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

Representations of Polynomial Rota-Baxter Algebras

A Rota--Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota-Baxter operator. We show that studying the modules over the polynomial Rota--Baxter algebra $(k[x],P)$ is equivalent to studying the modules over the Jordan plane, and we generalize the direct decomposability results for the $(k[x],P)$-modules in [Iy] from algebraically closed fields of characteristic zero to fields of characteristic zero. Furthermore, we provide a classification of Rota--Baxter modules up to isomorphism based on indecomposable $k[x]$-modules

The Octet Meson and Octet Baryon Interaction with strangeness and the Λ(1405)\Lambda(1405)

The octet meson and baryon interaction with strangeness $S=-1$ is studied fully relativistically with chiral Lagrangian. In this work, a Bethe-Salpeter equation approach with spectator quasipotential approximation is applied to study the reactions $K^-p\to MB$ with $MB=K^-p,\bar{K}^0n, \pi^-\Sigma^+,\pi^0\Sigma^0, \pi^+\Sigma^-$ and $\eta \Lambda$ with all possible partial waves and theoretical results are comparable with experimental data. It is found that the Weinberg-Tomozawa potential derived from the lowest order chiral Lagrangian only provides the contributions from partial waves with spin-parities $J^P=1/2^+$ and $1/2^-$. Two-pole structure of the $\Lambda(1405)$ is confirmed with poles at $1383+99 i$ and $1423+14i$ MeV. The lower and higher poles originate from $\Sigma \pi$ interaction as a resonance and $\bar{K}N$ interaction as a bound state, respectively.Comment: 13 pages, 4 figure

Understanding Y(4274) and X(4320) in the J/ψϕJ/\psi \phi invariant mass spectrum

In this work we study the structures near 4.3 GeV in the $J/\psi \phi$ invariant mass spectra in B meson decay process $B^+\to J/\psi \phi K^+$ and two photon fusion process $\gamma\gamma\to J/\psi \phi$. The Y(4274) as a $D_sD_{s0}$ $(2317)$ molecular is studied in the Bethe-Salpeter equation approach with quasipotential approximation. The absence of Y(4274) in $\gamma\gamma\to J/\psi \phi$ channel can be well explained by the decay widths of Y(4274) decaying to $\gamma\gamma$ and $J/\psi\phi$. The distribution of mass difference released by CMS collaboration is reproduced by two resonances near 4.3 GeV, Y(4274) and X(4320). The different production mechanism suggests X(4320) observed in the B decay should be the missing $3^3P_1$ charmonium state $\chi''_{c1}$ and different from X(4350) observed in two photon fusion which can be assigned as $\chi''_{c2}$.Comment: 17 pages, 8 figure

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

Dark matter halo assembly bias: environmental dependence in the non-Markovian excursion set theory

In the standard excursion set model for the growth of structure, the statistical properties of halos are governed by the halo mass and are independent of the larger scale environment in which the halos reside. Numerical simulations, however, have found the spatial distributions of halos to depend not only on their mass but also on the details of their assembly history and environment. Here we present a theoretical framework for incorporating this "assembly bias" into the excursion set model. Our derivations are based on modifications of the path integral approach of Maggiore & Riotto (2010) that models halo formation as a non-Markovian random walk process. The perturbed density field is assumed to evolve stochastically with the smoothing scale and exhibits correlated walks in the presence of a density barrier. We write down conditional probabilities for multiple barrier crossings, and derive from them analytic expressions for descendant and progenitor halo mass functions and halo merger rates as a function of both halo mass and the linear overdensity of the larger-scale environment of the halo. Our results predict a higher halo merger rate and higher progenitor halo mass function in regions of higher overdensity, consistent with the behavior seen in N-body simulations.Comment: 13 pages, 1 figure, ApJ in pres

Is the Radial Profile of the Phase-Space Density of Dark Matter Halos a Power-Law?

The latest cosmological N-body simulations find two intriguing properties for dark matter haloes: (1) their radial density profile, rho, is better fit by a form that flattens to a constant at the halo center (the Einasto profile) than the widely-used NFW form; (2) the radial profile of the pseudo-phase-space density, rho/sigma3, on the other hand, continues to be well fit by a power law, as seen in earlier lower-resolution simulations. In this paper we use the Jeans equation to argue that (1) and (2) cannot both be true at all radii. We examine the implied radial dependence of rho/sigma3 over 12 orders of magnitude in radius by solving the Jeans equation for a broad range of input rho and velocity anisotropy beta. Independent of beta, we find that rho/sigma3 is approximately a power law only over the limited range of halo radius resolvable by current simulations (down to ~0.1% of the virial radius), and rho/sigma3 deviates significantly from a power-law below this scale for both the Einasto and NFW rho. The same conclusion also applies to a more general density-velocity relation rho/sigma_D^epsilon. Conversely, when we enforce rho/sigma^3 r^{-eta} as an input, none of the physically allowed rho (occurring for the narrow range 1.8<eta<1.9444) follows the Einasto form. We expect the next generation simulations with better spatial resolution to settle the debate: either the Einasto profile will continue to hold and rho/sigma3 will deviate from a power law, or rho/sigma3 will continue as a power law and rho will deviate from its current parameterizations.Comment: 6 pages, 4 figure

Personalized QoS Prediction of Cloud Services via Learning Neighborhood-based Model

The explosion of cloud services on the Internet brings new challenges in service discovery and selection. Particularly, the demand for efficient quality-of-service (QoS) evaluation is becoming urgently strong. To address this issue, this paper proposes neighborhood-based approach for QoS prediction of cloud services by taking advantages of collaborative intelligence. Different from heuristic collaborative filtering and matrix factorization, we define a formal neighborhood-based prediction framework which allows an efficient global optimization scheme, and then exploit different baseline estimate component to improve predictive performance. To validate the proposed methods, a large-scale QoS-specific dataset which consists of invocation records from 339 service users on 5,825 web services on a world-scale distributed network is used. Experimental results demonstrate that the learned neighborhood-based models can overcome existing difficulties of heuristic collaborative filtering methods and achieve superior performance than state-of-the-art prediction methods

High Quality of Electromagnetically Induced Transparency Spectroscopy of 87Rb in a Buffer Gas Cell with Magnetic Field

We have studied the phenomenon of electromagnetically induced transparency (EIT) of 87Rb vapor with a buffer gas in a magnetic field at room temperature. It is found that the spectral lines caused by the velocity selective optical pump effects get much weaker and wider when the sample cell mixed with a 5-Torr N2 gas while the EIT signal kept almost unchanged. A weighted least-square fit is also developed to remove the Doppler broadening completely. This spectral method provides a way to measure the Zeeman splitting with high resolution, for example, the {\Lambda}-type EIT resonance splits into four peaks on the D2 line of 87Rb in the thermal 2-cm vapor cell with a magnetic field along the electric field of the linearly polarized coupling laser. The high resolution spectrum can be used to lock the laser to a given frequency by tuning the magnetic field.Comment: 6 figure