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Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Standard : CrossProduct
U\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}V
[{U}_{2}{V}_{3}-{U}_{3}{V}_{2},-{U}_{1}{V}_{3}+{U}_{3}{V}_{1},{U}_{1}{V}_{2}-{U}_{2}{V}_{1}]
This function is part of the LinearAlgebra package, and so it can be used in the form CrossProduct(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[CrossProduct](..).
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
\mathrm{V1}≔〈1,2,3〉
\textcolor[rgb]{0,0,1}{\mathrm{V1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{V2}≔〈2,3,4〉
\textcolor[rgb]{0,0,1}{\mathrm{V2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\end{array}]
\mathrm{CrossProduct}\left(\mathrm{V1},\mathrm{V2}\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{-1}\end{array}]
\mathrm{V1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{V2}
[\begin{array}{c}\textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{-1}\end{array}]
\mathrm{CrossProduct}\left(\mathrm{V1},\mathrm{V2},\mathrm{datatype}=\mathrm{float}\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{-1.}\\ \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{-1.}\end{array}]
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Toward An Ultrasonic Sensor for Pressure Vessels | J. Pressure Vessel Technol. | ASME Digital Collection
J. S. Sandman,
J. S. Sandman
J. Pressure Vessel Technol. May 2008, 130(2): 021501 (5 pages)
Sandman, J. S., and Tittmann, B. R. (March 17, 2008). "Toward An Ultrasonic Sensor for Pressure Vessels." ASME. J. Pressure Vessel Technol. May 2008; 130(2): 021501. https://doi.org/10.1115/1.2892030
The focus of this paper is an ultrasonic position indication system that is capable of determining one-dimensional target location in a high-temperature steel container with gaseous medium. The combination of the very high acoustical impedance of steel
(45.4MRayl)
and the very low impedance of a gas, for example, ambient air
(0.0004MRayl)
, causes significant reflections at the interfaces. The strategy of this investigation was to develop an ultrasonic transducer capable of replacing a small portion of pressure vessel wall. In building such a transducer, acoustic matching layers for the steel-gas interface, a mechanically and acoustically competent housing, an efficient piezoelectric element, and appropriate backing materials are developed and tested. The results include a successful housing design, high- temperature acoustic matching layers, and subsequent successful wave forms with good signal-to-noise ratio. Target location through
9.6in.
(24.5cm)
of ambient air was possible, with a steel pressure boundary
0.456in.
(1.160cm)
thick, and the use of one matching layer. Our transducer was tested repeatedly to
340°C
without apparent degradation. In addition to the experimental results, this investigation includes numerical simulations. Sample wave forms were predicted one dimensionally with the coupled acoustic piezoelectric analysis, a finite element program that predicts wave forms based on Navier’s equation for elastic wave propagation.
elastic waves, finite element analysis, piezoelectric materials, pressure vessels, ultrasonic transducers, air-coupled ultrasound, matching layer, transducer, elevated temperature, pressure vessel, gas medium, position indicator
Acoustics, High temperature, Pressure vessels, Temperature, Transducers, Ultrasonic transducers, Steel, Waves, Pressure, Finite element analysis, Design, Simulation, Elastic waves
Acoustic Waves: Devices, Imaging, & Analog Signal Processing
Acoustic Matching and Backing Layers for Medical Ultrasonic Transducers
Development of an Air-Coupled Ultrasonic Sensor for High Pressure, and Temperature Applications
Electromechanical Transducers and Wave Filters
Brekhovskikh
Simulation of Piezoelectric Devices by Two- and Three-Dimensional Finite Elements
Electromechanical Modeling Using Explicit Time-Domain Finite Elements
Optimization of the Transmitting Characteristics of a Tonpilz-Type Transducer by Proper Choice of Impedance Matching Layers
Piezoelectric Composite Materials for Ultrasonic Transducer Applications
,” Ph.D. thesis, The Pennsylvania State University.
Ultrasonic Absorption in Gases
Improvement of Ultrasonic Transducers by Using Multilayer Front Face
The Effects of Backing and Matching on the Performance of Piezoelectric Ceramic Transducers
Ultrasonic Field Modeling in Multilayered Fluid Structures Using the Distributed Point Source Method Technique
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Transfer function estimate - MATLAB tfestimate
Transfer Function Between Two Sequences
Transfer Function of MIMO System
txy = tfestimate(x,y) finds a transfer function estimate, txy, given an input signal, x, and an output signal, y.
If one of the signals is a matrix and the other is a vector, then the length of the vector must equal the number of rows in the matrix. The function expands the vector and returns a matrix of column-by-column transfer function estimates.
If x and y are matrices with the same number of rows but different numbers of columns, then txy is a multi-input/multi-output (MIMO) transfer function that combines all input and output signals. txy is a three-dimensional array. If x has m columns and y has n columns, then txy has n columns and m pages. See Transfer Function for more information.
If x and y are matrices of equal size, then tfestimate operates column-wise: txy(:,n) = tfestimate(x(:,n),y(:,n)). To obtain a MIMO estimate, append 'mimo' to the argument list.
txy = tfestimate(x,y,window) uses window to divide x and y into segments and perform windowing.
txy = tfestimate(x,y,window,noverlap) uses noverlap samples of overlap between adjoining segments.
txy = tfestimate(x,y,window,noverlap,nfft) uses nfft sampling points to calculate the discrete Fourier transform.
txy = tfestimate(___,'mimo') computes a MIMO transfer function for matrix inputs. This syntax can include any combination of input arguments from previous syntaxes.
[txy,w] = tfestimate(___) returns a vector of normalized frequencies, w, at which the transfer function is estimated.
[txy,f] = tfestimate(___,fs) returns a vector of frequencies, f, expressed in terms of the sample rate, fs, at which the transfer function is estimated. fs must be the sixth numeric input to tfestimate. To input a sample rate and still use the default values of the preceding optional arguments, specify these arguments as empty [].
[txy,w] = tfestimate(x,y,window,noverlap,w) returns the transfer function estimate at the normalized frequencies specified in w.
[txy,f] = tfestimate(x,y,window,noverlap,f,fs) returns the transfer function estimate at the frequencies specified in f.
[___] = tfestimate(x,y,___,freqrange) returns the transfer function estimate over the frequency range specified by freqrange. Valid options for freqrange are 'onesided', 'twosided', and 'centered'.
[___] = tfestimate(___,'Estimator',est) estimates transfer functions using the estimator est. Valid options for est are 'H1' and 'H2'.
tfestimate(___) with no output arguments plots the transfer function estimate in the current figure window.
Compute and plot the transfer function estimate between two sequences, x and y. The sequence x consists of white Gaussian noise. y results from filtering x with a 30th-order lowpass filter with normalized cutoff frequency
0.2\pi
rad/sample. Use a rectangular window to design the filter. Specify a sample rate of 500 Hz and a Hamming window of length 1024 for the transfer function estimate.
Use fvtool to verify that the transfer function approximates the frequency response of the filter.
Obtain the same result by returning the transfer function estimate in a variable and plotting its absolute value in decibels.
Estimate the transfer function for a simple single-input/single-output system and compare it to the definition.
\mathit{m}
\mathit{a}
{\mathit{F}}_{\mathit{s}}=1
Hz. A damper impedes the motion of the mass by exerting on it a force proportional to speed, with damping constant
\mathit{b}=0.01
Generate 2000 time samples. Define the sampling interval
\Delta \mathit{t}=1/{\mathit{F}}_{\mathit{s}}
\begin{array}{c}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right),\end{array}
\mathit{x}={\left[\begin{array}{cc}\mathit{r}& \mathit{v}\end{array}\right]}^{\mathit{T}}
\mathit{r}
\mathit{v}
are respectively the position and velocity of the mass,
\mathit{u}
is the driving force, and
\mathit{y}=\mathit{a}
is the measured output. The state-space matrices are
A=\mathrm{exp}\left({A}_{c}\Delta t\right),\phantom{\rule{1em}{0ex}}B={A}_{c}^{-1}\left(A-I\right){B}_{c},\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cc}-1& -b\end{array}\right],\phantom{\rule{1em}{0ex}}D=1,
\mathit{I}
2×2
identity, and the continuous-time state-space matrices are
{A}_{c}=\left[\begin{array}{cc}0& 1\\ -1& -b\end{array}\right],\phantom{\rule{1em}{0ex}}{B}_{c}=\left[\begin{array}{c}0\\ 1\end{array}\right].
The mass is driven by random input for half of the measurement interval. Use the state-space model to compute the time evolution of the system starting from an all-zero initial state. Plot the acceleration of the mass as a function of time.
Estimate the transfer function of the system as a function of frequency. Use 2048 DFT points and specify a Kaiser window with a shape factor of 15. Use the default value of overlap between adjoining segments.
The frequency-response function of a discrete-time system can be expressed as the Z-transform of the time-domain transfer function of the system, evaluated at the unit circle. Verify that the estimate computed by tfestimate coincides with this definition.
Plot the estimate using the built-in functionality of tfestimate.
Estimate the transfer function for a simple multi-input/multi-output system.
An ideal one-dimensional oscillating system consists of two masses,
{\mathit{m}}_{1}
{\mathit{m}}_{2}
, confined between two walls. The units are such that
{\mathit{m}}_{1}=1
{\mathit{m}}_{2}=\mu
. Each mass is attached to the nearest wall by a spring with an elastic constant
\mathit{k}
. An identical spring connects the two masses. Three dampers impede the motion of the masses by exerting on them forces proportional to speed, with damping constant
\mathit{b}
. Sensors sample
{\mathit{a}}_{1}
{\mathit{a}}_{2}
, the accelerations of the masses, at
{\mathit{F}}_{\mathit{s}}=50
Generate 30000 time samples, equivalent to 600 seconds. Define the sampling interval
\Delta \mathit{t}=1/{\mathit{F}}_{\mathit{s}}
\begin{array}{c}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right),\end{array}
x={\left[\begin{array}{cccc}{r}_{1}& {v}_{1}& {r}_{2}& {v}_{2}\end{array}\right]}^{T}
{r}_{i}
{v}_{i}
are respectively the location and the velocity of the
i
u={\left[\begin{array}{cc}{u}_{1}& {u}_{2}\end{array}\right]}^{T}
is the vector of input driving forces, and
y={\left[\begin{array}{cc}{a}_{1}& {a}_{2}\end{array}\right]}^{T}
is the output vector. The state-space matrices are
A=\mathrm{exp}\left({A}_{c}\Delta t\right),\phantom{\rule{1em}{0ex}}B={A}_{c}^{-1}\left(A-I\right){B}_{c},\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cccc}-2k& -2b& k& b\\ k/\mu & b/\mu & -2k/\mu & -2b/\mu \end{array}\right],\phantom{\rule{1em}{0ex}}D=\left[\begin{array}{cc}1& 0\\ 0& 1/\mu \end{array}\right],
\mathit{I}
4×4
{A}_{c}=\left[\begin{array}{cccc}0& 1& 0& 0\\ -2k& -2b& k& b\\ 0& 0& 0& 1\\ k/\mu & b/\mu & -2k/\mu & -2b/\mu \end{array}\right],\phantom{\rule{1em}{0ex}}{B}_{c}=\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 0\\ 0& 1/\mu \end{array}\right].
\mathit{k}=400
\mathit{b}=0
\mu =1/10
Use the input and output data to estimate the transfer function of the system as a function of frequency. Specify the 'mimo' option to produce all four transfer functions. Use a 5000-sample Hann window to divide the signals into segments. Specify 2500 samples of overlap between adjoining segments and
{2}^{14}
DFT points. Plot the estimates.
Plot the theoretical transfer functions and their corresponding estimates.
The transfer functions have maxima at the expected values,
{\omega }_{1,2}/2\pi
\omega
are the eigenvalues of the modal matrix.
Add damping to the system by setting
\mathit{b}=0.1
. Compute the time evolution of the damped system with the same driving forces. Compute the
{\mathit{H}}_{2}
estimate of the MIMO transfer function using the same window and overlap. Plot the estimates using the tfestimate functionality.
Compare the estimates to the theoretical predictions.
Output signal, specified as a vector or matrix.
Window, specified as an integer or as a row or column vector. Use window to divide the signal into segments.
If window is an integer, then tfestimate divides x and y into segments of length window and windows each segment with a Hamming window of that length.
If window is a vector, then tfestimate divides x and y into segments of the same length as the vector and windows each segment using window.
If you specify window as empty, then tfestimate uses a Hamming window such that x and y are divided into eight segments with noverlap overlapping samples.
If you specify noverlap as empty, then tfestimate uses a number that produces 50% overlap between segments.
Number of DFT points, specified as a positive integer. If you specify nfft as empty, then tfestimate sets this argument to max(256,2p), where p = ⌈log2 N⌉ for input signals of length N and the ⌈ ⌉ symbols denote the ceiling function.
freqrange — Frequency range for transfer function estimate
Frequency range for the transfer function estimate, specified as a one of 'onesided', 'twosided', or 'centered'. The default is 'onesided' for real-valued signals and 'twosided' for complex-valued signals.
'onesided' — Returns the one-sided estimate of the transfer function between two real-valued input signals, x and y. If nfft is even, txy has nfft/2 + 1 rows and is computed over the interval [0,π] rad/sample. If nfft is odd, txy has (nfft + 1)/2 rows and the interval is [0,π) rad/sample. If you specify fs, the corresponding intervals are [0,fs/2] cycles/unit time for even nfft and [0,fs/2) cycles/unit time for odd nfft.
'twosided' — Returns the two-sided estimate of the transfer function between two real-valued or complex-valued input signals, x and y. In this case, txy has nfft rows and is computed over the interval [0,2π) rad/sample. If you specify fs, the interval is [0,fs) cycles/unit time.
'centered' — Returns the centered two-sided estimate of the transfer function between two real-valued or complex-valued input signals, x and y. In this case, txy has nfft rows and is computed over the interval (–π,π] rad/sample for even nfft and (–π,π) rad/sample for odd nfft. If you specify fs, the corresponding intervals are (–fs/2, fs/2] cycles/unit time for even nfft and (–fs/2, fs/2) cycles/unit time for odd nfft.
est — Transfer function estimator
'H1' (default) | 'H2'
Transfer function estimator, specified as 'H1' or 'H2'.
Use 'H1' when the noise is uncorrelated with the input signals.
Use 'H2' when the noise is uncorrelated with the output signals. In this case, the number of input signals must equal the number of output signals.
See Transfer Function for more information.
txy — Transfer function estimate
Transfer function estimate, returned as a vector, matrix, or three-dimensional array.
Cyclical frequencies, returned as a real-valued column vector.
The relationship between the input x and output y is modeled by the linear, time-invariant transfer function txy. In the frequency domain, Y(f) = H(f)X(f).
For a single-input/single-output system, the H1 estimate of the transfer function is given by
{H}_{1}\left(f\right)=\frac{{P}_{yx}\left(f\right)}{{P}_{xx}\left(f\right)},
where Pyx is the cross power spectral density of x and y, and Pxx is the power spectral density of x. This estimate assumes that the noise is not correlated with the system input.
For multi-input/multi-output (MIMO) systems, the H1 estimator becomes
{H}_{1}\left(f\right)={P}_{YX}\left(f\right){P}_{XX}^{-1}\left(f\right)=\left[\begin{array}{cccc}{P}_{{y}_{1}{x}_{1}}\left(f\right)& {P}_{{y}_{1}{x}_{2}}\left(f\right)& \cdots & {P}_{{y}_{1}{x}_{m}}\left(f\right)\\ {P}_{{y}_{2}{x}_{1}}\left(f\right)& {P}_{{y}_{2}{x}_{2}}\left(f\right)& \cdots & {P}_{{y}_{2}{x}_{m}}\left(f\right)\\ ⋮& ⋮& \ddots & ⋮\\ {P}_{{y}_{n}{x}_{1}}\left(f\right)& {P}_{{y}_{n}{x}_{2}}\left(f\right)& \cdots & {P}_{{y}_{n}{x}_{m}}\left(f\right)\end{array}\right]\text{\hspace{0.17em}}{\left[\begin{array}{cccc}{P}_{{x}_{1}{x}_{1}}\left(f\right)& {P}_{{x}_{1}{x}_{2}}\left(f\right)& \cdots & {P}_{{x}_{1}{x}_{m}}\left(f\right)\\ {P}_{{x}_{2}{x}_{1}}\left(f\right)& {P}_{{x}_{2}{x}_{2}}\left(f\right)& \cdots & {P}_{{x}_{2}{x}_{m}}\left(f\right)\\ ⋮& ⋮& \ddots & ⋮\\ {P}_{{x}_{m}{x}_{1}}\left(f\right)& {P}_{{x}_{m}{x}_{2}}\left(f\right)& \cdots & {P}_{{x}_{m}{x}_{m}}\left(f\right)\end{array}\right]}^{-1}
for m inputs and n outputs, where:
Pyixk is the cross power spectral density of the kth input and the ith output.
Pxixk is the cross power spectral density of the kth and ith inputs.
For two inputs and two outputs, the estimator is the matrix
{H}_{1}\left(f\right)=\frac{\left[\begin{array}{cc}{P}_{{y}_{1}{x}_{1}}\left(f\right){P}_{{x}_{2}{x}_{2}}\left(f\right)-{P}_{{y}_{1}{x}_{2}}\left(f\right){P}_{{x}_{2}{x}_{1}}\left(f\right)& {P}_{{y}_{1}{x}_{2}}\left(f\right){P}_{{x}_{1}{x}_{1}}\left(f\right)-{P}_{{y}_{1}{x}_{1}}\left(f\right){P}_{{x}_{1}{x}_{2}}\left(f\right)\\ {P}_{{y}_{2}{x}_{1}}\left(f\right){P}_{{x}_{2}{x}_{2}}\left(f\right)-{P}_{{y}_{2}{x}_{2}}\left(f\right){P}_{{x}_{2}{x}_{1}}\left(f\right)& {P}_{{y}_{2}{x}_{2}}\left(f\right){P}_{{x}_{1}{x}_{1}}\left(f\right)-{P}_{{y}_{2}{x}_{1}}\left(f\right){P}_{{x}_{1}{x}_{2}}\left(f\right)\end{array}\right]}{{P}_{{x}_{1}{x}_{1}}\left(f\right){P}_{{x}_{2}{x}_{2}}\left(f\right)-{P}_{{x}_{1}{x}_{2}}\left(f\right){P}_{{x}_{2}{x}_{1}}\left(f\right)}.
{H}_{2}\left(f\right)=\frac{{P}_{yy}\left(f\right)}{{P}_{xy}\left(f\right)},
where Pyy is the power spectral density of y and Pxy = P*yx is the complex conjugate of the cross power spectral density of x and y. This estimate assumes that the noise is not correlated with the system output.
For MIMO systems, the H2 estimator is well-defined only for equal numbers of inputs and outputs: n = m. The estimator becomes
{H}_{2}\left(f\right)={P}_{YY}\left(f\right){P}_{XY}^{-1}\left(f\right)=\left[\begin{array}{cccc}{P}_{{y}_{1}{y}_{1}}\left(f\right)& {P}_{{y}_{1}{y}_{2}}\left(f\right)& \cdots & {P}_{{y}_{1}{y}_{n}}\left(f\right)\\ {P}_{{y}_{2}{y}_{1}}\left(f\right)& {P}_{{y}_{2}{y}_{2}}\left(f\right)& \cdots & {P}_{{y}_{2}{y}_{n}}\left(f\right)\\ ⋮& ⋮& \ddots & ⋮\\ {P}_{{y}_{n}{y}_{1}}\left(f\right)& {P}_{{y}_{n}{y}_{2}}\left(f\right)& \cdots & {P}_{{y}_{n}{y}_{n}}\left(f\right)\end{array}\right]\text{\hspace{0.17em}}{\left[\begin{array}{cccc}{P}_{{x}_{1}{y}_{1}}\left(f\right)& {P}_{{x}_{1}{y}_{2}}\left(f\right)& \cdots & {P}_{{x}_{1}{y}_{n}}\left(f\right)\\ {P}_{{x}_{2}{y}_{1}}\left(f\right)& {P}_{{x}_{2}{y}_{2}}\left(f\right)& \cdots & {P}_{{x}_{2}{y}_{n}}\left(f\right)\\ ⋮& ⋮& \ddots & ⋮\\ {P}_{{x}_{n}{y}_{1}}\left(f\right)& {P}_{{x}_{n}{y}_{2}}\left(f\right)& \cdots & {P}_{{x}_{n}{y}_{n}}\left(f\right)\end{array}\right]}^{-1},
Pyiyk is the cross power spectral density of the kth and ith outputs.
Pxiyk is the complex conjugate of the cross power spectral density of the ith input and the kth output.
tfestimate uses Welch's averaged periodogram method. See pwelch for details.
Arguments specified using name value pairs must be compile time constants.
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Design and Verification of the Risø-B1 Airfoil Family for Wind Turbines | J. Sol. Energy Eng. | ASME Digital Collection
Peter Fuglsang,
Wind Energy Department, Risø National Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark
e-mail: peter.fuglsang@risoe.dk
Christian Bak,
Mac Gaunaa,
Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received by the ASME Solar Division Jan. 2004; final revision Mar. 2004. Associate Editor: P. Chaviaropoulous.
Fuglsang, P., Bak , C., Gaunaa , M., and Antoniou, I. (November 18, 2004). "Design and Verification of the Risø-B1 Airfoil Family for Wind Turbines ." ASME. J. Sol. Energy Eng. November 2004; 126(4): 1002–1010. https://doi.org/10.1115/1.1766024
This paper presents the design and experimental verification of the Risø-B1 airfoil family for MW-size wind turbines with variable speed and pitch control. Seven airfoils were designed with thickness-to-chord ratios between 15% and 53% to cover the entire span of a wind turbine blade. The airfoils were designed to have high maximum lift and high design lift to allow a slender flexible blade while maintaining high aerodynamic efficiency. The design was carried out with a Risø in-house multi disciplinary optimization tool. Wind tunnel testing was done for Risø-B1-18 and Risø-B1-24 in the VELUX wind tunnel, Denmark, at a Reynolds number of
1.6×106.
For both airfoils the predicted target characteristics were met. Results for Risø-B1-18 showed a maximum lift coefficient of 1.64. A standard case of zigzag tape leading edge roughness caused a drop in maximum lift of only 3.7%. Cases of more severe roughness caused reductions in maximum lift between 12% and 27%. Results for the Risø-B1-24 airfoil showed a maximum lift coefficient of 1.62. The standard case leading edge roughness caused a drop in maximum lift of 7.4%. Vortex generators and Gurney flaps in combination could increase maximum lift up to 2.2 (32%).
wind turbines, rotors, wind tunnels, aerodynamics
Airfoils, Design, Flow (Dynamics), Wind turbines, Wind tunnels, Surface roughness, Chords (Trusses)
Tangler, J. L., and Somers, D. M., 1995, “NREL Airfoil Families for HAWT’s,” Proc. WINDPOWER’95, Washington D.C., pp. 117–123.
Timmer, W. A., van Rooij, A., R.P.J.O.M., 2003, “Summary of the Delft University Wind Turbine Dedicated Airfoils,” AIAA-2003-0352.
Bjo¨rk, A., 1990, “Coordinates and Calculations for the FFA-W1-xxx, FFA-W2-xxx and FFA-W3-.xxx Series of Airfoils for Horizontal Axis Wind Turbines,” FFA TN 1990-15, Stockholm, Sweden.
Hill, D. M., and Garrad, A. D., 1988, “Design of Aerofoils for Wind Turbine Use,” Proc. IEA Symposium on Aerodynamics of Wind Turbines, Lyngby, Denmark.
Design of Optimized Profiles for Stall Regulated HAWTs Part 1: Design Concepts and Method Formulation
Hoadley, D., Madsen, H. A., and Bouras, B., 1993, “Aerofoil Section Design and Assessment,” Final Rep. Contract JOUR 0079, The European Comisio´n DGXII.
Fuglsang, P., and Bak, C., 2004, “Development of the Risø Wind Turbine Airfoils,” J. Wind Energy, 7, pp. 145–162.
Fuglsang, P., and Dahl, K. S., 1997, “Multipoint Optimization of Thick High Lift Airfoil Wind Turbines,” Proc. EWEC’97, Dublin, Ireland, pp. 468–471.
Hicks, R. M., Murman, E. M., and Vanderplaats, G. N., 1974, “An Assessment of Airfoil Design by Numerical Optimization,” Tech. rep., NASA TM X-3092.
Fuglsang, P., Bak, C., Gaunaa, M., and Antoniou, I., 2003, “Wind tunnel Tests of Risø-B1-18 and Risø-B1-24,” Risø-R-1375(EN), Risø National Laboratory, Denmark, January.
Bak, C., Fuglsang, P., Gaunaa, M., and Antoniou, I., 2003, “Wind Tunnel Measurements on Two Risø-B1 Airfoils,” Proc. EWEC’2003, Madrid.
Fuglsang, P., Antoniou, I., Sørensen, N. N., and Madsen H., 1998, “Validation of a Wind Tunnel Testing Facility for Blade Surface Pressure Measurements,” Risø-R-981(EN), Risø National Laboratory, Denmark.
Drela, M., 1989, “XFOIL, An Analysis and Design system for Low Reynolds Number Airfoils,” Low Reynolds Number Aerodynamics, 54 In Springer-Verlag Lec. Notes in Eng.
Michelsen, J. A., 1992, “Basis3D—A Platform for Development of Multiblock PDE Solvers,” Technical Report AFM 92-05, Technical University of Denmark.
Michelsen, J. A., 1994, “Block Structured Multigrid Solution of 2D and 3D elliptic PDE’s.” Technical Report AFM 94-06, Technical University of Denmark.
Sørensen, N. N., 1995, “General Purpose Flow Solver Applied to Flow over Hills,” Risø-R-827(EN), Risø National Laboratory, Denmark.
Menter, F. R., 1993, “Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows.” AIAA-932906.
Bertagnolio, F., Sørensen, N. N., Johansen, J., and Fuglsang, P., 2001, “Wind Turbine Airfoil Catalogue,” Risø-R-1280(EN), Risø National Laboratory, Denmark, pp. 152.
Sørensen, N. N., Michelsen, J. A., and Schreck, S., Navier-Stokes Predictions of the NREL Phase VI Rotor in the NASA Ames 80ft×120ft Wind Tunnel. J. Wind Energy 2002; 5:151–169.
Tangler, J. L., and Somers, D. M., 1987, “Status of the Special Purpose Airfoil Families,” Proc. WINDPOWER’87, San Fransisco.
Bjo¨rk, A., 1989, “Airfoil Design for Variable rpm Horizontal Axis Wind Turbines,” Proc., EWEC’89, Glasgow, Scotland.
Fuglsang, P., 2002, “Aerodynamic Design Guidelines For Wind Turbine Rotors,” Proc. 4th GRACM Congress on Computational Mechanics GRACM 2002, Patra, CD-rom.
Brooks, T. F., and Marcolini, M. A., 1984, “Airfoil Trailing Edge Flow Measurements and Comparison with Theory Incorporating Open Wind Tunnel Corrections,” AIAA-84-2266 AIAA/NASA 9th Aeroacoustic Conference.
Bubble Induced Unsteadiness on Wind Turbine Airfoils
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Lemma 27.8.1. Let $S$ be a graded ring. Let $f \in S$ homogeneous of positive degree.
If $g\in S$ homogeneous of positive degree and $D_{+}(g) \subset D_{+}(f)$, then
$f$ is invertible in $S_ g$, and $f^{\deg (g)}/g^{\deg (f)}$ is invertible in $S_{(g)}$,
$g^ e = af$ for some $e \geq 1$ and $a \in S$ homogeneous,
there is a canonical $S$-algebra map $S_ f \to S_ g$,
there is a canonical $S_0$-algebra map $S_{(f)} \to S_{(g)}$ compatible with the map $S_ f \to S_ g$,
the map $S_{(f)} \to S_{(g)}$ induces an isomorphism
\[ (S_{(f)})_{g^{\deg (f)}/f^{\deg (g)}} \cong S_{(g)}, \]
these maps induce a commutative diagram of topological spaces
\[ \xymatrix{ D_{+}(g) \ar[d] & \{ \mathbf{Z}\text{-graded primes of }S_ g\} \ar[l] \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(S_{(g)}) \ar[d] \\ D_{+}(f) & \{ \mathbf{Z}\text{-graded primes of }S_ f\} \ar[l] \ar[r] & \mathop{\mathrm{Spec}}(S_{(f)}) } \]
where the horizontal maps are homeomorphisms and the vertical maps are open immersions,
there are compatible canonical $S_ f$ and $S_{(f)}$-module maps $M_ f \to M_ g$ and $M_{(f)} \to M_{(g)}$ for any graded $S$-module $M$, and
the map $M_{(f)} \to M_{(g)}$ induces an isomorphism
\[ (M_{(f)})_{g^{\deg (f)}/f^{\deg (g)}} \cong M_{(g)}. \]
Any open covering of $D_{+}(f)$ can be refined to a finite open covering of the form $D_{+}(f) = \bigcup _{i = 1}^ n D_{+}(g_ i)$.
Let $g_1, \ldots , g_ n \in S$ be homogeneous of positive degree. Then $D_{+}(f) \subset \bigcup D_{+}(g_ i)$ if and only if $g_1^{\deg (f)}/f^{\deg (g_1)}, \ldots , g_ n^{\deg (f)}/f^{\deg (g_ n)}$ generate the unit ideal in $S_{(f)}$.
Proof. Recall that $D_{+}(g) = \mathop{\mathrm{Spec}}(S_{(g)})$ with identification given by the ring maps $S \to S_ g \leftarrow S_{(g)}$, see Algebra, Lemma 10.57.3. Thus $f^{\deg (g)}/g^{\deg (f)}$ is an element of $S_{(g)}$ which is not contained in any prime ideal, and hence invertible, see Algebra, Lemma 10.17.2. We conclude that (a) holds. Write the inverse of $f$ in $S_ g$ as $a/g^ d$. We may replace $a$ by its homogeneous part of degree $d\deg (g) - \deg (f)$. This means $g^ d - af$ is annihilated by a power of $g$, whence $g^ e = af$ for some $a \in S$ homogeneous of degree $e\deg (g) - \deg (f)$. This proves (b). For (c), the map $S_ f \to S_ g$ exists by (a) from the universal property of localization, or we can define it by mapping $b/f^ n$ to $a^ nb/g^{ne}$. This clearly induces a map of the subrings $S_{(f)} \to S_{(g)}$ of degree zero elements as well. We can similarly define $M_ f \to M_ g$ and $M_{(f)} \to M_{(g)}$ by mapping $x/f^ n$ to $a^ nx/g^{ne}$. The statements writing $S_{(g)}$ resp. $M_{(g)}$ as principal localizations of $S_{(f)}$ resp. $M_{(f)}$ are clear from the formulas above. The maps in the commutative diagram of topological spaces correspond to the ring maps given above. The horizontal arrows are homeomorphisms by Algebra, Lemma 10.57.3. The vertical arrows are open immersions since the left one is the inclusion of an open subset.
The open $D_{+}(f)$ is quasi-compact because it is homeomorphic to $\mathop{\mathrm{Spec}}(S_{(f)})$, see Algebra, Lemma 10.17.10. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology.
The third statement follows directly from Algebra, Lemma 10.17.2. $\square$
In 1.8, replace "are a compatible" by "are compatible", and "
S_f
S_f
-".
In the second last paragraph of the proof, the reference should point to 00E8.
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Lemma 13.5.4 (05R5)—The Stacks project
Section 13.5: Localization of triangulated categories
Lemma 13.5.4. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor between a pre-triangulated category and an abelian category. Let
\[ S = \{ f \in \text{Arrows}(\mathcal{D}) \mid H^ i(f)\text{ is an isomorphism for all }i \in \mathbf{Z}\} \]
Then $S$ is a saturated (see Categories, Definition 4.27.20) multiplicative system compatible with the triangulated structure on $\mathcal{D}$.
Proof. We have to prove axioms MS1 – MS6, see Categories, Definitions 4.27.1 and 4.27.20 and Definition 13.5.1. MS1, MS4, and MS5 are direct from the definitions. MS6 follows from TR3 and the long exact cohomology sequence (13.3.5.1). By Lemma 13.5.2 we conclude that MS2 holds. To finish the proof we have to show that MS3 holds. To do this let $f, g : X \to Y$ be morphisms of $\mathcal{D}$, and let $t : Z \to X$ be an element of $S$ such that $f \circ t = g \circ t$. As $\mathcal{D}$ is additive this simply means that $a \circ t = 0$ with $a = f - g$. Choose a distinguished triangle $(Z, X, Q, t, g, h)$ using TR1 and TR2. Since $a \circ t = 0$ we see by Lemma 13.4.2 there exists a morphism $i : Q \to Y$ such that $i \circ g = a$. Finally, using TR1 again we can choose a triangle $(Q, Y, W, i, j, k)$. Here is a picture
\[ \xymatrix{ Z \ar[r]_ t & X \ar[r]_ g \ar[d]^1 & Q \ar[r] \ar[d]^ i & Z[1] \\ & X \ar[r]_ a & Y \ar[d]^ j \\ & & W } \]
OK, and now we apply the functors $H^ i$ to this diagram. Since $t \in S$ we see that $H^ i(Q) = 0$ by the long exact cohomology sequence (13.3.5.1). Hence $H^ i(j)$ is an isomorphism for all $i$ by the same argument, i.e., $j \in S$. Finally, $j \circ a = j \circ i \circ g = 0$ as $j \circ i = 0$. Thus $j \circ f = j \circ g$ and we see that LMS3 holds. The proof of RMS3 is dual. $\square$
Well, this is mostly repetition of the previous lemma. Can it be deduced by setting
\mathcal D' = Ch(A)
F = H
in the previous lemma?
@#376: First of all, I don't think it can be deduced in that way. Try it!
General remark: Often it does make sense to repeat very similar arguments, because then you see what is the key thing you have to change. We can do this also because we are not writing a paper, and hence there is no restriction on the number of pages. But of course, if you get many similar arguments, then it makes sense to set up a kind of "machine" that does all of them at once.
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Position_angle Knowpia
In astronomy, position angle (usually abbreviated PA) is the convention for measuring angles on the sky. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images, this is a counterclockwise measure relative to the axis into the direction of positive declination.
An illustration of how position angle is estimated through a telescope eyepiece; the primary star is at center.
In the case of observed visual binary stars, it is defined as the angular offset of the secondary star from the primary relative to the north celestial pole.
As the example illustrates, if one were observing a hypothetical binary star with a PA of 135°, that means an imaginary line in the eyepiece drawn from the north celestial pole to the primary (P) would be offset from the secondary (S) such that the NCP-P-S angle would be 135°.
When graphing visual binaries, the NCP is, as in the illustration, normally drawn from the center point (origin) that is the Primary downward–that is, with north at bottom–and PA is measured counterclockwise. Also, the direction of the proper motion can, for example, be given by its position angle.
The definition of position angle is also applied to extended objects like galaxies, where it refers to the angle made by the major axis of the object with the NCP line.
NauticsEdit
The concept of the position angle is inherited from nautical navigation on the oceans, where the optimum compass course is the course from a known position s to a target position t with minimum effort. Setting aside the influence of winds and ocean currents, the optimum course is the course of smallest distance between the two positions on the ocean surface. Computing the compass course is known as the inverse problem of geodesics.
This article considers only the abstraction of minimizing the distance between s and t traveling on the surface of a sphere with some radius R: In which direction angle p relative to North should the ship steer to reach the target position?
Global geocentric coordinate systemEdit
The position angle of the point t at the point s is the angle at which the green and the dashed great circles intersect at s. The unit directions uE, uN and the rotation axis ω are marked by arrows.
Detailed evaluation of the optimum direction is possible if the sea surface is approximated by a sphere surface. The standard computation places the ship at a geodetic latitude φs and geodetic longitude λs, where φ is considered positive if north of the equator, and where λ is considered positive if east of Greenwich. In the global coordinate system centered at the center of the sphere, the Cartesian components are
{\displaystyle {\mathbf {s} }=R\left({\begin{array}{c}\cos \varphi _{s}\cos \lambda _{s}\\\cos \varphi _{s}\sin \lambda _{s}\\\sin \varphi _{s}\end{array}}\right)}
and the target position is
{\displaystyle {\mathbf {t} }=R\left({\begin{array}{c}\cos \varphi _{t}\cos \lambda _{t}\\\cos \varphi _{t}\sin \lambda _{t}\\\sin \varphi _{t}\end{array}}\right).}
The North Pole is at
{\displaystyle {\mathbf {N} }=R\left({\begin{array}{c}0\\0\\1\end{array}}\right).}
The minimum distance d is the distance along a great circle that runs through s and t. It is calculated in a plane that contains the sphere center and the great circle,
{\displaystyle d_{s,t}=R\theta _{s,t}}
where θ is the angular distance of two points viewed from the center of the sphere, measured in radians. The cosine of the angle is calculated by the dot product of the two vectors
{\displaystyle \mathbf {s} \cdot \mathbf {t} =R^{2}\cos \theta _{s,t}=R^{2}(\sin \varphi _{s}\sin \varphi _{t}+\cos \varphi _{s}\cos \varphi _{t}\cos(\lambda _{t}-\lambda _{s}))}
If the ship steers straight to the North Pole, the travel distance is
{\displaystyle d_{s,N}=R\theta _{s,N}=R(\pi /2-\varphi _{s})}
If a ship starts at t and swims straight to the North Pole, the travel distance is
{\displaystyle d_{t,N}=R\theta _{t,n}=R(\pi /2-\varphi _{t})}
Brief DerivationEdit
The cosine formula of spherical trigonometry[1] yields for the angle p between the great circles through s that point to the North on one hand and to t on the other hand
{\displaystyle \cos \theta _{t,N}=\cos \theta _{s,t}\cos \theta _{s,N}+\sin \theta _{s,t}\sin \theta _{s,N}\cos p.}
{\displaystyle \sin \varphi _{t}=\cos \theta _{s,t}\sin \varphi _{s}+\sin \theta _{s,t}\cos \varphi _{s}\cos p.}
The sine formula yields
{\displaystyle {\frac {\sin p}{\sin \theta _{t,N}}}={\frac {\sin(\lambda _{t}-\lambda _{s})}{\sin \theta _{s,t}}}.}
Solving this for sin θs,t and insertion in the previous formula gives an expression for the tangent of the position angle,
{\displaystyle \sin \varphi _{t}=\cos \theta _{s,t}\sin \varphi _{s}+{\frac {\sin(\lambda _{t}-\lambda _{s})}{\sin p}}\cos \varphi _{t}\cos \varphi _{s}\cos p;}
{\displaystyle \tan p={\frac {\sin(\lambda _{t}-\lambda _{s})\cos \varphi _{t}\cos \varphi _{s}}{\sin \varphi _{t}-\cos \theta _{s,t}\sin \varphi _{s}}}.}
Long DerivationEdit
Because the brief derivation gives an angle between 0 and π which does not reveal the sign (west or east of north ?), a more explicit derivation is desirable which yields separately the sine and the cosine of p such that use of the correct branch of the inverse tangent allows to produce an angle in the full range -π≤p≤π.
The computation starts from a construction of the great circle between s and t. It lies in the plane that contains the sphere center, s and t and is constructed rotating s by the angle θs,t around an axis ω. The axis is perpendicular to the plane of the great circle and computed by the normalized vector cross product of the two positions:
{\displaystyle \mathbf {\omega } ={\frac {1}{R^{2}\sin \theta _{s,t}}}\mathbf {s} \times \mathbf {t} ={\frac {1}{\sin \theta _{s,t}}}\left({\begin{array}{c}\cos \varphi _{s}\sin \lambda _{s}\sin \varphi _{t}-\sin \varphi _{s}\cos \varphi _{t}\sin \lambda _{t}\\\sin \varphi _{s}\cos \lambda _{t}\cos \varphi _{t}-\cos \varphi _{s}\sin \varphi _{t}\cos \lambda _{s}\\\cos \varphi _{s}\cos \varphi _{t}\sin(\lambda _{t}-\lambda _{s})\end{array}}\right).}
A right-handed tilted coordinate system with the center at the center of the sphere is given by the following three axes: the axis s, the axis
{\displaystyle \mathbf {s} _{\perp }=\omega \times {\frac {1}{R}}\mathbf {s} ={\frac {1}{\sin \theta _{s,t}}}\left({\begin{array}{c}\cos \varphi _{t}\cos \lambda _{t}(\sin ^{2}\varphi _{s}+\cos ^{2}\varphi _{s}\sin ^{2}\lambda _{s})-\cos \lambda _{s}(\sin \varphi _{s}\cos \varphi _{s}\sin \varphi _{t}+\cos ^{2}\varphi _{s}\sin \lambda _{s}\cos \varphi _{t}\sin \lambda _{t})\\\cos \varphi _{t}\sin \lambda _{t}(\sin ^{2}\varphi _{s}+\cos ^{2}\varphi _{s}\cos ^{2}\lambda _{s})-\sin \lambda _{s}(\sin \varphi _{s}\cos \varphi _{s}\sin \varphi _{t}+\cos ^{2}\varphi _{s}\cos \lambda _{s}\cos \varphi _{t}\cos \lambda _{t})\\\cos \varphi _{s}[\cos \varphi _{s}\sin \varphi _{t}-\sin \varphi _{s}\cos \varphi _{t}\cos(\lambda _{t}-\lambda _{s})]\end{array}}\right)}
and the axis ω. A position along the great circle is
{\displaystyle \mathbf {s} (\theta )=\cos \theta \mathbf {s} +\sin \theta \mathbf {s} _{\perp },\quad 0\leq \theta \leq 2\pi .}
The compass direction is given by inserting the two vectors s and s⊥ and computing the gradient of the vector with respect to θ at θ=0.
