Oscillator Variations of the Classical Theorem on Harmonic Polynomials

We study two-parameter oscillator variations of the classical theorem on harmonic polynomials, associated with noncanonical oscillator representations of sl(n) and o(n). We find the condition when the homogeneous solution spaces of the variated Laplace equation are irreducible modules of the concerned algebras and the homogeneous subspaces are direct sums of the images of these solution subspaces under the powers of the dual differential operator. This establishes a local (sl(2),sl(n)) and (sl(2),o(n)) Howe duality, respectively. In generic case, the obtained irreducible o(n)-modules are infinite-dimensional non-unitary modules without highest-weight vectors. As an application, we determine the structure of noncanonical oscillator representations of sp(2n). When both parameters are equal to the maximal allowed value, we obtain an infinite family of explicit irreducible (G,K)-modules for o(n) and sp(2n). Methodologically we have extensively used partial differential equations to solve representation problems.Comment: 49pages; This paper is a two-parameter extension of the first author's work "arXiv:0804.0305v2[math.RT]," which is equivalent to the one-paratemter special case of this work. The approaches have changed. The mistakes there have been correcte

Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter \mbb{Z}^2-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra $gl(n|m)$. Then we extend the result to two-parameter \mbb{Z}-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras $osp(2n|2m)$ and $osp(2n+1|2m)$.Comment: 35pages. This an extension and replacement of the first author's article arXiv:1001.3474v1[Math.RT

Volume Fluctuation and Autocorrelation Effects in the Moment Analysis of Net-proton Multiplicity Distributions in Heavy-Ion Collisions

Moments (Variance ($\sigma^2$), Skewness($S$), Kurtosis($\kappa$)) of multiplicity distributions of conserved quantities, such as net-baryon,net-charge and net-strangeness, are predicted to be sensitive to the correlation length of the system and connected to the thermodynamic susceptibilities computed in Lattice QCD and Hadron Resonance Gas (HRG) model. In this paper, we present several measurement artifacts that could lead to volume fluctuation and auto-correlation effects in the moment analysis of net-proton multiplicity distributions in heavy-ion collisions using the UrQMD model. We discuss methods to overcome these artifacts so that the extracted moments could be used to obtain physical conclusions. In addition we present methods to properly estimate the statistical errors in moment analysis.Comment: 8 pages, 12 figure

Tunable and switchable multi-wavelength erbium-doped fiber ring laser based on a modified dual-pass Mach-Zehnder interferometer

A tunable and switchable multi-wavelength erbium-doped fiber ring laser based on a new type tunable comb filter is proposed and demonstrated. By adjusting the polarization controllers, dual-function operation of the channel spacing tunability and the wavelength switching (interleaving) can be readily achieved. Up to 29 stable lasing lines with 0.4 nm spacing and 14 lasing wavelengths with 0.8 nm spacing in 3 dB bandwidth were obtained at room temperature. In addition, the lasing output, including the number of the lasing lines, the lasing evenness and the lasing locations, can also be flexibly adjusted through the wavelength-dependent polarization rotation mechanism.Comment: 11 pages, 6 figure

Benefits from Superposed Hawkes Processes

The superposition of temporal point processes has been studied for many years, although the usefulness of such models for practical applications has not be fully developed. We investigate superposed Hawkes process as an important class of such models, with properties studied in the framework of least squares estimation. The superposition of Hawkes processes is demonstrated to be beneficial for tightening the upper bound of excess risk under certain conditions, and we show the feasibility of the benefit in typical situations. The usefulness of superposed Hawkes processes is verified on synthetic data, and its potential to solve the cold-start problem of recommendation systems is demonstrated on real-world data

Baseline for the cumulants of net-proton distributions at STAR

We present a systematic comparison between the recently measured cumulants of the net-proton distributions by STAR for 0-5% central Au+Au collisions at $\sqrt{s_{NN}}$=7.7-200 GeV and two kinds of possible baseline measures, the Poisson and Binomial baselines. These baseline measures are assuming that the proton and anti-proton distributions independently follow Poisson statistics or Binomial statistics. The higher order cumulant net-proton data are observed to deviate from all the baseline measures studied at 19.6 and 27 GeV. We also compare the net-proton with net-baryon fluctuations in UrQMD and AMPT model, and convert the net-proton fluctuations to net-baryon fluctuations in AMPT model by using a set of formula.Comment: 4 pages, 5 figures, Proceedings of XXIV International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2014), May 19-24, 2014, Darmstadt, German

Completely Positive Tensors and Multi-Hypergraphs

Completely positive graphs have been employed to associate with completely positive matrices for characterizing the intrinsic zero patterns. As tensors have been widely recognized as a higher-order extension of matrices, the multi-hypergraph, regarded as a generalization of graphs, is then introduced to associate with tensors for the study of complete positivity. To describe the dependence of the corresponding zero pattern for a special type of completely positive tensors--the $\{0,1\}$ completely positive tensors, the completely positive multi-hypergraph is defined. By characterizing properties of the associated multi-hypergraph, we provide necessary and sufficient conditions for any $(0,1)$ associated tensor to be $\{0,1\}$ completely positive. Furthermore, a necessary and sufficient condition for a uniform multi-hypergraph to be completely positive multi-hypergraph is proposed as well

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs

Sequential Analysis of Cox Model under Response Dependent Allocation

Sellke and Siegmund (1983) developed the Brownian approximation to the Cox partial likelihood score as a process of calendar time, laying the foundation for group sequential analysis of survival studies. We extend their results to cover situations in which treatment allocations may depend on observed outcomes. The new development makes use of the entry time and calendar time along with the corresponding $\sigma$-filtrations to handle the natural information accumulation. Large sample properties are established under suitable regularity conditions

Some new properties of Confluent Hypergeometric Functions

The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!(\gamma)_{n}}z^{n}\ (\gamma\neq 0,-1,-2,\cdots)$, which are of many properties and great applications in statistics, mathematical physics, engineering and so on, have been given. In this paper, we investigate some new properties of${}_{1}F_{1}(\alpha;\gamma;z)$from the perspective of value distribution theory. Specifically, two different growth orders are obtained for$\alpha\in \mathbb{Z}_{\leq 0}$and$\alpha\not\in \mathbb{Z}_{\leq 0}$, which are corresponding to the reduced case and non-degenerated case of${}_{1}F_{1}(\alpha;\gamma;z)$. Moreover, we get an asymptotic estimation of characteristic function$T(r,{}_{1}F_{1}(\alpha;\gamma;z))$and a more precise result of$m\left(r, \frac{{}_{1}F_{1}'(\alpha;\gamma;z)}{{}_{1}F_{1}(\alpha;\gamma;z)}\right)$, compared with the Logarithmic Derivative Lemma. Besides, the distribution of zeros of the confluent hypergeometric functions is discussed. Finally, we show how a confluent hypergeometric function and an entire function are uniquely determined by their$c$-values.Comment: 20 pages. Submitted to Journal of Mathematical Analysis and Application