $\Lambda_c(2860)^+$ is possibly the $D$-wave $\check\Lambda_{c1}^{2}(\frac{3}{2}^+)$ which has a predicted branching ratio $R=\Gamma(\Sigma_c(2520)\pi)/\Gamma(\Sigma_c(2455)\pi)=2.8$. (4), $\Lambda_c(2880)^+$ is impossibly a $1P$-wave or $2S$-wave charmed baryon, it may be a $D$-wave $\check\Lambda_{c3}^{2}(\frac{5}{2}^+)$ with $\Gamma_{total}=1.3$ MeV. The predicted branching ratio $R=\Gamma(\Sigma_c(2520)\pi)/\Gamma(\Sigma_c(2455)\pi)=0.35$, which is consistent with experiment. (5), $\Lambda_c(2940)^+$ is the $P$-wave $\tilde\Lambda_{c2}^{ }(\frac{3}{2}^-)$ or $\tilde\Lambda_{c2}^{ }(\frac{5}{2}^-)$, it is also possibly the $D$-wave $\check\Lambda_{c3}^{2}(\frac{5}{2}^+)$ or $\check\Lambda_{c3}^{2}(\frac{7}{2}^+)$.Comment: 13 pages, 2 figures, 17 tables, RevTe

Molten-Salt Depleted-Uranium Reactor

The supercritical, reactor core melting and nuclear fuel leaking accidents have troubled fission reactors for decades, and greatly limit their extensive applications. Now these troubles are still open. Here we first show a possible perfect reactor, Molten-Salt Depleted-Uranium Reactor which is no above accident trouble. We found this reactor could be realized in practical applications in terms of all of the scientific principle, principle of operation, technology, and engineering. Our results demonstrate how these reactors can possess and realize extraordinary excellent characteristics, no prompt critical, long-term safe and stable operation with negative feedback, closed uranium-plutonium cycle chain within the vessel, normal operation only with depleted-uranium, and depleted-uranium high burnup in reality, to realize with fission nuclear energy sufficiently satisfying humanity long-term energy resource needs, as well as thoroughly solve the challenges of nuclear criticality safety, uranium resource insufficiency and low-carbon development. They could provide safe, cheap, abundant, and clean energy resource and electric power lasting thousands years for humanity.Comment: 4 pages, 0 figur

Intelligent Reflecting Surface: A Programmable Wireless Environment for Physical Layer Security

In this paper, we introduce an intelligent reflecting surface (IRS) to provide a programmable wireless environment for physical layer security. By adjusting the reflecting coefficients, the IRS can change the attenuation and scattering of the incident electromagnetic wave so that it can propagate in a desired way toward the intended receiver. Specifically, we consider a downlink multiple-input single-output (MISO) broadcast system where the base station (BS) transmits independent data streams to multiple legitimate receivers and keeps them secret from multiple eavesdroppers. By jointly optimizing the beamformers at the BS and reflecting coefficients at the IRS, we formulate a minimum-secrecy-rate maximization problem under various practical constraints on the reflecting coefficients. The constraints capture the scenarios of both continuous and discrete reflecting coefficients of the reflecting elements. Due to the non-convexity of the formulated problem, we propose an efficient algorithm based on the alternating optimization and the path-following algorithm to solve it in an iterative manner. Besides, we show that the proposed algorithm can converge to a local (global) optimum. Furthermore, we develop two suboptimal algorithms with some forms of closed-form solutions to reduce the computational complexity. Finally, the simulation results validate the advantages of the introduced IRS and the effectiveness of the proposed algorithm

Removal of surface plasmon polariton eigenmodes degeneracy

The effect of the tilt angle of metal film on the transmissivity of subwavelength holes in optically thick metal film is investigated. The transmission efficiency is found can be highly dependent on the tilt angle. It is also found that when the input photons are polarized not along the eigenmode directions of surface plasmon polariton, a birefringent phenomenon is observed when the array with periodic of subwavelength holes is tilted. Linear polarization states can be changed to elliptical polarization states and a phase can be added between two eigenmodes. The phase is changed with the tilt angle. A model based on surface plasmon polariton eigenmodes degeneracy is presented to explain these experimental results.Comment: 7 pages, 5 figure

Monotonic quantum-to-classical transition enabled by positively-correlated biphotons

Multiparticle interference is a fundamental phenomenon in the study of quantum mechanics.It was discovered in a recent experiment [Ra, Y.-S. et al, Proc. Natl Acad. Sci. USA \textbf{110}, 1227(2013)] that spectrally uncorrelated biphotons exhibited a nonmonotonic quantum-to-classical transition in a four-photon Hong-Ou-Mandel (HOM) interference. In this work, we consider the same scheme with spectrally correlated photons.By theoretical calculation and numerical simulation, we found the transition not only can be nonmonotonic with negative-correlated or uncorrelated biphotons, but also can be monotonic with positive-correlated biphotons. The fundamental reason for this difference is that the HOM-type multi-photon interference is a differential-frequency interference. Our study may shed new light on understanding the role of frequency entanglement in multi-photon behavior.Comment: 8 pages, 2 figure