{\displaystyle {\frac {\partial }{\partial \theta }}\mathbf {s} _{\mid \theta =0}=\mathbf {s} _{\perp }.}
The angle p is given by splitting this direction along two orthogonal directions in the plane tangential to the sphere at the point s. The two directions are given by the partial derivatives of s with respect to φ and with respect to λ, normalized to unit length:
{\displaystyle \mathbf {u} _{N}=\left({\begin{array}{c}-\sin \varphi _{s}\cos \lambda _{s}\\-\sin \varphi _{s}\sin \lambda _{s}\\\cos \varphi _{s}\end{array}}\right);}
{\displaystyle \mathbf {u} _{E}=\left({\begin{array}{c}-\sin \lambda _{s}\\\cos \lambda _{s}\\0\end{array}}\right);}
{\displaystyle \mathbf {u} _{N}\cdot \mathbf {s} =\mathbf {u} _{E}\cdot \mathbf {u} _{N}=0}
uN points north and uE points east at the position s. The position angle p projects s⊥ into these two directions,
{\displaystyle \mathbf {s} _{\perp }=\cos p\,\mathbf {u} _{N}+\sin p\,\mathbf {u} _{E}}
where the positive sign means the positive position angles are defined to be north over east. The values of the cosine and sine of p are computed by multiplying this equation on both sides with the two unit vectors,
{\displaystyle \cos p=\mathbf {s} _{\perp }\cdot \mathbf {u} _{N}={\frac {1}{\sin \theta _{s,t}}}[\cos \varphi _{s}\sin \varphi _{t}-\sin \varphi _{s}\cos \varphi _{t}\cos(\lambda _{t}-\lambda _{s})];}
{\displaystyle \sin p=\mathbf {s} _{\perp }\cdot \mathbf {u} _{E}={\frac {1}{\sin \theta _{s,t}}}[\cos \varphi _{t}\sin(\lambda _{t}-\lambda _{s})].}
Instead of inserting the convoluted expression of s⊥, the evaluation may employ that the triple product is invariant under a circular shift of the arguments:
{\displaystyle \cos p=(\mathbf {\omega } \times {\frac {1}{R}}\mathbf {s} )\cdot \mathbf {u} _{N}=\omega \cdot ({\frac {1}{R}}\mathbf {s} \times \mathbf {u} _{N}).}
If atan2 is used to compute the value, one can reduce both expressions by division through cos φt and multiplication by sin θs,t, because these values are always positive and that operation does not change signs; then effectively
{\displaystyle \tan p={\frac {\sin(\lambda _{t}-\lambda _{s})}{\cos \varphi _{s}\tan \varphi _{t}-\sin \varphi _{s}\cos(\lambda _{t}-\lambda _{s})}}.}
Birney, D. Scott; Gonzalez, Guillermo; Oesper, David (2007). Observational Astronomy. Cambridge University Press. p. 75. ISBN 978-0-521-85370-5.
^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 4.3.149". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
The Orbits of 150 Visual Binary Stars, by Dibon Smith (Accessed 2/26/06)
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E6B Knowpia
The E6B flight computer is a form of circular slide rule used in aviation and one of the very few analog calculating devices in widespread[citation needed] use in the 21st century.
The front of a metal E6B.
They are mostly used in flight training, because these flight computers have been replaced with electronic planning tools or software and websites that make these calculations for the pilots. These flight computers are used during flight planning (on the ground before takeoff) to aid in calculating fuel burn, wind correction, time en route, and other items. In the air, the flight computer can be used to calculate ground speed, estimated fuel burn and updated estimated time of arrival. The back is designed for wind vector solutions, i.e., determining how much the wind is affecting one's speed and course. They are frequently referred to by the nickname "whiz wheel."[1]
Flight computers are usually made out of aluminum, plastic or cardboard, or combinations of these materials. One side is used for wind triangle calculations using a rotating scale and a sliding panel. The other side is a circular slide rule. Extra marks and windows facilitate calculations specifically needed in aviation.
Electronic versions are also produced, resembling calculators, rather than manual slide rules. Aviation remains one of the few places that the slide rule is still in widespread use. Manual E6Bs/CRP-1s remain popular with some users and in some environments rather than the electronic ones because they are lighter, smaller, less prone to break, easy to use one-handed, quicker and do not require electrical power.
In flight training for a private pilot or instrument rating, mechanical flight computers are still often used to teach the fundamental computations. This is in part also due to the complex nature of some trigonometric calculations which would be comparably difficult to perform on a conventional scientific calculator. The graphic nature of the flight computer also helps in catching many errors which in part explains their continued popularity. The ease of use of electronic calculators means typical flight training literature[2] does not cover the use of calculators or computers at all. In the ground exams for numerous pilot ratings, programmable calculators or calculators containing flight planning software are permitted to be used.[3]
Many airspeed indicator (ASI) instruments have a movable ring built into the face of the instrument that is essentially a subset of the flight computer. Just like on the flight computer, the ring is aligned with the air temperature and the pressure altitude, allowing the true airspeed (TAS) to be read at the needle.
In addition, computer programs emulating the flight computer functions are also available, both for computers and smartphones.
Instructions for ratio calculations and wind problems are printed on either side of the computer for reference and are also found in a booklet sold with the computer. Also, many computers have Fahrenheit to Celsius conversion charts and various reference tables.
The front side of the flight computer is a logarithmic slide rule that performs multiplication and division. Throughout the wheel, unit names are marked (such as gallons, miles, kilometers, pounds, minutes, seconds, etc.) at locations that correspond to the constants that are used when going from one unit to another in various calculations. Once the wheel is positioned to represent a certain fixed ratio (for example, pounds of fuel per hour), the rest of the wheel can be consulted to utilize that same ratio in a problem (for example, how many pounds of fuel for a 2.5-hour cruise?) This is one area where the E6B and CRP-1 are different. Since the CRP-1s are made for the UK market, they can be used to perform the added conversions of Imperial to Metric units.
The wheel on the back of the calculator is used for calculating the effects of wind on cruise flight. A typical calculation done by this wheel answers the question: "If I want to fly on course A at a speed of B, but I encounter wind coming from direction C at a speed of D, then how many degrees must I adjust my heading, and what will my ground speed be?" This part of the calculator consists of a rotatable semi-transparent wheel with a hole in the middle, and a slide on which a grid is printed, that moves up and down underneath the wheel. The grid is visible through the transparent part of the wheel.
To solve this problem with a flight computer, first the wheel is turned so the wind direction (C) is at the top of the wheel. Then a pencil mark is made just above the hole, at a distance representing the wind speed (D) away from the hole. After the mark is made, the wheel is turned so that the course (A) is now selected at the top of the wheel. The ruler then is slid so that the pencil mark is aligned with the true airspeed (B) seen through the transparent part of the wheel. The wind correction angle is determined by matching how far right or left the pencil mark is from the hole, to the wind correction angle portion of the slide's grid. The true ground speed is determined by matching the center hole to the speed portion of the grid.
The mathematical formulas that equate to the results of the flight computer wind calculator are as follows:
(desired course is d, ground speed is Vg, heading is a, true airspeed is Va, wind direction is w, wind speed is Vw. d, a and w are angles. Vg, Va and Vw are consistent units of speed.
{\displaystyle \pi }
is approximated as 355/113 or 22/7)
{\displaystyle \Delta a=\sin ^{-1}\left({\frac {V_{w}\sin(w-d)}{V_{a}}}\right)}
True ground speed:
{\displaystyle V_{g}={\sqrt {V_{a}^{2}+V_{w}^{2}-2V_{a}V_{w}\cos(d-w+\Delta a)}}}
Wind Correction Angle, in degrees, as it might be programmed into a computer (which includes conversion of degrees to radians and back):
{\displaystyle \Delta a={\frac {180\deg }{\pi }}\sin ^{-1}\left({\frac {V_{w}}{V_{a}}}\sin \left({\frac {\pi (w-d)}{180\deg }}\right)\right)}
True ground speed is calculated as:
{\displaystyle V_{g}={\sqrt {V_{a}^{2}+V_{w}^{2}-2V_{a}V_{w}\cos \left({\frac {\pi (d-w+\Delta a)}{180\deg }}\right)}}}
Modern-day E6BsEdit
Although digital E6Bs are faster to learn initially, many flight schools still require their students to learn on mechanical E6Bs,[4] and for FAA pilot written exams and checkrides pilots are encouraged to bring their mechanical E6Bs with them for necessary calculations.
Closeup photo of a metal E-6B
The device's original name is E-6B, but is often abbreviated as E6B, or hyphenated as E6-B for commercial purposes.
The E-6B was developed in the United States by Naval Lt. Philip Dalton (1903–1941) in the late 1930s. The name comes from its original part number for the U.S Army Air Corps, before its reorganization in June 1941.
Philip Dalton was a Cornell University graduate who joined the United States Army as an artillery officer, but soon resigned and became a Naval Reserve pilot from 1931 until he died in a plane crash with a student practicing spins. He, with P. V. H. Weems, invented, patented and marketed a series of flight computers.
Dalton's first popular computer was his 1933 Model B, the circular slide rule with true airspeed (TAS) and altitude corrections pilots know so well. In 1936 he put a double-drift diagram on its reverse to create what the U.S. Army Air Corps (USAAC) designated as the E-1, E-1A and E-1B.
A couple of years later he invented the Mark VII, again using his Model B slide rule as a focal point. It was hugely popular with both the military and the airlines. Even Amelia Earhart's navigator Fred Noonan used one on their last flight. Dalton felt that it was a rushed design, and wanted to create something more accurate, easier to use, and able to handle higher flight speeds.
Closeup photo of a cardboard E6B
So he came up with his now famous wind arc slide, but printed on an endless cloth belt moved inside a square box by a knob. He applied for a patent in 1936 (granted in 1937 as 2,097,116). This was for the Model C, D and G computers widely used in World War II by the British Commonwealth (as the "Dalton Dead Reckoning Computer"), the U.S. Navy, copied by the Japanese, and improved on by the Germans, through Siegfried Knemeyer's invention of the disc-type Dreieckrechner device, somewhat similar to the eventual E6B's backside compass rose dial in general appearance, but having the compass rose on the front instead for real-time calculations of the wind triangle at any time while in flight. These are commonly available on collectible auction web sites.
The U.S. Army Air Corps decided the endless belt computer cost too much to manufacture, so later in 1937 Dalton morphed it to a simple, rigid, flat wind slide, with his old Model B circular slide rule included on the reverse. He called this prototype his Model H; the Army called it the E-6A.
In 1938 the Army wrote formal specifications, and had him make a few changes, which Weems called the Model J. The changes included moving the "10" mark to the top instead of the original "60". This "E-6B" was introduced to the Army in 1940, but it took Pearl Harbor for the Army Air Forces (as the former "Army Air Corps" was renamed on June 20, 1941) to place a large order. Over 400,000 E-6Bs were manufactured during World War II, mostly of a plastic that glows under black light (cockpits were illuminated this way at night).
The base name "E-6" was fairly arbitrary, as there were no standards for stock numbering at the time. For example, other USAAC computers of that time were the C-2, D-2, D-4, E-1 and G-1, and flight pants became E-1s as well. Most likely they chose "E" because Dalton's previously combined time and wind computer had been the E-1. The "B" simply meant it was the production model.
The designation "E-6B" was officially marked on the device only for a couple of years. By 1943 the Army and Navy changed the marking to their joint standard, the AN-C-74 (Army/Navy Computer 74). A year or so later it was changed to AN-5835, and then to AN-5834 (1948). The USAF called later updates the MB-4 (1953) and the CPU-26 (1958), but navigators and most instruction manuals continued using the original E-6B name. Many just called it the "Dalton Dead Reckoning Computer", one of its original markings.
Frontside of the military 6B/345
Backside of the military 6B/345
After Dalton's death, Weems[5] updated the E-6B and tried calling it the E-6C, E-10, and so forth, but finally fell back on the original name, which was so well known by 50,000 World War II Army Air Force navigator veterans. After the patent ran out, many manufacturers made copies, sometimes using a marketing name of "E6-B" (note the moved hyphen). An aluminium version was made by the London Name Plate Mfg. Co. Ltd. of London and Brighton and was marked "Computer Dead Reckoning Mk. 4A Ref. No. 6B/2645" followed by the arrowhead of UK military stores.
During World War II and into the early 1950s, The London Name Plate Mfg. Co. Ltd. produced a "Height & True Airspeed Computer Mk. IV" with the model reference "6B/345". The tool provided for calculation of the True Air Speed on the front side and Time-Speed calculations in relation to the altitude on the backside. They were still in use throughout the 1960s and 1970s in several European Air Forces, such as the German Air Force, until modern avionics made them obsolete.
Siegfried Knemeyer, inventor of the similar, contemporary Dreieckrechner flight calculator[6]
^ "E6B Flight Computer Tutorial PDF". 12 July 2021.
^ Pratt, Jeremy M. (2003). The Private Pilots License Course: Navigation & Meteorology. Airplan Flight Equipment Ltd. ISBN 978-1-874783-18-3. Retrieved 2014-01-21.
^ "Provision and Conduct of Ground Examinations for the Private Pilot Licence Aeroplanes & Helicopters". UK Civil Aviation Authority.
^ E6B Computer: Celebrating 75 Years Of Flight – InformationWeek
^ Weems Plath Story
^ Ronald van Riet's "Knemeyer Dreiechrechner" PDF document, chronicling the history of Knemeyer's own "whiz wheel" invention from 1936
Wikimedia Commons has media related to ASA E6B.
E6BX.com Online E6B – web-based E6B flight computer with illustrations
A Tale of Two Whiz Wheels: E6-B versus CR Wind Solutions
Free downloadable E6B[dead link] – requires Java Archived 30 December 2020 at the Wayback Machine
Free web based E6B aviation calculator
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Methods for Measuring and Computing the Adiabatic-Wall Temperature | GT | ASME Digital Collection
James Peck,
Peck, J, Liu, J, Bryden, KM, & Shih, TI. "Methods for Measuring and Computing the Adiabatic-Wall Temperature." Proceedings of the ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition. Volume 7B: Heat Transfer. Virtual, Online. September 21–25, 2020. V07BT12A004. ASME. https://doi.org/10.1115/GT2020-14169
For convective heat transfer involving multiple sources of different temperatures in the flow field such as in film cooling, the adiabatic-wall temperature, Tad, is used as the reference temperature to define the heat-transfer coefficient (HTC). Studies based on computational Fluid Dynamics (CFD) have always obtained Tad by requiring the cooled or heated surface to be adiabatic. Similarly, most experimental studies that measured Tad have sought to mimic adiabatic wall by using solids with very low thermal conductivity. Other experimental studies have obtained Tad by making two assumptions: (1) Tad at any given point on the cooled or heated surface is independent of the surface temperature, Ts, and the surface heat flux,
qs″
, at that point and (2) the HTC, had, which equals
qs″
/(Ts – Tad), depends only on the ratio of
qs″
to Ts – Tad. With these two assumptions, measuring
qs″
at two different Ts or vice versa at a point yields Tad and had at that point. In this study, CFD simulations, based on steady RANS, were performed to assess the assumptions invoked by CFD and experimental studies that seek to obtain Tad and had. The assessment was made by studying film cooling of a flat plate with an adiabatic wall and with isothermal walls, where the temperature of the isothermal wall, Ts, ranged from the lowest to the highest temperatures in the flow. Results from this study show Tad obtained by enforcing an adiabatic wall does not satisfy the requirement: where and when Ts – Tad = 0,
qs″
= 0. Therefore, had approaches infinity where Ts – Tad is either zero or nearly zero, but
qs″
≠ 0. Also, obtaining Tad and had by measuring two sets of (Ts,
qs″
) was found to yield non-unique values that depended strongly upon the pair of (Ts,
qs″
) chosen. To overcome the shortcomings of existing methods, a new method was developed in this study to obtain Tad that does satisfy the requirement: Ts – Tad = 0 where and when
qs″
= 0. Also, the method developed yields an had that is continuous across Ts – Tad = 0. By using the new method developed, errors in Tad and had obtained by existing methods were assessed.
film cooling, adiabatic wall temperature, heat transfer coefficient
Film cooling, Temperature, Computational fluid dynamics, Flow (Dynamics), Convection, Engineering simulation, Errors, Flat plates, Heat flux, Heat transfer, Heat transfer coefficients, Reynolds-averaged Navier–Stokes equations, Simulation, Solids, Thermal conductivity, Wall temperature
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Intersecting_secants_theorem Knowpia
The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
{\displaystyle \triangle PAC\sim \triangle PBD}
{\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}
For two lines AD and BC that intersect each other in P and some circle in A and D respective B and C the following equation holds:
{\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}
The theorem follows directly from the fact, that the triangles PAC and PBD are similar. They share
{\displaystyle \angle DPC}
{\displaystyle \angle ADB=\angle ACB}
as they are inscribed angles over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above:
{\displaystyle {\frac {PA}{PC}}={\frac {PB}{PD}}\Leftrightarrow |PA|\cdot |PD|=|PB|\cdot |PC|}
Next to the intersecting chords theorem and the tangent-secant theorem the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem.
Secant Secant Theorem at proofwiki.org
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Isoparametric Graded Finite Elements for Nonhomogeneous Isotropic and Orthotropic Materials | J. Appl. Mech. | ASME Digital Collection
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801
G. H. Paulino, Mem. ASME
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, July 2, 2001; final revision Nov. 14, 2001. Associate Editor: M.-J. Pindera. Discussion on the paper should be addressed to the Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the THE AMERICAN SOCIETY OF MECHANICAL ENGINEERSJOURNAL OF APPLIED MECHANICS.
J. Appl. Mech. Jul 2002, 69(4): 502-514 (13 pages)
Kim , J., and Paulino, G. H. (June 20, 2002). "Isoparametric Graded Finite Elements for Nonhomogeneous Isotropic and Orthotropic Materials ." ASME. J. Appl. Mech. July 2002; 69(4): 502–514. https://doi.org/10.1115/1.1467094
Graded finite elements are presented within the framework of a generalized isoparametric formulation. Such elements possess a spatially varying material property field, e.g. Young’s modulus
E
ν
for isotropic materials; and principal Young’s moduli
E11,E22,
G12,
ν12
for orthotropic materials. To investigate the influence of material property variation, both exponentially and linearly graded materials are considered and compared. Several boundary value problems involving continuously nonhomogeneous isotropic and orthotropic materials are solved, and the performance of graded elements is compared to that of conventional homogeneous elements with reference to analytical solutions. Such solutions are obtained for an orthotropic plate of infinite length and finite width subjected to various loading conditions. The corresponding solutions for an isotropic plate are obtained from those for the orthotropic plate. In general, graded finite elements provide more accurate local stress than conventional homogeneous elements, however, such may not be the case for four-node quadrilateral (Q4) elements. The framework described here can serve as the basis for further investigations such as thermal and dynamic problems in functionally graded materials.
functionally graded materials, finite element analysis, Young's modulus, Poisson ratio, shear modulus, boundary-value problems
Finite element analysis, Functionally graded materials, Stress, Tension, Stress concentration, Materials properties, Young's modulus, Boundary-value problems, Poisson ratio
Hirai, T., 1996, “Functionally Graded Materials,” Materials Science and Technology: Processing of Ceramics, Part 2, R. J. Brook, ed., VCH Verlagsgesellschaft mbH, Weinheim, Germany, 17B, pp. 292–341.
Miyamoto, Y., Kaysser, W. A., Rabin, B. H., Kawasaki, A., and Ford, R. G., 1999, Functionally Graded Materials: Design, Processing, and Applications, Kluwer, MA.
Suresh, S., and Mortensen, A., 1998, Fundamentals of Functionally Graded Materials, IOM Communications, London.
Koizumi, M., 1993, “The concept of FGM,” Proceedings of the Second International Symposium on Functionally Gradient Materials, Ceramic Transactions, J. B. Holt et al., eds., Westerville, Ohio, The American Ceramic Society, 34, pp. 3–10.
Thermomechanical Analysis of Functionally Graded Thermal Barrier Coatings With Different Microstructural Scales
Development of Large-Size Ceramic/Metal Bulk FGM Fabricated by Spark Plasma Sintering
Thermal Spray Processing of FGMs
M.R.S. Bull.
FGM Research Activities in Europe
Modelling Studies Applied to Functionally Graded Materials
Residual/Thermal Stresses in FGM and Laminated Thermal Barrier Coatings
Kurihara, K., Sasaki, K., and Kawarada, M., 1990, “Adhesion Improvement of Diamond Films,” Proceedings of the First International Symposium on Functionally Gradient Materials, M. Yamanouchi et al., eds., Tokyo, Japan.
Enhanced Thermal Stress Resistance of Structural Ceramics With Thermal Conductivity Gradient
Fracture Testing and Analysis of a Layered Functionally Graded Ti/TiB Beam in 3-Point Bending
Mode I Crack Problem in an Inhomogeneous Orthotropic Medium
The Mixed Mode Crack Problem in an Inhomogeneous Orthotropic Medium
Higher-Order Theory for Functionally Graded Materials
Evaluation of the Higher-Order Theory for Functionally Graded Materials Via the Finite-Element Method
Thermal Analysis of a Functionally Graded Material Subject to a Thermal Gradient Using the Boundary Element Method
Compos. Methods Appl. Mech. Eng.
Fracture of Nonhomogeneous Materials
Finite Element Analysis of Thermal Residual Stresses at Graded Ceramic-Metal Interfaces, Part I: Model Description and Geometric Effects
Finite Element Analysis of Thermal Residual Stresses at Graded Ceramic-Metal Interfaces, Part II: Interface Optimization for Residual Stress Reduction
A Simplified Method for Calculating the Crack-Tip Field of Functionally Graded Materials Using the Domain Integral
A Micromechanical Study of Residual Stresses in Functionally Graded Materials
Numerical Calculation of Stress Intensity Factors in Functionally Graded Materials
Finite Element Evaluation of Mixed Mode Stress Intensity Factors in Functionally Graded Materials
The Surface Crack Problem for a Plate with Functionally Graded Properties
Paulino, G. H., and Kim, J.-H., “The Weak Patch Test for Nonhomogeneous Materials Modeled With Graded Finite Elements” (submitted for publication).
Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Finite Element Evaluation of Quasi-Static Crack Growth in Functionally Graded Materials Using a Novel Cohesive Zone Fracture Model
Kim, J.-H., “Quasi-Static Crack Propagation in Functionally Graded Materials,” Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL.
Hibbitt, Karlson, & Sorensen, Inc., 2000, ABAQUS/Standard User’s Manual, Vol. II, Pawtucket, RI, Version 6.1 (p. 14.1.1-14).
Gullerud, A. S., Koppenhoefer, K. C., Roy, A., Roychowdhury, S., and Dodds, Jr., R. H., 2001, WARP3D-Release 13.11, University of Illinois, UILU-ENG-95-2012.
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times - Maple Help
Home : Support : Online Help : Connectivity : MTM Package : times
times(M1,M2)
The times(M1,M2) function performs element-wise multiplication of M1 * M2. The result, R, is formed as R[i,j] = M1[i,j] * M2[i,j].
\mathrm{with}\left(\mathrm{MTM}\right):
\mathrm{M1}≔\mathrm{Matrix}\left(2,3,'\mathrm{fill}'=3\right):
\mathrm{M2}≔\mathrm{Matrix}\left(2,3,'\mathrm{fill}'=2\right):
\mathrm{times}\left(\mathrm{M1},\mathrm{M2}\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{6}\end{array}]
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Shear in Structural Stability: On the Engesser–Haringx Discord | J. Appl. Mech. | ASME Digital Collection
Professor Emeritus of Structural Mechanics
, Private: Klinkenbergerweg 74, 6711 ML Ede, The Netherlands
e-mail: j.blaauwendraad@tudelft.nl
Johan Blaauwendraad Professor Emeritus of Structural Mechanics
Blaauwendraad, J. (February 4, 2010). "Shear in Structural Stability: On the Engesser–Haringx Discord." ASME. J. Appl. Mech. May 2010; 77(3): 031005. https://doi.org/10.1115/1.3197142
Since Haringx introduced his stability hypothesis for the buckling prediction of helical springs over 60 years ago, discussion is on whether or not the older hypothesis of Engesser should be replaced in structural engineering for stability studies of shear-weak members. The accuracy and applicability of both theories for structures has been subject of study in the past by others, but quantitative information about the accuracy for structural members is not provided. This is the main subject of this paper. The second goal is to explain the experimental evidence that the critical buckling load of a sandwich beam-column surpasses the shear buckling load
GAs
, which is commonly not expected on basis of the Engesser hypothesis. The key difference between the two theories regards the relationship, which is adopted in the deformed state between the shear force in the beam and the compressive load. It is shown for a wide range of the ratio of shear and flexural rigidity to which extent the two theories agree and/or conflict with each other. The Haringx theory predicts critical buckling loads which are exceeding the value
GAs
, which is not possible in the Engesser approach. That sandwich columns have critical buckling loads larger than
GAs
does, however, not imply the preference of the Haringx hypothesis. This is illustrated by the introduction of the thought experiment of a compressed cable along the central axis of a beam-column in deriving governing differential equations and finding a solution for three different cases of increasing complexity: (i) a compressed member of either flexural or shear deformation, (ii) a compressed member of both flexural and shear deformations, and (iii) a compressed sandwich column. It appears that the Engesser hypothesis leads to a critical buckling load larger than
GAs
for layered cross section shapes and predicts the sandwich behavior very satisfactory, whereas the Haringx hypothesis then seriously overestimates the critical buckling load. The fact that the latter hypothesis is perfectly confirmed for helical springs (and elastomeric bearings) has no meaning for shear-weak members in structural engineering. Then, the Haringx hypothesis should be avoided. It is strongly recommended to investigate the stability of the structural members on the basis of the Engesser hypothesis.
beams (structures), bending, buckling, sandwich structures, shear deformation, structural engineering, stability, Engesser theory, Haringx theory, shear-weak member, shear deformation, thick-faced sandwich, structural engineering
Buckling, Shear (Mechanics), Stress, Shear deformation, Structural engineering, Springs, Stiffness, Deformation
On the Buckling and the Lateral Rigidity of Helical Compression Springs, I
Proc. K. Ned. Akad. Wet.
On Highly Compressible Helical Springs and Rubber Bars and on Their Application in Vibration Isolation
,” Doctoral thesis, Delft University of Technology, The Netherlands.
Die Knickung von Schraubenfedern
Biezeno
Zuschrift an den Herausgeber
Die Knickfestigkeit gerader Stäbe
Arguments for and Against Engesser’s Buckling Formulas
Sandwich Buckling Formulas and Applicability of Standard Computational Algorithm for Finite Strain
Stability and Finite Strain of Homogenized Structures Soft in Shear: Sandwich or Fiber Composites, and Layered Bodies
Column Buckling With Shear Deformations—A Hyperelastic Formulation
Sandwich Column Buckling —A Hyperelastic Formulation
End Compression of Sandwich Columns
Ingenieurbauten 3, Theorie und Praxis
Bending and Buckling of Sandwich Beams
Sandwich Column Buckling Experiments
, Proceedings of the 20th Australasian Conference on the Mechanics of Structures and Materials, Toowoomba, University of Southern Queensland, Australia, pp.
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Lemma 10.134.13 (00S7)—The Stacks project
Lemma 10.134.13. Let $A \to B$ be a ring map. Let $S \subset B$ be a multiplicative subset. The canonical map $\mathop{N\! L}\nolimits _{B/A} \otimes _ B S^{-1}B \to \mathop{N\! L}\nolimits _{S^{-1}B/A}$ is a quasi-isomorphism.
Proof. We have $S^{-1}B = \mathop{\mathrm{colim}}\nolimits _{g \in S} B_ g$ where we think of $S$ as a directed set (ordering by divisibility), see Lemma 10.9.9. By Lemma 10.134.12 each of the maps $\mathop{N\! L}\nolimits _{B/A} \otimes _ B B_ g \to \mathop{N\! L}\nolimits _{B_ g/A}$ are quasi-isomorphisms. The lemma follows from Lemma 10.134.9. $\square$
(Why) Isn't the canonical map in the lemma a homotopy equivalence?
It is indeed a homotopy equivalence. We can deduce this from a general result on two term complexes of the form
\ldots \to 0 \to L^{-1} \to L^0 \to 0 \to \ldots
L^0
projective, see Lemma 15.84.4. It would be a bit annoying (I think) to prove it here in the commutative algebra chapter and having the quasi-isomorphism statement suffices for applications, I think and hope.
10 comment(s) on Section 10.134: The naive cotangent complex
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Inequalities With Polynomial and Rational Functions | Boundless Algebra | Course Hero
Polynomials can be expressed as inequalities, the solutions for which can be determined from the polynomial's zeros.
Solve for the zeros of a polynomial inequality to find its solution
To solve a polynomial inequality, first rewrite the polynomial in its factored form to find its zeros.
For each zero, input the value of the zero in place of
x
in the polynomial. Determine the sign (positive or negative) of the polynomial as it passes the zero in the rightward direction.
Determine the intervals between these roots which satisfy the inequality.
inequality: A statement that of two quantities, one is specifically less than or greater than another. Symbols: < or ≤ or > or ≥, as appropriate.
Like any other function, a polynomial may be written as an inequality, giving a large range of solutions.
The best way to solve a polynomial inequality is to find its zeros. The easiest way to find the zeros of a polynomial is to express it in factored form. At these points, the polynomial's value goes from negative to positive or positive to negative. This knowledge can then be used to determine the solutions of the inequality. Much of the work involved with solving inequalities is based in observation and judgement of a particular mathematical situation, and is therefore best demonstrated with an example.
Consider the polynomial inequality:
x^3+2x^2-5x-6>0
This can be expressed as the product of three terms:
(x-2)(x+1)(x+3)>0
The three terms reveal zeros at
x=-3
x=-1
x=2
. We know that the lower limit of the inequality crosses the x-axis at each of these
x
values, but now have to determine which direction (positive or negative) it takes at each crossing.
x+3>0
x>-3
x+1>0
x>-1
x-2>0
x>2
Thus, as the polynomial crosses the x-axis at
x=-3
(x+3)
equals zero, becoming positive to the right. At the same point,
(x+1)
(x-2)
are negative. The product of a positive and two negatives is positive, so we can conclude that the polynomial becomes positive as it passes
x=-3
The next zero is at
x=-1
. From the explanation above, we know that the polynomial is positive as it approaches its next zero, but we can use the same reasoning for proof. At
x=-1
(x+1)
equals zero, becoming positive to the right. The term
(x+3)
is positive, while
(x-2)
is negative. The product of two positives and a negative is negative, so we can conclude that the polynomial becomes negative as it passes
x=-1
The same process can be used to show that the polynomial becomes positive again at
x=2
Recalling the initial inequality, we can now determine the solution of exactly where the polynomial is greater than zero. Because there is no zero to the left of
x=-3
, we can assume that the polynomial is negative for all
x
-\infty
-3
. The polynomial is positive from
x=-3
x=-1
before becoming negative once more. It becomes positive at
x=2
, and because there are no more zeros to the right, we can assume the polynomial remains positive as
x
\infty
(-3,-1),(2,\infty)
For inequalities that are not expressed relative to zero, expressions can be added or subtracted from each side to take it into the desired form.
Rational inequalities can be solved much like polynomial inequalities.
Solve for the zeros and asymptotes of a rational inequality to find its solution
First factor the numerator and denominator polynomial to reveal the zeros in each.
x
with a zero (root) to determine whether the rational function is positive or negative to the right of that point. Repeat for all zeros.
The intervals that satisfy the inequality symbol will be the answer. Note that for any
\geq
\leq
, the interval will only be closed to include the zero if the zero is found in the numerator. If the zero is found in the denominator, that point is undefined, and cannot be included in the solution.
zero: Also known as a root, a zero is an
x
value at which the function of
x
inequality: A statement that of two quantities one is specifically less than or greater than another. Symbols:
<
\leq
>
\geq
As with solving polynomial inequalities, the first step to solving rational inequalities is to find the zeros. Because a rational expression consists of the ratio of two polynomials, the zeroes for both polynomials will be needed.
The zeros in the numerator are
x
-values at which the rational inequality crosses from negative to positive or from positive to negative. The zeros in the denominator are
x
-values are at which the rational inequality is undefined, the result of dividing by zero.
Consider the rational inequality:
\frac{x^2+2x-3}{x^2-4}>0
This equation can be factored to give:
\frac{(x+3)(x-1)}{(x+2)(x-2)}\geq 0
x
crosses rightward past
-3
(x+3)
becomes positive. At that same point,
(x-1)
(x+2)
(x-2)
are all negative. The product of a positive and three negatives is negative, so the rational expression becomes negative as it crosses
x=-3
in the rightward direction.
The same process can be used to determine that the rational expression is positive after passing the zero at
x=-2
, is negative after passing
x=1
, and is positive after passing
x=2
Thus we can conclude that for
x
values on the open interval from
-\infty
-3
, the rational expression is negative. From
-3
-2
, it is positive; from
-2
1
it is negative; from
1
2
it is positive, and from
2
\infty
Because the inequality is written as
\geq0
>0
, we will need to evaluate the
x
values at zeros to determine whether the function is defined.
x=-2
x=2
, the rational function has a denominator equal to zero and becomes undefined.
x=-3
x=1
, the rational function has a numerator equal to zero, which makes the function overall equal to zero, making it inclusive in the solution.
Thus, the full solution is:
[-3, -2), [1, 2)
Inequality (mathematics). Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Inequality_(mathematics). License: CC BY-SA: Attribution-ShareAlike
Boundless. Provided by: Boundless Learning. Located at: http://www.boundless.com//algebra/definition/zero. License: CC BY-SA: Attribution-ShareAlike
Review of Polynomials and Rational Functions
MATH 2202 • Kennesaw State University
2.6 and 2.7 Rational Functions and Polynomial and Rational Inequalities-2.pdf
MATH AP • East Lake High School
Reflect on the concept of polynomial and rational functions.docx
Concept of Polynomial and Rational Functions.docx
MANAGEMENT 517 • California State University, Sacramento
Application of polynomials and rational functions in banking and finance.edited.docx
SOCIOLOGY 345 • Maseno University
Lesson 3-1 Limits of Polynomial and Rational Functions.docx
MATH 4 • Mayfield High School, Cleveland
Lesson 7.3 - Applications of Polynomials and Rational Functions.pdf
STATICS 101 • De La Salle University
4.21.21 APC March 21st Practice with Polynomials and Rational Functions.pdf
MATH PRECALC • Parkview High School
A rational function is a fraction with polynomials in the numerator and denominator.docx
MATH MISC • Harvard University
LIMITS OF POLYNOMIALS AND RATIONAL FUNCTIONS AS x → a Example 5 Find....pdf
MATH 2306 • University of San Carlos - Main Campus
lesson-6-limits-of-polynomial-and-rational-functions.docx
MATH 11 • Bishop Allen Academy
Copy_of_lesson-3-limits-of-polynomial-and-rational-functions.docx
MATH 1 • Alternative Scarborough Education 1
The application of polynomials and rational functions.docx
MATH algebra • Harvard University
MATH 1271 • University of Minnesota-Twin Cities
Notes for Polynomials and Rational Functions (Michelle).docx
MATH MISC • Mindanao State University - Iligan Institute of Technology
CHAPTER THREE GRAPH OF POLYNOMIAL AND RATIONAL FUNCTIONS.pdf
MAT 421 • Universiti Teknologi Mara
Key_Features_of_Polynomial_and_Rational_Functions
MATH 172 • Beaufort County Community College
IXL - Determine end behavior of polynomial and rational functions (Calculus practice).pdf
GFNCH VBHNHCGNHGNHGG THBGBGXHNF • Greenforest Mccalep Christian Ac
Polynomials and Rational Functions.docx
2.3 Polynomial and Rational Functions.pptx
MATH 150 • University of KwaZulu-Natal- Westville Campus
4.1 - Review of Sketching of Polynomial and Rational Functions.pdf
ENGLISH 101 • Irvine High School
MA1200 Chapter 3 Polynomials and Rational Functions
MA 1200 • City University of Hong Kong
Study Guide on Polynomial and Rational Inequalities
MAC MISC • University of Miami
CA Chapter 3 T - Polynomial and Rational Functions Test REVIEW.pdf
MATH 1314 • Lone Star College System, Woodlands
MAC 1140 • University of Florida
Polynomial and Rational Functions Summ
MATH 110 • Northern Illinois University
Unit test- Polynomials and Rational Functions
MIE 221 • University of Toronto
Polynomial and rational functions can be used to model a wide variety of phenomena of science.docx
Polynomial_and_Rational_Functions.pdf
MTH 1101 • University of Guyana
|
刘迪：Stata空间溢出效应的动态图形-空间计量| 连享会主页
原文：How to create animated graphics to illustrate spatial spillover effects
中文版PPT 在线浏览
作者：Di Liu, Senior Econometrician[1]
This post is organized as follows. First, I estimate the parameters of a SAR model. Second, I show why a SAR model can produce spatial spillover effects. Finally, I show how to create an animated graph that illustrates the spatial spillover effects.
A SAR model
I want to analyze the homicide rate in Texas counties as a function of unemployment. I suspect that the homicide rate in one county affects the homicide rate in neighboring counties.
I want to answer two questions.
How can I set up a model that explicitly allows the homicide rate in one county to depend on the homicide rate in neighboring counties?
Given my model, if the unemployment rate in Dallas increases to 10%, how would the homicide rate change in the neighboring counties of Dallas ?
Fit a SAR model
A standard linear model for the homicide rate in county
i
\left({\mathbf{h}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}_{i}\right)
as a function of the unemployment rate in that county’s
{\mathbf{u}\mathbf{n}\mathbf{e}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{o}\mathbf{y}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}}_{i}
{\mathbf{h}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}_{i}={\beta }_{0}+{\beta }_{1}{\mathbf{u}\mathbf{n}\mathbf{e}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{o}\mathbf{y}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}}_{i}+{ϵ}_{i}
A SAR model allows
\left({\mathbf{h}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}_{i}\right)
to depend on the homicide rate in neighboring counties. I need some new notation to write down a SAR model. I let
\left({W}_{i,j}\right)
be a positive number if county
j
is a neighbor of county
i
, zero if the
j
is not a neighbor of
i
, and zero if
j=i
, because no county can border itself.
Given this notation, a SAR model that allows the homicide rate in county
i
to depend on the homicide rate in neighboring counties can be written as
{\mathbf{h}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}_{i}={\gamma }_{1}\sum _{j=1}^{N}{W}_{i,j}{\mathbf{h}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}_{j}+{\beta }_{1}{\mathbf{u}\mathbf{n}\mathbf{e}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{o}\mathbf{y}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}}_{i}+{\beta }_{0}+{ϵ}_{i}
\left({W}_{i,j}\right)
defines the closeness between county
i
and county
j
\sum _{j=1}^{N}{W}_{i,j}{\mathbf{h}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}_{j}
is a weighted sum of the homicide rates in county
i
’s neighboring counties, and it specifies how the homicide rates in neighboring counties affect the homicide rate in county
i
Stacking the neighborhood information in
\left({W}_{i,j}\right)
for each county
i
produces a matrix
\mathbf{W}
that records the neighbor information for each county
i
\mathbf{W}
is known as a spatial-weighting matrix.
The spatial-weighting matrix that we are using has a special structure; each element is either a value
c
or zero, where
c
is greater than zero. This type of spatial-weighting matrix is known as a normalized contiguity matrix.
In Stata, we use spmatrix to create a spatial-weighting matrix, and we use spregress to fit a cross-sectional SAR model.
I begin by downloading some data on the homicide rates of U.S. counties from the Stata website and creating a subsample that uses only data on counties in Texas.
. /* Get data for Texas counties' homicide rate */
. copy http://www.stata-press.com/data/r15/homicide1990.dta ., replace
. use homicide1990
(S.Messner et al.(2000), U.S southern county homicide rates in 1990)
. keep if sname == "Texas"
. save texas, replace
file texas.dta saved
Intuitively, a file that specifies the borders of all the places of interest is known as a shape file texas.dta is linked to the Stata version of a shape file that specifies the borders of all the counties in Texas. I now download that dataset from the Stata website and use spset to show that they are linked.
. copy http://www.stata-press.com/data/r15/homicide1990_shp.dta, replace
. spset
Sp dataset texas.dta
linked shapefile: homicide1990_shp.dta
I now use spmatrix to create a normalized contiguity spatial-weighting matrix.
. /* Create a spatial contiguity matrix */
Now that I have my data and my spatial-weighting matrix, I can estimate the model parameters.
. /* Estimate SAR model parameters */
. spregress hrate unemployment, dvarlag(W) gs2sls
(254 observations (places) used)
(weighting matrix defines 254 places)
GS2SLS estimates Wald chi2(2) = 14.23
hrate | Coef. Std. Err. z P>|z| [95% Conf. Interval]
hrate |
unemployment | .4584241 .152503 3.01 0.003 .1595237 .7573245
_cons | 2.720913 1.653105 1.65 0.100 -.5191143 5.960939
W |
hrate | .3414964 .1914865 1.78 0.075 -.0338103 .7168031
Now we are ready to answer the second question. Based on our estimation results from spregress, we can proceed in three steps.
Predict the homicide rate using original data.
Change Dallas’s unemployment rate to 10% and predict the homicide rate again.
Compute the difference between two predictions and map it.
. preserve /* save data temporarily */
. /* Step 1: predict homicide rate using original data */
. predict y0
(option rform assumed; reduced-form mean)
. /* Step 2: change Dallas unemployment rate to 10%, and predict again*/
. replace unemployment = 10 if cname == "Dallas"
. /* Step 3: Compute the prediction difference and map it*/
. generate double y_diff = y1 - y0
. grmap y_diff, title("Global spillover")
. restore /* return to original data */
The above graph shows that a change in the unemployment rate in Dallas changes the homicide rates in the counties that are near to Dallas, in addition to the homicide rate in Dallas. The change in Dallas spills over to the nearby counties, and the effect is known as a spillover effect.
