Normal Probability Distribution

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Jan Kruger - One of the best experts on this subject based on the ideXlab platform.

  • condition for a bivariate Normal Probability Distribution in phase space to be a quantum state
    Physical Review A, 1992
    Co-Authors: Jan Kruger
    Abstract:

    In order that a bivariate Normal Probability Distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner Distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$${\mathit{p}}^{2}$\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.

  • condition for a bivariate Normal Probability Distribution in phase space to be a quantum state
    Physical Review A, 1992
    Co-Authors: Jan Kruger
    Abstract:

    In order that a bivariate Normal Probability Distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner Distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$${\mathit{p}}^{2}$\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.

Saeid Saboori - One of the best experts on this subject based on the ideXlab platform.

  • assessing wind uncertainty impact on short term operation scheduling of coordinated energy storage systems and thermal units
    Renewable Energy, 2016
    Co-Authors: Reza Hemmati, Hedayat Saboori, Saeid Saboori
    Abstract:

    Renewable resources, especially wind power, are widely integrated into the power systems nowadays. Managing uncertainty of the large scale wind power is often known as one of the most challenging issues in the power system operation scheduling. Additionally, energy storage systems (ESSs) have been widely investigated in the power systems owing to their valuable applications, especially renewable energy smoothing and time shift. In this paper, a stochastic unit commitment (UC) model is proposed to assess the impact of the wind uncertainty impact on ESSs and thermal units schedule in UC problem. Wind uncertainty is modeled based on the two measures. First, the wind penetration level is changed with respect to the basic level. Second, the wind forecasting error is modeled through a Normal Probability Distribution function with different variances. The ESSs are modeled based on several technical characteristics and optimally scheduled considering different levels of the wind penetration and forecasting accuracies. The proposed formulation is a stochastic mixed integer linear programming (SMILP) and solved using GAMS software. Simulation results demonstrate that the wind uncertainty have a considerable impact on operation cost and ESSs schedule while proposed optimum storage scheduling through the stochastic programming will reduce the daily operational cost considerably.

Reza Hemmati - One of the best experts on this subject based on the ideXlab platform.

  • assessing wind uncertainty impact on short term operation scheduling of coordinated energy storage systems and thermal units
    Renewable Energy, 2016
    Co-Authors: Reza Hemmati, Hedayat Saboori, Saeid Saboori
    Abstract:

    Renewable resources, especially wind power, are widely integrated into the power systems nowadays. Managing uncertainty of the large scale wind power is often known as one of the most challenging issues in the power system operation scheduling. Additionally, energy storage systems (ESSs) have been widely investigated in the power systems owing to their valuable applications, especially renewable energy smoothing and time shift. In this paper, a stochastic unit commitment (UC) model is proposed to assess the impact of the wind uncertainty impact on ESSs and thermal units schedule in UC problem. Wind uncertainty is modeled based on the two measures. First, the wind penetration level is changed with respect to the basic level. Second, the wind forecasting error is modeled through a Normal Probability Distribution function with different variances. The ESSs are modeled based on several technical characteristics and optimally scheduled considering different levels of the wind penetration and forecasting accuracies. The proposed formulation is a stochastic mixed integer linear programming (SMILP) and solved using GAMS software. Simulation results demonstrate that the wind uncertainty have a considerable impact on operation cost and ESSs schedule while proposed optimum storage scheduling through the stochastic programming will reduce the daily operational cost considerably.

M Roger - One of the best experts on this subject based on the ideXlab platform.