SAR model and spatial spillover
In this section, I show why a SAR model generates a spillover effect. In the process, I provide a formula for this effect that I use to create the animated graph.
The matrix form for a SAR model is
\mathbf{y}=\lambda \mathbf{W}\mathbf{y}+\mathbf{X}\beta +ϵ
\mathbf{y}
\mathbf{y}=\left(\mathbf{I}–\lambda \mathbf{W}{\right)}^{-1}\mathbf{X}\beta +ϵ
\mathbf{y}
\mathbf{X}
is known as the the expectation of
\mathbf{y}
\mathbf{X}
ϵ
\mathbf{X}
\mathbf{y}
\mathbf{X}
E\left(\mathbf{y}|\mathbf{X}\right)=\left(\mathbf{I}–\lambda \mathbf{W}{\right)}^{-1}\mathbf{X}\beta
Note that this conditional expectation specifies the mean for each county in Texas because
\mathbf{y}
is a vector.
We use this equation to define the effect of going from one set of values for
\mathbf{X}
to another set. In the case at hand, I let
{\mathbf{X}}_{\mathbf{0}}
contain the covariate values in the observed data and let
{\mathbf{X}}_{\mathbf{1}}
contain the same values except that the unemployment rate in Dallas has been set to 10%. With this notation, I see that going from
{\mathbf{X}}_{\mathbf{0}}
{\mathbf{X}}_{\mathbf{1}}
causes the mean homicide rates for each county in Texas to change by
\begin{array}{l}E\left(\mathbf{y}|{\mathbf{X}}_{1}\right)-E\left(\mathbf{y}|{\mathbf{X}}_{0}\right)\\ =\left(\mathbf{I}-\lambda \mathbf{W}{\right)}^{-1}{\mathbf{X}}_{1}\beta -\left(\mathbf{I}-\lambda \mathbf{W}{\right)}^{-1}{\mathbf{X}}_{0}\beta \\ =\left(\mathbf{I}-\lambda \mathbf{W}{\right)}^{-1}\mathrm{\Delta }\mathbf{X}\beta \end{array}\phantom{\rule{1em}{0ex}}\left(1\right)
\mathrm{\Delta }\mathbf{X}={\mathbf{X}}_{\mathbf{1}}–{\mathbf{X}}_{\mathbf{0}}
I now show that a technical condition assumed in SAR models produces an expression for the animated graph. SAR models are widely used because they satisfy a stability condition. Intuitively, this stability condition says that the inverse matrix
\left(\mathbf{I}–\lambda \mathbf{W}{\right)}^{-1}
can be written as a sum of terms that decrease in size exponentially fast. This condition is that
\left(\mathbf{I}–\lambda \mathbf{W}{\right)}^{-1}=\left(\mathbf{I}+\lambda \mathbf{W}+{\lambda }^{2}{\mathbf{W}}^{2}+{\lambda }^{3}{\mathbf{W}}^{3}+\dots \right)\phantom{\rule{1em}{0ex}}\left(2\right)
Plugging the formula from (2) into the effect in (1) yields
\begin{array}{ll}E\left(\mathbf{y}\mid {\mathbf{X}}_{1}\right)-E\left(\mathbf{y}\mid {\mathbf{X}}_{0}\right)& \\ =\left(\mathbf{I}-\lambda \mathbf{W}{\right)}^{-1}\mathrm{\Delta }\mathbf{X}\beta \\ =\left(\mathbf{I}+\lambda \mathbf{W}+{\lambda }^{2}{\mathbf{W}}^{2}+{\lambda }^{3}{\mathbf{W}}^{3}+\dots \right)\mathrm{\Delta }\mathbf{X}\beta \\ =\mathrm{\Delta }\mathbf{X}\beta +\lambda \mathbf{W}\mathrm{\Delta }\mathbf{X}\beta +{\lambda }^{2}{\mathbf{W}}^{2}\mathrm{\Delta }\mathbf{X}\beta +{\lambda }^{3}{\mathbf{W}}^{3}\mathrm{\Delta }\mathbf{X}\beta +\dots \phantom{\rule{1em}{0ex}}\left(3\right)\end{array}
which is the expression for the effect that I use to generate the animated graph.
Each term in (3) has some intuition, which is most easily presented in terms of my example. The first term (
\mathrm{\Delta }\mathbf{X}\beta
) is the initial effect of the change, and it affects only the homicide rate in Dallas. The second term (
\lambda \mathbf{W}\mathrm{\Delta }\mathbf{X}\beta
) is the effect of the change on the outcome in those places that are neighbors of Dallas. The third term (
{\lambda }^{2}{\mathbf{W}}^{2}\mathrm{\Delta }\mathbf{X}\beta
) is the effect of the change on the outcome in those places that are neighbors of neighbors of Dallas. The intuition continues in the pattern for the remaining terms.
Create animated graphs for spillover effects
I now describe how I generate the animated graph. Each graph plots the change using a subset of the terms in (3). The first graph plots the change computed from the first term only. The second graph plots the change computed from the first and second terms only. The third graph plots the change computed from the first three terms only. And so on.
The first four steps of the code do the following.
It computes and plots
\mathrm{\Delta }\mathbf{X}\beta
\mathrm{\Delta }\mathbf{X}\beta +\lambda \mathbf{W}\mathrm{\Delta }\mathbf{X}\beta
It compute and plots
\mathrm{\Delta }\mathbf{X}\beta +\lambda \mathbf{W}\mathrm{\Delta }\mathbf{X}\beta +{\lambda }^{2}{\mathbf{W}}^{2}\mathrm{\Delta }\mathbf{X}\beta
\mathrm{\Delta }\mathbf{X}\beta +\lambda \mathbf{W}\mathrm{\Delta }\mathbf{X}\beta +{\lambda }^{2}{\mathbf{W}}^{2}\mathrm{\Delta }\mathbf{X}\beta +{\lambda }^{3}{\mathbf{W}}^{3}\mathrm{\Delta }\mathbf{X}\beta
Steps 5 through 20 perform the analogous operations.
Finally, combine graphs from step 1 to step 20, and create an animated graph.
Here is the code that implements this process.
1 /* get estimate of spatial lag parameter lambda */
2 local lambda = _b[W:hrate]
4 /* xb based on original data */
5 predict xb0, xb
7 /* xb based on modified data */
8 replace unemployment = 10 if cname == "Dallas"
11 /* compute the outcome change in the first step */
12 generate dy = xb1 - xb0
13 format dy %9.2f
15 /* Initialize Wy, lamWy, */
16 generate Wy = dy
17 generate lamWy = dy
19 /* map the outcome change in step 1 */
20 grmap dy
21 graph export dy_0.png, replace
22 local input dy_0.png
24 /* compute the outcome change from step 2 to 11 */
25 forvalues p=1/20 {
26 spgenerate tmp = W*Wy
27 replace lamWy = `lambda'^`p'*tmp
28 replace Wy = tmp
29 replace dy = dy + lamWy
30 grmap dy
31 graph export dy_`p'.png, replace
32 local input `input' dy_`p'.png
33 drop tmp
36 /* convert graphs into a animated graph */
37 shell convert -delay 150 -loop 0 `input' glsp.gif
39 /* delete the generated pgn file */
40 shell rm -fR *.png
This code uses the ereturn results produced by spregress above and its corresponding predictcommand.
Line 2 puts the estimate of
\lambda
in the local macro lambda.
Lines 5, 7, 8, and 9 compute
\mathbf{X}\beta
{\mathbf{X}}_{\mathbf{0}}
{\mathbf{X}}_{\mathbf{1}}
and store them in xb0 and xb1, respectively.
Line 12 computes the first term (
\mathrm{\Delta }\mathbf{X}\beta
) and stores it in dy.
Lines 16 and 17 store the initial values for
{\mathbf{W}}^{p}\mathbf{y}
{\lambda }^{p}{\mathbf{W}}^{p}\mathbf{y}
p=0
Lines 20–22 produce the first plot in the animated graph. The local macro input will contain all the plots used to create the animated graph when the code finishes.
Lines 25–34 compute the terms and create the plots for the remaining terms. Line 26 usesspgenerate to compute
{\mathbf{W}}^{p}\mathbf{y}
. Line 27–33 perform operations analogous to those of dy.
In Line 37, I use a Linux tool “convert” to combine the graphs to produce an animated graph. On Windows, I can use software such as FFmpeg and Camtasia. For more details, see How to create animated graphics using Stata by Chuck Huber.
Line 40 deletes all the unnecessary .png files.
Here is the animated graph created by this code.
In this post, I discussed spillover effects and why SAR models produce them in the context of an example using the counties in Texas. I also showed how the effects can be computed as an accumulated sum. I used the accumulated sum to create an animated graph that illustrates how the effects spill over in the counties in Texas.
Posts by Di Liu, Senior Econometrician: https://blog.stata.com/author/dliu/
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Lemma 5.12.10 (005F)—The Stacks project
Lemma 5.12.10. Let $X$ be a topological space. Assume
the intersection of two quasi-compact opens is quasi-compact.
For any $x \in X$ the connected component of $X$ containing $x$ is the intersection of all open and closed subsets of $X$ containing $x$.
Proof. Let $T$ be the connected component containing $x$. Let $S = \bigcap _{\alpha \in A} Z_\alpha $ be the intersection of all open and closed subsets $Z_\alpha $ of $X$ containing $x$. Note that $S$ is closed in $X$. Note that any finite intersection of $Z_\alpha $'s is a $Z_\alpha $. Because $T$ is connected and $x \in T$ we have $T \subset S$. It suffices to show that $S$ is connected. If not, then there exists a disjoint union decomposition $S = B \amalg C$ with $B$ and $C$ open and closed in $S$. In particular, $B$ and $C$ are closed in $X$, and so quasi-compact by Lemma 5.12.3 and assumption (1). By assumption (2) there exist quasi-compact opens $U, V \subset X$ with $B = S \cap U$ and $C = S \cap V$ (details omitted). Then $U \cap V \cap S = \emptyset $. Hence $\bigcap _\alpha U \cap V \cap Z_\alpha = \emptyset $. By assumption (3) the intersection $U \cap V$ is quasi-compact. By Lemma 5.12.6 for some $\alpha ' \in A$ we have $U \cap V \cap Z_{\alpha '} = \emptyset $. Since $X \setminus (U \cup V)$ is disjoint from $S$ and closed in $X$ hence quasi-compact, we can use the same lemma to see that $Z_{\alpha ''} \subset U \cup V$ for some $\alpha '' \in A$. Then $Z_\alpha = Z_{\alpha '} \cap Z_{\alpha ''}$ is contained in $U \cup V$ and disjoint from $U \cap V$. Hence $Z_\alpha = U \cap Z_\alpha \amalg V \cap Z_\alpha $ is a decomposition into two open pieces, hence $U \cap Z_\alpha $ and $V \cap Z_\alpha $ are open and closed in $X$. Thus, if $x \in B$ say, then we see that $S \subset U \cap Z_\alpha $ and we conclude that $C = \emptyset $. $\square$
Comment #636 by Wei Xu on June 01, 2014 at 06:54
Dear stacks project,
There is a possible small gap between "for some
\alpha \in A
U \cap V \cap Z_\alpha = \emptyset
." and "Hence
Z_\alpha = U \cap Z_\alpha \coprod V \cap Z_\alpha
". Possibly we might need to add words like "(with some argurements) we may aslo assume this
Z_{\alpha}\subset (U\cup V)
" before the sentense "Hence ..."
Yes, I agree one needs an argument there. I added something here. Thanks!
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Neusis_construction Knowpia
The neusis (from Ancient Greek: νεῦσις from νεύειν neuein "incline towards"; plural: νεύσεις neuseis) is a geometric construction method that was used in antiquity by Greek mathematicians.
Geometric constructionEdit
The neusis construction consists of fitting a line element of given length (a) in between two given lines (l and m), in such a way that the line element, or its extension, passes through a given point P. That is, one end of the line element has to lie on l, the other end on m, while the line element is "inclined" towards P.
Point P is called the pole of the neusis, line l the directrix, or guiding line, and line m the catch line. Length a is called the diastema (διάστημα; Greek for "distance").
A neusis construction might be performed by means of a marked ruler that is rotatable around the point P (this may be done by putting a pin into the point P and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye with crosshairs: this is the origin of the scale division on the ruler. A second marking on the ruler (the blue eye) indicates the distance a from the origin. The yellow eye is moved along line l, until the blue eye coincides with line m. The position of the line element thus found is shown in the figure as a dark blue bar.
Neusis trisection of an angle θ "> 135° to find φ = θ/3, using only the length of the ruler. The radius of the arc is equal to the length of the ruler. For angles θ < 135° the same construction applies, but with P extended beyond AB.
Use of the neusisEdit
Neuseis have been important because they sometimes provide a means to solve geometric problems that are not solvable by means of compass and straightedge alone. Examples are the trisection of any angle in three equal parts, and the doubling of the cube.[1][2] Mathematicians such as Archimedes of Syracuse (287–212 BC) and Pappus of Alexandria (290-350 AD) freely used neuseis; Sir Isaac Newton (1642-1726) followed their line of thought, and also used neusis constructions.[3] Nevertheless, gradually the technique dropped out of use.
Regular PolygonsEdit
In 2002, A. Baragar that showed that every point constructible with marked ruler and compass lies in a tower of fields over
{\displaystyle \mathbb {Q} }
{\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K}
, such that the degree of the extension at each step is no higher than 6. Of all prime-power polygons below the 100-gon, this is enough to show that the regular 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, and 89-gons cannot be constructed with neusis. (If a regular p-gon is constructible, then
{\displaystyle \zeta _{p}=e^{\frac {2\pi i}{p}}}
is constructible, and in these cases p − 1 has a prime factor higher than 5.) The 3-, 4-, 5-, 8-, 16-, 17-, 32-, and 64-gons can be constructed with only a straightedge and compass, and the 7-, 9-, 13-, 19-, 27-, 37-, 73-, 81-, and 97-gons with angle trisection. However, it is not known in general if all quintics (fifth-order polynomials) that are solvable by radicals have neusis-constructible roots, which is relevant for the 11-, 25-, 31-, 41-, and 61-gons.[4] Benjamin and Snyder showed in 2014 that the regular 11-gon is neusis-constructible;[1] the 25-, 31-, 41-, and 61-gons remain open problems. More generally, the constructibility of all powers of 5 greater than 5 itself by marked ruler and compass is an open problem, along with all primes greater than 11 of the form p = 2r3s5t + 1 where t > 0 (all prime numbers that are greater than 11 and equal to one more than a regular number that is divisible by 10).[4]
Waning popularityEdit
T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides (ca. 440 BC) was the first to put compass-and-straightedge constructions above neuseis. The principle to avoid neuseis whenever possible may have been spread by Hippocrates of Chios (ca. 430 BC), who originated from the same island as Oenopides, and who was—as far as we know—the first to write a systematically ordered geometry textbook. One hundred years after him Euclid too shunned neuseis in his very influential textbook, The Elements.
The next attack on the neusis came when, from the fourth century BC, Plato's idealism gained ground. Under its influence a hierarchy of three classes of geometrical constructions was developed. Descending from the "abstract and noble" to the "mechanical and earthly", the three classes were:
constructions with straight lines and circles only (compass and straightedge);
constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas);
constructions that needed yet other means of construction, for example neuseis.
In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution. Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other construction methods might have been used was branded by the late Greek mathematician Pappus of Alexandria (ca. 325 AD) as "a not inconsiderable error".
Tomahawk (geometry)
^ a b Benjamin, Elliot; Snyder, C (May 2014). "On the construction of the regular hendecagon by marked ruler and compass". Mathematical Proceedings of the Cambridge Philosophical Society. 156 (3): 409–424. doi:10.1017/S0305004113000753. Archived from the original on September 26, 2020. Retrieved 26 September 2020.
^ Weisstein, Eric W. "Neusis Construction." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NeusisConstruction.html
^ Guicciardini, Niccolò (2009). Isaac Newton on Mathematical Certainty and Method, Issue 4. M.I.T Press. p. 68. ISBN 9780262013178.
^ a b Arthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151-164, doi:10.1080/00029890.2002.11919848
R. Boeker, 'Neusis', in: Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa red. (1894–), Supplement 9 (1962) 415–461.–In German. The most comprehensive survey; however, the author sometimes has rather curious opinions.
T. L. Heath, A history of Greek Mathematics (2 volumes; Oxford 1921).
H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum [= The Theory of Conic Sections in Antiquity] (Copenhagen 1886; reprinted Hildesheim 1966).
MathWorld page
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4.3 Measurement devices: Technology of irradiance transducers | EME 810: Solar Resource Assessment and Economics
J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 8: Measure & Estimation of the Solar Resource (Focus on instrumentation.)
Sengupta et al. (2015) Best Practices Handbook for the Collection and Use of Solar Resource Data for Solar Energy Applications. NREL/TP-5D00-63112: p. 19-62: Chapter 3. Measuring Solar Radiation
OK, so the eyes don't have it. So, how do we go about measuring the solar resource? The Chapter represents a beginning overview to the subject of measurement. The assigned white paper from the National Renewable Energy Laboratory covers our topic in much greater depth.
Measurement is an important aspect of all scientific endeavors. It is especially important in the proper and efficient design of solar energy collection systems. Proper solar assessment involves metrological and climate data, and correct measurement of global (beam and diffuse) radiation is essential to any solar design effort. Without adequate and precise measurement of the solar resources, system designers and engineers would essentially be "flying blind." In this section, we will discuss the equipment used to perform the required measurements.
Pyranometers and Pyrheliometers
The Pyranometer: Global Irradiation Measurements
Pyranometers act as solar energy transducers, in that they collect irradiance signals and transform them into electrical information signals. That information is passed on to a data logger and computer, and then we either present the data in short bursts (1 second) or integrate and average the data over longer periods of 1 minute to 1 hour.
Research grade pyranometers use a film of opaque material to collect thermal energy. The thermal energy diffuses into a thermal transducer called a thermopile (a stack of thermal devices) that produces a small current proportional to the temperature. We should note that metals (in general) are very good reflectors, making them also very poor absorbers. So, how do we get a material that functions on thermal gradients to make use of the radiation from the sun?
The key is in the absorber material: Parson's black is a paint with very low reflectance across shortwave and longwave bands of light (~300-50,000 nm; making it an effective blackbody). However, if covered by glass (a selective surface), the "window" of light acceptance from the Sun is about 300-2800 nm. This system assembly forms a shortwave (band) global (component) pyranometer. Now imagine, if we develop a thermopile with a thin coating of a black absorber, but replace the glass with a material that is transparent in the longwave band (many organopolymers/plastics), we will have created a longwave (band) global (component) pyranometer.
On the other hand, inexpensive pyranometers can use photodiodes. Photodiodes are photovoltaics (just small). They are semiconductor films that directly convert shortwave band radiation into electrical signals (no thermal conversion step necessary). While the cutoff for a silicon photodiode is <1100nm, the integrated power response is fairly comparable to that of a Parson's black-coated thermopile detector. However, they do not perform as well (relative to thermopile detectors) near sunrise and sunset due to a cosine response error.
Cosine Response Error
Remember the cosine projection effect that we discussed in Lesson 2? It matters here for solar measurement. In the morning and evening, at low solar altitude angles (
{\alpha }_{s}
), some of the radiation incident on the detector is reflected, which produces a reading less than it should be. Some correction can be made for this using a black cylinder casing and a small white plastic disk cover (with a low reflectance at low angles to minimize the cosine error).
Review of a pyranometer in operation:
For standard research, technicians mount pyranometers in a horizontal orientation. Pyranometers produce a voltage in response to incident solar radiation. Provided that a pyranometer uses a thermopile (thermoelectric detector), the device acts as an "integrator" of all components and bands of light. In the case of a glass enclosure, even a thermopile detector will operate only in the shortwave band. Pyranometers based on photodiodes are used only for shortwave global radiation measurements. The following two images are explained in detail at the University of Oregon's Solar Radiation Monitoring Laboratory (maintained by Dr. Frank Vignola). The left image is a LI-COR pyranometer, which uses a silicon photodiode to measure irradiance (a little PV cell). The right image, which looks like a flying saucer from the 1950s, is an Eppley Precision Spectral Pyranometer (PSP). The Eppley is a First Class Radiometer, and uses a thermopile to measure irradiance. The white ring is to reflect stray light away, such that the system does not heat up and so that the influence of the ground reflectance (the albedo) is minimal.
Figure 4.2: LI-COR pyranometer (left), Eppley Precision Spectral Pyranometer (PSP) (right)
Credit: UO SRML
Standard pyranometers are designed to be mounted horizontally in shadow-free areas, with the normal vector relative to the surface of the collector (which is horizontal) pointing vertically. Measurements of downwelling shortwave band irradiance from a horizontal pyranometer collect Global Horizontal Irradiance, or GHI. However, through a simple modification, a pyranometer may also be used to measure diffuse irradiance. By using an occulting disk or band, beam radiation can be blocked from the sensor surface of the pyranometer, leaving only diffuse radiation to be measured.
The Pyrheliometer: Beam Component Measurements
If we wished to measure only the direct component of downwelling irradiation, we would use a pyrheliometer. The device is a combination of a long tube with a thermopile at the base of the tube and a two-axis tracking system to always point the aperture of the device directly normal to the surface of the Sun. A measure of irradiance from a pyrheliometer is therefore called Direct Normal Irradiance (DNI) (Gb,n) data. An Eppley Normal Incidence Pyrheliometer is displayed below on the left, while an Eppley Solar Tracker is displayed on the right.
Figure 4.3: Left: Eppley Normal Pyrheliometer (tracking system not displayed). Right: Eppley Solar Tracker system (2-axis tracking ability). The Eppley Normal Incidence Pyrheliometer is to be mounted on the Solar Tracker.
Curious side note: The World Meteorological Organization (WMO) has a definition for "sunshine." Sunshine means irradiance conditions of >120 W/m2 from the direct component of solar radiation. Really, sunshine has a definition!
Until now, we have assumed that measurements of GHI or DNI will come from surface-based measurement methods. By reading Ch. 4 of the CSP Best Practices, we also see that satellites can be used to retrieve GHI (not typically DNI). Geostationary Satellites are used to collect GHI data.
GOES-West (
\lambda =-115°
) located to observe the eastern Pacific and the western half of the United States. The actual satellite is GOES-15 (in place as of late 2011, also, soon to be replaced in 2015).
GOES-East (
\lambda =-75°
) located in a good spot to keenly observe Atlantic weather systems and weather over the eastern half of the United States. The actual satellite is GOES-13 (in place as of 2010, soon to be replaced in 2015).
Meteosat-9 (
\lambda =0°
\lambda =+57.5°
MTSAT (
\lambda =+140°
) Australia/Asia
In the United States, the National Oceanic and Atmospheric Administration's geostationary satellites go by the name of "GOES," which is an acronym for "Geostationary Operational Environmental Satellite." Two operational geostationary satellites, GOES-13 and GOES-11, currently orbit over the equator at 75 and 135 degrees longitude West, respectively. As an aside, GOES-12 is currently drifting east toward
\varphi =-60°
, where it will provide images of South America.
To access images from GOES or geostationary weather satellites operated by other countries visit:
University of Wisconsin's Website.
NOAA's GOES Satellite Server. This is operated by the National Environmental Satellite, Data and Information Service (NESDIS).
Geostationary satellites are far from perfect. Consider that images of clouds at high latitudes will become highly distorted due to the cosine projection effect, or from viewing the Earth at increasingly oblique angles. For latitudes poleward of approximately 70 degrees, geostationary satellites become essentially useless. But, this is also where the solar resource becomes quite limited. Polar-orbiting satellites can therefore collect at high latitudes where geostationary satellites are not efficient. Each polar orbiter has its cycle effectively fixed in space, completing 14 orbits per day while the Earth rotates.
‹ 4.2 Vision is a significant bias to assessing the Solar Resource up 4.4 Empirical Correlation for Estimating Components of Light ›
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networks(deprecated)/graphical - Maple Help
Home : Support : Online Help : networks(deprecated)/graphical
tests whether a list of integers is graphical
graphical(intlist)
graphical(intlist, 'MULTI')
specifies that multigraph is permitted
This procedure tests whether intlist is the degree sequence of a simple graph (or multigraph with no loops if 'MULTI' is specified).
If intlist is graphical (multigraphical) the procedure call returns a list of edges for one realization of intlist as a graph. Otherwise it returns FAIL.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[graphical](...).
\mathrm{with}\left(\mathrm{networks}\right):
\mathrm{new}\left(G\right):
\mathrm{addvertex}\left(1,2,3,4,5,6,7,G\right):
\mathrm{graphical}\left([6,4,6,4,4,4,6]\right)
[{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}]
\mathrm{addedge}\left(,G\right):
\mathrm{degreeseq}\left(G\right)
[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]
\mathrm{new}\left(H\right):
\mathrm{addvertex}\left(1,2,3,4,5,6,7,H\right):
\mathrm{graphical}\left([3,3,6,6,6,3,3],'\mathrm{MULTI}'\right)
[{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}}]
\mathrm{addedge}\left(,H\right):
\mathrm{vdegree}\left(3,H\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{graphical}\left([5,4,3,2,1,1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
networks(deprecated)[degreeseq]
|
Using Modal Algorithms - MATLAB & Simulink - MathWorks 日本
In many cases, a model's
j\mathrm{Ï}
-axis poles are important to keep after model reduction, e.g., rigid body dynamics of a flexible structure plant or integrators of a controller. A unique routine, modreal, serves the purpose nicely.
modreal puts a system into its modal form, with eigenvalues appearing on the diagonal of its A-matrix. Real eigenvalues appear in 1-by-1 blocks, and complex eigenvalues appear in 2-by-2 real blocks. All the blocks are ordered in ascending order, based on their eigenvalue magnitudes, by default, or descending order, based on their real parts. Therefore, specifying the number of
j\mathrm{Ï}
-axis poles splits the model into two systems with one containing only
j\mathrm{Ï}
-axis dynamics, the other containing the remaining dynamics.
G = rss(30,1,1); % random 30-state model
[Gjw,G2] = modreal(G,1); % only one rigid body dynamics
G2.D = Gjw.D; % put DC gain of G into G2
Gjw.D = 0;
sigma(Gjw)
ylabel('Rigid Body')
sigma(G2)
ylabel('Nonrigid Body')
Further model reduction can be done on G2 without any numerical difficulty. After G2 is further reduced to Gred, the final approximation of the model is simply Gjw+Gred.
This process of splitting
j\mathrm{Ï}
-axis poles has been built in and automated in all the model reduction routines balancmr, schurmr, hankelmr, bstmr, and hankelsv, so that users need not worry about splitting the model.
Examine the Hankel singular value plot.
Calculate an eighth-order reduced model.
[gr,info] = reduce(G,8);
bode(G,'b-',gr,'r--')
The default algorithm balancmr of reduce has done a great job of approximating a 30-state model with just eight states. Again, the rigid body dynamics are preserved for further controller design.
modreal | balancmr | schurmr | hankelmr | bstmr | hankelsv
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Transfer_principle Knowpia
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0.
An incipient form of a transfer principle was described by Leibniz under the name of "the Law of Continuity".[1] Here infinitesimals are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of the principle of permanence. Similar tendencies are found in Cauchy, who used infinitesimals to define both the continuity of functions (in Cours d'Analyse) and a form of the Dirac delta function.[1]: 903
In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system. Its most common use is in Abraham Robinson's nonstandard analysis of the hyperreal numbers, where the transfer principle states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.
Transfer principle for the hyperrealsEdit
The transfer principle concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreal numbers. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical realisation of a project initiated by Leibniz.
The idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets. As Robinson put it, the sentences of [the theory] are interpreted in *R in Henkin's sense.[2]
The theorem to the effect that each proposition valid over R, is also valid over *R, is called the transfer principle.
There are several different versions of the transfer principle, depending on what model of nonstandard mathematics is being used. In terms of model theory, the transfer principle states that a map from a standard model to a nonstandard model is an elementary embedding (an embedding preserving the truth values of all statements in a language), or sometimes a bounded elementary embedding (similar, but only for statements with bounded quantifiers).
The transfer principle appears to lead to contradictions if it is not handled correctly. For example, since the hyperreal numbers form a non-Archimedean ordered field and the reals form an Archimedean ordered field, the property of being Archimedean ("every positive real is larger than 1/n for some positive integer n") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than 1/n for some positive integer n" is false; however the correct interpretation is "every positive hyperreal is larger than 1/n for some positive hyperinteger n". In other words, the hyperreals appear to be Archimedean to an internal observer living in the nonstandard universe, but appear to be non-Archimedean to an external observer outside the universe.
A freshman-level accessible formulation of the transfer principle is Keisler's book Elementary Calculus: An Infinitesimal Approach.
{\displaystyle x}
{\displaystyle x\geq \lfloor x\rfloor ,}
{\displaystyle \lfloor \,\cdot \,\rfloor }
is the integer part function. By a typical application of the transfer principle, every hyperreal
{\displaystyle x}
{\displaystyle x\geq {}^{*}\!\lfloor x\rfloor ,}
{\displaystyle {}^{*}\!\lfloor \,\cdot \,\rfloor }
is the natural extension of the integer part function. If
{\displaystyle x}
is infinite, then the hyperinteger
{\displaystyle {}^{*}\!\lfloor x\rfloor }
is infinite, as well.
Generalizations of the concept of numberEdit
Historically, the concept of number has been repeatedly generalized. The addition of 0 to the natural numbers
{\displaystyle \mathbb {N} }
was a major intellectual accomplishment in its time. The addition of negative integers to form
{\displaystyle \mathbb {Z} }
already constituted a departure from the realm of immediate experience to the realm of mathematical models. The further extension, the rational numbers
{\displaystyle \mathbb {Q} }
, is more familiar to a layperson than their completion
{\displaystyle \mathbb {R} }
, partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by
{\displaystyle \mathbb {Q} }
. Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer. The necessity for such an extension stems not from physical observation but rather from the internal requirements of mathematical coherence. The infinitesimals entered mathematical discourse at a time when such a notion was required by mathematical developments at the time, namely the emergence of what became known as the infinitesimal calculus. As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries. Keisler wrote:
"In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."
The self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:
{\displaystyle \forall x\in \mathbb {R} \quad \exists y\in \mathbb {R} \quad x<y.}
The same will then also hold for hyperreals:
{\displaystyle \forall x\in {}^{\star }\mathbb {R} \quad \exists y\in {}^{\star }\mathbb {R} \quad x<y.}
Another example is the statement that if you add 1 to a number you get a bigger number:
{\displaystyle \forall x\in \mathbb {R} \quad x<x+1}
which will also hold for hyperreals:
{\displaystyle \forall x\in {}^{\star }\mathbb {R} \quad x<x+1.}
The correct general statement that formulates these equivalences is called the transfer principle. Note that, in many formulas in analysis, quantification is over higher-order objects such as functions and sets, which makes the transfer principle somewhat more subtle than the above examples suggest.
Differences between R and *REdit
The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that
{\displaystyle 1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots }
but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like ω is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.
The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.
Constructions of the hyperrealsEdit
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. Vladimir Kanovei and Shelah[3] give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it.
In its most general form, transfer is a bounded elementary embedding between structures.
The ordered field *R of nonstandard real numbers properly includes the real field R. Like all ordered fields that properly include R, this field is non-Archimedean. It means that some members x ≠ 0 of *R are infinitesimal, i.e.,
{\displaystyle \underbrace {\left|x\right|+\cdots +\left|x\right|} _{n{\text{ terms}}}<1{\text{ for every finite cardinal number }}n.}
The only infinitesimal in R is 0. Some other members of *R, the reciprocals y of the nonzero infinitesimals, are infinite, i.e.,
{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ terms}}}<\left|y\right|{\text{ for every finite cardinal number }}n.}
The underlying set of the field *R is the image of R under a mapping A ↦ *A from subsets A of R to subsets of *R. In every case
{\displaystyle A\subseteq {^{*}\!A},}
with equality if and only if A is finite. Sets of the form *A for some
{\displaystyle \scriptstyle A\,\subseteq \,\mathbb {R} }
are called standard subsets of *R. The standard sets belong to a much larger class of subsets of *R called internal sets. Similarly each function
{\displaystyle f:A\rightarrow \mathbb {R} }
extends to a function
{\displaystyle {^{*}\!f}:{^{*}\!A}\rightarrow {^{*}\mathbb {R} };}
these are called standard functions, and belong to the much larger class of internal functions. Sets and functions that are not internal are external.
The importance of these concepts stems from their role in the following proposition and is illustrated by the examples that follow it.
The transfer principle:
Suppose a proposition that is true of *R can be expressed via functions of finitely many variables (e.g. (x, y) ↦ x + y), relations among finitely many variables (e.g. x ≤ y), finitary logical connectives such as and, or, not, if...then..., and the quantifiers
{\displaystyle \forall x\in \mathbb {R} {\text{ and }}\exists x\in \mathbb {R} .}
For example, one such proposition is
{\displaystyle \forall x\in \mathbb {R} \ \exists y\in \mathbb {R} \ x+y=0.}
Such a proposition is true in R if and only if it is true in *R when the quantifier
{\displaystyle \forall x\in {^{*}\!\mathbb {R} }}
{\displaystyle \forall x\in \mathbb {R} ,}
{\displaystyle \exists }
Suppose a proposition otherwise expressible as simply as those considered above mentions some particular sets
{\displaystyle \scriptstyle A\,\subseteq \,\mathbb {R} }
. Such a proposition is true in R if and only if it is true in *R with each such "A" replaced by the corresponding *A. Here are two examples:
{\displaystyle [0,1]^{\ast }=\{\,x\in \mathbb {R} :0\leq x\leq 1\,\}^{\ast }}
{\displaystyle \{\,x\in {^{*}\mathbb {R} }:0\leq x\leq 1\,\},}
including not only members of R between 0 and 1 inclusive, but also members of *R between 0 and 1 that differ from those by infinitesimals. To see this, observe that the sentence
{\displaystyle \forall x\in \mathbb {R} \ (x\in [0,1]{\text{ if and only if }}0\leq x\leq 1)}
is true in R, and apply the transfer principle.
The set *N must have no upper bound in *R (since the sentence expressing the non-existence of an upper bound of N in R is simple enough for the transfer principle to apply to it) and must contain n + 1 if it contains n, but must not contain anything between n and n + 1. Members of
{\displaystyle {^{*}\mathbb {N} }\setminus \mathbb {N} }
are "infinite integers".)
Suppose a proposition otherwise expressible as simply as those considered above contains the quantifier
{\displaystyle \forall A\subseteq \mathbb {R} \dots {\text{ or }}\exists A\subseteq \mathbb {R} \dots \ .}
Such a proposition is true in R if and only if it is true in *R after the changes specified above and the replacement of the quantifiers with
{\displaystyle [\forall {\text{ internal }}A\subseteq {^{*}\mathbb {R} }\dots ]}
{\displaystyle [\exists {\text{ internal }}A\subseteq {^{*}\mathbb {R} }\dots ]\ .}
Three examplesEdit
The appropriate setting for the hyperreal transfer principle is the world of internal entities. Thus, the well-ordering property of the natural numbers by transfer yields the fact that every internal subset of
{\displaystyle \mathbb {N} }
has a least element. In this section internal sets are discussed in more detail.
Every nonempty internal subset of *R that has an upper bound in *R has a least upper bound in *R. Consequently the set of all infinitesimals is external.
The well-ordering principle implies every nonempty internal subset of *N has a smallest member. Consequently the set
{\displaystyle {^{*}\mathbb {N} }\setminus \mathbb {N} }
of all infinite integers is external.
If n is an infinite integer, then the set {1, ..., n} (which is not standard) must be internal. To prove this, first observe that the following is trivially true:
{\displaystyle \forall n\in \mathbb {N} \ \exists A\subseteq \mathbb {N} \ \forall x\in \mathbb {N} \ [x\in A{\text{ iff }}x\leq n].}
{\displaystyle \forall n\in {^{*}\mathbb {N} }\ \exists {\text{ internal }}A\subseteq {^{*}\mathbb {N} }\ \forall x\in {^{*}\mathbb {N} }\ [x\in A{\text{ iff }}x\leq n].}
As with internal sets, so with internal functions: Replace
{\displaystyle \forall f:A\rightarrow \mathbb {R} \dots }
{\displaystyle \forall {\text{ internal }}f:{^{*}\!A}\rightarrow {^{*}\mathbb {R} }\dots }
when applying the transfer principle, and similarly with
{\displaystyle \exists }
{\displaystyle \forall }
For example: If n is an infinite integer, then the complement of the image of any internal one-to-one function ƒ from the infinite set {1, ..., n} into {1, ..., n, n + 1, n + 2, n + 3} has exactly three members by the transfer principle. Because of the infiniteness of the domain, the complements of the images of one-to-one functions from the former set to the latter come in many sizes, but most of these functions are external.
This last example motivates an important definition: A *-finite (pronounced star-finite) subset of *R is one that can be placed in internal one-to-one correspondence with {1, ..., n} for some n ∈ *N.
^ a b Keisler, H. Jerome. "Elementary Calculus: An Infinitesimal Approach". p. 902.
^ Robinson, A. The metaphysics of the calculus, in Problems in the Philosophy of Mathematics, ed. Lakatos (Amsterdam: North Holland), pp. 28–46, 1967. Reprinted in the 1979 Collected Works. Page 29.
^ Kanovei, Vladimir; Shelah, Saharon (2004), "A definable nonstandard model of the reals" (PDF), Journal of Symbolic Logic, 69: 159–164, arXiv:math/0311165, doi:10.2178/jsl/1080938834
Hardy, Michael: "Scaled Boolean algebras". Adv. in Appl. Math. 29 (2002), no. 2, 243–292.
Kanovei, Vladimir; Shelah, Saharon (2004), "A definable nonstandard model of the reals", Journal of Symbolic Logic, 69: 159–164, arXiv:math/0311165, doi:10.2178/jsl/1080938834
Keisler, H. Jerome (2000). "Elementary Calculus: An Infinitesimal Approach".
Kuhlmann, F.-V. (2001) [1994], "Transfer principle", Encyclopedia of Mathematics, EMS Press
Łoś, Jerzy (1955) Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres. Mathematical interpretation of formal systems, pp. 98–113. North-Holland Publishing Co., Amsterdam.
Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3, MR 0205854
|
Steps to obtain a transformation. (1) put a i at one of the
O\left({\left(\frac{1}{\epsilon }\right)}^{3}\right)
grid points d-ball of b i′ . (2) put a j at a grid point on the intersection of the sphere centered at a i with radius |a i a j | and d-ball of b j′ . There are at most
O\left({\left(\frac{1}{\epsilon }\right)}^{2}\right)
grid points on the intersection. (3) use a i and a j as the rotation axis.
|
Simulation and Experimental Analysis of a Solar Driven Absorption Chiller With Partially Wetted Evaporator | J. Sol. Energy Eng. | ASME Digital Collection
, Marchstraße 18, D-10587 Berlin, Germany
e-mail: jan.albers@tu-berlin.de
Giovanni Nurzia,
Giovanni Nurzia
Albers, J., Nurzia, G., and Ziegler, F. (January 11, 2010). "Simulation and Experimental Analysis of a Solar Driven Absorption Chiller With Partially Wetted Evaporator." ASME. J. Sol. Energy Eng. February 2010; 132(1): 011016. https://doi.org/10.1115/1.4000331
The efficient operation of a solar cooling system strongly depends on the chiller behavior under part load conditions, since driving energy and cooling load are never constant. For this reason, the performance of a single stage, hot water driven 30 kW
H2O/LiBr
-absorption chiller employed in a solar cooling system with a field of
350 m2
evacuated tube collector has been analyzed under part load conditions with both simulations and experiments. A simulation model has been developed for the whole absorption chiller (Type Yazaki WFC-10), where all internal mass and energy balances are solved. The connection to the external heat reservoirs of hot, chilled, and cooling water is done by lumped and distributed UA values for the main heat exchangers. In addition to an analytical evaporator model—which is described in detail—experimental correlations for UA values have been used for the condenser, generator, and solution heat exchanger. For the absorber, a basic model based on the Nusselt theory has been employed. The evaporator model was developed, taking into account the distribution of refrigerant on the tube bundle, as well as the change in operation from a partially dry to an overflowing evaporator. A linear model is derived to calculate the wetted area. The influence of these effects on cooling capacity and coefficient of performance (COP) is calculated for three different combinations of hot and cooling water temperature. The comparison to experimental data shows a good agreement in the various operational modes of the evaporator. The model is able to predict the transition from partially dry to an overflowing evaporator quite well. The present deviations in the domain with high refrigerant overflow can be attributed to the simple absorber model and the linear wetted area model. Nevertheless, the results of this investigation can be used to improve control strategies for new and existing solar cooling systems.
cooling, heat exchangers, refrigerants, solar absorber-convertors
Absorption, Condensers (steam plant), Cooling, Flow (Dynamics), Heat exchangers, Refrigerants, Simulation, Solar energy, Stress, Temperature, Water, Generators, Experimental analysis, Heat transfer, Simulation models
Analysis of the Part Load Behaviour of Sorption Chillers With Thermally Driven Solution Pumps
Proceedings of the 21st International Congress of Refrigeration
, IIR/IIF, WA, Paper No. ICR0570.
Investigation Into the Influence of the Cooling Water Temperature on the Operating Conditions of Thermosyphon Generators
, Denver, CO, June 22–24, Paper No. ISHPC-053-2005.