  • analytical models of the wall pressure spectrum under a turbulent boundary layer with adverse pressure gradient
    Journal of Fluid Mechanics, 2019
    Co-Authors: G Grasso, Prateek Jaiswal, Hao Wu, Stephane Moreau, M Roger
    Abstract:

    This paper presents a comprehensive analytical approach to the modelling of wall-pressure fluctuations under a turbulent boundary layer, unifying and expanding the analytical models that have been proposed over many decades. The Poisson equation governing pressure fluctuations is Fourier transformed in the wavenumber domain to obtain a modified Helmholtz equation, which is solved with a Green’s function technique. The source term of the differential equations is composed of turbulence–mean shear and turbulence–turbulence interaction terms, which are modelled separately within the hypothesis of a joint Normal Probability Distribution of the turbulent field. The functional expression of the turbulence statistics is shown to be the most critical point for a correct representation of the wall-pressure spectrum. The effect of various assumptions on the shape of the longitudinal correlation function of turbulence is assessed in the first place with purely analytical considerations using an idealised flow model. Then, the effect of the hypothesis on the spectral Distribution of boundary-layer turbulence on the resulting wall-pressure spectrum is compared with the results of direct numerical simulation computations and pressure measurements on a controlled-diffusion aerofoil. The boundary layer developing over the suction side of this aerofoil in test conditions is characterised by an adverse pressure gradient. The final part of the paper discusses the numerical aspect of wall-pressure spectrum computation. A Monte Carlo technique is used for a fast evaluation of the multi-dimensional integral formulation developed in the theoretical part.

Takeshi Matsuura - One of the best experts on this subject based on the ideXlab platform.

  • morphological study of fluorinated asymmetric polyetherimide ultrafiltration membranes by surface modifying macromolecules
    Journal of Membrane Science, 2003
    Co-Authors: M Khayet, C Feng, Takeshi Matsuura
    Abstract:

    Phase inversion polyetherimide (PEI) flat sheet membranes were surface modified using fluorinated surface modifying macromolecules (SMMs) additives. Two SMM formulations were used. Each SMM was blended into polyetherimide casting solutions containing the solvent N,N-dimethyleacetamide and the non-solvent hydroxybutyric acid γ-lactone (GBL). The effects of the SMM and the PEI base polymer concentrations on the morphological properties of the prepared membranes have been investigated. Contact angle measurements indicate that PEI membrane surface becomes more hydrophobic after adding the SMMs to the PEI casting solutions, while X-ray photoelectron spectroscopy analysis shows enrichment of fluorine on the PEI membrane surfaces. The SMM modified and unmodified PEI membranes were also characterized by means of atomic force microscopy (AFM), gas permeation tests and ultrafiltration experiments using aqueous solutions of polyethylene glycol (PEG) and polyethylene oxide (PEO) of various molecular weights. Pore sizes and nodule sizes obtained from AFM images were remarkably fitted to the log-Normal Probability Distribution curves. Mean pore sizes, pore size Distributions, nodule sizes, nodule size Distributions and roughness parameters of the membranes were determined. It was found that SMM actively migrated to the air surface and changed the surface properties of the PEI membranes. The mean pore size and the molecular weight cut-off (MWCO) of the SMM modified PEI membranes were lower than those corresponding to the unmodified membranes, while the nodule sizes were larger. The surface roughness parameters were reduced when SMMs were added.

  • membrane characterization by solute transport and atomic force microscopy
    Journal of Membrane Science, 1998
    Co-Authors: S Singh, K C Khulbe, Takeshi Matsuura, P Ramamurthy
    Abstract:

    Various ultrafiltration and nanofiltration membranes were characterized by solute transport and also by atomic force microscope (AFM). The molecular weight cut-off (MWCO) of the membranes studied were found to be between 3500 and 98,000 Daltons. The mean pore size (μp) and the geometric standard deviation (σp) around mean ranged from 0.7 to 11.12 nm and 1.68 to 3.31, respectively, when calculated from the solute transport data. Mean pore sizes measured by AFM were about 3.5 times larger than calculated from the solute transport. Pore sizes measured by AFM were remarkably fitted to the log-Normal Probability Distribution curve. Pore sizes of the membranes with low MWCO (20,000 Daltons and lower) could not be measured by AFM because of indistinct pores. In most cases, the pore density ranged from 38 to 1291 pores/μm2. In general, the pore density was higher for the membrane having lower MWCO. Surface porosity was around 0.5–1.0% as measured from the solute transport and was 9.5–12.9% as obtained from AFM images. When membranes were coated with a thin layer of sulfonated polyphenylene oxide, mean pore sizes were reduced for all the membranes. Surface roughness was also reduced on coating.