C. -D.
Berichtsteil II: Simulation und experimentelle Identifikation des Betriebsverhaltens von kleinen heißwasserbetriebenen Absorptionskältemaschinen
, Institut für Energietechnik und Fernwärme-Forschungsinstitut, Hannover eV, Koordinierter Schlussbericht AiF Fördernummer 11259 B.
‘The Intertube Falling Film: Part 1—Flow Characteristics, Mode Transitions and Hysteresis
Falling Film Evaporation Heat Transfer of Water/Salt Mixtures From Roll Worked Enhanced Tubes and Tube Bundle
Validation of a Model for the Absorption Process of H2O(vap) by a LiBr(aq) in a Horizontal Tube Bundle, Using a Multi-Factorial Analysis
The Effect of Microsurface Treatment on Heat and Mass Transfer Performance for a Falling Film H2O/LiBr Absorber
The Characteristic Equation of Sorption Chillers
Proceedings of the International Sorption Heat Pump Conference (ISHPC 99)
, Munich, Germany, March 24–26, pp.
A Simple Method for Modeling the Operating Characteristics of Absorption Chillers
Proceedings of the Eurotherm Seminar No.59
, Nancy-Ville, France, Vols.
A Modular Computer Simulation of Absorption Systems
, Part 2B, pp.
Experimental Evaluation of the Performances of a H2O–LiBr Absorption Refrigerator Under Different Service Conditions
Simulation of a Solar Absorption Air Conditioning System
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School of Management, Department of Accounting, Jinan University, Guangzhou, China.
Wang, R. (2018) Strategic Deviance and Accounting Conservatism. American Journal of Industrial and Business Management, 8, 1197-1228. doi: 10.4236/ajibm.2018.85082.
AC{C}_{i,t}={\beta }_{0}+{\beta }_{1}CF{O}_{i,t}+{\beta }_{2}DCF{O}_{i,t}+{\beta }_{3}CF{O}_{i,t}\times DCF{O}_{i,t}+{\epsilon }_{i,t}
\begin{array}{c}ACC={\beta }_{0}+{\beta }_{1}CFO+{\beta }_{2}DCFO+{\beta }_{3}CFO\times DCFO+{\beta }_{4}SD+{\beta }_{5}SD\times CFO\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{6}SD\times DCFO+{\beta }_{7}SD\times CFO\times DCFO+{\beta }_{8}SIZE+{\beta }_{9}SIZE\times CFO\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{10}SIZE\times DCFO+{\beta }_{11}SIZE\times CFO\times DCFO+{\beta }_{12}LEV+{\beta }_{13}LEV\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times CFO+{\beta }_{14}LEV\times DCFO+{\beta }_{15}LEV\times CFO\times DCFO+{\beta }_{16}MTB\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{17}MTB\times CFO+{\beta }_{18}MTB\times DCFO+{\beta }_{19}MTB\times CFO\times DCFO+{\epsilon }_{i,t}\end{array}
\Delta N{I}_{t}={\beta }_{0}+{\beta }_{1}\Delta N{I}_{t-1}+{\beta }_{2}D\Delta N{I}_{t-1}+{\beta }_{3}\Delta N{I}_{t-1}\times D\Delta N{I}_{t-1}+{\epsilon }_{i,t}
\begin{array}{c}\Delta {\text{NI}}_{t}={\beta }_{0}+{\beta }_{1}\Delta {\text{NI}}_{t-1}+{\beta }_{2}D\Delta {\text{NI}}_{t-1}+{\beta }_{3}\Delta N{I}_{t-1}\times D\Delta N{I}_{t-1}+{\beta }_{4}SD+{\beta }_{5}SD\times \Delta N{I}_{t-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{6}SD\times D\Delta N{I}_{t-1}+{\beta }_{7}SD\times \Delta N{I}_{t-1}\times D\Delta N{I}_{t-1}+{\beta }_{8}SIZE+{\beta }_{9}SIZE\times \Delta N{I}_{t-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{10}SIZE\times D\Delta N{I}_{t-1}+{\beta }_{11}SIZE\times \Delta N{I}_{t-1}\times D\Delta N{I}_{t-1}+{\beta }_{12}LEV+{\beta }_{13}LEV\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times \Delta N{I}_{t-1}+{\beta }_{14}LEV\times D\Delta N{I}_{t-1}+{\beta }_{15}LEV\times \Delta N{I}_{t-1}\times D\Delta N{I}_{t-1}+{\beta }_{16}MTB\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{17}MTB\times \Delta N{I}_{t-1}+{\beta }_{18}MTB\times D\Delta N{I}_{t-1}+{\beta }_{19}MTB\times \Delta N{I}_{t-1}\times D\Delta N{I}_{t-1}+{\epsilon }_{i,t}\end{array}
[1] Meyer, J.W. and Rowan, B. (1977) Institutionalized Organizations: Formal Structure as Myth and Ceremony. American Journal of Sociology, 83, 340-363. https://doi.org/10.1086/226550
[2] Carpenter, M.A. (2000) The Price of Change: The Role of CEO Compensation in Strategic Variation and Deviation from Industry Strategy Norms. Journal of Management, 26, 1179-1198. https://doi.org/10.1177/014920630002600606
[3] Thompson, J.D. (1967) Organizations in Action: Social Science Bases of Administrative Theory. Transaction Publishers, Piscataway, New Jersey.
[4] Walton, R.E. and Dutton, J.M. (1969) The Management of Interdepartmental Conflict: A Model and Review. Administrative Science Quarterly, 14, 73-84. https://doi.org/10.2307/2391364
[5] Geletkanycz, M.A. and Hambrick, D.C. (1997) The External Ties of Top Executives: Implications for Strategic Choice and Performance. Administrative Science Quarterly, 42, 654-681. https://doi.org/10.2307/2393653
[6] Deephouse, D.L. (1999) To Be Different, or to Be the Same? It’s a Question (and Theory) of Strategic Balance. Strategic Management Journal, 20, 147-166. https://doi.org/10.1002/(SICI)1097-0266(199902)20:2<147::AID-SMJ11>3.0.CO;2-Q
[7] Cannon, W.B., McIver, M.A. and Bliss, S.W. (1924) Studies on the Conditions of Activity in Endocrine Glands: XIII. A Sympathetic and Adrenal Mechanism for Mobilizing Sugar in Hypoglycemia. American Journal of Physiology-Legacy Content, 69, 46-66. https://doi.org/10.1152/ajplegacy.1924.69.1.46
[8] Beaver, W.H. and Ryan, S.G. (2005) Conditional and Unconditional Conservatism: Concepts and Modeling. Review of Accounting Studies, 10, 269-309. https://doi.org/10.1007/s11142-005-1532-6
[9] Basu, S. (1997) The Conservatism Principle and the Asymmetric Timeliness of Earnings1. Journal of Accounting and Economics, 24, 3-37. https://doi.org/10.1016/S0165-4101(97)00014-1
[10] Li, Z.Q. and Lu, W.B. (2003) The Robustness of Accounting Earnings: Discovery and Implications. Accounting Research, 2, 19-27.
[11] Ball, R. and Shivakumar, L. (2005) Earnings Quality in UK Private Firms: Comparative Loss Recognition Timeliness. Journal of Accounting and Economics, 39, 83-128. https://doi.org/10.1016/j.jacceco.2004.04.001
[13] Tang, J., Crossan, M. and Rowe, W.G. (2011) Dominant CEO, Deviant Strategy, and Extreme Performance: The Moderating Role of a Powerful Board. Journal of Management Studies, 48, 1479-1503. https://doi.org/10.1111/j.1467-6486.2010.00985.x
[14] Zhao, F., Wang, T.N. and Zhang, L. (2012) Empirical Research on the Effect of Diversification Strategy on Firm Performance. China Soft Science, 11, 111-122.
[15] Chen, S., Xiao, X.X., Yang, Y. and Zou, C. (2014) CEO Power, Strategic Differences and Firm Performance: A Regulatory Effect Based on Environmental Dynamics. Research in Finance and Trade, 25, 7-16.
[16] Lei, H., Wang, Y.N., Nie, S.S. and Ouyang, L.P. (2015). Research on Time Lag Effect of Competitive Strategy Performance Based on Financial Performance Composite Index. Accounting Research, 5, 64-71.
[17] Ye, K.T., Zhang, H. and Zhang, Y.X. (2014) Value Correlation between Corporate Strategy Differences and Accounting Information. Accounting Research, 5, 44-51.
[18] Ma, X.M. (2015) Corporate Strategic Differences, Information Disclosure Quality and Accounting Information Value Relevance. Doctoral Dissertation, Zhejiang Gongshang University.
[19] Ye, K.T., Dong, X.Y. and Cui, Y.J. (2015) Corporate Strategic Positioning and Accounting Earnings Management Behavior Selection. Accounting Research, 2015, 23-29.
[20] Liu X. ,et al. (2016)Does the Type of Firm’s Strategy Affect the Characteristics of Earnings? A Survey of the Perspective of Accounting Conservatism Nankai Business Review 19, 111-121.
[21] Watts, R.L. (1993) A Proposal for Research on Conservatism.
[22] Watts, R.L. (2003) Conservatism in Accounting Part I: Explanations and Implications. Accounting Horizons, 17, 207-221. https://doi.org/10.2308/acch.2003.17.3.207
[23] Ahmed, A.S., Billings, B.K., Morton, R.M. and Stanford-Harris, M. (2002) The Role of Accounting Conservatism in Mitigating Bondholder-Shareholder Conflicts over Dividend Policy and in Reducing Debt Costs. The Accounting Review, 77, 867-890. https://doi.org/10.2308/accr.2002.77.4.867
[24] Nikolaev, V.V. (2010) Debt Covenants and Accounting Conservatism. Journal of Accounting Research, 48, 51-89. https://doi.org/10.1111/j.1475-679X.2009.00359.x
[25] Liu, F.W. and Wang, Y. (2006) Empirical Study of the Influence of Corporate Governance Mechanisms on Accounting Conservatism. Journal of Shanghai Lixin University of Commerce, 20, 16-22.
[26] Chen, X.D. and Huang, D.S. (2007) Corporate Governance and Accounting Conservatism: Empirical Research Based on Listed Companies. Securities Market Report, 3, 10-17.
[27] Liu, Y.G., Wu, X.M. and Jiang, T. (2010) Nature of Property Rights, Debt Financing and Accounting Conservatism: Empirical Evidence from Chinese Listed Companies. Accounting Research, 1, 43-50.
[28] LaFond, R. (2005) The Influence of Ownership Structure on Earnings Conservatism and the Informativeness of Stock Prices: An International Comparison. Sloan School of Management. Working Paper.
[29] Cao, Y., Li, L. and Sun, J. (2005) Empirical Study of the Influence of Corporate Control on the Soundness of Accounting Earnings. Economic Management, 14, 34-42.
[30] Fang, H.X. and Zhang, Z.P. (2012) Internal Control Quality and Accounting Conservatism: Evidence from the 2007-2010 Annual Report of Shenzhen A Share Company. Auditing and Economic Research, 5.
[31] Ahmed, A.S. and Duellman, S. (2007) Accounting Conservatism and Board of Director Characteristics: An Empirical Analysis. Journal of Accounting and Economics, 43, 411-437. https://doi.org/10.1016/j.jacceco.2007.01.005
[32] LaFond, R. and Watts, R.L. (2008) The Information Role of Conservatism. The Accounting Review, 83, 447-478. https://doi.org/10.2308/accr.2008.83.2.447
[33] Zhang, Z.G., Liu, Y.L. and Tan, D.J. (2011) Managerial Background Characteristics and Accounting Conservatism: Empirical Evidence from Chinese Listed Companies. Accounting Research, 7, 11-18.
[34] Ahmed, A.S. and Duellman, S. (2013) Managerial Overconfidence and Accounting Conservatism. Journal of Accounting Research, 51, 1-30. https://doi.org/10.1111/j.1475-679X.2012.00467.x
[35] Sun, G.G. and Zhao, J.Y. (2014) The Nature of Property Rights Differences, Management Overconfidence and Accounting Conservatism. Accounting Research, 5, 52-58.
[36] Shen, Y.J., Liang, S.K. and Chen, D.H. (2013) Employee Compensation and Accounting Conservatism-Based on Empirical Evidence from Chinese Listed Companies. Accounting Research, 4, 73-80.
[37] Kellogg, R.L. (1984) Accounting Activities, Security Prices, and Class Action Lawsuits. Journal of Accounting and Economics, 6, 185-204. https://doi.org/10.1016/0165-4101(84)90024-7
[38] Ball, R., Robin, A. and Wu, J.S. (2003) Incentives versus Standards: Properties of Accounting Income in Four East Asian Countries. Journal of Accounting and Economics, 36, 235-270. https://doi.org/10.1016/j.jacceco.2003.10.003
[39] Zhou, Z.J. and Du, X.Q. (2012) Tax Burden, Accounting Conservatism and Compensation Performance Sensitivity. Financial Research, 10, 167-179.
[40] Che F. ,et al. (2012)Income Tax Reform, Accounting-Tax Differences and Accounting Conservatism Journal of Zhongnan University of Economics and Law 6, 93-99.
[41] Zhu, C.F. and Li, Z.W. (2008) Research on the Effect of State Holding on Accounting Conservatism. Accounting Research, 5, 38-45.
[42] Du, X.Q., Lei, Y. and Guo, J.H. (2009) Political Connections, Political Contacts, and Accounting Conservatism of Private Listed Companies. China Industrial Economy, 7, 87-97.
[43] Chen, Y.Y., Tan, Y. and Tan, J.S. (2013) Political Connections and Accounting Conservatism. Nankai Management Review, 1, 33-40.
[44] Jiang, Y. and Tian, K.R. (2013) Internal Characteristics of State-Controlled Listed Companies, Government Grants and Accounting Conservatism. Auditing and Economic Research, 1, 77-86.
[45] Lin, H. (2006) Accounting Discretion and Managerial Conservatism: An Intertemporal Analysis. Contemporary Accounting Research, 23, 1017-1041. https://doi.org/10.1506/0343-6720-V320-4730
[46] Wang, R.Z., ó Hogartaigh, C. and van Zijl, T. (2009) A Signaling Theory of Accounting Conservatism.
[47] Stiglitz, J.E. and Weiss, A. (1981) Credit Rationing in Markets with Imperfect Information. The American Economic Review, 71, 393-410.
[48] Mintzberg, H. (1978) Patterns in Strategy Formation. Management Science, 24, 934-948. https://doi.org/10.1287/mnsc.24.9.934
[49] Yang, H.J. (2007) A Review of Research on Accounting Conservatism. Accounting Research, 1, 4.
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Pressure-Loss Coefficient of 90 deg Sharp-Angled Miter Elbows | J. Fluids Eng. | ASME Digital Collection
Pressure-Loss Coefficient of 90 deg Sharp-Angled Miter Elbows
Wameedh T. M. Al-Tameemi,
Wameedh T. M. Al-Tameemi
Sheffield S1 4ET, UK;
Reconstruction and Projects Office,
e-mail: wtal-tameemi1@sheffield.ac.uk
Sheffield S1 4ET, UK
e-mail: p.ricco@sheffield.ac.uk
Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 26, 2017; final manuscript received December 12, 2017; published online January 30, 2018. Assoc. Editor: Moran Wang.
Al-Tameemi, W. T. M., and Ricco, P. (January 30, 2018). "Pressure-Loss Coefficient of 90 deg Sharp-Angled Miter Elbows." ASME. J. Fluids Eng. June 2018; 140(6): 061102. https://doi.org/10.1115/1.4038986
The pressure drop across
90deg
sharp-angled miter elbows connecting straight circular pipes is studied in a bespoke experimental facility by using water and air as working fluids flowing in the range of bulk Reynolds number
500<Re<60,000
. To the best of our knowledge, the dependence on the Reynolds number of the pressure drop across the miter elbow scaled by the dynamic pressure, i.e., the pressure-loss coefficient
K
, is reported herein for the first time. The coefficient is shown to decrease sharply with the Reynolds number up to about
Re=20,000
and, at higher Reynolds numbers, to approach mildly a constant
K=0.9
, which is about 20% lower than the currently reported value in the literature. We quantify this relation and the dependence between
K
and the straight-pipe friction factor at the same Reynolds number through two new empirical correlations, which will be useful for the design of piping systems fitted with these sharp elbows. The pressure drop is also expressed in terms of the scaled equivalent length, i.e., the length of a straight pipe that would produce the same pressure drop as the elbow at the same Reynolds number.
Duct, Pressure drop, Channel, Pipe flows
Pipes, Pressure, Pressure drop, Water, Reynolds number, Flow (Dynamics), Fluids, Friction
Flow Characteristics in the Curved Rectangular Channels: Visualization of Secondary Flow
Prediction of Pressure Drop for Turbulent Fluid Flow in 90 Bends
An Experimental Investigation Into the Pressure Drop for Turbulent Flow in 90 Elbow Bends
, “A Standard Method to Determine Loss Coefficients of Conduit Components Based on the Second Law of Thermodynamics,”
Loss Coefficients for Periodically Unsteady Flows in Conduit Components: Illustrated for Laminar Flow in a Circular Duct and a 90 Degree Bend
Losses Due to Conduit Components: An Optimization Strategy and Its Application
, Vol. 21, United States Department of Commerce, Washington, DC.
, “Loss of Energy in Miter Bends,” Transactions of the Munich Hydraulic Institute, American Society of Mechanical Engineers, New York, Bulletin No. 3.
Energy Loss in Smooth-and Rough-Surfaced Bends and Curves in Pipe Lines
Trans. Hydraul. Inst. Munich Tech. Univ.
Prediction of Compressible Flow Pressure Losses in 30–150 Deg Sharp-Cornered Bends
Aekula
CFD Predictions and Experimental Comparisons of Pressure Drop Effects of Turning Vanes in 90 Duct Elbows
Flow of Fluids Through Valves, Fittings, and Pipes
, Newyork.
Dev. Chem. Eng. Miner. Process.
The Flow of Fluids Through Commercial Pipe Lines
, “NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP. 9.0,” National Institute of Standards and Technology, Boulder, CO.
, Sausalito, CA.
Friction-Factor Equation Spans All Fluid-Flow Regimes
.https://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0ahUKEwiytpn08uPYAhVGXKwKHVK7BWMQFggwMAI&url=http%3A%2F%2Ffiles.engineering.com%2Fdownload.aspx%3Ffolder%3D85c0f3a6-a102-4a22-9d35-f15858c0dd2b%26file%3DCEM_-_Friction-factor_equation_(1977).pdf&usg=AOvVaw0wy1cyeZUu01eJFafbQ6sX
Pipe Flow Measurements Over a Wide Range of Reynolds Numbers Using Liquid Helium and Various Gases
A Direct Numerical Simulation Study on the Mean Velocity Characteristics in Turbulent Pipe Flow
Turbulent Drag Reduction by Biopolymers in Large Scale Pipes
A Mechanistic Heat Transfer Correlation for Non-Boiling Two-Phase Flow in Horizontal, Inclined and Vertical Pipes
Calculations of Pressure Drop in a Circular Duct Using the Traditional Method Compared to That Using the Constricted Flow Diameter
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Signal Visualization and Measurements in MATLAB - MATLAB & Simulink - MathWorks Australia
Signal Visualization in Time and Frequency Domains
Create a sine wave with a frequency of 100 Hz sampled at 1000 Hz. Generate five seconds of the 100 Hz sine wave with additive
N\left(0,0.0025\right)
white noise in one-second intervals. Send the signal to a time scope and spectrum analyzer for display and measurement.
SampPerFrame = 1000;
SW = dsp.SineWave('Frequency',100,...
'SamplesPerFrame',SampPerFrame);
'YLimits',[-2, 2],...
'Method','welch','AveragingMethod','exponential');
sigData = SW() + 0.05*randn(SampPerFrame,1);
SA(sigData);
TS(sigData);
Using the time scope, you can make a number of time-domain signal measurements.
Cursor Measurements - Puts screen cursors on all scope displays.
Signal Statistics - Displays maximum, minimum, peak-to-peak difference, mean, median, RMS values of a selected signal, and the times at which the maximum and minimum occur.
Bilevel Measurements - Displays information about a selected signal's transitions, aberrations, and cycles.
Peak Finder - Displays maxima and the times at which they occur.
You can enable and disable these measurements from the Measurements tab.
To illustrate the use of measurements in the time scope, simulate an ECG signal. Use the ecg function to generate 2700 samples of the signal. Use a Savitzky-Golay filter to smooth the signal and periodically extend the data to obtain approximately 11 periods.
x = 3.5*ecg(2700).';
y = repmat(sgolayfilt(x,0,21),[1 13]);
sigData = y((1:30000) + round(2700*rand(1))).';
Display the signal in the time scope and use the Peak Finder and Data Cursor measurements. Assume a sample rate of 4 kHz.
TS_ECG = timescope('SampleRate',4000,...
'TimeSpanSource','Auto',...
'ShowGrid', true);
TS_ECG(sigData);
TS_ECG.YLimits = [-4, 4];
Enable Peak Measurements from the Measurements tab by clicking the corresponding toolstrip button. The Peaks pane appears at the bottom of the time scope window. For the Num Peaks property, enter 8 and press Enter. In the Peaks pane, the time scope displays a list of 8 peak amplitude values and the times at which they occur.
There is a constant time difference of 0.675 seconds between each heartbeat. Therefore, the heart rate of the ECG signal is given by the following equation:
\frac{60\phantom{\rule{0.16666666666666666em}{0ex}}sec/min}{0.675\phantom{\rule{0.16666666666666666em}{0ex}}sec/beat}=88.89\phantom{\rule{0.16666666666666666em}{0ex}}beats/min\phantom{\rule{0.16666666666666666em}{0ex}}\left(bpm\right)
Enable Cursor Measurements from the Measurements tab by clicking the corresponding toolstrip button. The cursors appear on the time scope with a box showing the change in time and value between the two cursors. You can drag the cursors and use them to measure the time between events in the waveform. As you drag a cursor, the time an value at the cursor appears. This figure shows how to use cursors to measure the time interval between peaks in the ECG waveform. The
\Delta T
measurement in the cursor box demonstrates that the time interval between the two peaks is 0.675 seconds corresponding to a heart rate of 1.482 Hz or 88.9 beats/min.
Signal Statistics and Bilevel Measurements
You can also select Signal Statistics and various bilevel measurements from the Measurements tab. Signal Statistics can be used to determine the signal's minimum and maximum values as well as other metrics like the peak-to-peak, mean, median, and RMS values. Bilevel measurements can be used to determine information about rising and falling transitions, transition aberrations, overshoot and undershoot information, settling time, pulse width, and duty cycle. To read more about these measurements, see Configure Time Scope MATLAB Object.
This section explains how to make frequency domain measurements with the spectrum analyzer.
The spectrum analyzer provides the following measurements:
Cursor Measurements - places cursors on the spectrum display.
Peak Finder - displays maxima and the frequencies at which they occur.
Channel Measurements - displays occupied bandwidth and ACPR channel measurements.
Distortion Measurements - displays harmonic and intermodulation distortion measurements.
You can enable and disable these measurements from the spectrum analyzer toolstrip.
To illustrate the use of measurements with the spectrum analyzer, create a 2.5 kHz sine wave sampled at 48 kHz with additive white Gaussian noise. Evaluate a high-order polynomial (9th degree) at each signal value to model non-linear distortion. Display the signal in a spectrum analyzer.
SW = dsp.SineWave('Frequency',2500,...
SA_Distortion = spectrumAnalyzer('SampleRate',Fs,...
'PlotAsTwoSidedSpectrum',false);
y = [1e-6 1e-9 1e-5 1e-9 1e-6 5e-8 0.5e-3 1e-6 1 3e-3];
x = SW() + 1e-8*randn(SampPerFrame,1);
sigData = polyval(y, x);
SA_Distortion(sigData);
release(SA_Distortion);
Enable the harmonic distortion measurements by selecting the Distortion button on the Measurements tab of the spectrum analyzer toolstrip. In the Distortion section, change the value for Num Harmonics to 9 and check the Label Harmonics checkbox. In the Harmonic Distortion panel at the bottom of the spectrum analyzer window, you see the value of the fundamental close to 2500 Hz and 8 harmonics as well as their SNR, SINAD, THD and SFDR values, which are referenced with respect to the fundamental output power.
You can track time-varying spectral components by using the Peak Finder measurements. You can show and optionally label up to 100 peaks. To invoke the Peak Finder, select the Peak Finder button on the Measurements tab of the spectrum analyzer toolstrip.
To illustrate the use of Peak Finder, create a signal consisting of the sum of three sine waves with frequencies of 5, 15, and 25 kHz and amplitudes of 1, 0.1, and 0.01 respectively. The data is sampled at 100 kHz. Add
N\left(0,1{0}^{-8}\right)
white Gaussian noise to the sum of sine waves and display the one-sided power spectrum in the spectrum analyzer.
SW1 = dsp.SineWave(1e0,5e3,0,...
SW2 = dsp.SineWave(1e-1,15e3,0,...
SA_Peak = spectrumAnalyzer('SampleRate',Fs,...
sigData = SW1() + SW2() + SW3() + 1e-4*randn(SampPerFrame,1);
SA_Peak(sigData);
release(SA_Peak);
Enable the Peak Finder to label the three sine wave frequencies. The frequency values and powers in dBm are displayed below the plot. You can increase or decrease the maximum number of peaks, specify a minimum peak distance, and change other settings in the Peaks section of the Measurements tab.
To learn more about the use of measurements with the spectrum analyzer, see the Spectrum Analyzer Measurements example.
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Gaussian_curvature Knowpia
From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).
{\displaystyle K=\kappa _{1}\kappa _{2}.}
If both principal curvatures are of the same sign: κ1κ2 > 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
If the principal curvatures have different signs: κ1κ2 < 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
If one of the principal curvatures is zero: κ1κ2 = 0, the Gaussian curvature is zero and the surface is said to have a parabolic point.
Relation to geometriesEdit
Relation to principal curvaturesEdit
{\displaystyle K={\frac {{\bigl \langle }(\nabla _{2}\nabla _{1}-\nabla _{1}\nabla _{2})\mathbf {e} _{1},\mathbf {e} _{2}{\bigr \rangle }}{\det g}},}
{\displaystyle K(\mathbf {p} )=\det S(\mathbf {p} ),}
where S is the shape operator.
Total curvatureEdit
{\displaystyle \sum _{i=1}^{3}\theta _{i}=\pi +\iint _{T}K\,dA.}
A more general result is the Gauss–Bonnet theorem.
Important theoremsEdit
Theorema egregiumEdit
Gauss–Bonnet theoremEdit
Surfaces of constant curvatureEdit
Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. Minding also raised the question of whether a closed surface with constant positive curvature is necessarily rigid.
Liebmann's theorem (1900) answered Minding's question. The only regular (of class C2) closed surfaces in R3 with constant positive Gaussian curvature are spheres.[2] If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature.[3]
Hilbert's theorem (1901) states that there exists no complete analytic (class Cω) regular surface in R3 of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class C2 immersed in R3, but breaks down for C1-surfaces. The pseudosphere has constant negative Gaussian curvature except at its singular cusp.[4]
There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution
{\displaystyle (\phi (v)\cos(u),\phi (v)\sin(u),\psi (v))}
{\displaystyle \phi (v)=C\cos v}
{\displaystyle \psi (v)=\int _{0}^{v}{\sqrt {1-C^{2}\sin ^{2}v'}}\ dv'}
(an incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for
{\displaystyle C\neq 1}
either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is pseudosphere.[5]
Alternative formulasEdit
Gaussian curvature of a surface in R3 can be expressed as the ratio of the determinants of the second and first fundamental forms II and I:
{\displaystyle K={\frac {\det(\mathrm {I\!I} )}{\det(\mathrm {I} )}}={\frac {LN-M^{2}}{EG-F^{2}}}.}
The Brioschi formula (after Francesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form:
{\displaystyle K={\frac {{\begin{vmatrix}-{\frac {1}{2}}E_{vv}+F_{uv}-{\frac {1}{2}}G_{uu}&{\frac {1}{2}}E_{u}&F_{u}-{\frac {1}{2}}E_{v}\\F_{v}-{\frac {1}{2}}G_{u}&E&F\\{\frac {1}{2}}G_{v}&F&G\end{vmatrix}}-{\begin{vmatrix}0&{\frac {1}{2}}E_{v}&{\frac {1}{2}}G_{u}\\{\frac {1}{2}}E_{v}&E&F\\{\frac {1}{2}}G_{u}&F&G\end{vmatrix}}}{\left(EG-F^{2}\right)^{2}}}}
For an orthogonal parametrization (F = 0), Gaussian curvature is:
{\displaystyle K=-{\frac {1}{2{\sqrt {EG}}}}\left({\frac {\partial }{\partial u}}{\frac {G_{u}}{\sqrt {EG}}}+{\frac {\partial }{\partial v}}{\frac {E_{v}}{\sqrt {EG}}}\right).}
For a surface described as graph of a function z = F(x,y) with
{\displaystyle F_{x}(P)=0}
{\textstyle F_{y}(P)=0}
, Gaussian curvature at
{\displaystyle P}
{\displaystyle K={\frac {F_{xx}\cdot F_{yy}-F_{xy}^{2}}{\left(1+F_{x}^{2}+F_{y}^{2}\right)^{2}}}}
For an implicitly defined surface, F(x,y,z) = 0, the Gaussian curvature can be expressed in terms of the gradient ∇F and Hessian matrix H(F):[8][9]
{\displaystyle K=-{\frac {\begin{vmatrix}H(F)&\nabla F^{\mathsf {T}}\\\nabla F&0\end{vmatrix}}{|\nabla F|^{4}}}=-{\frac {\begin{vmatrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\\end{vmatrix}}{|\nabla F|^{4}}}}
For a surface with metric conformal to the Euclidean one, so F = 0 and E = G = eσ, the Gauss curvature is given by (Δ being the usual Laplace operator):
{\displaystyle K=-{\frac {1}{2e^{\sigma }}}\Delta \sigma .}
Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:[10]
{\displaystyle K=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}}
Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane:[10]
{\displaystyle K=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}}
Gaussian curvature may be expressed with the Christoffel symbols:[11]
{\displaystyle K=-{\frac {1}{E}}\left({\frac {\partial }{\partial u}}\Gamma _{12}^{2}-{\frac {\partial }{\partial v}}\Gamma _{11}^{2}+\Gamma _{12}^{1}\Gamma _{11}^{2}-\Gamma _{11}^{1}\Gamma _{12}^{2}+\Gamma _{12}^{2}\Gamma _{12}^{2}-\Gamma _{11}^{2}\Gamma _{22}^{2}\right)}
Earth's Gaussian radius of curvature
^ Porteous, I. R. (1994). Geometric Differentiation. Cambridge University Press. ISBN 0-521-39063-X.
^ Kühnel, Wolfgang (2006). Differential Geometry: Curves, Surfaces, Manifolds. American Mathematical Society. ISBN 0-8218-3988-8.
^ Gray, Alfred (1997). "28.4 Hilbert's Lemma and Liebmann's Theorem". Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.). CRC Press. pp. 652–654. ISBN 9780849371646. .
^ "Hilbert theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ Carmo, Manfredo Perdigão do (2016) [First published 1976]. Differential geometry of curves and surfaces (2nd ed.). Mineola, NY: Dover Publications. p. 171. ISBN 978-0-486-80699-0 – via zbMATH.
^ Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. p. 228. ISBN 0-8284-1087-9.
^ "General investigations of curved surfaces of 1827 and 1825". [Princeton] The Princeton university library. 1902.
^ Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces". Computer Aided Geometric Design. 22 (7): 632–658. CiteSeerX 10.1.1.413.3008. doi:10.1016/j.cagd.2005.06.005.
^ Spivak, M. (1975). A Comprehensive Introduction to Differential Geometry. Vol. 3. Boston: Publish or Perish.
^ a b Bertrand–Diquet–Puiseux theorem
^ Struik, Dirk (1988). Lectures on Classical Differential Geometry. Courier Dover Publications. ISBN 0-486-65609-8.
Grinfeld, P. (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 978-1-4614-7866-9.
"Gaussian curvature", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Solubility of carbon and nitrogen in a sulfur-bearing iron melt: Constraints for siderophile behavior at upper mantle conditions | American Mineralogist | GeoScienceWorld
V.S. Sobolev Institute of Geology and Mineralogy, Siberian Branch of the Russian Academy of Sciences, 3 Koptyug Avenue, Novosibirsk, 630090 Russia
Novosibirsk State University, 2 Pirogov Street, Novosibirsk, 630090 Russia
* E-mail: sokola@igm.nsc.ru. Orcid 0000-0002-7721-5152.
Alexander F. Khokhryakov;
Yuri M. Borzdov;
Igor N. Kupriyanov;
Alexander G. Sokol, Alexander F. Khokhryakov, Yuri M. Borzdov, Igor N. Kupriyanov, Yuri N. Palyanov; Solubility of carbon and nitrogen in a sulfur-bearing iron melt: Constraints for siderophile behavior at upper mantle conditions. American Mineralogist 2019;; 104 (12): 1857–1865. doi: https://doi.org/10.2138/am-2019-7103
Carbon solubility in a liquid iron alloy containing nitrogen and sulfur has been studied experimentally in a carbon-saturated Fe-C-N-S-B system at pressures of 5.5 and 7.8 GPa, temperatures of 1450 to 1800 °C, and oxygen fugacities from the IW buffer to log fO2 ΔIW-6 (ΔIW is the logarithmic difference between experimental fO2 and that imposed by the coexistence of iron and wüstite). Carbon saturation of Fe-rich melts at 5.5 and 7.8 GPa maintains crystallization of flaky graphite and diamond. Diamond containing 2100–2600 ppm N and 130–150 ppm B crystallizes in equilibrium with BN within the diamond stability field at 7.8 GPa and 1600 to 1800 °C, while graphite forms at other conditions. The solubility of carbon in the C-saturated metal melt free from nitrogen and sulfur is 6.2 wt% C at 7.8 GPa and 1600 °C and decreases markedly with increasing nitrogen. A 1450–1600 °C graphite-saturated iron melt with 6.2–8.8 wt% N can dissolve: 3.6–3.9 and 1.4–2.5 wt% C at 5.5 and 7.8 GPa, respectively. However, the melt equilibrated with boron nitride and containing 1–1.7 wt% sulfur and 500–780 ppm boron dissolves twice less nitrogen while the solubility of carbon remains relatively high (3.8–5.2 wt%). According to our estimates, nitrogen partitions between diamond and the iron melt rich in volatiles at
DNDm/Met=0.013−0.024
. The pressure increase in the Fe-C-N system affects iron affinity of N and C: it increases in nitrogen but decreases in carbon. The reduction of C solubility in a Fe-rich melt containing nitrogen and sulfur may have had important consequences in the case of imperfect equilibration between the core and the mantle during their separation in the early Earth history. The reduction of C solubility allowed C supersaturation of the liquid iron alloy and crystallization of graphite and diamond. The carbon phases could float in the segregated core liquid and contribute to the carbon budget of the overlying silicate magma ocean. Therefore, the process led to the formation of graphite and diamond, which were the oldest carbon phases in silicate mantle.
Discussion of “Oxygen, iron, and sulfur geochemical cycles on early Earth: Paradigms and contradictions” (Ohmoto et al.)
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Controller-driven DC-DC inverting or four-switch step-up or step-down voltage regulator - MATLAB - MathWorks Deutschland
LC Parameters
Controller-driven DC-DC inverting or four-switch step-up or step-down voltage regulator
The Buck-Boost Converter block represents a DC-DC converter that can either step up or step down DC voltage from one side of the converter to the other as driven by an attached controller and gate-signal generator. Buck-boost converters are also known as step-up/step-down voltage regulators because they can increase or decrease voltage magnitude.
The block can also invert voltage so that the polarity of the output voltage is the opposite of the polarity of the input voltage. The magnitude of the output voltage depends on the duty cycle.
The Buck-Boost Converter block allows you to model an inverting buck-boost converter with one switching device or a buck-boost converter with four switching devices. Options for the type of switching devices are:
You can model this converter as an inverting buck-boost converter with a physical signal gate control port or with two electrical control ports, or as a four-switch buck-boost converter with an electrical control port. To select the gate control port, set the Modeling option parameter to either:
PS control port — Inverting buck-boost converter with a physical signal port.
Electrical control ports — Inverting buck-boost converter with one positive and one negative electrical conserving ports. To control switching device gates using Simscape™ Electrical™ blocks, select this option.
Four-switch converter — Four-switch buck-boost converter with an electrical conserving port.
The inverting converter models contain a switching device, a diode, an inductor, and an output capacitor.
The four-switch converter model contains four switching devices, an inductor, and an output capacitor.
You can include a snubber circuit for each switching device. Snubber circuits contain a series-connected resistor and capacitor. They protect switching devices against high voltages that inductive loads produce when the device turns off the voltage supply to the load. Snubber circuits also prevent excessive rates of current change when a switching device turns on.
Multiplex the converted gate-control signals into a single vector using a Four-Pulse Gate Multiplexer
To enable this port, set Modeling option to Four-switch converter.
Modeling option — Whether to model inverting or four-switch buck-boost converter
PS control port (default) | Electrical control ports | Four-switch converter
Whether to model an inverting or a four-switch buck-boost converter and specify physical or electrical control ports for the switching device gate.
Switching device type for the converter. For the four-switch model, the switches are identical.
This table shows how the visibility of Diode parameters depends on how you configure the Model dynamics and Reverse recovery time parameterization parameters. To learn how to read this table, see Parameter Dependencies.
Diode with no dynamics Diode with charge dynamics
Forward voltage Forward voltage
On resistance On resistance
Off conductance Off conductance
Specify stretch factor Specify reverse recovery time directly Specify reverse recovery charge
Reverse recovery time stretch factor Reverse recovery time, trr Reverse recovery charge, Qrr
Diode with no dynamics (default) | Diode with charge dynamics
Diode with no dynamics — Select this option to prioritize simulation speed using the Diode block.
-\frac{{i}^{2}{}_{RM}}{2a}
Capacitance — Capacitance
[2] Xiaoyong, R., Z. Tang, X. Ruan, J. Wei and G. Hua. Four Switch Buck-Boost Converter for Telecom DC-DC power supply applications. Twenty-Third Annual IEEE Applied Power Electronics Conference and Exposition. Austin, TX: 2008, pp 1527-1530.
Average-Value DC-DC Converter | Bidirectional DC-DC Converter | Buck Converter | Boost Converter | Converter | GTO | IGBT (Ideal Switching) | MOSFET (Ideal Switching) | Ideal Semiconductor Switch | PWM Generator | PWM Generator (Three-phase, Two-level) | Six-Pulse Gate Multiplexer | Three-Level Converter (Three-Phase) | Thyristor (Piecewise Linear)
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Home : Support : Online Help : gamma
The names Γ and γ may refer to any of several distinct quantities or operations.
For the number 0.577215... (Euler's constant), see gamma.
For the Gamma function, see GAMMA.
For the Gamma distribution in statistics, see Statistics[Distributions][Gamma].
For adjusting the gamma of an image, see ImageTools[Gamma].
For the Dirac gamma matrices, see Physics[Dgamma].
For the Gamma process in finance, see Finance[GammaProcess].
For the risk measure Gamma under the Black-Scholes model, see Finance[BlackScholesGamma].
For the Digamma and Polygamma functions, see Psi.
\mathrm{limit}\left(\left(\mathrm{sum}\left(\frac{1}{k},k=1..n\right)\right)-\mathrm{log}\left(n\right),n=\mathrm{\infty }\right)
\textcolor[rgb]{0,0,1}{\mathrm{\gamma }}
\mathrm{evalf}[20]\left(\mathrm{\gamma }\right)
\textcolor[rgb]{0,0,1}{0.57721566490153286061}
\mathrm{diff}\left(\mathrm{\Gamma }\left(x\right),x\right)
\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{GammaDistribution}\left(b,c\right)\right)
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_R}}
\mathrm{Variance}\left(X\right)
{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
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2013 Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection
Xinchao Yang, Xiju Zong, Xingong Cheng, Zhenlai Han
The stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. The existence of the Hopf bifurcation with delay
\tau
at the endemic equilibria is established by analyzing the distribution of the characteristic values. The explicit formulae which determine the direction of the bifurcations, stability, and the other properties of the bifurcating periodic solutions are given by using the normal form theory and the center manifold theorem. Numerical simulation verifies the theoretical results.
Xinchao Yang. Xiju Zong. Xingong Cheng. Zhenlai Han. "Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection." J. Appl. Math. 2013 1 - 15, 2013. https://doi.org/10.1155/2013/875783
Xinchao Yang, Xiju Zong, Xingong Cheng, Zhenlai Han "Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-15, (2013)
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Spatiotemporal Analysis of the Foreshock–Mainshock–Aftershock Sequence of the 6 July 2017 Mw 5.8 Lincoln, Montana, Earthquake | Seismological Research Letters | GeoScienceWorld
Spatiotemporal Analysis of the Foreshock–Mainshock–Aftershock Sequence of the 6 July 2017 Mw 5.8 Lincoln, Montana, Earthquake
Nicole D. McMahon;
Nicole D. McMahon
Department of Geosciences, Warner College of Natural Resources, Colorado State University, 1482 Campus Delivery, Fort Collins, Colorado 80523‐1482 U.S.A., Nicole.McMahon@colostate.edu, Rick.Aster@colostate.edu
Also at National Earthquake Information Center, U.S. Geological Survey, Denver Federal Center, P.O. Box 25046, MS 966, Golden, Colorado 80225 U.S.A.
William L. Yeck;
William L. Yeck
National Earthquake Information Center, U.S. Geological Survey, Denver Federal Center, P.O. Box 25046, MS 966, Golden, Colorado 80225 U.S.A., wyeck@usgs.gov, benz@usgs.gov
Michael C. Stickney;
Montana Bureau of Mines and Geology, Montana College of Mineral Science and Technology, 1300 West Park Street, Butte, Montana 59701‐8997 U.S.A., mstickney@mtech.edu
Richard C. Aster;
Department of Geosciences, University of Montana, 32 Campus Drive #1296, Missoula, Montana 59812‐1296 U.S.A., Hilary.Martens@mso.umt.edu
Harley M. Benz
Nicole D. McMahon, William L. Yeck, Michael C. Stickney, Richard C. Aster, Hilary R. Martens, Harley M. Benz; Spatiotemporal Analysis of the Foreshock–Mainshock–Aftershock Sequence of the 6 July 2017
Mw
5.8 Lincoln, Montana, Earthquake. Seismological Research Letters 2018;; 90 (1): 131–139. doi: https://doi.org/10.1785/0220180180
Mw
5.8 earthquake occurred on 6 July 2017 at 12.2‐km depth, 11 km southeast of Lincoln in west‐central Montana. No major damage or injuries were reported; however, the widely felt mainshock generated a prolific aftershock sequence with more than 1200 located events through the end of 2017. The Lincoln event is the latest in a series of moderate to large earthquakes that have affected western Montana. We characterize the spatiotemporal evolution of the sequence using matched filter detection and multiple event relocation techniques. Moment tensor solutions and aftershock locations indicate faulting occurred on a 9‐km‐long north‐northeast‐striking, near‐vertical, strike‐slip fault antithetic to the Lewis and Clark Line, the main through‐going fault system. Seismicity primarily occurs between 6‐ and 16‐km depth, consistent with seismicity in the Intermountain Seismic Belt. We estimate a fault rupture area of
∼64 km2
∼30 cm
of average fault displacement. We identified four foreshocks during the three days before and 3005 aftershocks in the three weeks after the mainshock. The supplemented catalog frequency–magnitude distribution has a
b
‐value of 0.79 and a minimum magnitude of completeness of 0.7. The overall decay rate is consistent with a modified Omori decay law
p
‐value of 0.76 and
c
‐value of 0.32. This event demonstrates that unmapped faults antithetic to major geologic structures play a role in accommodating regional strain in western Montana and can host significant earthquakes.
Lincoln earthquake 2017
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Lemma 65.4.6 (03BX)—The Stacks project
Lemma 65.4.6. Let $S$ be a scheme. There exists a unique topology on the sets of points of algebraic spaces over $S$ with the following properties:
if $X$ is a scheme over $S$, then the topology on $|X|$ is the usual one (via the identification of Lemma 65.4.2),
for every morphism of algebraic spaces $X \to Y$ over $S$ the map $|X| \to |Y|$ is continuous, and
for every étale morphism $U \to X$ with $U$ a scheme the map of topological spaces $|U| \to |X|$ is continuous and open.
Proof. Let $X$ be an algebraic space over $S$. Let $p : U \to X$ be a surjective étale morphism where $U$ is a scheme over $S$. We define $W \subset |X|$ is open if and only if $|p|^{-1}(W)$ is an open subset of $|U|$. This is a topology on $|X|$ (it is the quotient topology on $|X|$, see Topology, Lemma 5.6.2).
Let us prove that the topology is independent of the choice of the presentation. To do this it suffices to show that if $U'$ is a scheme, and $U' \to X$ is an étale morphism, then the map $|U'| \to |X|$ (with topology on $|X|$ defined using $U \to X$ as above) is open and continuous; which in addition will prove that (3) holds. Set $U'' = U \times _ X U'$, so that we have the commutative diagram
\[ \xymatrix{ U'' \ar[r] \ar[d] & U' \ar[d] \\ U \ar[r] & X } \]
As $U \to X$ and $U' \to X$ are étale we see that both $U'' \to U$ and $U'' \to U'$ are étale morphisms of schemes. Moreover, $U'' \to U'$ is surjective. Hence we get a commutative diagram of maps of sets
\[ \xymatrix{ |U''| \ar[r] \ar[d] & |U'| \ar[d] \\ |U| \ar[r] & |X| } \]
The lower horizontal arrow is surjective (see Lemma 65.4.4 or Lemma 65.4.5) and continuous by definition of the topology on $|X|$. The top horizontal arrow is surjective, continuous, and open by Morphisms, Lemma 29.36.13. The left vertical arrow is continuous and open (by Morphisms, Lemma 29.36.13 again.) Hence it follows formally that the right vertical arrow is continuous and open.
To finish the proof we prove (2). Let $a : X \to Y$ be a morphism of algebraic spaces. According to Spaces, Lemma 64.11.6 we can find a diagram
\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ X \ar[r]^ a & Y } \]
where $U$ and $V$ are schemes, and $p$ and $q$ are surjective and étale. This gives rise to the diagram
\[ \xymatrix{ |U| \ar[d]_ p \ar[r]_\alpha & |V| \ar[d]^ q \\ |X| \ar[r]^ a & |Y| } \]
where all but the lower horizontal arrows are known to be continuous and the two vertical arrows are surjective and open. It follows that the lower horizontal arrow is continuous as desired. $\square$
Comment #2333 by Wessel on December 20, 2016 at 13:47
I'm probably being too pedantic, but shouldn't the lemma include a phrase along the lines of "if X is a scheme, then |X| is endowed with the Zariski topology"? Otherwise, technically, we could endow everything with the discrete topology. A similar remark applies to Tag 04XL.
OK, it is not pedantic as much as cautious... technically you are correct... but I am going to leave it as is... for now...
Comment #4214 by David Holmes on May 14, 2019 at 04:55
I came here to make the same comment as Wessel, but then I saw Wessel's comment, so I won't (maybe I just did...). So maybe count this as another vote for changing it??
OK, I have added it as a condition in the lemma. But I am leaving the case of Lemma 99.4.7 alone for now because the typography suggests that we are dealing with the space associated to
\mathcal{X}
X
in different ways (for one we are trying to define the topology and for the second we already have it). See changes here.
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ULP Considerations of Native Floating-Point Operators - MATLAB & Simulink
Adherence of Native Floating Point Operators to IEEE-754 Standard
ULP Values of Floating Point Operators
The representation of infinitely real numbers with a finite number of bits requires an approximation. This approximation can result in rounding errors in floating-point computation. To measure the rounding errors, the floating-point standard uses relative error and ULP (Units in the Last Place) error. To learn about relative error, see Relative Accuracy and ULP Considerations.
If the exponent range is not upper-bounded, Units in Last Place (ULP) of a floating-point number x is the distance between two closest straddling floating-point numbers a and b nearest to x. The IEEE-754 standard requires that the result of an elementary arithmetic operation such as addition, multiplication, and division is correctly round. A correctly rounded result means that the rounded result is within 0.5 ULP of the exact result.
Native floating point technology in HDL Coder™ follows IEEE standard of floating-point arithmetic. Basic arithmetic operations such as addition, subtraction, multiplication, division, and reciprocal are mandated by IEEE to have zero ULP error. When you perform these operations in native floating-point mode, the numerical results obtained from the generated HDL code match the original Simulink® model.
Certain advanced math operations such as exponential, logarithm, and trigonometric operators have machine-specific implementation behaviors because these operators use recurring taylor series and remez expression based implementations. When you use these operators in native floating-point mode, there can be relatively small differences in numerical results between the Simulink model and the generated HDL code.
You can measure the difference in numerical results as a relative error or ULP. A nonzero ULP for these operators does not mean noncompliance with the IEEE standard. A ULP of one is equivalent to a relative error of 10^-7. You can ignore such relatively small errors by specifying a custom tolerance value for the ULP when generating a HDL test bench. For example, you can specify a custom floating-point tolerance of one ULP to ignore the error when verifying the generated code. For more information, see Floating-Point Tolerance Parameters.
The table enumerates the ULP of floating-point operators that have a nonzero ULP. In addition to these operators, the HDL Reciprocal block has a ULP of five.
Units in the Last Place (ULP) error
10^u 1
hypot 1
asinh 2
For certain floating-point input values, some blocks can produce simulation results that vary from the MATLAB® simulation results. To see the difference in results, before you generate code, enable generation of the validation model. In the Configuration Parameters dialog box, on the HDL Code Generation pane, select the Generate validation model check box.
If you perform computations that involve complex numbers and an exception such as Inf or NaN, the HDL simulation result with native floating point can potentially vary from the Simulink simulation result. For example, if you multiply a complex input with Inf, the Simulink simulation result is Infi whereas the HDL simulation result is NaN+Infi.
HDL Coder does not generate a mismatch error between reference and native floating-point values if both values are NaN.
If you compute the square root or logarithm of a negative number, the HDL simulation result with native floating point is 0. This result matches the simulation result when you verify the design with a SystemVerilog DPI test bench. In Simulink, the result obtained is NaN. According to the IEEE-754 standard, if you compute the square root or logarithm of a negative number, the result is that number itself.
If the input to the Direct Lookup Table (n-D) is of floating-point data type, but the elements of the table use a smaller data type, the generated HDL code can be potentially incorrect. For example, the input is of single type and the elements use uint8 type. To obtain accurate HDL simulation results, use the same data type for the input signal and the elements of the lookup table.
If you use the Cosine block with the inputs -7.729179E28 or 7.729179E28, the generated HDL code has a ULP of 4. For all other inputs, the ULP is 2.
When you use a Math Function block to compute mod(a,b) or rem(a,b), where a is the dividend and b is the divisor, the simulation result in native floating-point point mode varies from the MATLAB simulation result in these cases:
b\text{ is integer and }\frac{a}{b}>{2}^{32}
, the simulation result in native floating-point mode is zero. For such significant difference in magnitude between the numbers a and b, this implementation saves area on the target FPGA device.
\frac{a}{b}\text{ is close to }{2}^{23}
, the simulation result in native floating-point mode can potentially vary from the MATLAB simulation results.
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Sufficient Dilated LMI Conditions for H ∞ Static Output Feedback Robust Stabilization of Linear Continuous-Time Systems
2012 Sufficient Dilated LMI Conditions for
{H}_{\infty }
Static Output Feedback Robust Stabilization of Linear Continuous-Time Systems
Kamel Dabboussi, Jalel Zrida
New sufficient dilated linear matrix inequality (LMI) conditions for the
{H}_{\infty }
static output feedback control problem of linear continuous-time systems with no uncertainty are proposed. The used technique easily and successfully extends to systems with polytopic uncertainties, by means of parameter-dependent Lyapunov functions (PDLFs). In order to reduce the conservatism existing in early standard LMI methods, auxiliary slack variables with even more relaxed structure are employed. It is shown that these slack variables provide additional flexibility to the solution. It is also shown, in this paper, that the proposed dilated LMI-based conditions always encompass the standard LMI-based ones. Numerical examples are given to illustrate the merits of the proposed method.
Kamel Dabboussi. Jalel Zrida. "Sufficient Dilated LMI Conditions for
{H}_{\infty }
Static Output Feedback Robust Stabilization of Linear Continuous-Time Systems." J. Appl. Math. 2012 1 - 13, 2012. https://doi.org/10.1155/2012/812920
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Duke Math. J. 171 (7), (15 May 2022)
Approximation by juntas in the symmetric group, and forbidden intersection problems
David Ellis, Noam Lifshitz
Duke Math. J. 171 (7), 1417-1467, (15 May 2022) DOI: 10.1215/00127094-2021-0050
KEYWORDS: forbidden intersections, Symmetric group, Erdos–Ko–Rado problems, pseudorandomness, juntas, 05D05
A family of permutations
\mathcal{F}\subset {S}_{n}
is said to be t-intersecting if any two permutations in
\mathcal{F}
agree on at least t points. It is said to be
\left(t-1\right)
-intersection-free if no two permutations in
\mathcal{F}
agree on exactly
t-1
points. If
S,T\subset \left\{1,2,\dots ,n\right\}
|S|=|T|
\mathrm{\pi }:S\to T
is a bijection, then the π-star in
{S}_{n}
is the family of all permutations that agree with π on S. An s-star is a π-star such that π is a bijection between sets of size s.
Friedgut and Pilpel, and independently the first author, showed that if
\mathcal{F}\subset {S}_{n}
is t-intersecting, and n is sufficiently large depending on t, then
|\mathcal{F}|\le \left(n-t\right)!
; this proved a conjecture of Deza and Frankl from 1977.
Here, we prove a considerable strengthening of the Deza–Frankl conjecture, namely, that if n is sufficiently large depending on t, and
\mathcal{F}\subset {S}_{n}
\left(t-1\right)
-intersection-free, then
|\mathcal{F}|\le \left(n-t\right)!
, with equality iff
\mathcal{F}
is a t-star.
The main ingredient of our proof is a “junta approximation” result, namely, that any
\left(t-1\right)
-intersection-free family of permutations is essentially contained in a t-intersecting junta (a “junta” being a union of boundedly many
O\left(1\right)
-stars). The proof of our junta approximation result relies, in turn, on (i) a weak regularity lemma for families of permutations (which outputs a junta whose stars are intersected by
\mathcal{F}
in a weakly pseudorandom way), (ii) a combinatorial argument that “bootstraps” the weak notion of pseudorandomness into a stronger one, and finally (iii) a spectral argument for highly pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature and one being algebraic. The connection we demonstrate between these notions of pseudorandomness may find further applications.
Planes in four-space and four associated CM points
Menny Aka, Manfred Einsiedler, Andreas Wieser
KEYWORDS: equidistribution, planes in four-space, CM points, ergodic theory, number theory, glue group, 37A17, 11H55, 11F85
To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian
\mathrm{Gr}\left(2,4\right)
and four shapes of lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement, and the second two essentially arise from the accidental local isomorphism between
\mathrm{SO}\left(4\right)
\mathrm{SO}\left(3\right)×\mathrm{SO}\left(3\right)
. As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings, we prove simultaneous equidistribution of all of these objects under two splitting conditions.
Closed hypersurfaces of low entropy in
{\mathbb{R}}^{4}
are isotopically trivial
Jacob Bernstein, Lu Wang
KEYWORDS: self-expander, self-shrinker, Mean curvature flow, isotopy, 53C44, 53A10, 35J20, 35K93, 57Q37
We show that any closed connected hypersurface in
{\mathbb{R}}^{4}
with entropy less than or equal to that of the round cylinder is smoothly isotopic to the standard three-sphere.
Honda–Tate theory for Shimura varieties
Mark Kisin, Keerthi Madapusi Pera, Sug Woo Shin
KEYWORDS: Shimura varieties, abelian varieties, p-divisible groups, 11G18, 14G35
A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are nonempty provided that the corresponding group G is quasisplit at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda–Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.
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networks(deprecated)/neighbors - Maple Help
Home : Support : Online Help : networks(deprecated)/neighbors
find neighboring vertices, treating all edges as undirected
neighbors(v, G)
neighbors(vset, G)
neighbors(G)
set of vertices of G
Important:The networks package has been deprecated. Use the superseding command GraphTheory[Neighbors]instead.
Given a vertex v of a graph G, this routine returns the set of vertices which are at the ends of any edges incident with v, independent of direction.
Given a set of vertices, vset, the neighbors of the subgraph induced by vset are computed.
Directional information can be retrieved through use of the commands head() and tail(). Alternatively, the commands arrivals() and departures() provide information about ``neighbors'' with respect to incoming or outgoing directed edges.
When called with just a graph the actual neighbors table, indexed by vertices, is returned. Caution is required in this instance as this neighbors table is fully maintained by the network primitives such as addedge() and delete(). Specifically, direct assignments are not normally made to this table by the user.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[neighbors](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{complete}\left(4\right):
\mathrm{addvertex}\left(0,G\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{connect}\left(0,1,G,\mathrm{directed}\right)
\textcolor[rgb]{0,0,1}{\mathrm{e7}}
\mathrm{arrivals}\left(0,G\right)
\textcolor[rgb]{0,0,1}{\varnothing }
\mathrm{departures}\left(0,G\right)
{\textcolor[rgb]{0,0,1}{1}}
\mathrm{neighbors}\left(0,G\right)
{\textcolor[rgb]{0,0,1}{1}}
\mathrm{arrivals}\left(1,G\right)
{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}
\mathrm{departures}\left(1,G\right)
{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}
\mathrm{neighbors}\left(1,G\right)
{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}
\mathrm{arrivals}\left(G\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\varnothing }\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}]\right)
\mathrm{departures}\left(G\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}]\right)
\mathrm{neighbors}\left(G\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}]\right)
GraphTheory[MinimumDegree]
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Characteristics of Seismicity inside and outside the Salton Sea Geothermal FieldCharacteristics of Seismicity inside and outside the Salton Sea Geothermal Field | Bulletin of the Seismological Society of America | GeoScienceWorld
Characteristics of Seismicity inside and outside the Salton Sea Geothermal Field
Yifang Cheng;
Department of Earth Science, University of Southern California, 3651 Trousdale Parkway, ZHS 117, Los Angeles, California 90089, chengyif@usc.edu
ConocoPhilips School of Geology and Geophysics, University of Oklahoma, 100 East Boyd Street, RM 710, Norman, Oklahoma 73019
Yifang Cheng, Xiaowei Chen; Characteristics of Seismicity inside and outside the Salton Sea Geothermal Field. Bulletin of the Seismological Society of America 2018;; 108 (4): 1877–1888. doi: https://doi.org/10.1785/0120170311
The number of human‐made earthquakes is on the rise in recent years and has led to increasing attention on the associated hazards. At the Salton Sea Geothermal field (SSGF), one of the largest geothermal field in southern California, a local borehole seismic network, has improved monitoring of small earthquakes and allows us to better understand the seismogenic response to injection operations and corresponding earthquake hazard. We analyze the spatial distribution of seismicity and
b
‐value for both inside and outside of the geothermal operation field from 2008 to 2014. Comparing with areas outside the geothermal production field, there are five times more small earthquakes (
M<2
) inside the production field at 2‐ to 5‐km depth range with high
b
‐value. But the seismic rate and depth range of large earthquakes (
M>3
) do not have an obvious relationship with injection activities, and most of them are located within low
b
‐value area. Then we investigate the characteristics of 48 spatiotemporal isolated earthquake clusters. The analysis reveals a wide distribution of swarms and aftershock sequences across the whole region. Meanwhile, there is a concentration of small‐magnitude‐mainshock swarms and mixture‐type clusters (small mainshock, short‐duration bursts with high aftershock productivity) within the high
b
‐value geothermal operation area. The differences suggest that geothermal operation has clear influence on seismicity characteristics, indicating a possibility of distinguishing induced seismicity from natural seismicity.
Salton Sea Field
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Token Value - Documentation
ARENA Security Tokens are a very unique financial instruments designed to deliver massive rewards for a limited period of time, in exchange for the agreement of all Holders to sell back no later than at a specific point in time in the future.
One could call it a next gen "evolution" of formerly popular financial instruments, combining the best of Securities & Bonds with the advantages of cutting edge blockchain technology.
These Security Tokens are key to our plans, as they allows us to combine both a mighty attractive investment opportunity, and a genuine for good long-term mission focused on consumer surplus, that'll provide us with lasting competitive advantages and viral growth.
This includes the full value of Company assets, including everything except for direct Dividends, BBB and Intangible Assets such as intellectual property.
Includes multiple products & revenue streams, as well as any profit from ownership shares in other separate organizations such as the CATs DAO.
All revenue from all sources contributes to growing the size distributions to all allocations in varying proportions, as determined by the proportion of Security Tokens held by each of the three primary distribution allocations, at the moment in time in which distribution happens.
This means both your direct profit as a Holder, and the cumulative collective benefit through platform revenue distributions. And of course, the more the public revenue distributions grow, the more it will become attractive to participate to get a piece, for both existing and new Users, leading to further growth.
Are paid out continuously and automatically through our systems to ARENA Security Token Holders' on-platform wallets. You can customize frequency and assets to be received in payouts. Default option is a quarterly payout featuring a mix of currencies among those supported by CryptoArena, but can be customized at will. (for example: everything paid in BTC monthly. Conversion fees apply for customizations.)
Is a totally unique and novel mechanism meant to appropriately reward investors for the value of their agreement to our unique terms and long-term mission, and the reason why you really want some ARENA Security Tokens. Find out more in the dedicated section.
A semi-permanent small discount (-0.01%) on all applicable trading fees of Holder accounts. A marginal benefit mostly meant for differentiation. Discount is lost when you sell your Security Tokens.
Real Securities, Real Rights.
ARENA Tokens are Securities, and as such, bestow their Holders with all standard Shareholder's Rights:
Right to proportional ownership means that owning 1% of token supply equates to owning 1% of the whole Company including all assets. As such, the price value of the tokens is derived from CryptoArena's book value, including the relative value of any type of property that the Company owns or will own in the future, including buildings, machinery, and more.
Right to transfer ownership refers to the ability to sell your tokens on a market. This might seem mundane, but the liquidity provided by exchanges is in fact a big deal. The speed with which you can convert assets into liquid currency is one of the major advantages of Securities over other investment options such as real estate, which can take months or more in some cases, to be able to be converted into something you can spend.
Right to dividends means that whenever the Company registers a net profit, you have the right to receive your proportional share of it. If you own 1% and the Company makes $ 100, you get $ 1 paid out to your account.
As a Token Holder, you are a part owner of CryptoArena, and as such you also may elect to participate in major company decisions through pro rata democratic voting.
Public Companies are compelled by law to disclose their financial documentation on a periodic basis. CryptoArena will publish a yearly Prospectus containing all performance information and strategic decisions, as well as all Auditing results and documentation, and quarterly financial statements.
Further, CryptoArena believes in the "don't trust, verify!" philosophy of the blockchain world. Chain data correctness and integrity are publicly verifiable 24/7.
The right to sue for wrongful acts means that if the Company or its duly appointed representatives act maliciously or negligently, the Company may be held liable for damages to its investors.
Additionally to the above, ARENA Security Tokens will also grant their bearers with:
Right to BuyBack Bonus
This is an added revenue streams meant to compensate investors for the value of their agreement to the terms and processes governing self-decentralization. Meaning this compensation is on top of face value, not in lieu of it.
Incremental basis Rewards
The value of the BBB is shared pro rata only by the proportion of outstanding token holders. CryptoArena Foundation, whom also keeps tokens, has no claim on these funds. This means that whenever a ARENA Holder "exits" and thus claims its BBB, the amount of outstanding tokens decreases, rendering all the ones remaining more valuable, as their claims on all future accruals in the BBB will become proportionally larger. Find more info & examples here.
How much is this BBB worth?
For every 1 $ you make in Dividends, you accrue 0.50 $ in BBB. To start with. Increases over time.
In practical terms, this means shares are worth more than their weight, thus, if you own 4% of total supply, your share will actually earn 6% of net profits, but you won’t be able to access the BBB portion (that extra 2%) unless / until you Exit.
Escalating scarcity & Price support
Even better, the extra weight portion increases permanently whenever another Holder exits.
Tokens repurchased through the BBB mechanism are automatically and permanently allocated to revenue distributions, which also means all future accruals within the fund will belong to a numerically smaller number of tokens, leading to greater and greater shares the more others exit.
ARENA Tokens are designed to reward endurance - those who hodl the longest will benefit most.
Simplifying, it works somewhat like the children’s game “musical chairs” - Whenever a “chair” is taken out of the game (tokens bought back), the remaining others become more valuable. Also, increases to allocations other than your own individual share, also benefit you, both directly and indirectly. Unsold Tokens contribute to ongoing cash position, as long as they remain unsold.
1) Justifying the extra weight
Uses Starting Allocations
Company has 100 shares makes 100€ in net revenue in previous period and distributes it. There are 50 Holders with equal shares. Bob is one of them
Only ARENA Security Holders claim the buybacks, thus, in this example, there are 50 people with have an equal claim on the 25 € of the BuyBack.
€ 0.50 = 25 * (1/50)
This means that Bob earned € 1.50 in total from his 1% share, rather than € 1.
2) Growth Over Time
Now let's dive into the how the weight of your shares grows over time, while holding the same numerical amount of tokens. ARENA Security Tokens are designed to reward endurance. Those who hodl the longest, get most.
Token Holders may, at any time, decide to sell off their tokens through the BuyBack mechanism, this is an automated process that repurchases Tokens and allocates them to general public revenue distributions.
Setting aside the public distributions themselves, this means that the amount of outstanding tokens with a claim on the funds in the BuyBack decreases over time. There are now 45 people with an equal claim on the 25 € that was accrued by the buyback fund in this hypothetical period:
€ 0.55 = 25 * (1/45)
This means that now Bob earns € 1.55 from holding the same numerical amount of tokens, on all future accruals of funds in the BuyBack. The more others exit, the more this will increase.
Engineered for Hodlers
Our Security Tokens are specifically designed to reward the endurance of investors. The Dividend component allows you to obtain a running passive income to either live on or reinvest, while the BBB's unique rewards accumulate incrementally in secure storage.
Our advise is simple: HODL it until you can.
p.s. founders tokens are locked all the way to the end to demonstrate commitment to our promises.
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The Age of the Universe | Astronomy 801: Planets, Stars, Galaxies, and the Universe
If we agree that Hubble's Law tells us that the universe is expanding, it also implies that in the past the universe was much smaller than it is today. If we assume that the expansion's apparent velocity (that is, how fast the galaxies appear to be moving apart) has been constant over the history of the universe, we can calculate how long ago the galaxies began their separation. This should tell us the time that the expansion began, which should give us an estimate of the age of the universe.
If the expansion of the universe is happening rapidly, then we expect the universe to be relatively young, because it has taken only a short time for the galaxies to expand to large distances. If, on the other hand, the universal expansion is progressing at a slow speed, then the age of the universe should be relatively old, because it has taken a long time for the galaxies to reach large distances from each other. We know how fast the universe is expanding, because we know the value of Hubble's constant (H0 ). The faster the universe is expanding, the faster the galaxies will appear to be moving away from each other.
You can actually calculate an estimate for the age of the Universe from Hubble's Law. The distance between two galaxies is D. The apparent velocity with which they are separating from each other is v. At some point, the galaxies were touching, and we can consider that time the moment of the Big Bang. If you take the separation between the two galaxies (D) and divide that by the apparent velocity (v), that will leave you with how long it took for the galaxies to reach their current separation. The standard analogy here is to consider that you are now 300 miles from home. You drove 60 mph the entire time, so how long did it take you to get here? Well, 300 miles / 60 mph = 5 hours.
So, the time it has taken for the galaxies to reach their current separations is
t=D/v
But, from Hubble's Law, we know that
v={H}_{0}D
\quad t=D/v=D/\left({H}_{0}\times D\right)=1/{H}_{0}
. So, you can take
1/{H}_{0}
as an estimate for the age of the Universe.
The best estimate for
{H}_{0}=73km/s/Mpc
. To turn this into an age, we'll have to do a unit conversion.
1Mpc=3.08×{10}^{19}km
{H}_{0}\quad =\text{ }\left(73\text{ }km/s/Mpc\right)\text{ }x\text{ }(1\text{ }Mpc/3.08\text{ }x\text{ }{10}^{19}\quad km)\text{ }=\text{ }2.37\text{ }x\text{ }{10}^{-18}\quad 1/s
So, the age of the Universe is
\quad t\text{ }=\text{ }1/{H}_{0}\quad =\text{ }1\text{ }/\text{ }2.37\text{ }x\text{ }{10}^{-18}\quad 1/s\text{ }=\text{ }4.22\text{ }x\text{ }{10}^{17}\quad s\text{ }=\text{ }13.4\text{ }billion\text{ }years
From stellar evolution, we have estimated the ages of the oldest globular clusters to be approximately 12-13 billion years old. These are the oldest objects we have identified, and it is a nice check on our estimates for the age of the Universe that they are consistent. It would have been strange if we were unable to find any objects roughly as old as the Universe or if we found anything significantly older than the estimated age of the Universe. For many years, until about 10 years ago, however, there was a controversy over the age of the universe derived from Hubble's Constant. The best theories available at the time were estimating that the stars at the Main Sequence Turn Off in many globular clusters had ages of 15 billion years old or more. This creates a problem. How can the universe contain an object older than itself? Recently, however, advances in our understanding of the stars have led us to refine the ages of the stars in globular clusters, and we now estimate them to be about 13 billion years old. This means, though, that the stars in the globular clusters must have formed within the first several hundred million years of the universe's existence!
‹ The Implications of Hubble's Law: An Expanding Universe up The Large Scale Structure of the Universe ›
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Aod ntuple - Atlas Wiki
Aod ntuple
This page contains basic prescriptions to get physics objects from the AOD and the AOD-based Root ntuple (from now on defined as Woutuple).
Some comments on quality selection cuts will be added as work progresses.
--Barison 18:12, 19 May 2005 (MET DST)
1 Monte Carlo Truth Particles
7 Truth Jets
8 BJet
9 Missing Et
Monte Carlo Truth Particles
The ntuples contain the full MC truth information ("truth") that is contained in the AOD. For the T1 (ttbar) sample there is an additional block ("hardtruth") with the hard scatter information remade from the ESD by Eric Cogneras. In the Analysis Skeleton there is an example where some of the information is printed to the screen.
For each truth_particle the 4-vector, PDG code, status code and decay tree navigation information: references to decay products of any given truth particle are stored in tidx_dg0_truth through tidx_dg9_truth, which contain the indices of the decay products in the truth ntuple block. The actual number of daughters for a given decay is specified by numdg_truth. tidx_moth_truth contains the index of the mother particle in the truth ntuple block. If a given link does not exists (e.g. a particle has no mother), then the link index value is set to -1. For convenience, direct 'jump links' exist to the top quark and anti-top quark in the event (if present) in tidx_top_truth and tidx_antitop_truth. Direct jump links to the decay producs of the top (a W and a b quark) are also stored in tidx_Wplus_truth, tidx_Wminus_truth, tidx_b_truth and tidx_antib_truth
Note: Due to some problems with the decoding of the HERWIG truth block it is not always very clear how to extract the kinematics from the top, the W and the b correctly. Please check this information before you use it and, if necessary, get the truth from these particles yourself from the full block.
Electron in AOD/Woutuple
AOD Container Name AOD Variable Woutuple Variable Variable Type Comment
ElectronCollection (*elecTES)->size() n_elec Int Number of electrons in the Woutuple
(*elecItr)->hlv().x() px_elec Double Px
(*elecItr)->hlv().y() py_elec Double Py
(*elecItr)->hlv().z() pz_elec Double Pz
(*elecItr)->hlv().perp() pt_elec Double Pt
(*elecItr)->hlv().eta() eta_elec Double Eta
(*elecItr)->hlv().phi() phi_elec Double Phi
(*elecItr)->isEM() isem_elec Int isEM flag (see below)
(*elecItr)->hasTrack() hastrk_elec Int (bool) HasTrack flag: presence of charged track in the Inner Detector
(*elecItr)->z0wrtPrimVtx() z0vtx_elec Double Intersection (z) of track with the beam axis
(*elecItr)->d0wrtPrimVtx() d0vtx_elec Double Transverse impact parameter d0
(*elecItr)->numberOfBLayerHits() nblayerhits_elec Int Number of hits in the Pixel B-layer
(*elecItr)->numberOfPixelHits() npixelhits_elec Int Number of hits in the Pixel detector
(*elecItr)->numberOfSCTHits() nscthits_elec Int Number of hits in the SCT
(*elecItr)->numberOfTRTHits() ntrthits_elec Int Number of hits in the TRT
(*elecItr)->numberOfTRTHighThresholdHits() ntrththits_elec Int Number of high threshold hits in the TRT
(*elecItr)->author() auth_elec Int (enum) Algorithm used to create the electron: unknown=0, egamma=1, softe=2
(*elecItr)->parameter(ElectronParameters::EoverP) eoverp_elec Double E/P ratio
(*elecItr)->parameter(ElectronParameters::etcone) etcone_elec Double Energy deposition in a cone dR=0.45 around the electron cluster
(*elecItr)->parameter(ElectronParameters::etcone20) etcone20_elec Double Energy deposition in a cone dR=0.20 around the electron cluster. Standard cone size for ATLFAST
(*elecItr)->parameter(ElectronParameters::etcone30) etcone30_elec Double Energy deposition in a cone dR=0.30 around the electron cluster. Currently empty
(*elecItr)->parameter(ElectronParameters::etcone40) etcone40_elec Double Energy deposition in a cone dR=0.40 around the electron cluster
(*elecItr)->parameter(ElectronParameters::emWeight) emwgt_elec Double Weight for electrons (see below)
(*elecItr)->parameter(ElectronParameters::pionWeight) piwgt_elec Double Weight for pions (see below)
There are 3 types of quality cuts you can perform on the electron candidates:
Cuts based on the isEM flag
Cuts based on likelihood
Cuts based on NeuralNet output
1. The isEM flag uses both calorimeter and tracking information in addition to TRT information. The flag is a bit field which marks whether the candidate passed or not some safety checks.
The bit field marks the following checks:
Cluster based egamma
ClusterEtaRange = 0,
ClusterHadronicLeakage = 1,
ClusterMiddleSampling = 2,
ClusterFirstSampling = 3,
Track based egamma
TrackEtaRange = 8,
TrackHitsA0 = 9,
TrackMatchAndEoP = 10,
TrackTRT = 11
In 9.0.4 there is a problem with TRT simulation so one has to mask TRT bit to recover the lost efficiency.
To get the flag in your AOD analysis you should use:
(*elec)->isEM()
To mask the TRT bits you should use: (*elec)->isEM()&0x7FF==0
If you use isEM then you will select electrons with an overall efficiency of about 80% in the barrel but much lower in the crack and endcap.
2. The likelihood ratio is constructed using the following variables: energy in different calorimeter samplings, shower shapes in both eta and phi and E/P ration. No TRT information is used here. You need to access two variables called emweight and pionweight then you can construct the likelihood ratio, defined by: emweight/(emweight+pionweight).
In AOD, you use the following code:
ElecEMWeight = elec*->parameter(ElectronParameters::emWeight); ElecPiWeight = elec*->parameter(ElectronParameters::pionWeight);
Then form the variable: X = ElecEMWeight/(ElecEMWeight+ElecPiWeight);
Requiring X > 0.6 will give you more than 90% efficiency for electrons.
3. The NeuralNet variable uses as inputs the same variables used for likelihood. To use it in AOD you should proceed as follow:
ElecepiNN = elec*->parameter(ElectronParameters::epiNN);
Requiring ElecepiNN > 0.6 will give you about 90% eff for electrons.
However, you should be aware that the NN was trained in full eta range while the likelihood was computed in 3 bins in eta: barrel, crack and endcap. So I would suggest to use likelihood for now.
To require an isolated electron, you have to cut on the energy deposited in the cone around the electron cluster. ATLFAST for example requires Et<10 GeV in a cone of dR=0.2. You can simulate the ATLFAST cut by requiring etcone20<10.*GeV
We did not investigate this.
Muon in AOD/Woutuple
MuonCollection (*muonTES)->size() n_muon Int Number of muons in the Woutuple
(*muonItr)->hlv().x() px_muon Double Px
(*muonItr)->hlv().y() py_muon Double Py
(*muonItr)->hlv().z() pz_muon Double Pz
(*muonItr)->hlv().perp() pt_muon Double Pt
(*muonItr)->hlv().eta() eta_muon Double Eta
(*muonItr)->hlv().phi() phi_muon Double Phi
(*muonItr)->author() auth_muon Int (enum) Algorithm used to create the muon: unknown=0, highPt=1, lowPt=2
(*muonItr)->chi2() chi2_muon Double Chi2 of the track fit. Empty for now (see below)
(*muonItr)->getConeIsol()[0] coneiso0_muon Double
(*muonItr)->getEtIsol()[0] etiso0_muon Double
(*muonItr)->hasCombinedMuon() hascombi_muon Int (bool)
(*muonItr)->hasInDetTrackParticle() hasindettp_muon Int (bool)
(*muonItr)->hasMuonSpectrometerTrackParticle() hasmuspectp_muon Int (bool)
(*muonItr)->hasMuonExtrapolatedTrackParticle() hasmmuextrtp_muon Int (bool)
(*muonItr)->hasCombinedMuonTrackParticle() hascombimutp_muon Int (bool)
(*muonItr)->hasCluster() hasclus_muon Int (bool)
(*muonItr)->isHighPt() ishipt_muon Int (bool) Is the muon produced with the highPt algorithm? (see also author())
(*muonItr)->isLowPt() islopt_muon Int (bool) Is the muon produced with the lowPt algorithm? (see also author())
(*muonItr)->numberOfBLayerHits() nblayerhits_muon Int
(*muonItr)->numberOfPixelHits( npixelhits_muon Int
(*muonItr)->numberOfSCTHits() nscthits_muon Int
(*muonItr)->numberOfTRTHits() ntrthits_muon Int
(*muonItr)->numberOfMDTHits() nmdthits_muon Int
(*muonItr)->numberOfCSCEtaHits() ncscetahits_muon Int
(*muonItr)->numberOfCSCPhiHits() ncscphihits_muon Int
(*muonItr)->numberOfRPCEtaHits() nrpcetahits_muon Int
(*muonItr)->numberOfRPCPhiHits( nrpcphihits_muon Int
(*muonItr)->numberOfTGCEtaHits() ntgcetahits_muon Int
(*muonItr)->numberOfTGCPhiHits() ntgcphihits_muon Int
(*muonItr)->z0wrtPrimVtx() z0vtx_muon Double
(*muonItr)->d0wrtPrimVtx() d0vtx_muon Double
temporary: The muons have highPt and lowPt algorithms. The overlap is removed, but you may want to only use the highPt ones. The chi2() method is always 0 in 10.0.1, so you will have to access the CombinedMuon through something like
const Rec::TrackParticle* cbndMuon = (*muonItr)->get_CombinedMuonTrackParticle();
if( cbndMuon ) {
double chi2 = cbndMuon->fitQuality()->chiSquared();
int ndof = cbndMuon->fitQuality()->numberDoF();
if( ndof > 0 ) chi2 = chi2/ndof;
return chi2;
There are three collections of jets which are all stored in the ntuple. The ntuple variable algo_jet tells you which jet clustering algorithm created the jet. Be sure to cut on algo_jet when you loop over the jets, otherwise you'll see many jets 3 times.
KtTowerParticleJets (Kt algorithm with parameter D=1, algo_jet=0)
Cone4TowerParticleJets (Cone algorithm with R=0.4, algo_jet=1)
ConeTowerParticleJets (Cone algorithm with R=0.7, algo_jet=2)
All three alogrithms have a common data structure.
You can get the energy contributions to a jet from different calorimeter samplings. Each calorimeter (Barrel and Endcap) has up to 4 samplings:
PreSamplerB, EMB1, EMB2, EMB3, // LAr barrel
PreSamplerE, EME1, EME2, EME3, // LAr EM endcap
HEC0, HEC1, HEC2, HEC3, // Hadronic end cap cal.
TileBar0, TileBar1, TileBar2, // Tile barrel
TileGap1, TileGap2, TileGap3, // Tile gap (ITC & scint)
TileExt0, TileExt1, TileExt2, // Tile extended barrel
FCAL0, FCAL1, FCAL2, // Forward EM endcap
Overlapping jets and fake jets: A trick used in CDF is to neglect jets which are close to an Electron (typically dR<0.7). The same trick should be applied for Photons and Taus.
Jet in AOD/Woutuple
KtTowerParticleJets Cone4TowerParticleJets ConeTowerParticleJets (*jetTES)->size() n_jet Int Number of jets in the Woutuple
(*jetItr)->hlv().x() px_jet Double Px
(*jetItr)->hlv().y() py_jet Double Py
(*jetItr)->hlv().z() pz_jet Double Pz
(*jetItr)->hlv().perp() pt_jet Double Pt
(*jetItr)->hlv().eta() eta_jet Double Eta
(*jetItr)->hlv().phi() phi_jet Double Phi
(*jetItr)->pCalo().x() px_calo_jet Double
(*jetItr)->pCalo().y() py_calo_jet Double
(*jetItr)->pCalo().z() pz_calo_jet Double
(*jetItr)->etEM(0) etem0_jet Double Et from EM_Calo Sample_1(B+E) and Presampler (B+E)
(*jetItr)->etEM(1) etem1_jet Double Et from EM_Calo Sample_2(B+E)
(*jetItr)->etHad(0) ethad0_jet Double Et from HAD_Calo Sample_0(B+E)
(*jetItr)->energyInSample(CaloSampling::PreSamplerB) epresb_jet Double Energy in PreSampler (Barrel)
(*jetItr)->energyInSample(CaloSampling::EMB1) eemb1_jet Double Energy in EM_Calo Sample_1 (Barrel)
(*jetItr)->energyInSample(CaloSampling::PreSamplerE) eprese_jet Double Energy in PreSampler (Endcap)
(*jetItr)->energyInSample(CaloSampling::EME1) eeme1_jet Double Energy in EM_Calo Sample_1 (Endcap)
(*jetItr)->energyInSample(CaloSampling::HEC0) ehec0_jet Double Energy in HAD_Calo Sample_0 (Endcap)
(*jetItr)->energyInSample(CaloSampling::TileBar0) etilebar0_jet Double Energy in HAD_Calo Sample_0 (Barrel)
(*jetItr)->energyInSample(CaloSampling::TileGap1) etilegap1_jet Double Energy in Tile_gap Sample_1
(*jetItr)->energyInSample(CaloSampling::TileExt0) etileext0_jet Double Energy in Tile Extended Barrel Sample_0
(*jetItr)->energyInSample(CaloSampling::FCAL0) efcal0_jet Double Energy in FCAL Sample_0
(*jetItr)->energyInSample(CaloSampling::Unknown) eunknown_jet Double WTF???
(*jetItr)->energyInCryostat() ; ecryo_jet Double Energy lost in the cryostat — empiric evaluation sqrt(EMB3 * TileBar0)
Truth jets are formed by running the jet reconstruction algorithm on final Truth particles from the simulation. Jets created this way do not contain the effects of detector energy resolution and other experimental issues. The Woutuple contains truth jets generated with the same three jet algorithms as for reconstruction level jets (Cone,Cone4 and Kt). The author of each jet in the trujet block can be determined from the algo_trujet integer, which has the same meaning as the algo_jet variable in the reco-level jet block
The default b-tagging algorithm is run on cone jets with R=0.7
It is possible to run a b-tagging refit completely on AOD. Andi Wildauer has done a lot of work on this and documented it on:
BTagging refit on AOD
For the top group this was done by Eric (T1 sample only) which is why for the T1 sample there is also a block containing the btag information for R=0.4 jets. The user can choose this by setting the bjet_algo to 1 in the Analysis Skeleton.
Missing Et
There are seven(!) Missing Et objects available in AOD. The Woutuple contains all of them
Missing Et objects available in AOD
Type AOD Container Name Include File Comment
Missing Et MET_Base MissingEtCalo.h uncalibrated ETMiss
Missing Et calibrated MET_Calib calibrated ETMiss
Missing Et Truth MET_Truth MissingEtTruth.h ETMiss from Truth
Missing Et Muon MET_Muon MissingET.h ETMiss from Muons
Missing Et Final MET_Final ETMiss for Physics analysis : calib+muons+cryostat correction
Missing Et Cryostat correction MET_Cryo Cryostat term
Missing Et Topological Clusters MET_Topo ETMiss from Topological Jet Clusters
The calibration of the MET is obtained by using a H1 algorithm. This algorithm corrects the cell energy as a function of the energy density in the cell:
{\displaystyle weight={\frac {\ln \left(|E_{cell}/V|\right)}{\ln(2)}}+26}
The weights are stored in a lookup table here: Code for H1 calibration
Missing Et objects available in Woutuple
MET_Base et_base_etm Double_t Missing Et (uncalibrated)
ht_base_etm Double_t Total Ht (uncalibrated)
px_base_etm Double_t Missing Px (uncalibrated)
py_base_etm Double_t Missing Py (uncalibrated)
compet_base_etm[7] Double_t* Missing Et (uncalibrated)
7 calorimeter samples
see MissingEtCalo.h
comppx_base_etm[7] Double_t* Missing Px (uncalibrated)
comppy_base_etm[7] Double_t* Missing Py (uncalibrated)
MET_Calib et_calib_etm Double_t Missing Et (calibrated)
ht_calib_etm Double_t Total Ht (calibrated)
px_calib_etm Double_t Missing Px (calibrated)
py_calib_etm Double_t Missing Py (calibrated)
compet_calib_etm[7] Double_t* Missing Et (calibrated)
comppx_calib_etm[7] Double_t* Missing Px (calibrated)
comppy_calib_etm[7] Double_t* Missing Py (calibrated)
MET_Truth et_truth_etm Double_t Missing Et (MC Truth)
ht_truth_etm Double_t Total Ht (MC Truth)
px_truth_etm Double_t Missing Px (MC Truth)
py_truth_etm Double_t Missing Py (MC Truth)
truet_truth_etm[6] Double_t* Missing Et (MC Truth)
see MissingEtTruth.h
trupx_truth_etm[6] Double_t* Missing Px (MC Truth)
trupy_truth_etm[6] Double_t* Missing Py (MC Truth)
MET_Muon et_muon_etm Double_t Missing Et (Muon Spectrometer)
ht_muon_etm Double_t Total Ht (Muon Spectrometer)
px_muon_etm Double_t Missing Px (Muon Spectrometer)
py_muon_etm Double_t Missing Py (Muon Spectrometer)
MET_Final et_final_etm Double_t Missing Et
(Final — default for physical analysis)
ht_final_etm Double_t Total Ht
px_final_etm Double_t Missing Px
py_final_etm Double_t Missing Py
MET_Cryo et_cryo_etm Double_t Missing Et (Cryostat)
ht_cryo_etm Double_t Total Ht (Cryostat)
px_cryo_etm Double_t Missing Px (Cryostat)
py_cryo_etm Double_t Missing Py (Cryostat)
MET_Topo et_topo_etm Double_t Missing Et (Topological clustering)
ht_topo_etm Double_t Total Ht (Topological clustering)
px_topo_etm Double_t Missing Px (Topological clustering)
py_topo_etm Double_t Missing Py (Topological clustering)
The best variables for evaluating missing Et are MET_Final or a vector sum of (MET_Topo+MET_Cryo+MET_Muon)
Container/Object Names for AOD
Storegate Keys For AOD 10.x
Particle Preselection Cuts
Electron.h
Muon.h
ParticleJet.cxx
Retrieved from "https://wiki.nikhef.nl/atlas/index.php?title=Aod_ntuple&oldid=4682"
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Home : Support : Online Help : Programming : ImageTools Package : Draw Subpackage : Circle
ImageTools:-Draw Circle Primitive
ImageTools:-Draw Solid Circle Primitive
Circle( image, xCenter, yCenter, diameter, opts )
SolidCircle( image, xCenter, yCenter, diameter, opts )
The xCenter and yCenter arguments give the coordinates of the circle's center, and the diameter argument specifies its diameter in pixels.
The thickness option specifies the thickness of the boundary line. The entire boundary will be rendered within the specified diameter. Increasing the boundary thickness will not make the exterior of the circle any larger; all the variation in thickness is towards the interior.
The color and thickness options are best provided as keyword equations, but can also be provided as positional arguments in the 5th and 6th positions respectively.
This function is part of the ImageTools:-Draw package, so it can be used in the short form Circle(..) only after executing the command with(ImageTools:-Draw). However, it can always be accessed through the long form of the command by using ImageTools:-Draw:-Circle(..).
\mathrm{with}\left(\mathrm{ImageTools}\right):
\mathrm{with}\left(\mathrm{ImageTools}:-\mathrm{Draw}\right):
\mathrm{img}≔\mathrm{Create}\left(240,320,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{Circle}\left(\mathrm{img},160,120,240-64,\mathrm{color}=0.5,\mathrm{thickness}=3\right)
\mathrm{Embed}\left(\mathrm{img}\right)
for the next step, ensure that you have appropriate permissions for the directory you will write to.
\mathrm{Write}\left("circ1.png",\mathrm{img}\right)
\textcolor[rgb]{0,0,1}{6056}
\mathrm{img}≔\mathrm{Create}\left(240,320,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{SolidCircle}\left(\mathrm{img},160,120,240-64,\mathrm{color}=0.5\right)
\mathrm{Embed}\left(\mathrm{img}\right)
\mathrm{img4}≔\mathrm{Create}\left(62,80,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{img}≔\mathrm{Create}\left(15,20,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{Circle}\left(\mathrm{img},10,7,10,\mathrm{color}=0.5,\mathrm{thickness}=1\right)
\mathrm{img4}[9..23,11..30]≔\mathrm{img}:
\mathrm{img}≔\mathrm{Scale}\left(\mathrm{img},16,\mathrm{method}=\mathrm{nearest}\right):
\mathrm{gridFill}\left(\mathrm{img},16\right)
\mathrm{crossHair}\left(\mathrm{img},10,7,16\right)
\mathrm{crossHair}\left(\mathrm{img},5,2,16\right)
\mathrm{crossHair}\left(\mathrm{img},15,12,16\right)
\mathrm{Embed}\left(\mathrm{img}\right)
\mathrm{img}≔\mathrm{Create}\left(15,20,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{Circle}\left(\mathrm{img},10,7,10,\mathrm{color}=0.5,\mathrm{thickness}=2.5\right)
\mathrm{img4}[9..23,51..70]≔\mathrm{img}:
\mathrm{img}≔\mathrm{Scale}\left(\mathrm{img},16,\mathrm{method}=\mathrm{nearest}\right):
\mathrm{gridFill}\left(\mathrm{img},16\right)
\mathrm{crossHair}\left(\mathrm{img},10,7,16\right)
\mathrm{crossHair}\left(\mathrm{img},5,2,16\right)
\mathrm{crossHair}\left(\mathrm{img},15,12,16\right)
\mathrm{Embed}\left(\mathrm{img}\right)
\mathrm{img}≔\mathrm{Create}\left(15,20,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{SolidCircle}\left(\mathrm{img},10,7,10,\mathrm{color}=0.5\right)
\mathrm{img4}[40..54,11..30]≔\mathrm{img}:
\mathrm{img}≔\mathrm{Scale}\left(\mathrm{img},16,\mathrm{method}=\mathrm{nearest}\right):
\mathrm{gridFill}\left(\mathrm{img},16\right)
\mathrm{crossHair}\left(\mathrm{img},10,7,16\right)
\mathrm{crossHair}\left(\mathrm{img},5,2,16\right)
\mathrm{crossHair}\left(\mathrm{img},15,12,16\right)
\mathrm{Embed}\left(\mathrm{img}\right)
\mathrm{img}≔\mathrm{Create}\left(15,20,\mathrm{channels}=3,\mathrm{background}=\mathrm{white}\right):
\mathrm{SolidCircle}\left(\mathrm{img},10.3,7.4,11,\mathrm{color}=0.5\right)
\mathrm{img4}[40..54,51..70]≔\mathrm{img}:
\mathrm{img}≔\mathrm{Scale}\left(\mathrm{img},16,\mathrm{method}=\mathrm{nearest}\right):
\mathrm{gridFill}\left(\mathrm{img},16\right)
\mathrm{crossHair}\left(\mathrm{img},10.3,7.4,16\right)
\mathrm{crossHair}\left(\mathrm{img},4.8,1.9,16\right)
\mathrm{crossHair}\left(\mathrm{img},15.8,12.9,16\right)
\mathrm{Embed}\left(\mathrm{img}\right)
\mathrm{Embed}\left(\mathrm{img4}\right)
The ImageTools[Draw][Circle] and ImageTools[Draw][SolidCircle] commands were introduced in Maple 2018.
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New Zealand general election forecasts
This page provides probabilistic predictions for the 2017 New Zealand General Election from “Model A”. The preferred approach in these times of volatility is a hybrid model that combines Model A with Model B.
Both models draws on multiple opinion polls, but go a step beyond a straightforward poll aggregator in that the estimated voting intention from successive polls is used to forecast the chances of each party to actually win seats on election day, taking into account uncertainty. Polling results are also adjusted to take into account different polling firms’ past performance in predicting different parties’ results.
This page is updated periodically as more data become available.
All material changes are described in this changelog.
Source code for the analysis, including all committed changes, is available in the nz-election-forecast repository on GitHub
Source code for the write up, including all committed changes, is available in my blog repository on GitHub
An alternative approach using state space modelling is available as Model B.
The scenarios outlined in the graphic are defined as:
“National led coalition similar to 2014” - Under this outcome, the National, Māori, ACT and United Future parties between them would have a majority of seats in Parliament (as per the post-2014 government); but National do not have enough to govern by themselves. This scenario does not distinguish between the many possible subsets of this outcome (eg many of these simulated results involve one of the coalition partners not being essential to form a government).
“NZ First needed to make government” - In this scenario, neither the National/Māori/ACT/United Future nor the Labour/Green combination would have a majority of seats, but one or (more frequently) either of them could form a majority with support from New Zealand First. Note that this includes some scenarios where Labour + Greens + New Zealand First seats exactly tie with National + Māori + ACT + United Future.
Other scenarios should be self-explanatory
For simplicity of presentation, no attempt has been made to identify separately all possible scenarios encompassed in the outcomes described above. You can play at building your own coalition with an interactive web app which also lets you tweak the assumptions of allocation of nine key electorate seats (see below).
Here are the actual projected seat counts. Note that there is correlation between the predicted seat counts of various parties, which stops us from just adding up the likely values of the various parties. For example, if Labour does particularly well, it will be at least to some extent at the expense of the Greens (going on past results). the projections at the top of this page take this into account, but it isn’t visible in the histograms in the next chart:
The predictions are simulations based on a model which smooths all polling numbers since the 2014 election and projects the trend forward. The model controls for “house effects”, estimates of the amount that different polling firms over- or under-estimate the party vote of different political parties, based on polling firms’ performance in the four previous elections. This adjustment process (amongst other minor changes) generally slightly increases the expected vote for New Zealand First, and decreases it for the Greens, compared to published poll numbers.
The graphic below shows this smoothing model, and the forecasts up to election day, of the underlying tendency for party vote for each party:
The model provides an estimate of the range of party vote outcomes for each of nine political parties for which there are sufficient polling data to make predictions. Voting outcomes are uncertain not just because of the sampling error (this is the margin of error, typically around 3.1% for this sort of variable, usually quoted with survey results in the media and is the uncertainty associated with random sampling with no other sources of error), but because of hard-to-determine non-sampling error, and because of genuine changes in voting intention over time. As more polls become available closer to the election, the prediction intervals provided in the graphic above are expected to narrow.
Party vote is strongly (usually negatively) correlated between parties because parties are competing for the same voters. I get around this by modelling vote as a multivariate normal distribution on a logit scale. The mean and covariance matrix of the distribution on election day are estimated from a generalized additive model implemented in Simon Woods’ mgcv R package. Simulations are done with mvrnorm from Bill Venables’ and Brian Ripley’s MASS R package.
The simulations take into account not only the uncertainty in the forecast of the underlying party vote, but the randomness associated with individual election days. This is the individual level randomness from our statistical model, even after we have an uncertain estimate of the expected value of vote for each party.
The conversion of party vote to seats depends on whether parties exceed the 5% threshold and/or hold an electorate seat. The key electorate seat outcomes are simulated by a very primitive basis:
Ohariu is assumed to go to United Future 60% of the time, Labour 40%
Epsom is assumed to go to ACT 80% of the time, National 20%
The seven Māori electorates are estimated to go to Labour proportionately to their 2014 vote in each electorate with probability
\frac{Labour2014}{Labour2014 + 0.9 (Mana2014 + Maori2014)}
, and to Mana (Te Tai Tokerau) or the Māori party (the other six electorates) other times
These assumptions are not terribly data-driven, but are better than simply assuming that existing electorate representatives stay the same. Any improvements welcomed; as far as I’m aware there simply aren’t data available to do much better than arbitrary assumptions for these individual electorates.
Seat allocation computation from the simulated party vote results uses the Sainte-Lague allocation formula as implemented in my nzelect R package, which I am confident matches the approach used by the Electoral Commission. All the simulated seat allocations are available for download.
The graphic below shows the simulated outcomes in terms of seats for the various parties in relation to eachother. The numbers in green in the upper right of the chart are correlation coefficients between outcomes in the different simulations; for example, number of Labour and National seats are strongly negatively related to eachother: simulations where National get lots of seats generally means Labour do badly, and vice versa.
You can explore a wider range of coalition possibilities, and tweak the assumptions for individual electorates, with this interactive web app.
Predicting the 2014 election from March 2014 data?
The test for any forecasting method is how it goes at predicting real life results, pretending to come from a position of ignorance. So I used the same method to predict the results of the 20 September 2014 election, limiting myself to data up to 20 March 2014. This meant repeating the house effects estimation with a smaller dataset, refitting the models, etc. I cut a few corners, particularly on the Māori electorates where I just allocated them 50/50 to Labour or someone else; I can’t realistically say what arbitrary guess I would have made three years ago, but I don’t think it makes that much difference.
The results aren’t too bad for a six-month out prediction. In the end, the Green party exceeded these expectations with 10.7% of the party vote, and New Zealand First got 8.7% (out performing the polls materially). Labour under-performed compared to this retrospective prediction, getting only 25.1% of the party vote. The downwards curve in intent to vote for Labour in that election cycle was only just becoming apparent six months in advance - see the chart below. The National Party final party vote in 2014 was 47.0%, within the prediction interval.
If I’d applied this method in March 2014, I would have identified the actual results (which were, very narrowly, a National-led coalition) as the most probable outcome:
I strongly recommend reading Nate Silver’s reflections on The Real Story of 2016 about polls, forecasts and political analysis in the 2016 US Presidential election, much of which is relevant in other electoral situations. It includes this gem:
“…there are real shortcomings in how American politics are covered, including pervasive groupthink among media elites, an unhealthy obsession with the insider’s view of politics, a lack of analytical rigor, a failure to appreciate uncertainty, a sluggishness to self-correct when new evidence contradicts pre-existing beliefs, and a narrow viewpoint that lacks perspective from the longer arc of American history.”
This page is not associated with any political party, media or commentator. I have made every effort to provide a transparent, technical probabilistic forecast of the election results and limit any subjective judgement to technical matters relating to model building. No political judgement or interpretation is to be inferred from these forecasts. Even more than is always the case, this page has no connection whatsoever to my day job.
Non-politicised corrections, reactions or suggestions are welcomed - use the comments section below or log an issue with the source code repository on GitHub. There are some known areas for follow up already such as:
weight polling observations proportionately to sample size - I only haven’t done this because I don’t have the sample size data convenient to hand. If/when I add sample sizes to the nzelect R package I will incorporate it into the analysis here too.
compare this simple approach to a more formally specified Bayesian latent state space model. I’m keen on doing this but it’s less familiar territory for me; and I’m suspicious that it will be extremely computationally intensive (eg multiple days of processing). Will get around to it sooner or later.
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Power of a lens — lesson. Science State Board, Class 10.
Relation between the power of lens and focal length
The ability of a ray of light to converge or diverge when it strikes a lens is determined by the focal length of the lens. The power of a lens refers to its ability to converge (convex lens) or diverge (concave lens). As a result, the power of a lens can be defined as the degree of light ray convergence or divergence. The reciprocal of a lens' focal length is used to calculate its power.
P=\frac{1}{f}
The dioptre is the SI unit of lens power. It is denoted by the letter D. The power of the lens is expressed in ‘D' if the focal length is expressed in ‘m'. As a result, the power of a lens with a focal length of one metre is 1D.
The power of a convex lens is assumed to be positive, whereas the power of a concave lens is assumed to be negative.
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Banana Function Minimization - MATLAB & Simulink - MathWorks Benelux
Optimization Without Derivatives
Optimization with Estimated Derivatives
Optimization with Steepest Descent
Optimization with Analytic Gradient
Optimization with Analytic Hessian
Optimization with a Least Squares Solver
Optimization with a Least Squares Solver and Jacobian
This example shows how to minimize Rosenbrock's "banana function":
f\left(x\right)=100\left(x\left(2\right)-x\left(1{\right)}^{2}{\right)}^{2}+\left(1-x\left(1\right){\right)}^{2}.
f\left(x\right)
is called the banana function because of its curvature around the origin. It is notorious in optimization examples because of the slow convergence most methods exhibit when trying to solve this problem.
f\left(x\right)
has a unique minimum at the point
x=\left[1,1\right]
f\left(x\right)=0
. This example shows a number of ways to minimize
f\left(x\right)
starting at the point
x0=\left[-1.9,2\right]
The fminsearch function finds a minimum for a problem without constraints. It uses an algorithm that does not estimate any derivatives of the objective function. Rather, it uses a geometric search method described in fminsearch Algorithm.
Minimize the banana function using fminsearch. Include an output function to report the sequence of iterations.
The fminunc function finds a minimum for a problem without constraints. It uses a derivative-based algorithm. The algorithm attempts to estimate not only the first derivative of the objective function, but also the matrix of second derivatives. fminunc is usually more efficient than fminsearch.
Minimize the banana function using fminunc.
If you attempt to minimize the banana function using a steepest descent algorithm, the high curvature of the problem makes the solution process very slow.
You can run fminunc with the steepest descent algorithm by setting the hidden HessUpdate option to the value 'steepdesc' for the 'quasi-newton' algorithm. Set a larger-than-default maximum number of function evaluations, because the solver does not find the solution quickly. In this case, the solver does not find the solution even after 600 function evaluations.
If you provide a gradient, fminunc solves the optimization using fewer function evaluations. When you provide a gradient, you can use the 'trust-region' algorithm, which is often faster and uses less memory than the 'quasi-newton' algorithm. Reset the HessUpdate and MaxFunctionEvaluations options to their default values.
If you provide a Hessian (matrix of second derivatives), fminunc can solve the optimization using even fewer function evaluations. For this problem the results are the same with or without the Hessian.
The recommended solver for a nonlinear sum of squares is lsqnonlin. This solver is even more efficient than fminunc without a gradient for this special class of problems. To use lsqnonlin, do not write your objective as a sum of squares. Instead, write the underlying vector that lsqnonlin internally squares and sums.
As in the minimization using a gradient for fminunc, lsqnonlin can use derivative information to lower the number of function evaluations. Provide the Jacobian of the nonlinear objective function vector and run the optimization again.
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Dimension theory and nonstable $K$-theory for net groups
Dimension theory and nonstable
K
-theory for net groups
Bak, Anthony ; Stepanov, Alexei
author = {Bak, Anthony and Stepanov, Alexei},
title = {Dimension theory and nonstable $K$-theory for net groups},
AU - Bak, Anthony
AU - Stepanov, Alexei
TI - Dimension theory and nonstable $K$-theory for net groups
Bak, Anthony; Stepanov, Alexei. Dimension theory and nonstable $K$-theory for net groups. Rendiconti del Seminario Matematico della Università di Padova, Tome 106 (2001), pp. 207-253. http://www.numdam.org/item/RSMUP_2001__106__207_0/
[Bk1] A. Bak, Nonabelian K-theory: The nilpotent class of K1 and general stability, K-Theory, 4 (1991), pp. 363-397. | MR 1115826 | Zbl 0741.19001
[Bk2] A. Bak, K-theory of forms, Annals Math. Studies 98, Princeton Univ. Press Princeton, N.J. 1981. | MR 632404 | Zbl 0465.10013
[Bk3] A. Bak, Finite completions, Unpublished.
[Bk4] A. Bak, Lectures on dimension theory, algebraic homotopy theory, and nonabelian K-theory, Lecture Notes, Buenos-Aires, 1995.
[Bk5] A. Bak, Dimension theory and group valued functors, preprint.
[BkV] A. Bak - N.A. Vavilov, Structure of hyperbolic unitary groups I. Elementary subgroups, Algebra Colloquium, 7:2 (2000), pp. 159-196. | MR 1810843 | Zbl 0963.20024
[BV1] Z.I. Borewicz - N.A. Vavilov, The distribution of subgroups containing a group of block diagonal matrices in the general linear group over a ring, Sov. Math. - Izv. VUZ, 26:11 (1982), pp. 13-18. | Zbl 0521.20033
[BV1] Z.I. Borewich - -N.A. Vavilov, The distribution of subgroups in the general linear group over a commutative ring, Proc. Steklov. Inst. Math., 3 (1985), pp. 27-46. | Zbl 0653.20048
[G] I.Z. Golubchik, On the subgroups of the general linear group GLn(R) over an associative ring, R. Russian Math. Surveys, 39:1 (1984), pp. 157-158. | MR 733962 | Zbl 0572.20031
[H] R. Hazrat, Dimension theory and nonstable K1 of quadratic modules, K-Theory, to appear. | MR 1962906 | Zbl 1020.19001
[M] J. Milnor, Introduction to algebraic K-theory, Annals Math. Studies Princeton Univ. Press Princeton, N.J., 1971. | MR 349811 | Zbl 0237.18005
[Mu] A. Mundkur, Dimension theory and nonstable K1, Algebras and Representation Theory, to appear. | MR 1890592 | Zbl 1010.19001
[S1] A.V. Stepanov, On the distribution of subgroups normalized by a given subgroup, J. Sov. Math., 64 (1993), pp. 769-776. | MR 1164862 | Zbl 0790.20069
[S2] A.V. Stepanov, Description of subgroups of the general linear group over a ring with the use of stability conditions, Rings and linear groups, Krasnodar Univ. Press, Krasnodar, Russian, 1988, 82-91. | MR 1206033
[SuTu] A.A. Suslin - M.S. Tulenbaev, A theorem on stabilization for Milnor's K2-functor, J. Sov. Math., 17 (1981), pp. 1804-1819. | Zbl 0461.18008
[T] G. Tang, Hermitian Groups and K-Theory, K-Theory, 13:3 (1998), pp. 209-267. | MR 1609905 | Zbl 0899.19003
[Tu] M.S. Tulenbaev, The Schur multiplier of the group of elementary matrices of finite order, J. Sov. Math., 17:4 (1981), pp. 2062-2067. | Zbl 0459.20042
[Vs1] L.N. Vaserstein, On normal subgroups of GLn over a ring, Lecture Notes Math., 854 (1981), pp. 456-465. | MR 618316 | Zbl 0464.20030
[Vv1] N.A. Vavilov, Subgroups of the general linear group over a ring that contain a group of block-triangular matrices, Transl. Amer. Math. Soc., 2nd Ser., 132 (1986), pp. 103-104. | Zbl 0594.20040
[Vv2] N.A. Vavilov, Structure of Chevalley groups over commutative rings, Proc. Conf. Non-associative algebras and related topics, Hiroshima, 1990 World Sci. Publ. Singapore et al. (1991), 219-335. | MR 1150262 | Zbl 0799.20042
[VvS] N.A. Vavilov - A.V. Stepanov, Subgroups of the general linear group over rings satisfying stability conditions, Sov. Math., Izv. VUZ, 33:10 (1989), pp. 23-31. | MR 1044472 | Zbl 0702.20033
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Frank–Wolfe algorithm - Wikipedia
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method,[1] reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956.[2] In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function (taken over the same domain).
4 Lower bounds on the solution value, and primal-dual analysis
{\displaystyle {\mathcal {D}}}
is a compact convex set in a vector space and
{\displaystyle f\colon {\mathcal {D}}\to \mathbb {R} }
is a convex, differentiable real-valued function. The Frank–Wolfe algorithm solves the optimization problem
{\displaystyle f(\mathbf {x} )}
{\displaystyle \mathbf {x} \in {\mathcal {D}}}
A step of the Frank–Wolfe algorithm
Initialization: Let
{\displaystyle k\leftarrow 0}
{\displaystyle \mathbf {x} _{0}\!}
be any point in
{\displaystyle {\mathcal {D}}}
Step 1. Direction-finding subproblem: Find
{\displaystyle \mathbf {s} _{k}}
{\displaystyle \mathbf {s} ^{T}\nabla f(\mathbf {x} _{k})}
{\displaystyle \mathbf {s} \in {\mathcal {D}}}
(Interpretation: Minimize the linear approximation of the problem given by the first-order Taylor approximation o{\displaystyle f}
{\displaystyle \mathbf {x} _{k}\!}
Step 2. Step size determination: Set
{\displaystyle \alpha \leftarrow {\frac {2}{k+2}}}
, or alternatively find
{\displaystyle \alpha }
{\displaystyle f(\mathbf {x} _{k}+\alpha (\mathbf {s} _{k}-\mathbf {x} _{k}))}
{\displaystyle 0\leq \alpha \leq 1}
Step 3. Update: Let
{\displaystyle \mathbf {x} _{k+1}\leftarrow \mathbf {x} _{k}+\alpha (\mathbf {s} _{k}-\mathbf {x} _{k})}
{\displaystyle k\leftarrow k+1}
While competing methods such as gradient descent for constrained optimization require a projection step back to the feasible set in each iteration, the Frank–Wolfe algorithm only needs the solution of a linear problem over the same set in each iteration, and automatically stays in the feasible set.
The convergence of the Frank–Wolfe algorithm is sublinear in general: the error in the objective function to the optimum is
{\displaystyle O(1/k)}
after k iterations, so long as the gradient is Lipschitz continuous with respect to some norm. The same convergence rate can also be shown if the sub-problems are only solved approximately.[3]
The iterates of the algorithm can always be represented as a sparse convex combination of the extreme points of the feasible set, which has helped to the popularity of the algorithm for sparse greedy optimization in machine learning and signal processing problems,[4] as well as for example the optimization of minimum–cost flows in transportation networks.[5]
If the feasible set is given by a set of linear constraints, then the subproblem to be solved in each iteration becomes a linear program.
While the worst-case convergence rate with
{\displaystyle O(1/k)}
can not be improved in general, faster convergence can be obtained for special problem classes, such as some strongly convex problems.[6]
Lower bounds on the solution value, and primal-dual analysis[edit]
{\displaystyle f}
is convex, for any two points
{\displaystyle \mathbf {x} ,\mathbf {y} \in {\mathcal {D}}}
{\displaystyle f(\mathbf {y} )\geq f(\mathbf {x} )+(\mathbf {y} -\mathbf {x} )^{T}\nabla f(\mathbf {x} )}
This also holds for the (unknown) optimal solution
{\displaystyle \mathbf {x} ^{*}}
{\displaystyle f(\mathbf {x} ^{*})\geq f(\mathbf {x} )+(\mathbf {x} ^{*}-\mathbf {x} )^{T}\nabla f(\mathbf {x} )}
. The best lower bound with respect to a given point
{\displaystyle \mathbf {x} }
{\displaystyle {\begin{aligned}f(\mathbf {x} ^{*})&\geq f(\mathbf {x} )+(\mathbf {x} ^{*}-\mathbf {x} )^{T}\nabla f(\mathbf {x} )\\&\geq \min _{\mathbf {y} \in D}\left\{f(\mathbf {x} )+(\mathbf {y} -\mathbf {x} )^{T}\nabla f(\mathbf {x} )\right\}\\&=f(\mathbf {x} )-\mathbf {x} ^{T}\nabla f(\mathbf {x} )+\min _{\mathbf {y} \in D}\mathbf {y} ^{T}\nabla f(\mathbf {x} )\end{aligned}}}
The latter optimization problem is solved in every iteration of the Frank–Wolfe algorithm, therefore the solution
{\displaystyle \mathbf {s} _{k}}
of the direction-finding subproblem of the
{\displaystyle k}
-th iteration can be used to determine increasing lower bounds
{\displaystyle l_{k}}
during each iteration by setting
{\displaystyle l_{0}=-\infty }
{\displaystyle l_{k}:=\max(l_{k-1},f(\mathbf {x} _{k})+(\mathbf {s} _{k}-\mathbf {x} _{k})^{T}\nabla f(\mathbf {x} _{k}))}
Such lower bounds on the unknown optimal value are important in practice because they can be used as a stopping criterion, and give an efficient certificate of the approximation quality in every iteration, since always
{\displaystyle l_{k}\leq f(\mathbf {x} ^{*})\leq f(\mathbf {x} _{k})}
It has been shown that this corresponding duality gap, that is the difference between
{\displaystyle f(\mathbf {x} _{k})}
and the lower bound
{\displaystyle l_{k}}
, decreases with the same convergence rate, i.e.
{\displaystyle f(\mathbf {x} _{k})-l_{k}=O(1/k).}
^ Levitin, E. S.; Polyak, B. T. (1966). "Constrained minimization methods". USSR Computational Mathematics and Mathematical Physics. 6 (5): 1. doi:10.1016/0041-5553(66)90114-5.
^ Frank, M.; Wolfe, P. (1956). "An algorithm for quadratic programming". Naval Research Logistics Quarterly. 3 (1–2): 95–110. doi:10.1002/nav.3800030109.
^ Dunn, J. C.; Harshbarger, S. (1978). "Conditional gradient algorithms with open loop step size rules". Journal of Mathematical Analysis and Applications. 62 (2): 432. doi:10.1016/0022-247X(78)90137-3.
^ Clarkson, K. L. (2010). "Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm". ACM Transactions on Algorithms. 6 (4): 1–30. CiteSeerX 10.1.1.145.9299. doi:10.1145/1824777.1824783.
^ Fukushima, M. (1984). "A modified Frank-Wolfe algorithm for solving the traffic assignment problem". Transportation Research Part B: Methodological. 18 (2): 169–177. doi:10.1016/0191-2615(84)90029-8.
^ Bertsekas, Dimitri (1999). Nonlinear Programming. Athena Scientific. p. 215. ISBN 978-1-886529-00-7.
Jaggi, Martin (2013). "Revisiting Frank–Wolfe: Projection-Free Sparse Convex Optimization". Journal of Machine Learning Research: Workshop and Conference Proceedings. 28 (1): 427–435. (Overview paper)
The Frank–Wolfe algorithm description
Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-30303-1. .
Marguerite Frank giving a personal account of the history of the algorithm
Retrieved from "https://en.wikipedia.org/w/index.php?title=Frank–Wolfe_algorithm&oldid=1079921729"
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A parametric curve is determined by two real functions
f
g
of a parameter
t
. This curve is therefore the set of points
\left(x=f\left(t\right),y=g\left(t\right)\right)
for different values of t. On the other hand, we also have the graphs of the two coordinate functions
f
g
which are curves.
This exercise can either present the two curves of
g
and ask you to recognize the parametric curve
\left(x=f\left(t\right),y=g\left(t\right)\right)
among others, or present the parametric curve and ask you to recognize the curves of
g
Style randomly determined by the server recognize the graphs of f and g from the parametric curve recognize the parametric curve from the graphs of
g
Description: recognize a parametric curve by the graphs of its coordinate functions. serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
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Probing Microstructure Dynamics With X-Ray Diffraction Microscopy | J. Eng. Mater. Technol. | ASME Digital Collection
R. M. Suter,
R. M. Suter
Department of Physics and Materials Science and Engineering,
e-mail: suter@andrew.cmu.edu
C. M. Hefferan,
C. M. Hefferan
e-mail: cheffera@andrew.cmu.edu
S. F. Li
e-mail: sfli@andrew.cmu.edu
D. Hennessy,
e-mail: hennessy@anl.gov
e-mail: changshi.xiao@gmail.com
U. Lienert,
U. Lienert
e-mail: lienert@aps.anl.gov
B. Tieman
e-mail: tieman@aps.anl.gov
Suter, R. M., Hefferan, C. M., Li, S. F., Hennessy, D., Xiao, C., Lienert, U., and Tieman, B. (March 12, 2008). "Probing Microstructure Dynamics With X-Ray Diffraction Microscopy." ASME. J. Eng. Mater. Technol. April 2008; 130(2): 021007. https://doi.org/10.1115/1.2840965
We describe our recent work on developing X-ray diffraction microscopy as a tool for studying three dimensional microstructure dynamics. This is a measurement technique that is demanding of experimental hardware and presents a challenging computational problem to reconstruct the sample microstructure. A dedicated apparatus exists at beamline 1-ID of the Advanced Photon Source for performing these measurements. Submicron mechanical precision is combined with focusing optics that yield
≈2μm
high×1.3mm
wide line focused beam at
50keV
. Our forward modeling analysis approach generates diffraction from a simulated two dimensional triangular mesh. Each mesh element is assigned an independent orientation by optimizing the fit to experimental data. The method is computationally demanding but is adaptable to parallel computation. We illustrate the state of development by measuring and reconstructing a planar section of an aluminum polycrystal microstructure. An orientation map of
∼90
grains is obtained along with a map showing the spatial variation in the quality of the fit to the data. Sensitivity to orientation variations within grains is on the order of
0.1deg
. Volumetric studies of the response of microstructures to thermal or mechanical treatment will soon become practical. It should be possible to incorporate explicit treatment of defect distributions and to observe their evolution.
aluminium, crystal microstructure, crystal orientation, X-ray diffraction, X-ray microscopy, X-ray optics
Aluminum, Dynamics (Mechanics), Microscopy, X-ray diffraction, Diffraction, Sensors, Photons, Resolution (Optics), X-rays
Forward Modeling Method for Microstructure Reconstruction Using X-Ray Diffraction Microscopy: Single Crystal Verification
Formation and Subdivision of Deformation Structures During Plastic Deformation
Tracking: A Method for Structural Characterization of Grains in Powders or Polycrystals
Three-Dimensional Maps of Grain Boundaries and the Stress-State of Individual Grains
Focusing Optics for High-Energy X-ray Diffraction
El Dasher
Systematical Characterization of Material Response to Microscale Laser Shock Peening
Evaluation of Fatigue Damage in Coarse-Grained Aluminum With Scanning X-Ray Energy Dispersive Diffraction Microscope
Metal Microstructures in Four Dimensions
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Model RF to RF modulator - Simulink - MathWorks France
LO to Out isolation
Modulator Icons
Modulator block icon updated
Model RF to RF modulator
The Modulator block models an RF to RF modulator. The Modulator block mask icons are dynamic and indicate the current state of the applied noise parameter. For more information, see Modulator Icons.
Available power gain — Relates the ratio of power of a single sideband (SSB) at the output to the input power. This calculation assumes a matched load and source termination.
Available power gain — Ratio of power of SSB at the output to input power
Ratio of power of SSB at output to input power, specified as a scalar in dB or a unitless ratio. For a unitless ratio, select None.
For example, the vector [a0,a1,a2,a3] specifies the relation Vout = a0 + a1Vin + a2Vin2 + a3Vin3. Trailing zeros are omitted. So, [a0,a1,a2] defines the same polynomial as [a0,a1,a2,0].
The default value is [0,1], corresponding to the linear relation Vout = Vin.
Input impedance (Ohm) — Input impedance of modulator
Input impedance of modulator, specified as a scalar in Ohms.
Output impedance (Ohm) — Output impedance of modulator
Output impedance of modulator, specified as a scalar in Ohms.
Ground and hide negative terminals — Ground and hide negative terminals
Edit System — Break modulator block links and replace internal variables by appropriate values
Use this button to break modulator links to the library. The internal variables are replaced by their values which are estimated using modulator parameters. The Modulator becomes a simple subsystem masked only to keep the icon.
Use Edit System to edit the internal variables without expanding the subsystem. Use Expand System to expand the subsystem in Simulink™ canvas and to edit the subsystem.
LO to Out isolation — Ratio of magnitude of LO voltage to leaked voltage at output port (RF)
Ratio of magnitude of LO voltage to leaked voltage at output port (RF), specified as a scalar in dB, or a unitless ratio. For a unitless ratio, select None.
Single-sideband noise figure of mixer, specified as a scalar in dB.
To model noise in a circuit envelope model with a Modulator block, you must select the Simulate noise check box in the Configuration block dialog box.
Select this parameter to add phase noise to your modulator system.
Phase noise level, specified as a scalar, vector, or matrix with each element in dBc/Hz.
Even and odd order: The Modulator can produce second-order and third-order intermodulation frequencies, in addition to a linear term.
Odd order: The Modulator generates only "odd-order" intermodulation frequencies.
The linear gain determines the linear a1 term. The block calculates the remaining terms from the values specified in IP3, 1-dB gain compression power, Output saturation power, and Gain compression at saturation. The number of constraints you specify determines the order of the model. The figure shows the graphical definition of the nonlinear Modulator parameters.
Intercept points convention, specified as Input (input referred) or Output (output referred). Use this specification for the intercept points IP2, IP3, the 1-dB gain compression power, and the Output saturation power.
Second-order intercept point, specified as a scalar in dBm, W, mW, or dBW. The default value, inf dBm, corresponds to an unspecified point.
Third-order intercept point, specified as a scalar in dBm, W, mW, or dBW. The default value, inf dBm, corresponds to an unspecified point.
1-dB gain compression power, specified as a scalar in dBm, W, mW, or dBW.
Gain compression at saturation, specified as scalar in dBm, W, mW, or dBW.
Frequency domain: Model a filter using convolution with an impulse response. The Design method is specified as ideal. The impulse response is computed independently for each carrier frequency to capture the ideal filtering response. When a transition between full transmission and full reflection of the ideal filter occurs within the envelope band around a carrier, the frequency-domain implementation captures this transition correctly up to a frequency resolution specified in Impulse response duration.
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}>{R}_{\text{ratio}}
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}<\frac{1}{{R}_{\text{ratio}}}
{R}_{\text{ratio}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\sqrt{1+{\epsilon }^{2}}+\epsilon }{\sqrt{1+{\epsilon }^{2}}-\epsilon }
\epsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{10}^{\left(0.1{R}_{\text{p}}\right)}-1}
To enable this parameter, select Implement using filter order and set Design method to Butterworth or Chebyshev.
[2.1 2.9]GHz (default) | 2-tuple vector
Impulse response duration used to simulate phase noise, specified as a scalar in s, ms, us or ns. You cannot specify impulse response if the amplifier is nonlinear.
The phase noise profile resolution in frequency is limited by the duration of the impulse response used to simulate it. Increase this duration to improve the accuracy of the phase noise profile. A warning message appears if the phase noise frequency offset resolution is too high for a given impulse response duration. The message also specifies the minimum duration suitable for the required resolution
Constant per carrier: Model a filter with either full transmission or full reflection set as constant throughout the entire envelope band around each carrier.The Design method is specified as ideal.
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}>{R}_{\text{ratio}}
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}<\frac{1}{{R}_{\text{ratio}}}
{R}_{\text{ratio}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\sqrt{1+{\epsilon }^{2}}+\epsilon }{\sqrt{1+{\epsilon }^{2}}-\epsilon }
\epsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{10}^{\left(0.1{R}_{\text{p}}\right)}-1}
[2] Grob, Siegfried, and Lindner, Jurgen. "Polynomial Model Derivation of Nonlinear Amplifiers." Department of Information Technology, University of Ulm, Germany.
R2021b: Modulator block icon updated
Starting in R2021b, the Modulator block icon has updated. The block icons are now dynamic and show the current state of the noise parameter.
When you open a model created before R2021b containing a Modualtor block, the software replaces the block icon with the R2021b version.
Demodulator | IQ Modulator | Mixer
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Multivariate and Rational Splines - MATLAB & Simulink - MathWorks Switzerland
f\left(x,y,z\right)=\sum _{u=1}^{U}\sum _{v=1}^{V}\sum _{w=1}^{W}{B}_{u,k}\left(x\right){B}_{v,l}\left(y\right){B}_{w,m}\left(z\right){a}_{u,v,w}
f\left(x\right)=\sum _{j=1}^{n-3}\Psi \left(x-{c}_{j}\right){a}_{j}+x\left(1\right){a}_{n-2}+x\left(2\right){a}_{n-1}+{a}_{n}
p\sum _{i=1}^{n-3}|{y}_{i}-f{c}_{i}{}^{2}|+\left(1-p\right)\int \left({|{D}_{1}{D}_{1}f|}^{2}+2{|{D}_{1}{D}_{2}f|}^{2}+{|{D}_{2}{D}_{2}f|}^{2}\right)
{D}^{m}s=\sum _{j=0}^{m}\left(\begin{array}{c}m\\ j\end{array}\right){D}^{j}w{D}^{m-j}r
\left(\left(\left(v\left(1:\text{end}-1,m+1\right)-\sum _{j=1}^{m}\left(\begin{array}{c}m\\ j\end{array}\right)v\left(\text{end},j+1\right)v\left(1:\text{end}-1,j+1\right)\right)/v\left(\text{end},1\right)\right)
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Định luật Titius–Bode – Wikipedia tiếng Việt
Định luật Titius–Bode
Đề nghị bài viết này hợp nhất với Quy luật Titius-Bode. (Thảo luận)
Định luật Titius–Bode (đôi khi còn được gọi ngắn gọn là Đinh luật Bode) là một giả thuyết cũ nhằm xác định quỹ đạo của các hành tinh khi quay quanh một thiên thể khác, bao gồm cả quỹ đạo của Mặt trời và quỹ đạo tại Bán trục lớn của các hành tinh trong Hệ Mặt Trời được miêu tả bởi công thức truy hồi ở dưới. Giả thuyết này đã dự đoán chính xác quỹ đạo của Tiểu hành tinh Ceres và Sao Thiên Vương (Uranus), nhưng lại thất bại trong việc dự đoán quỹ đạo của Sao Hải Vương (Neptune) và Sao Diêm Vương (Pluto). Định luật này được đặt theo tên của hai nhà khoa học Johann Daniel Titius và Johann Elert Bode.
Solar System diagram showing planetary spacing in whole numbers, when the Sun-Neptune distance is normalized to 100. The numbers listed are distinct from the Bode sequence, but can give an appreciation for the harmonic resonances that are generated by the gravitational "pumping" action of the gas giants.
4.1 Lunar systems and other planetary systems
Định luật liên quan tới bán trục lớn
{\displaystyle a}
của các hành tinh trong Hệ Mặt Trời tính theo đơn vị bằng 10 lần Đơn vị thiên văn - coi bán trục lớn của Trái Đất có giá trị là 10:
{\displaystyle a=4+n}
{\displaystyle n=0,3,6,12,24,48...}
ngoại trừ số đầu tiên, mỗi giá trị sau đều bằng hai lần giá trị liền trước.
Một cách khác để biểu diễn công thức:
{\displaystyle a=1.5\times 2^{(n-1)}+4}
{\displaystyle n=-\infty ,2,3,4...}
Hoặc kết quả có thể chia cho 10 để chuyển sang Đơn vị thiên văn (AU), với công thức sau:
{\displaystyle a=0.4+0.3\times 2^{m}}
{\displaystyle m=-\infty ,0,1,2...}
Với các hành tinh bên ngoài, mỗi hành tinh được dự đoán là có khoảng cách từ Mặt Trời xa khoảng gấp 2 lần khoảng cách của hành tinh trước.
The first mention of a series approximating Bode's Law is found in David Gregory's The Elements of Astronomy, published in 1715. In it, he says, "...supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety six."[1] A similar sentence, likely paraphrased from Gregory,[1] appears in a work published by Christian Wolff in 1724.
In 1764, Charles Bonnet said in his Contemplation de la Nature that, "We know seventeen planets that enter into the composition of our solar system [that is, major planets and their satellites]; but we are not sure that there are no more."[1] To this, in his 1766 translation of Bonnet's work, Johann Daniel Titius added the following unattributed addition, removed to a footnote in later editions:
Take notice of the distances of the planets from one another, and recognize that almost all are separated from one another in a proportion which matches their bodily magnitudes. Divide the distance from the Sun to Saturn into 100 parts; then Mercury is separated by four such parts from the Sun, Venus by 4+3=7 such parts, the Earth by 4+6=10, Mars by 4+12=16. But notice that from Mars to Jupiter there comes a deviation from this so exact progression. From Mars there follows a space of 4+24=28 such parts, but so far no planet was sighted there. But should the Lord Architect have left that space empty? Not at all. Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope. Next to this for us still unexplored space there rises Jupiter's sphere of influence at 4+48=52 parts; and that of Saturn at 4+96=100 parts.
In 1772, Johann Elert Bode, aged only twenty-five, completed the second edition of his astronomical compendium Anleitung zur Kenntniss des gestirnten Himmels, into which he added the following footnote, initially unsourced, but credited to Titius in later versions:[2]
When originally published, the law was approximately satisfied by all the known planets — Mercury through Saturn — with a gap between the fourth and fifth planets. It was regarded as interesting, but of no great importance until the discovery of Uranus in 1781 which happens to fit neatly into the series. Based on this discovery, Bode urged a search for a fifth planet. Ceres, the largest object in the asteroid belt, was found at Bode's predicted position in 1801. Bode's law was then widely accepted until Neptune was discovered in 1846 and found not to satisfy Bode's law. Simultaneously, the large number of known asteroids in the belt resulted in Ceres no longer being considered a planet at that time. Bode's law was discussed as an example of fallacious reasoning by the astronomer and logician Charles Sanders Peirce in 1898.[3]
The discovery of Pluto in 1930 confounded the issue still further. While nowhere near its position as predicted by Bode's law, it was roughly at the position the law had predicted for Neptune. However, the subsequent discovery of the Kuiper belt, and in particular of the object Eris, which is larger than Pluto yet does not fit Bode's law, have further discredited the formula.[4]
Đây là khoảng cách của các hành tinh trong Hệ Mặt Trời, theo con số tính toán và con số thực tế:
Graphical plot using data from table to the left
T-B rule distance (AU)
Real distance (AU)
% error (using real distance as the accepted value)
Mercury 0 0.4 0.39 2.56%
Venus 1 0.7 0.72 2.78%
Earth 2 1.0 1.00 0.00%
Mars 4 1.6 1.52 5.26%
Ceres1 8 2.8 2.77 1.08%
Jupiter 16 5.2 5.20 0.00%
Saturn 32 10.0 9.54 4.82%
Uranus 64 19.6 19.2 2.08%
Neptune 128 38.8 30.06 29.08%
Pluto2 256 77.22 39.44 95.75%
1 Ceres was considered a small planet from 1801 until the 1860s. Pluto was considered a planet from 1930 to 2006. Both are now classified as dwarf planets.
2 While the difference between the T-B rule distance and real distance seems very large here, if Neptune is 'skipped,' the T-B rule's distance of 38.8 is quite close to Pluto's real distance with an error of only 1.62%.
There is no solid theoretical explanation of the Titius–Bode law, but if there is one it is possibly a combination of orbital resonance and shortage of degrees of freedom: any stable planetary system has a high probability of satisfying a Titius–Bode-type relationship. Since it may simply be a mathematical coincidence rather than a "law of nature", it is sometimes referred to as a rule instead of "law".[5] However, astrophysicist Alan Boss states that it is just a coincidence, and the planetary science journal Icarus no longer accepts papers attempting to provide improved versions of the law.[4]
Orbital resonance from major orbiting bodies creates regions around the Sun that are free of long-term stable orbits. Results from simulations of planetary formation support the idea that a randomly chosen stable planetary system will likely satisfy a Titius–Bode law.
Dubrulle and Graner[6][7] have shown that power-law distance rules can be a consequence of collapsing-cloud models of planetary systems possessing two symmetries: rotational invariance (the cloud and its contents are axially symmetric) and scale invariance (the cloud and its contents look the same on all scales), the latter being a feature of many phenomena considered to play a role in planetary formation, such as turbulence.
Lunar systems and other planetary systemsSửa đổi
There is a decidedly limited number of systems on which Bode's law can presently be tested. Two of the solar planets have a number of big moons that appear possibly to have been created by a process similar to that which created the planets themselves. The four big satellites of Jupiter and the biggest inner satellite, Amalthea, cling to a regular, but non-Bode, spacing with the four innermost locked into orbital periods that are each twice that of the next inner satellite. The big moons of Uranus have a regular, but non-Bode, spacing.[8] However, according to Martin Harwit, "a slight new phrasing of this law permits us to include not only planetary orbits around the Sun, but also the orbits of moons around their parent planets."[9] The new phrasing is known as Dermott's law.
Of the recent discoveries of extrasolar planetary systems, few have enough known planets to test whether similar rules apply to other planetary systems. An attempt with 55 Cancri suggested the equation a = 0.0142 e 0.9975 n, and predicts for n = 5 an undiscovered planet or asteroid field at 2 AU.[10] This is controversial.[11] Furthermore the orbital period and semimajor axis of the innermost planet in the 55 Cancri system have been significantly revised (from 2.817 days to 0.737 days and from 0.038 AU to 0.016 AU respectively) since the publication of these studies.[12]
Recent astronomical research suggests that planetary systems around some other stars may fit Titius–Bode-like laws.[13][14] Bovaird and Lineweaver[15] applied a generalized Titius-Bode relation to 68 exoplanet systems which contain four or more planets. They showed that 96% of these exoplanet systems adhere to a generalized Titius-Bode relation to a similar or greater extent than the Solar System does. The locations of potentially undetected exoplanets are predicted in each system.
Subsequent research managed to detect five planet candidates from predicted 97 planets from the 68 planetary systems. The study showed that the actual number of planets could be larger. The occurrence rate of Mars and Mercury sized planets are currently unknown so many planets could be missed due to their small size. Other reasons were accounted to planet not transiting the star or the predicted space being occupied by circumstellar disks. Despite this, the number of planets found with Titius-Bode law predictions were still lower than expected.[16]
^ a b c “Dawn: A Journey to the Beginning of the Solar System”. Space Physics Center: UCLA. 2005. Bản gốc lưu trữ ngày 24 tháng 5 năm 2012. Truy cập ngày 3 tháng 11 năm 2007.
^ Hoskin, Michael (ngày 26 tháng 6 năm 1992). “Bodes' Law and the Discovery of Ceres”. Observatorio Astronomico di Palermo "Giuseppe S. Vaiana". Truy cập ngày 5 tháng 7 năm 2007.
^ Pages 194-196 in
Peirce, Charles Sanders, Reasoning and the Logic of Things, The Cambridge Conference Lectures of 1898, Kenneth Laine Ketner, ed., intro., and Hilary Putnam, intro., commentary, Harvard, 1992, 312 pages, hardcover (ISBN 978-0674749665, ISBN 0-674-74966-9), softcover (ISBN 978-0-674-74967-2, ISBN 0-674-74967-7) HUP catalog page.
^ a b Alan Boss (tháng 10 năm 2006). “Ask Astro”. Astronomy. 30 (10): 70.
^ Carroll, Bradley W.; Ostlie, Dale A. (2007). An Introduction to Modern Astrophysics . Addison-Wesley. tr. 716–717. ISBN 0-8053-0402-9.
^ F. Graner, B. Dubrulle (1994). “Titius-Bode laws in the solar system. Part I: Scale invariance explains everything”. Astronomy and Astrophysics. 282: 262–268. Bibcode:1994A&A...282..262G.
^ B. Dubrulle, F. Graner (1994). “Titius–Bode laws in the solar system. Part II: Build your own law from disk models”. Astronomy and Astrophysics. 282: 269–276. Bibcode:1994A&A...282..269D.
^ Cohen, Howard L. “The Titius-Bode Relation Revisited”. Bản gốc lưu trữ ngày 28 tháng 9 năm 2007. Truy cập ngày 24 tháng 2 năm 2008.
^ Harwit, Martin. Astrophysical Concepts (Springer 1998), pages 27-29.
^ Arcadio Poveda and Patricia Lara (2008). “The Exo-Planetary System of 55 Cancri and the Titus-Bode Law” (PDF). Revista Mexicana de Astronomía y Astrofísica (44): 243–246.
^ Ivan Kotliarov (ngày 21 tháng 6 năm 2008). "The Titius-Bode Law Revisited But Not Revived". arΧiv:0806.3532 [physics.space-ph].
^ Rebekah I. Dawson, Daniel C. Fabrycky (2010). “Title: Radial velocity planets de-aliased. A new, short period for Super-Earth 55 Cnc e”. Astrophysical Journal. 722: 937–953. arXiv:1005.4050. Bibcode:2010ApJ...722..937D. doi:10.1088/0004-637X/722/1/937.
^ “The HARPS search for southern extra-solar planets” (PDF). ngày 23 tháng 8 năm 2010. Truy cập ngày 24 tháng 8 năm 2010. Section 8.2: "Extrasolar Titius-Bode-like laws?"
^ P. Lara, A. Poveda, and C. Allen. On the structural law of exoplanetary systems. AIP Conf. Proc. 1479, 2356 (2012); doi: 10.1063/1.4756667
^ Timothy Bovaird, Charles H. Lineweaver (2013). “Title: Exoplanet predictions based on the generalized Titius–Bode relation”. Monthly Notices of the Royal Astronomical Society. arXiv:1304.3341. Bibcode:2013MNRAS.tmp.2080B. doi:10.1093/mnras/stt1357.
^ “[1405.2259] Testing the Titius”. Truy cập 8 tháng 2 năm 2015.
Định luật Dermott
Phaeton (Hành tinh giả định)
Đường xoắn ốc Logarit
The ghostly hand that spaced the planets New Scientist ngày 9 tháng 4 năm 1994, p13
Plants and Planets: The Law of Titius-Bode explained by H.J.R. Perdijk
Lấy từ “https://vi.wikipedia.org/w/index.php?title=Định_luật_Titius–Bode&oldid=67991274”
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Find sources: "Flare" countermeasure – news · newspapers · books · scholar · JSTOR (June 2015) (Learn how and when to remove this template message)
An infrared guided AIM-9M Sidewinder missile hitting a flare
A US Army AH-64 Apache releasing decoy flares
Russian Knights fire their flares as a salute to Igor Tkachenko.
3 Decoying
4.1 Pyrotechnic flares
4.1.1 Blackbody payloads
4.1.2 Spectrally balanced payloads
4.2 Pyrophoric flares
4.3 Highly flammable payloads
In contrast to radar-guided missiles, IR-guided missiles are very difficult to find as they approach aircraft. They do not emit detectable radar, and they are generally fired from behind, directly toward the engines. In most cases, pilots have to rely on their wingmen to spot the missile's smoke trail and alert of a launch. Since IR-guided missiles have a shorter range than their radar-guided counterparts, good situational awareness of altitude and potential threats continues to be an effective defense. More advanced electro-optical systems can detect missile launches automatically from the distinct thermal emissions of a missile's rocket motor.
Once the presence of a "live" IR missile is indicated, flares are released by the aircraft in an attempt to decoy the missile. Some systems are automatic, while others require manual jettisoning of the flares. The aircraft would then pull away at a sharp angle from the flare (and the terminal trajectory of the missile) and reduce engine power in attempt to cool the thermal signature. Ideally the missile's seeker head is then confused by this change in temperature and flurry of new heat signatures, and starts to follow one of the flares rather than the aircraft.
More modern IR-guided missiles have sophisticated on-board electronics and secondary electro-optical sensors that help discriminate between flares and targets, reducing the effectiveness of flares as a reactionary countermeasure. A newer procedure involves preemptively deploying flares in anticipation of a missile launch, which distorts the expected image of the target should one be let loose. This "pre-flaring" increases the chances that the missile then follows the flares or the open sky in between, rather than a part of the actual defender.
As non-state combatants such as insurgents and terrorists gain access to anti-air missiles, there is an increasing trend to equip slower-moving helicopters with flare countermeasures. Consequently, flare dispensers have become a regular feature on military helicopters. Almost all of the UK's helicopters, whether they are transport or attack models, are equipped with flare dispensers or missile approach warning systems. Similarly, the US armed forces have adopted active defenses with their helicopters.[citation needed]
Apart from military use, some civilian aircraft are also equipped with countermeasure flares, against terrorism: the Israeli airline El Al, having been the target of the failed 2002 airliner attack, in which shoulder-launched surface-to-air missiles were fired at an airliner while taking off, began equipping its fleet with radar-based, automated flare release countermeasures from June 2004.[1][2] This caused concerns in some European countries, which proceeded to ban such aircraft from landing at their airports.[3]
DecoyingEdit
A C-130 Hercules deploying flares, sometimes referred to as Angel Flares[according to whom?] due to the characteristic shape seen from this angle
C-130 flare and chaff dispensers, 1997
Flares burn at thousands of degrees Celsius, which is much hotter than the exhaust of a jet engine. IR missiles seek out the hotter flame, believing it to be an aircraft in afterburner or the beginning of the engine's exhaust source.
As the more modern infrared seekers tend to have spectral sensitivity tailored to more closely match the emissions of airplanes and reject other sources (the so-called CCM, or counter-countermeasures), the modernized decoy flares have their emission spectrum optimized to also match the radiation of the airplane (mainly its engines and engine exhaust). In addition to spectral discrimination, the CCMs can include trajectory discrimination and detection of size of the radiation source.
The newest generation of the FIM-92 Stinger uses a dual IR and UV seeker head, which allows for a redundant tracking solution, effectively negating the effectiveness of modern decoy flares (according to the U.S. Department of Defense). While research and development in flare technology has produced an IR signature on the same wavelength as hot engine exhaust, modern flares still produce a notably (and immutably) different UV signature than an aircraft engine burning kerosene jet-fuel.
HMS Dragon's Westland Lynx helicopter fires flares during an exercise over the Type 45 destroyer
A Dutch Eurocopter AS532 Cougar fires its flares during a nightly exercise.
Polish Air Force MiG-29 at the 2014 Rome International Air Show
F-15E Strike Eagle releasing flares
For the infrared generating charge, two approaches are possible: pyrotechnic and pyrophoric as stored, chemical-energy-source IR-decoy flares contain pyrotechnic compositions, liquid or solid pyrophoric substances, or liquid or solid highly flammable substances.[4]
Upon ignition of the decoy flare, a strongly exothermal reaction is started, releasing infrared energy and visible smoke and flame, emission being dependent on the chemical nature of the payload used.
There is a wide variety of calibres and shapes available for aerial decoy flares. Due to volume storage restrictions on board platforms, many aircraft of American origin use square decoy flare cartridges. Nevertheless, cylindrical cartridges are also available on board American aircraft, such as MJU 23/B on the B-1 Lancer or MJU-8A/B on the F/A-18 Hornet; however, these are used mainly on board French aircraft and those of Russian origin, e.g. PPI-26 IW on the MiG 29.
Schematic view of a MJU-7A/B decoy flare cartridge : anodised aluminium cartridge (1); an electrical impulse cartridge (2), providing both expulsion and, in some cases, direct ignition of the payload; a pusher plate acting as a safe&arm device (3); the payload (4) with first fire layer (5); the wrapping self-adhesive polyester reinforced aluminum foil (6); and a front washer (7).
Square calibres and typical decoy flares:
1x1x8 Inch e.g. M-206, MJU-61, (Magnesium/Teflon/Viton (MTV) based) M-211, M-212 (spectral flares)
2x1x8 Inch e.g. MJU-7A/B (MTV based), MJU-59/B (spectral flare)
2x2.5x8 Inch e.g. MJU-10/B (MTV based)
Cylindrical calibres and typical decoy flares:
2.5 Inch e.g. MJU-23/B (MTV based)
1.5 Inch e.e. MJU 8 A/B (MTV based)
1 Inch e.g. PPI 26 IW
Pyrotechnic flaresEdit
Pyrotechnic flares use a slow-burning fuel-oxidizer mixture that generates intense heat. Thermite-like mixtures, e.g. Magnesium/Teflon/Viton (MTV), are common. Other combinations include ammonium perchlorate/anthracene/magnesium, or can be based on red phosphorus.
To adjust the emission characteristics to match closer the spectrum of jet engines, charges on the base of double base propellants. These compositions can avoid the metal content and achieve cleaner burning without the prominent smoke trail.
Blackbody payloadsEdit
Certain pyrotechnic compositions, for example MTV, give a great flame emission upon combustion and yield a temperature-dependent signature and can be understood as gray bodies of high emissivity (
{\displaystyle e}
~0.95). Such payloads are called blackbody payloads. Other payloads, like iron/potassium perchlorate pellets, only yield a low flame emission but also show temperature-dependent signature.[5] Nevertheless, the lower combustion temperature as compared to MTV results in a lower amount of energy released in the short-wavelength IR range. Other blackbody payloads include ammonium perchlorate/anthracene/magnesium and hydroxyl-terminated polybutadiene (HTPB) binder.[6]
Spectrally balanced payloadsEdit
A sectional of the typical LLU-2B ground illumination flare
Other payloads provide large amounts of hot carbon dioxide upon combustion and thus provide a temperature-independent selective emission in the wavelength range between 3 and 5 μm. Typical pyrotechnic payloads of this type resemble whistling compositions and are often made up from potassium perchlorate and hydrogen lean organic fuels.[7] Other spectrally balanced payloads are made up similarly as double base propellants and contain nitrocellulose (NC), and other esters of nitric acid [8] or nitro compounds as oxidizers such as hexanitroethane and nitro compounds and nitramines as high-energy fuels.[9]
Pyrophoric flaresEdit
Pyrophoric flares work on the principle of ejecting a special pyrophoric material out of an airtight cartridge, usually using a gas generator, e.g. a small pyrotechnic charge or pressurized gas. The material then self-ignites in contact with air. The materials can be solid, e.g. iron platelets coated with ultrafine aluminium, or liquid, often organometallic compounds; e.g. alkyl aluminium compounds, e.g. triethylaluminium. Pyrophoric flares may have reduced effectiveness at high altitudes, due to lower air temperature and lower availability of oxygen; however oxygen can be co-ejected with the pyrophoric fuel.[10]
The advantage of alkyl aluminium and similar compounds is the high content of carbon and hydrogen, resulting in bright emission lines similar to spectral signature of burning jet fuel. Controlled content of solid combustion products, generating continuous black-body radiation, allows further matching of emission characteristics to the net infrared emissions of fuel exhaust and hot engine components.
The flames of pyrophoric fuels can also reach the size of several metres, in comparison with about less than one metre flame of MTV flares. The trajectory can be also influenced by tailoring the aerodynamic properties of the ejected containers.[11]
Solid pyrophoric payloads are based on iron platelets coated with a porous aluminium layer. Based on the very high specific surface area of aluminium those platelets instantaneously oxidize upon contact with air. In contrast to triethylaluminium combustion, these platelets yield a temperature-dependent signature.
Highly flammable payloadsEdit
These payloads contain red phosphorus as an energetic filler. The red phosphorus is mixed with organic binders to give brushable pastes that can be coated on thin polyimide platelets. The combustion of those platelets yields a temperature-dependent signature. Endergonic additives such as highly dispersed silica or alkali halides may further lower the combustion temperature.[12]
^ Missile defense for El Al fleet, CNN, May 24, 2004. Accessed July 18, 2006.
^ "El Al Fits Fleet with Anti-Missile System". Reuters. 2006-02-16. Retrieved 2010-10-05.
^ Europe objects to El Al's anti-missile shield, Ynetnews, Feb 26, 2006. Accessed July 18, 2006.
^ E.-C. Koch, Pyrotechnic Countermeasures: II. Advanced Aerial Infrared Countermeasures, Prop., Expl., Pyrotech.31 2006, 3
^ J. Callaway, Expendable Infrared Radiating means, GB Patent 2 387 430, 2003, GB.
^ D. B. Nielson, D. M. Lester, Blackbody Decoy Flare Compositions for Thrusted Applications and Methods of Use, US Patent 5 834 680, 1998, USA.
^ J. Callaway, T. D. Sutlief, Infrared Emitting Decoy Flare, US Patent Application 2004/0011235 A1, 2004, GB.
^ R. Gaisbauer, V. Kadavanich, M. Fegg, C. Wagner, H. Bannasch, Explosive Body, WO2006/034746, 2006, DE
^ Koch, E.C. (2006). Infrarotleuchtmasse (in German). DE 1020040043991.
^ Davut B. Ebeoglu & C. W. Martin (May 1, 1974). "The Infrared Signature of Pyrophorics". Defense Technical Information Center. Retrieved 2010-10-05.
^ "Flame-stabilized pyrophoric IR decoy flare". PatentStorm LLC. Retrieved 2010-10-05.
^ H. Bannasch, M. Wegscheider, M. Fegg, H. Büsel, Spektrale Scheinzielanpassung und dazu verwendbare Flarewirkmasse, WO 95/05572, 1995, D.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Flare_(countermeasure)&oldid=1081816299"
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In this lesson, you looked for ways to convert between equivalent forms of fractions, decimals, and percents. Using the portions web, write the other forms of the number for each of the given portions below. Show your work so that a team member could understand your process.
\frac { 4 } { 5 }
as a decimal, as a percent, and with words/picture.
\text{This fraction can be written as }\left ( \frac{4}{5} \right )\left ( \frac{20}{20} \right )=\frac{80}{100}.
As a decimal and a percent, this is equivalent to
0.8
80\%
. Can you describe this portion in words?
0.30
as a fraction, as a percent, and with words/picture.
In words, this is written as three tenths or thirty hundredths. Now express it as a fraction and a percent.
85\%
as a fraction, as a decimal, and with words/picture.
Refer to part of the Math Notes box from Lesson 3.1.5 below for help converting percents to decimals and fractions.
\left. \begin{array} { l } { 78.6 \% = 78.6 \div 100 = 0.786 } \\ { \text { Percent to fraction: } } \\ { \text { Use } 100 \text { as the denominator. } } \\ { \text { the number in the percent as } } \\ { \text { numerator. Simplify as need } } \\ { 22 \% = \frac { 22 } { 100 } \cdot \frac { 1 / 2 } { 1 / 2 } = \frac { 11 } { 50 } } \end{array} \right.
0.85, \frac{85}{100}= \frac{17}{20},\text{ eighty-five hundredths}
Write one and twenty-three hundredths as a percent, as a decimal, and as a fraction.
One and twenty-three hundredths is greater than one, so each equivalent form should represent a portion greater than one.
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A Study (Trait) - Open Targets Genetics Documentation
Search by a Study
View loci associated with a trait in the selected study
Identify prioritised genes functionally implicated by each locus
View 95% credible sets (where available) and proxies at each locus
Summarises the details of the selected study, including the maximal N and link-outs to the PubMed record. Selecting 'Compare to Related Studies' allows you to check for overlap between loci reported by this study, and loci for other traits either prioritised by Open Targets Genetics based on shared architecture, or selected manually by the user. Details on this feature can be found here.
Independently-Associated Loci
Loci reported by the selected study are displayed in a simplified Manhattan view, with the line of genome-wide significance highlighted red. Our methods for locus definition are detailed here. Hover over a locus to view its details, including the top-ranked gene for the locus according to the Open Targets Genetics pipeline. Selecting a chromosome using the drop-down or by clicking the chromosome number on the x-axis will expand and zoom the view of loci located on that chromosome, and restrict the loci details displayed in the accompanying table, below. Options to download the plot as a vector are provided.
Full details of each locus are summarised in the table below the Manhattan view. Results can be sorted by a single column by clicking the corresponding header, and further details on column contents are displayed when hovering over the header '?' icon. Various download options for the full table contents are available. The top-ranked gene is defined as the gene with the largest weight of functional evidence across all sources and cell types linking it to the specified locus either directly or via a
V_T
. If more than one gene is scored equally at this locus, all genes with the maximum score are shown. If the selected study has full sumstats available, the credible set size for each locus is displayed alongside the number of
V_T
defined by LD with the lead.
Clicking the lead variant or top gene links will redirect to the corresponding variant or gene entity page. To view the Locus Plot with the corresponding
V_L
, gene and study selected, click through the 'Locus' button.
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EUDML | Some results on the unramified principal series of p-adic groups. EuDML | Some results on the unramified principal series of p-adic groups.
Some results on the unramified principal series of p-adic groups.
Li, Jian-Shu. "Some results on the unramified principal series of p-adic groups.." Mathematische Annalen 292.4 (1992): 747-761. <http://eudml.org/doc/164939>.
author = {Li, Jian-Shu},
keywords = {unramified quasisplit reductive -adic group; unramified character; minimal parabolic subgroup; induced principal series representation; Whittaker model; spherical functions; Whittaker functions},
title = {Some results on the unramified principal series of p-adic groups.},
AU - Li, Jian-Shu
TI - Some results on the unramified principal series of p-adic groups.
KW - unramified quasisplit reductive -adic group; unramified character; minimal parabolic subgroup; induced principal series representation; Whittaker model; spherical functions; Whittaker functions
p
L
J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, F. Shahidi, On lifting from classical groups to
G{L}_{N}
William Casselman, Freydoon Shahidi, On irreducibility of standard modules for generic representations
unramified quasisplit reductive
p
-adic group, unramified character, minimal parabolic subgroup, induced principal series representation, Whittaker model, spherical functions, Whittaker functions
Articles by Jian-Shu Li
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Marketing technology in macroeconomics | SpringerPlus | Full Text
Kenichi Tamegawa1
In this paper, we incorporate a marketing technology into a dynamic stochastic general equilibrium model by assuming a matching friction for consumption. An improvement in matching can be interpreted as an increase in matching technology, which we call marketing technology because of similar properties. Using a simulation analysis, we confirm that a positive matching technology shock can increase output and consumption.
The considerable progress in information technology (IT) since the late 1990s increased the productivity of goods and contributed to the IT boom in the economies of many countries in the 2000s. In economics, IT development is typically expressed as an increase in total factor productivity (TFP). This is a point of view from supply side of the economy. [Jorgenson (2001]) and Jorgenson et al. ([2008]) pointed out that nonfarm business productivity growth surged from 1997 to 2001.
In addition to the supply side effects of IT, we can consider that IT also affects the demand side. Through web sites such as Amazon.com, for example, the Internet enables us to buy numerous goods instantaneously. A recent development in IT, the so-called Web 2.0, which includes social networking services such as Facebook, has enabled firms to contact individual consumers and promote their products. Recent developments in mobile phone technology, for example the iPhone, provide opportunities for matching consumers and products. This is reflected in the worldwide increase in Internet users (see Figure 1), which in turn increases opportunities for matching. In convenience, we call the technology, which enhances matching opportunity, “marketing technology,” because it can easily match consumer needs with a firm’s products and therefore resembles the concept of marketing. Broadly speaking, IT may enhance productivity as stated above. In this paper, however, we limit the scope of marketing technology to that which provides greater opportunities for sales, since studies for supply side technology like TFP are plentiful.
Internet users across the globe. Note: The line depicts percentage per 100 inhabitants. Source: ITU World Telecommunication/ICT Indicators database.
The way that production technology affects business cycles is well known1, but research on the effects of technology such as marketing technology on the macroeconomy has not yet been undertaken at least within the framework of macroeconomics. It is therefore quite interesting to investigate the effects of marketing technology. Our goals are as follows: first, to incorporate the marketing sector into an economic model; second, to assess the effects of the positive shock of marketing technology on the macroeconomy. First of all, we express marketing technology in an economic model by employing a matching or search friction in the goods market. Researchers have frequently employed this assumption in the labor market on the basis of [Mortensen and Pissarides (1994]). Adopting the matching friction suits our purpose because progress in marketing technology can be modeled as a reduction of matching friction between consumers and firms.
To accomplish the second goal, we use a dynamic stochastic general equilibrium (DSGE) model, which is a useful tool in analyzing the macro economy. The model consists of identity equations and behavioral equations that are derived from agents’ optimization problems2. Our model is constructed on the basis of a standard real business cycle (RBC) model as described in King et al. ([1988])3. Of course, it can easily be extended to a New Keynesian model by adding a sticky price assumption, as used in Christiano et al. ([2005]).
In this paper, we show the effects of marketing technology by performing a numerical simulation. The main result is a positive response of output, which occurs because progress in marketing technology can increase matched consumption. In our settings, the sudden increase in households’ consumption provides an incentive to work more to smooth out the consumption path. Similar to increases in TFP, developments in IT technology that affect the demand side can also increase output and therefore income.
The remainder of our paper is organized as follows. Section 2 explains the key equation, which plays an important role in this paper. Section 3 constructs our model. Section 4 presents a simulation analysis of marketing technology. Section 5 discusses how incorporating the marketing sector into a DSGE model alters the model’s responses to shocks from other marketing technologies. Section 5 concludes the paper.
Matching friction for consumption
This section explains the key equation of our model: matching friction. Suppose that a consumer has a consumption plan, denoted by Ct, and that firms use some amount of resources denoted by a t to advertise their goods. We then assume that consumer needs are met through the following Cobb-Douglas type matching function:
{C}_{t}^{m}={e}^{{Z}_{t}^{C}}{\left({C}_{t}\right)}^{\gamma }{\left({a}_{t}\right)}^{1-\gamma }\text{,}
where C t m represents matched consumption. In the above equation, an increase in Z t C implies that the matching opportunity becomes bigger. We therefore call it marketing technology. High planned consumption and advertisement also facilitate the matching. The motivation of assuming Eq (1) stems from the study of matching friction for the labor market introduced by [Mortensen and Pissarides (1994])4. In their study, labor matching results from a combination of vacancies offered by firms and the labor force provided by households. This assumption is also useful in a consumption matching framework.
For the following simulation, we assume that log Z t C follows an AR(1) process. Note that under this setting,
{\theta }_{t}\equiv {C}_{t}^{m}/{C}_{t}={Z}_{t}^{C}{\left({a}_{t}/{C}_{t}\right)}^{1-\gamma }
can be interpreted as a matching probability. Moreover, in Eq (1), If γ = 1 and Z t C ≡ 0, the model constructed below is reduced to a standard RBC model.
In our model, firms have a marketing sector and a production sector, households live infinitely, and there exists a the government. The population is normalized to 1. We begin by explaining the matching friction.
Firms: Production sector
Firms in the production sector have the following Cobb-Douglas production function:
{Y}_{t}={e}^{{Z}_{t}^{Y}}{\left({K}_{t}\right)}^{\alpha }{\left({h}_{t}\right)}^{1-\alpha }\text{,}
where Y t represents output; K t , capital stock; h t , hours worked; And log Z t Y, a productivity shock with mean 0. With this technology of production, firms’ gross profits are as follows:
{Y}_{t}-{a}_{t}-{w}_{t}{h}_{t}-\left({R}_{t}-1+\delta \right){K}_{t}\text{,}
where a t is the goods used in advertising. The net output for firms is therefore Y t − a t . The first-order condition for profits maximization yields
{w}_{t}=\left(1-\alpha \right)\frac{{Y}_{t}}{{h}_{t}}
The gross rental rate is as follows:
{R}_{t}=\alpha \frac{{Y}_{t}}{{K}_{t}}+1-\delta
Firms: Marketing sector
The marketing sector receives a t from the production sector and conducts marketing activities. Consequently, their goods meet consumer needs through the consumption matching function. As stated above, to conduct this activity, we assume that the marketing sector needs a t . The marketing sector demands a t to maximize C t m − a t . The first-order condition is
\left(1-\gamma \right)\frac{{C}_{t}^{m}}{{a}_{t}}=1.
Note that C t m − a t is not profit but merely a hypothetical objective function.
This subsection explains the aggregated behavior of households. First note that households are subject to the following inter-temporal budget constraint5:
{D}_{t+1}={R}_{t}{D}_{t}+{w}_{t}{h}_{t}-{\theta }_{t}{C}_{t}-{T}_{t}\text{,}
where D t represents financial assets and T t denotes lump-sum tax. Assuming that temporal utility is log θ t C t , households decide their planned consumption and labor supply by maximizing the following utility function, given {θt}:
{E}_{0}\left[\sum _{t=0}^{\infty }{\beta }^{t}\left\{log{\theta }_{t}{C}_{t}+\tau log\left(1-{h}_{t}\right)\right\}\right]\text{,}
where β represents a discount rate. The first-order conditions are
\frac{1}{{C}_{t}}={E}_{t}\left[\frac{\beta {R}_{t+1}}{{C}_{t+1}}\right]\text{,}
\frac{\tau {h}_{t}}{1-{h}_{t}}=\frac{{w}_{t}}{{C}_{t}}\text{.}
Note that the consumption path is independent {θt} as shown in Eq (8).
Assuming that capital stock is accumulated As Kt + 1 = (1 − δ)K t + I t with a depreciation rate of δ and an investment of I t and that the primary balance for the government is always zero, an equilibrium condition K t+1 = D t+1 yields
{Y}_{t}={C}_{t}^{m}+{I}_{t}+{G}_{t}+{a}_{t}\text{,}
where G t represents government expenditure (which is equal to T t ).
For convenience of understanding the flow of goods, we provide Figure 2. First, firms produce goods using labor and capital goods that are provided from households. Households (consumers) consume the goods and pay tax to government. Advertisement is implemented through the goods that firms produce; in other words, advertisements are own consumption for firms.
How does the model behave against a positive marketing shock? First, since this shock provides a matching opportunity, matched consumption increases; consequently, saving decreases. This decrease in turn raises the rental rate and provides an incentive to work more. Therefore, output also increases. Planned consumption nevertheless decreases because rental rate increases. Although a matching improvement increases output over several periods, consumption later decreases because of a consumption-smoothing motive. On the other hand, an increase in saving reduces the rental rate and causes a decrease in the labor supply also decreases. Intuitively speaking, an increase in matching technology raises consumption; this forces households to work more to compensate for the increased consumption. As a result, output increases.
To confirm the above theoretical conjecture, we linearize and simulate the model. The parameter settings are [α β δ Cmh] = [1/3 0.99 0.02 0.6 1/3], where Cm and h denote the steady-state values. For γ, we consider γ = 0.95 and γ = 0.5. A persistency parameter for log Z t C is 0.9. The output share of advertising is 0.01 in the steady state. In Figure 3, we show impulse responses to the one percent shock for log Z t C. In the case of γ = 0.5, since consumption matching is strongly affected by advertising, responses to the marketing shock are volatile.
Impulse responses to the marketing technology shock. Note: The above lines are shown as the percentage deviations from the steady state.
As shown above, while a positive marketing shock can raise output, it decreases investment. This phenomenon seems to contrast with the experience of the late1990s. In the actual economy, however, IT can increase TFP. We can therefore consider that for this period, investment increases through a positive TFP shock. Of course, since there is a possibility that matching technology increase investment in the actual economy, careful empirical research is needed.
Discussion: Consumption matching friction neutrality
How does the consumption matching friction alter responses to a supply or demand shock other than by matching technology relative to a standard RBC model? In a linearized model, the answer is that the friction does not alter the other shock responses. This is because households know how much their needs are matched by goods produced by firms; in other words, they know the matching probability θ t . Households then know the amount of goods to consume under a given shock even though matching friction is assumed. This implies that C t m does not depend on the value of γ. Regardless of the value of γ, the responses to shocks other than the marketing technology shock are not altered.
This neutrality is not a drawback but an attraction from the empirical view point. Incorporating a consumption matching friction into a DSGE model may improve the results of empirical analyses such as that of [Smets and Wouters (2003]), since adding this assumption does not harm the model properties. Further, marketing technology is considered to be a new structural shock. With this new shock, the model can allow for richer dynamics, which helps reduce the problem of the degree of stochastic singularity (see Ruge-Murcia, [2007] and Tovar, [2009]).
In this paper, we incorporated a marketing technology into a DSGE model by assuming a matching friction for consumption. The improvement in matching could be interpreted as an increase in matching technology. Using a simulation analysis, we confirmed that positive matching technology shock can raise output and consumption.
Further implications of what this paper has demonstrated in theoretical results need to be assessed through empirical studies. Fortunately, methods of empirical research on the basis of a DSGE model, for example, the method that [Smets and Wouters (2003]) used, are now becoming more familiar to economists. To investigate the effects of marketing technology on the economy of the late 1990s is quite interesting, but this is left for the future.
1For example, see [Romer (2011]).
2The motivation for using the DSGE models in analyzing the macroeconomy is to avoid the famous critique by [Lucas (1976]): a model has to be described such that it is invariant to exogenous shock.
3Famous DSGE models are surveyed in [Tovar (2009]) and Mc[Candless (2008]).
4There are many studies that investigate the effects of labor market friction on business cycles. For example, see [Shimer (2010]).
5This expression of budget constraint can be archived from the law of large numbers for θ t .
Auto regression.
Christiano L, Eichenbaum M, Evans C: Nominal rigidities and the dynamic effects of a shock to monetary policy. J Polit Econ 2005, 113: 1-45. 10.1086/426038
Jorgenson DW: Information Technology and the U.S. Economy. Am Econ Rev 2001, 91: 1-32.
Jorgenson DW, Ho MS, Stiroh KJ: A Retrospective Look at the U.S. Productivity Growth Resurgence. J Econ Perspect 2008,22(1):3-24. 10.1257/jep.22.1.3
King R, Plosser C, Rebelo S: Production, growth and business cycles I: the basic neoclassical model. J Monet Econ 1988, 21: 195-232. 10.1016/0304-3932(88)90030-X
Lucas R: Econometric policy evaluation: a critique. Carnegie-Rochester Conf Ser Public Policy 1976, 1: 19-46.
McCandless G: The ABCs of RBCs. Harvard University Press, Cambridge; 2008.
Mortensen T, Pissarides C: Job creation and job destruction in the theory of unemployment. Rev Econ Stud 1994, 61: 397-415. 10.2307/2297896
Romer D: Advanced Macroeconomics. McGraw-Hill, Irwin; 2011.
Ruge-Murcia FJ: Methods to estimate dynamic stochastic general equilibrium models. J Econ Dyn Control 2007, 31: 2599-2636. 10.1016/j.jedc.2006.09.005
Shimer R: Labor Markets and Business Cycles. Princeton University Press, Princeton; 2010.
Smets F, Wouters R: An estimated dynamic stochastic general equilibrium model of the Euro area. J Eur Econ Assoc 2003, 20: 1123-1175.
Tovar CE: DSGE Models and Central Banks. Economics: The Open-Access, Open-Assessment E-Journal 2009., 3: http://dx.doi.org/10.5018/economics-ejournal.ja.2009-16
I am grateful to anonymous referees and Shin Fukuda for their helpful comments.
School of Commerce, Meiji University, 1-1 Kanda-Surugadai, Chiyoda-ku, Tokyo, 101-8301, Japan
Correspondence to Kenichi Tamegawa.
KT carried out constructing the model, conducted the simulation analysis and drafted the manuscript. All authors read and approved the final manuscript.
Tamegawa, K. Marketing technology in macroeconomics. SpringerPlus 1, 28 (2012). https://doi.org/10.1186/2193-1801-1-28
Matching friction
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\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):
R
R
R≔\mathrm{PolynomialRing}\left([x,y,u,v]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
f≔ux+v;
g≔vy+u
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}
Using GeneralConstruct, construct a constructible set from the common solutions of
g
{u}^{2}+{v}^{2}-1
\mathrm{cs}≔\mathrm{GeneralConstruct}\left([f,g],[{u}^{2}+{v}^{2}-1],R\right)
\textcolor[rgb]{0,0,1}{\mathrm{cs}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{lrs}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_system}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_system}}]
\mathrm{lrc}≔\mathrm{map}\left(\mathrm{RepresentingChain},\mathrm{lrs},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{lrc}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]
\mathrm{map}\left(\mathrm{Equations},\mathrm{lrc},R\right)
[[\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}]]
\mathrm{map}\left(\mathrm{RepresentingInequations},\mathrm{lrs},R\right)
[[{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{v}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[]]
{u}^{2}+{v}^{2}-1
1
{u}^{2}+{v}^{2}-1
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Senkou (Leading) Span B Definition
What Is Senkou (Leading) Span B?
Senkou Span B, also called Leading Span B, is one of five components of the Ichimoku Cloud indicator. Leading Span B works in conjunction with the Senkou Span A line to form a cloud formation known as a kumo. The cloud provides support and resistance levels. Both Senkou Span A and Span B are plotted 26 periods into the future, providing a glimpse into where support and resistance may form next.
Senkou Span B forms a cloud with Senkou Span A which shows potential areas of support or resistance.
When the price is above the cloud, the lines act as support and when the price is below the cloud, the lines act as resistance.
Leading Span B only uses historical data, yet it is called “leading” because its value is plotted 26 periods into the future.
Understanding Senkou (Leading) Span B
Senkou Span B and Senkou Span A form the cloud formation in an Ichimoku Kinko Hyo indicator, also called the Ichimoku Cloud. The Ichimoku Cloud includes five different lines that provide traders with different pieces of information.
Senkou Span B moves slower than Senkou Span A because Span B is calculated using 52 periods of data. Senkou Span A is calculated using 26 periods and nine periods. The fewer periods used in Span A mean that the indicator will react quicker to price changes.
If Senkou Span B is at the top of the cloud, then this is generally considered bearish. Short-term prices (Span A) have fallen below the longer-term price midpoint (Span B). The Senkou Span lines provide the midpoint of a price range because they are dividing the sum of the high and the low by two.
When Senkou Span A is forming the top of the cloud, it is considered bullish, since the shorter-term price (Span A) is moving above the longer-term midpoint price (Span B).
Leading Span A and Span B crossovers may signal a trend change. When Span A crosses above Span B, it may indicate the start of an uptrend. When Span A crosses below Span B, a downtrend or correction may be starting.
When the price is above Senkou Span A and/or Span B, some traders view them as providing potential support. If the price falls to these lines, then it may bounce off of them. When the price is below Leading Span A and/or Span B, these lines are viewed as providing resistance or possible areas to sell or short sell.
Senkou (Leading) Span B Calculation
\begin{aligned}&\text{Senkou Span B}=\frac {\text{52 Period High} +\text{ 52 Period Low}}{2}\\&\text{Plot value 26 periods into the future.}\end{aligned}
Senkou Span B=252 Period High+ 52 Period LowPlot value 26 periods into the future.
Find the high price during the last 52 periods.
Find the low price during the last 52 periods.
Add the high and low periods together, then divide by two.
Plot the value 26 periods into the future.
Repeat steps one through four when each period ends.
A simple moving average (SMA) sums up the closing prices over X number of periods, then divides the result by X to provide an average of all of the closing prices. Leading Span B doesn’t calculate an average; rather, it calculates the midpoint of a 52-period range. These two indicators will look quite different on a chart. The Senkou Spans also are plotted 26 periods into the future, and that isn’t the norm for an SMA.
Limitations of Senkou Span B
Senkou Span B is a lagging indicator, even though its value is plotted 26 periods into the future. The indicator can be slow to react to price changes, since it can take a long time for the price to generate a new high or low over 52 periods. Luckily, Senkou Span A reacts quicker, but also, sometimes it may not react quickly enough.
Crossovers may occur after a large price move already has occurred, making the crossover signal almost useless for trading purposes. Also, the span lines may not provide support or resistance, and the price may move right through them. However, this provides information about the trend and its direction.
Senkou Span B should be used in conjunction with other technical indicators and methodologies, such as price action trading, to help confirm or reject the information that Span B and the other Ichimoku indicators are providing.
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Patrons & Champions - Documentation
Patrons & Champions are the two special User categories involved in our unique Copy Trading solution, and their agreements are referred to as Patronages.
A "Smart Contract" is essentially just a contract with built-in payment and enforcement mechanisms.
A patronage agreement is a quid pro quo (something for something else), a customizable “trading contract” between two or more users (Patron/s & Champion/s), secured & enforced by smart contract.
A Champion performs trading decisions on behalf of Patrons in exchange for a success fee based on pre-agreed terms & conditions.
All patronages must meet the following basic condition to be considered successful:
Y > X + CF + ½ CF
Where X is the initial sum invested by the Patron, Y is the final sum accrued at the end of the patronage and CF is the custom Champion's Fee. If the final amount be lower than the initial one plus the Champion’s share, plus half as much, the patronage is automatically considered unsuccessful and any agreed upon Champion Fee will be nullified.
This means that any patronage must at least return a profit to be considered successful
$ 100 patronage featuring a 2% Champion fee would need to return at least $ 103 not to be failed.
This mechanism also helps preventing few large accounts from monopolizing the service, ensuring the diversification of Champions' strategies, pricing and targets.
Patrons and Champions, other than the above-stated minimum success condition, may customize the terms and conditions of their patronage agreement to their leisure. All patronages require the establishment and agreement of at least one termination condition to be negotiated & confirmed.
Both categories may publish, negotiate, veto or accept their own or the other party’s terms and conditions. Patronage agreements are fully customizable.
Patrons (copier)
Sit back & Collect
Patrons are Users investing in other Users, to trade on their behalf. Any user that invests in a Champion is automatically considered a Patron, no opt in is required.
Patrons are ranked based on the amount of volume they have committed to Patronages, this is solely a cosmetic difference, for the purpose of differentiation. No direct economic advantage is obtained from a higher tier, but for example, some Champions may elect to restrict their service only to big fish.
Volume committed to Patronages (yearly)
Don't know how to trade? Don't understand crypto? Or just lacking the time? Hire a Champion!
Champions (copied)
Champions are Users offering their time & trading expertise to other Users, in exchange for (at least) a success fee. Additional targets and conditions, with further potential fee structures, may be negotiated independently by the parties involved through the system.
Your skills are valuable! Get more out of your trades.
Champions are evaluated by potential backers based on two metrics: Glory Points and PPR (Positive Patronage Rate) - as is hopefully intuitive, this is the rate at which each Champion has successfully completed the patronages it has accepted thus far. Expressed in percentage format.
PPR = (Successful / Total)*100
By subscribing as a Champion, you agree to allow CryptoArena to collect data relating to your trading performance and choices on the platform, and to display it publicly, for the purposes of facilitating the evaluation of risks, and the investment decisions of Patrons considering your services. Your identity is protected through pseudonimity (public figures may request use of their real identity upon verification).
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4.5 Decoupling Beam and Diffuse: Clearness and Clear Sky Indices | EME 810: Solar Resource Assessment and Economics
J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 8: Measure & Estimation of the Solar Resource (Focus on Empirical Correlation for Components.)
Reindl, Beckman, and Duffie (1990) Diffuse Fraction Correlations. Solar Energy J. 45(1) 1-7.
[repeated] C. A. Gueymard (2008) From Global Horizontal to Global Tilted Irradiance: How accurate are solar energy engineering predictions in practice? Solar 2008 Conference, San Diego, CA, American Solar Energy Society
Optional: Liu and Jordan (1960) The Interrelationship and Characteristic Distribution of Direct, Diffuse, and Total Solar Radiation. Solar Energy J. 4(3), 1–19.
Please make sure you read all of Ch 8 in SECS for this lesson, again maintaining focus on the same section "Empirical Correlation for Components" and this page content. In the two additional readings, it is OK to scan the Reindl et al. paper and the Gueymard paper for key elements that are parallel with the page content.
System designers do not always have the benefit of designing SECS with horizontal surfaces. Many times, these surfaces are tilted at various angles and have various orientations. In such situations, designers and engineers must make estimations for tilted surfaces based on data for horizontal surfaces.
In order to estimate, we first have to break apart the beam horizontal component from the diffuse horizontal components. This has been achieved historically by a methodology established in the 1950s and 60s by Profs. Ben Liu and Richard Jordan (our supplemental reading that is included to add context and the entire line of research that has been applied from then until now).
The availability of solar data is very important when calculating the amount of radiation incident on a collector. Engineers and designers commonly make use of average hourly, daily, and monthly local data. However, the most common measurement available is the Global Horizontal Irradiance (GHI), which is then integrated through a data logger into hourly irradiation, or minute irradiation.
Estimation is an effective tool that involves the use of empirical models that were developed over the last 4-5 decades. The only tools we need are the equations for calculating hourly and daily extraterrestrial irradiance (Air Mass Zero, or AM0) and the integrated energy density (J/m2) gathered from a horizontally mounted pyranometer. These empirical methods to decouple beam and diffuse horizontal components are termed Liu and Jordan transformations, after the initial paper in 1960.
The Clearness Index
The linkage between the two data for horizontal orientation are the clearness indices (kT, KT, and
{\overline{K}}_{T}
). This index is simply a measure of the ratio of measured irradiation in a locale relative to the extraterrestrial irradiation calculated (AMo) at the given locale.
{k}_{T}=\frac{I}{{I}_{0}}
: the hourly clearness index for Total or global irradiation (that's what the "T" is for). This is a ratio of measured energy density against energy density for extraterrestrial solar in one hour.
{K}_{T}=\frac{H}{{H}_{0}}
: the daily clearness index for Total irradiation. This is a ratio of measured energy density against energy density for extraterrestrial solar in one day.
{\overline{K}}_{T}=\frac{\overline{H}}{\overline{{H}_{0}}}
: the monthly average daily clearness index for Total irradiation. This is a ratio of measured energy density averaged over the month as one day, against the energy density for extraterrestrial solar for an average day.
For KT →1: atmosphere is clear. For KT →0: atmosphere is cloudy. However, this measure incorporates both light scattering and light absorption. Keep in mind that a fraction is not a percentage, and in our case for a cumulative distribution, it is a decimal value between 0--1.
The Clear Sky Index
There is also an alternate indicator for the way that the atmosphere attenuates light on an hour to hour or day to day basis. This is the "clear sky index" (kc). Mathematically, the clear sky index is defined as
kc=\frac{\text{measured}}{\text{calculated clear sky}}
and it has been proposed that 1-kc is a very good indicator of the degree of "cloudiness" in the sky.
So, why do we use either the clearness index or the clear sky index? The answer at the moment is persistence. While it is likely that the clear sky index is more useful than the older clearness index in the long term, all the core research for the empirical calculations used in softwares like TRNSYS, Energy+, and SAM was based on kT.
The Historical Backdrop: the Clearness Index
In the 1960s, Liu and Jordan found that for different US locations with the same value of
{\overline{K}}_{T}
, the cumulative distribution curves of KT were identical, almost irrespective of latitude and elevation.\marginnote{A cumulative distribution describes the frequency or fraction of occurrence of days in the month below a given daily clearness index, KT}. This work was expanded into equations by Bendt et al.,\cite{Bendt81} using 20 years of real measurements in 90 locations in the USA. However, it was determined that the data sets were not so similar from region to region (e.g., the tropics had different correlations than the temperate USA, India was different from Africa, etc.) This work was followed by Hawas and Muneer for India and Lloyd for the UK, among others.\cite{Hawas85,Lloyd82}
Remember this! KT distributions are not universal---they are regional and empirically derived. For all of our future work, we will only rely on hourly kT values, and the manner in which kT is used to back out a value of Ib, the hourly beam irradiation component on a horizontal surface.
1. What is the ratio for the hourly clearness index?
ANSWER: kT is the ratio of measured hourly horizontal irradiance (I) per calculated hourly horizontal irradiation estimated for Air Mass Zero (AM0), or I0.
2. What is the ratio for the hourly clear sky index?
ANSWER: kc is the ratio of measured hourly horizontal irradiance (I) per calculated hourly horizontal irradiation estimated for clear sky conditions (via REST2, or Bird, or otherwise).
3. Which ratio is an argument in the typical Liu-Jordan style empirical correlations?
ANSWER: kT, the hourly clearness index
4. What are the results of using the clearness indices in Liu-Jordan empirical correlations?
ANSWER: Diffuse horizontal fraction, or the ratio of diffuse horizontal irradiation in an hour relative to hourly measured irradiation. Beam horizontal components can be inferred from the Id fractions.
5. Can we determine the contribution of the ground components or horizon diffuse components in this correlation step?
ANSWER: No, one must use an anisotropic sky model such as Perez et al. (1990) or HDKR in a follow-up step.
‹ 4.4 Empirical Correlation for Estimating Components of Light up 4.6 Using Components for a Tilted Aperture ›
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Williams %R, also known as the Williams Percent Range, is a type of momentum indicator that moves between 0 and -100 and measures overbought and oversold levels. The Williams %R may be used to find entry and exit points in the market. The indicator is very similar to the Stochastic oscillator and is used in the same way. It was developed by Larry Williams and it compares a stock’s closing price to the high-low range over a specific period, typically 14 days or periods.
An overbought or oversold reading doesn't mean the price will reverse. Overbought simply means the price is near the highs of its recent range, and oversold means the price is in the lower end of its recent range.
\begin{aligned} &\text{Wiliams \%}R=\frac{\text{Highest High}-\text{Close}}{\text{Highest High}-\text{Lowest Low}}\\ &\textbf{where}\\ &\text{Highest High} = \text{Highest price in the lookback}\\ &\text{period, typically 14 days.}\\ &\text{Close} = \text{Most recent closing price.}\\ &\text{Lowest Low} = \text{Lowest price in the lookback}\\ &\text{period, typically 14 days.} \end{aligned}
Wiliams %R=Highest High−Lowest LowHighest High−ClosewhereHighest High=Highest price in the lookbackperiod, typically 14 days.Close=Most recent closing price.Lowest Low=Lowest price in the lookback
The Williams %R is calculated based on price, typically over the last 14 periods.
On the 14th period, note the current price, the highest price, and lowest price. It is now possible to fill in all the formula variables for Williams %R.
Traders can also watch for momentum failures. During a strong uptrend, the price will often reach -20 or above. If the indicator falls, and then can't get back above -20 before falling again, that signals that the upward price momentum is in trouble and a bigger price decline could follow.
Overbought and oversold readings on the indicator don't mean a reversal will occur. Overbought readings actually help confirm an uptrend, since a strong uptrend should regularly see prices that are pushing to or past prior highs (what the indicator is calculating).
The indicator can also be too responsive, meaning it gives many false signals. For example, the indicator may be in oversold territory and starts to move higher, but the price fails to do so. This is because the indicator is only looking at the last 14 periods. As periods go by, the current price relative to the highs and lows in the lookback period changes, even if the price hasn't really moved.
Larry Williams CTI Publishing. "Original Williams %R." Accessed Nov. 19, 2020.
What are the main differences between Williams %R oscillator & The Relative Strength Index (RSI)?
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Control of Hopf Bifurcation in Autonomous System Based on Washout Filter
2013 Control of Hopf Bifurcation in Autonomous System Based on Washout Filter
Wenju Du, Yandong Chu, Jiangang Zhang, Yingxiang Chang, Jianning Yu, Xinlei An
In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibrium
{E}_{0}
undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibrium
{E}_{+}
. Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems.
Wenju Du. Yandong Chu. Jiangang Zhang. Yingxiang Chang. Jianning Yu. Xinlei An. "Control of Hopf Bifurcation in Autonomous System Based on Washout Filter." J. Appl. Math. 2013 1 - 16, 2013. https://doi.org/10.1155/2013/482351
Wenju Du, Yandong Chu, Jiangang Zhang, Yingxiang Chang, Jianning Yu, Xinlei An "Control of Hopf Bifurcation in Autonomous System Based on Washout Filter," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-16, (2013)
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Proof of the De Gennes formula for the superheating field in the weak $\kappa $ limit
\kappa
author = {Bolley, Catherine and Helffer, Bernard},
title = {Proof of the {De} {Gennes} formula for the superheating field in the weak $\kappa $ limit},
TI - Proof of the De Gennes formula for the superheating field in the weak $\kappa $ limit
Bolley, Catherine; Helffer, Bernard. Proof of the De Gennes formula for the superheating field in the weak $\kappa $ limit. Annales de l'I.H.P. Analyse non linéaire, Tome 14 (1997) no. 5, pp. 597-613. http://www.numdam.org/item/AIHPC_1997__14_5_597_0/
[1] C. Bolley and B. Helffer, Rigorous results on the Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field, Nonlinear Studies, Part I: Vol. 3, n° 1, 1996, pp. 1-29, Part II: Vol. 3, n° 2, 1996, pp. 1-32. | MR 1396033 | Zbl 0857.34006
[2] C. Bolley and B. Helffer, Sur les asymptotiques des champs critiques pour l'équation de Ginzburg-Landau. Séminaire Equations aux dérivées partielles de l'Ecole Polytechnique, November 1993. | Numdam | Zbl 0877.35120
[3] C. Bolley and B. Helffer, Rigorous results for the Ginzburg-Landau equations associated to a superconducting film in the weak κ-limit, Reviews in Math. Physics, Vol. 8, n° 1, 1996, pp. 43-83. | MR 1372515 | Zbl 0864.35097
[4] C. Bolley and B. Helffer, Superheating in a film in the weak κ limit: numerical results and approximate models, Dec. 1994 (Part I to appear in M2 AN).
[6] S.J. Chapman, Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model, Preprint, 1992. | MR 1359498
[7] V.P. Galaiko, Superheating critical field for superconductors of the first kind, Soviet Physics JETP, Vol. 27, n° 1, July 1968.
[8] P.G. De Gennes, Superconductivity, selected topics in solid state physics and theoretical Physics, Proc. of 8th Latin american school of physics, Caracas, 1966.
[9] V.L. Ginzburg, On the theory of superconductivity, Nuovo Cimento, Vol. 2, 1955, p. 1234. | Zbl 0067.23504
[10] V.L. Ginzburg, On the destruction and the onset of superconductivity in a magnetic field, Soviet Physics JETP, Vol. 7, 1958, p. 78. | Zbl 0099.44703
[11] V.L. Ginzburg and L.D. Landau, On the theory of superconductivity, Zh. Eksperim. i teor. Fiz., Vol. 20, 1950, pp. 1064-1082. English translation Men of Physics, L. D. LANDAU, I, Ed. by D. TerHaar, Pergamon oxford, 1965, pp. 138-167.
[12] S.P. Hastings, M.K. Kwong and W.C. Troy, The existence of multiple solutions for a Ginzburg-Landau type model of superconductivity, Preprint May 1995. | MR 1426209
[13] D. Saint James And P.G. De Gennes, Onset of superconductivity in decreasing fields, Phys. Lett., Vol. 7, 1963, p. 306.
[14] D. Saint James, G. Sarma and E.J. Thomas, Type II Superconductivity, Pergamon Press, 1969.
[15] H. Parr, Superconductive superheating field for finite κ, Z. Physik, Vol. B25, 1976, pp. 359-361.
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EUDML | A strong Liouville theorem for -harmonic functions on graphs. EuDML | A strong Liouville theorem for -harmonic functions on graphs.
A strong Liouville theorem for
p
-harmonic functions on graphs.
Holopainen, Ilkka, and Soardi, Paolo M.. "A strong Liouville theorem for -harmonic functions on graphs.." Annales Academiae Scientiarum Fennicae. Mathematica 22.1 (1997): 205-226. <http://eudml.org/doc/226694>.
author = {Holopainen, Ilkka, Soardi, Paolo M.},
keywords = {-harmonic functions on graphs; -Laplacian; Harnack inequality; Liouville theorem; -harmonic functions on graphs; -Laplacian},
title = {A strong Liouville theorem for -harmonic functions on graphs.},
AU - Soardi, Paolo M.
TI - A strong Liouville theorem for -harmonic functions on graphs.
KW - -harmonic functions on graphs; -Laplacian; Harnack inequality; Liouville theorem; -harmonic functions on graphs; -Laplacian
Thierry Delmotte, Harnack inequalities on graphs
Romain Tessera, Vanishing of the first reduced cohomology with values in an
{L}^{p}
p
-harmonic functions on graphs,
p
-Laplacian, Harnack inequality, Liouville theorem,
p
p
Articles by Soardi
|
Ozone Layer — lesson. Science CBSE, Class 9.
The triatomic molecule of oxygen is known as ozone (
{O}_{3}
). The ozone layer is also called the ozonosphere. It is the upper atmosphere region, which is \(9 \)and \(22 \)miles above the earth’s surface, containing relatively high ozone molecules concentrations. This layer is present in the stratosphere.
The ozone was discovered by French physicists Charles Fabry and Henri Buisson in \(1913\). According to seasons and geographic regions, the thickness of the ozone layer differs.
In the stratosphere, the temperature increases with an increase in height due to the absorption of solar radiation by the ozone layer.
The ozone layer blocks almost all solar radiation of wavelength less than \(290 \)nanometers from reaching the earth’s surface, including ultraviolet (UV) and other forms of radiation that could injure or kill most living things. Ozone is poisonous and unstable gas near the surface of the earth.
Fact: In our solar system, Venus also has a thin ozone layer.
The UV radiation is from \(100\)-\(240\)nm; this photon energy disrupts the bond in the oxygen molecule and breaks it. The following are the steps involved in ozone formation.
Step 1: The breaking down of chemical bonds within the oxygen molecules by high-energy solar photons results in the free oxygen atoms.
Step 2: The single oxygen atom reacts with the oxygen molecules forming ozone.
The overall reaction is . This process of ozone formation is called photodissociation.
The UV radiation is from \(240\)-\(315\)nm. This photon energy disrupts the bond in the ozone molecule and breaks it.
The breaking down of chemical bonds within the ozone molecules by high-energy solar photons and resulting in oxygen molecules.
The overall net reaction destruction of ozone is
The thinning of the ozone layer or depletion is caused due to various factors such as modernisation techniques, industrialisation etc.
1. Man-made cause
The chemicals that release chlorine and bromine when exposed to intense ultraviolet light in the stratosphere are called as ozone-depleting substances (ODS).
Chlorine and bromine are found to be stable and in the earth's atmosphere, but they can float and remain static in the stratosphere, and they can react at supersonic speeds, causing ozone depletion.
Fact: One chlorine atom has the capacity to break one lakh of ozone molecules.
The chemicals such as chlorofluorocarbons (CFCs), carbon tetrachloride, hydrochlorofluorocarbons (HCFCs), methyl chloroform, brominated fluorocarbons (halons) are known as ozone-depleting substances.
These ODS are commonly used for cold cleaning, vapour degreasing, chemical processing, adhesives, aerosols, fire extinguishers, solvents, etc. They are found in appliances such as freezers, refrigerators, air conditioners etc.
The volcanic eruptions, stratospheric winds, and sun-spots (dark in colour and cooler region of the sun) are natural phenomena for ozone depletion.
Still, the percentage of depletion is \(1-2\%\), significantly less than the man-made cause of depletion.
Effect of ozone depletion:
The effect of ozone depletion can lead to severe impacts for all life forms on earth. When harmful radiation enters the earth's surface, it can cause cancer, cataracts, and immune deficiency disorders.
Also, it affects materials such as plastics, wood, rubber etc., where it will degrade them to a more considerable extent.
Prevention of ozone depletion:
Prohibit the Use of harmful ozone-depleting substances or chemicals
Use of eco-friendly products for all needs
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Play Voice - Monogatari Documentation
Play a voice audio file
'play voice <voice_id> [with [properties]]'
The play voice action let's you, as it name says, play voice files so that you can make your characters speak. You can play as many voices as you want simultaneously.
To stop a voice, check out the Stop Voice documentation.
Action ID: Voice
The name of the voice file you want to play. These assets must be declared beforehand.
The following is a comprehensive list of the properties available for you to modify certain behaviors of this action.
The fade property let's you add a fade in effect to the voice, it accepts a time in seconds, representing how much time you want it to take until the voice reaches it's maximum volume.
The volume property let's you define how high the voice will be played.
Make the voice loop. This property does not require any value.
To play a voice, you must first add the file to your assets/voice/ directory and then declare it. To do so, Monogatari has an has a function that will let you declare all kinds of assets for your game.
'<voice_id>': 'voiceFileName'
The following will play the sound, and once the sound ends, it will simply stop.
'This is the dialog that the voice file is narrating',
The following will play the voice file, and once it ends, it will start over on an infinite loop until it is stopped using the Stop Voice Action.
The following will play the voice file, and will use a fade in effect.
'play voice dialog_001 with fade 3',
The following will set the volume of this voice to 73%.
'play voice dialog_001 with volume 73',
Please note however, that the user's preferences regarding volumes are always respected, which means that this percentage is taken from the current player preferences, meaning that if the player has set the volume to 50%, the actual volume value for the voice will be the result of:
50 * 0.73 = 36.5%
'play voice dialog_001 with volume 100 loop fade 20',
|
EUDML | Moments of characteristic polynomials enumerate two-rowed lexicographic arrays. EuDML | Moments of characteristic polynomials enumerate two-rowed lexicographic arrays.
Moments of characteristic polynomials enumerate two-rowed lexicographic arrays.
Strahov, E.. "Moments of characteristic polynomials enumerate two-rowed lexicographic arrays.." The Electronic Journal of Combinatorics [electronic only] 10.1 (2003): Research paper R24, 8 p.-Research paper R24, 8 p.. <http://eudml.org/doc/123339>.
@article{Strahov2003,
author = {Strahov, E.},
keywords = {random unitary matrix; moments of characteristic polynomials; Keating and Snaith conjecture; last passage percolation theory; lexicographic array; Riemann zeta function},
title = {Moments of characteristic polynomials enumerate two-rowed lexicographic arrays.},
AU - Strahov, E.
TI - Moments of characteristic polynomials enumerate two-rowed lexicographic arrays.
KW - random unitary matrix; moments of characteristic polynomials; Keating and Snaith conjecture; last passage percolation theory; lexicographic array; Riemann zeta function
random unitary matrix, moments of characteristic polynomials, Keating and Snaith conjecture, last passage percolation theory, lexicographic array, Riemann zeta function
L
\zeta \left(s\right)
L\left(s,\chi \right)
Articles by Strahov
|
Fourier Transform - MATLAB & Simulink - MathWorks Benelux
Relationship to the Fourier Transform
Visualizing the Discrete Fourier Transform
Frequency Response of Linear Filters
Perform Fast Convolution Using the Fourier Transform
Perform FFT-Based Correlation to Locate Image Features
The Fourier transform is a representation of an image as a sum of complex exponentials of varying magnitudes, frequencies, and phases. The Fourier transform plays a critical role in a broad range of image processing applications, including enhancement, analysis, restoration, and compression.
If f(m,n) is a function of two discrete spatial variables m and n, then the two-dimensional Fourier transform of f(m,n) is defined by the relationship
F\left({\omega }_{1},{\omega }_{2}\right)=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }f\left(m,n\right){e}^{-j{\omega }_{1}m}{e}^{-j{\omega }_{2}n}.
The variables ω1 and ω2 are frequency variables; their units are radians per sample. F(ω1,ω2) is often called the frequency-domain representation of f(m,n). F(ω1,ω2) is a complex-valued function that is periodic both in ω1 and ω2, with period
2\pi
. Because of the periodicity, usually only the range
-\pi \le {\omega }_{1},{\omega }_{2}\le \pi
is displayed. Note that F(0,0) is the sum of all the values of f(m,n). For this reason, F(0,0) is often called the constant component or DC component of the Fourier transform. (DC stands for direct current; it is an electrical engineering term that refers to a constant-voltage power source, as opposed to a power source whose voltage varies sinusoidally.)
The inverse of a transform is an operation that when performed on a transformed image produces the original image. The inverse two-dimensional Fourier transform is given by
f\left(m,n\right)=\frac{1}{4{\pi }^{2}}{\int }_{{\omega }_{1}=-\pi }^{\pi }{\int }_{{\omega }_{2}=-\pi }^{\pi }F\left({\omega }_{1},{\omega }_{2}\right){e}^{j{\omega }_{1}m}{e}^{j{\omega }_{2}n}d{\omega }_{1}d{\omega }_{2}.
Roughly speaking, this equation means that f(m,n) can be represented as a sum of an infinite number of complex exponentials (sinusoids) with different frequencies. The magnitude and phase of the contribution at the frequencies (ω1,ω2) are given by F(ω1,ω2).
To illustrate, consider a function f(m,n) that equals 1 within a rectangular region and 0 everywhere else. To simplify the diagram, f(m,n) is shown as a continuous function, even though the variables m and n are discrete.
The following figure shows, as a mesh plot, the magnitude of the Fourier transform,
|F\left({\omega }_{1},{\omega }_{2}\right)|,
of the rectangular function shown in the preceding figure. The mesh plot of the magnitude is a common way to visualize the Fourier transform.
Magnitude Image of a Rectangular Function
The peak at the center of the plot is F(0,0), which is the sum of all the values in f(m,n). The plot also shows that F(ω1,ω2) has more energy at high horizontal frequencies than at high vertical frequencies. This reflects the fact that horizontal cross sections of f(m,n) are narrow pulses, while vertical cross sections are broad pulses. Narrow pulses have more high-frequency content than broad pulses.
Another common way to visualize the Fourier transform is to display
\mathrm{log}|F\left({\omega }_{1},{\omega }_{2}\right)|
as an image, as shown.
Log of the Fourier Transform of a Rectangular Function
Using the logarithm helps to bring out details of the Fourier transform in regions where F(ω1,ω2) is very close to 0.
Examples of the Fourier transform for other simple shapes are shown below.
Fourier Transforms of Some Simple Shapes
Working with the Fourier transform on a computer usually involves a form of the transform known as the discrete Fourier transform (DFT). A discrete transform is a transform whose input and output values are discrete samples, making it convenient for computer manipulation. There are two principal reasons for using this form of the transform:
The input and output of the DFT are both discrete, which makes it convenient for computer manipulations.
There is a fast algorithm for computing the DFT known as the fast Fourier transform (FFT).
The DFT is usually defined for a discrete function f(m,n) that is nonzero only over the finite region
0\le m\le M-1
0\le n\le N-1
. The two-dimensional M-by-N DFT and inverse M-by-N DFT relationships are given by
F\left(p,q\right)=\sum _{m=0}^{M-1}\sum _{n=0}^{N-1}f\left(m,n\right){e}^{-j2\pi pm/M}{e}^{-j2\pi qn/N}\text{ }\begin{array}{c}p=0,\text{ }1,\text{ }...,\text{ }M-1\\ q=0,\text{ }1,\text{ }...,\text{ }N-1\end{array}
f\left(m,n\right)=\frac{1}{MN}\sum _{p=0}^{M-1}\sum _{q=0}^{N-1}F\left(p,q\right){e}^{j2\pi pm/M}{e}^{j2\pi qn/N}\text{ }\begin{array}{c}m=0,\text{ }1,\text{ }...,\text{ }M-1\\ \text{ }n=0,\text{ }1,\text{ }...,\text{ }N-1\end{array}
The values F(p,q) are the DFT coefficients of f(m,n). The zero-frequency coefficient, F(0,0), is often called the "DC component." DC is an electrical engineering term that stands for direct current. (Note that matrix indices in MATLAB® always start at 1 rather than 0; therefore, the matrix elements f(1,1) and F(1,1) correspond to the mathematical quantities f(0,0) and F(0,0), respectively.)
The MATLAB functions fft, fft2, and fftn implement the fast Fourier transform algorithm for computing the one-dimensional DFT, two-dimensional DFT, and N-dimensional DFT, respectively. The functions ifft, ifft2, and ifftn compute the inverse DFT.
The DFT coefficients F(p,q) are samples of the Fourier transform F(ω1,ω2).
\begin{array}{cc}F\left(p,q\right)=F\left({\omega }_{1},{\omega }_{2}\right){|}_{\begin{array}{l}{\omega }_{1}=2\pi p/M\\ {\omega }_{2}=2\pi q/N\end{array}}& \begin{array}{l}p=0,1,...,M-1\\ q=0,1,...,N-1\end{array}\end{array}
Construct a matrix f that is similar to the function f(m,n) in the example in Definition of Fourier Transform. Remember that f(m,n) is equal to 1 within the rectangular region and 0 elsewhere. Use a binary image to represent f(m,n).
f = zeros(30,30);
f(5:24,13:17) = 1;
imshow(f,'InitialMagnification','fit')
Compute and visualize the 30-by-30 DFT of f with these commands.
F = fft2(f);
F2 = log(abs(F));
imshow(F2,[-1 5],'InitialMagnification','fit');
colormap(jet); colorbar
Discrete Fourier Transform Computed Without Padding
This plot differs from the Fourier transform displayed in Visualizing the Fourier Transform. First, the sampling of the Fourier transform is much coarser. Second, the zero-frequency coefficient is displayed in the upper left corner instead of the traditional location in the center.
To obtain a finer sampling of the Fourier transform, add zero padding to f when computing its DFT. The zero padding and DFT computation can be performed in a single step with this command.
F = fft2(f,256,256);
This command zero-pads f to be 256-by-256 before computing the DFT.
imshow(log(abs(F)),[-1 5]); colormap(jet); colorbar
Discrete Fourier Transform Computed with Padding
The zero-frequency coefficient, however, is still displayed in the upper left corner rather than the center. You can fix this problem by using the function fftshift, which swaps the quadrants of F so that the zero-frequency coefficient is in the center.
F = fft2(f,256,256);F2 = fftshift(F);
imshow(log(abs(F2)),[-1 5]); colormap(jet); colorbar
The resulting plot is identical to the one shown in Visualizing the Fourier Transform.
This section presents a few of the many image processing-related applications of the Fourier transform.
The Fourier transform of the impulse response of a linear filter gives the frequency response of the filter. The function freqz2 computes and displays a filter's frequency response. The frequency response of the Gaussian convolution kernel shows that this filter passes low frequencies and attenuates high frequencies.
Frequency Response of a Gaussian Filter
See Design Linear Filters in the Frequency Domain for more information about linear filtering, filter design, and frequency responses.
This example shows how to perform fast convolution of two matrices using the Fourier transform. A key property of the Fourier transform is that the multiplication of two Fourier transforms corresponds to the convolution of the associated spatial functions. This property, together with the fast Fourier transform, forms the basis for a fast convolution algorithm.
Note: The FFT-based convolution method is most often used for large inputs. For small inputs it is generally faster to use the imfilter function.
Create two simple matrices, A and B. A is an M-by-N matrix and B is a P-by-Q matrix.
Zero-pad A and B so that they are at least (M+P-1)-by-(N+Q-1). (Often A and B are zero-padded to a size that is a power of 2 because fft2 is fastest for these sizes.) The example pads the matrices to be 8-by-8.
B(8,8) = 0;
Compute the two-dimensional DFT of A and B using the fft2 function. Multiply the two DFTs together and compute the inverse two-dimensional DFT of the result using the ifft2 function.
C = ifft2(fft2(A).*fft2(B));
Extract the nonzero portion of the result and remove the imaginary part caused by roundoff error.
C = C(1:5,1:5);
C = real(C)
7.0000 21.0000 30.0000 23.0000 9.0000
This example shows how to use the Fourier transform to perform correlation, which is closely related to convolution. Correlation can be used to locate features within an image. In this context, correlation is often called template matching.
Read a sample image into the workspace.
Create a template for matching by extracting the letter "a" from the image. Note that you can also create the template by using the interactive syntax of the imcrop function.
a = bw(32:45,88:98);
Compute the correlation of the template image with the original image by rotating the template image by 180 degrees and then using the FFT-based convolution technique. (Convolution is equivalent to correlation if you rotate the convolution kernel by 180 degrees.) To match the template to the image, use the fft2 and ifft2 functions. In the resulting image, bright peaks correspond to occurrences of the letter.
C = real(ifft2(fft2(bw) .* fft2(rot90(a,2),256,256)));
imshow(C,[]) % Scale image to appropriate display range.
To view the locations of the template in the image, find the maximum pixel value and then define a threshold value that is less than this maximum. The thresholded image shows the locations of these peaks as white spots in the thresholded correlation image. (To make the locations easier to see in this figure, the example dilates the thresholded image to enlarge the size of the points.)
max(C(:))
thresh = 60; % Use a threshold that's a little less than max.
D = C > thresh;
E = imdilate(D,se);
imshow(E) % Display pixels with values over the threshold.
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Fast Multipole Method for Large Structures - MATLAB & Simulink - MathWorks Switzerland
Relation Between Memory Used and Problem Size
Fast Multipole Method (FMM)
Combined Field Integral Equation (CFIE)
The fast multipole method (FMM) computational technique in Antenna Toolbox™ allows you to model and analyze antennas and arrays on large platforms like aircraft and automobiles.
V=ZI
Antenna Toolbox uses Method of Moments Solver for Metal and Dielectric Structures to calculate the interaction matrix and solve system equations.
To calculate the surface currents on the antenna structure, you first define Rao-Wilton-Glisson (RWG) basis functions. A RWG basis function is a pair of triangles that share an edge, and is shown in the figure.
{t}_{n}^{+}
{t}_{n}^{-}
{A}_{n}^{+}
{A}_{n}^{-}
{l}_{n}
\begin{array}{l}{\stackrel{\to }{f}}_{n}\left(\stackrel{\to }{r}\right)=\left\{\begin{array}{cc}\frac{{l}_{n}}{2{A}_{n}^{+}}{\stackrel{\to }{\rho }}_{n}^{+S},& \stackrel{\to }{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{+}\\ \frac{{l}_{n}}{2{A}_{n}^{-}}{\stackrel{\to }{\rho }}_{n}^{-S},& \stackrel{\to }{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{-}\end{array}\text{ }\\ \end{array}
{\stackrel{\to }{\rho }}_{n}^{+}=\stackrel{\to }{r}-{\stackrel{\to }{r}}_{n}^{+}
{t}_{n}^{+}
\stackrel{\to }{r}
{\stackrel{\to }{\rho }}_{n}^{-}={\stackrel{\to }{r}}_{n}^{+}-\stackrel{\to }{r}
{t}_{n}^{-}
\nabla \cdot {\stackrel{\to }{f}}_{n}\left(\stackrel{\to }{r}\right)=\left\{\begin{array}{cc}\frac{{l}_{n}}{{A}_{n}^{+}},& \stackrel{\to }{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{+}\\ -\frac{{l}_{n}}{{A}_{n}^{-}},& \stackrel{\to }{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{-}\end{array}
{t}_{n}^{+}
{t}_{n}^{-}
The interaction matrix Z is a complex dense symmetric matrix. It is a square N-by-N matrix, where N is the number of basis functions, that is, the number of interior edges in the structure. Consider the scenario of a large structure like an aircraft or a ship. Typical narrow-band antennas like the dipole or patch are half-wavelength in size, but ships or aircraft can often be at least 100 wavelengths or more in size. To solve for the electromagnetic effects of either radiation or scattering from this structure using a full-wave solver, the first step is to mesh the structure and then form the basis functions. Doing so generates more than 50,000 triangles. Since the memory requirement for the direct solver is of the order of O(N2), in basis function space, the growth is as shown in this plot.
Under any of the following conditions the number of the unknowns become very large:
Structure refined with a finer mesh
Analysis of a physically large structure
The acceleration achieved by the FMM algorithm is due to its ability to subdivide the problem into successively smaller spatial regions, thereby ensuring that a given pair of source and target clusters are distant enough for the interaction to be computed using multipole expansions. The following figure illustrates that.
This approach fits well with the need to accelerate the computation of interactions between separated pairs of basis functions, that is, source and target dipole pairs. The problem of determining the electromagnetic potential at a given set of target points in a Helmholtz type of problem can be expressed as:
u\left(r\right)=\sum _{n=1}^{N}{c}_{n}\frac{\mathrm{exp}\left(jk|r-{r}_{n}|\right)}{|r-{r}_{n}|}-{v}_{n}·\nabla \left(\frac{\mathrm{exp}\left(jk|r-{r}_{n}|\right)}{|r-{r}_{n}|}\right)
wherein, cn and vn represent the collection of charge and dipole strengths, respectively, k is the wavenumber, and u(r) is the potential computed by FMM in 3-D space.
FMM speeds up the computation of the matrix-vector product by substantially accelerating the computation of point-to-point interactions mediated by the Green's function. The original current and charge distributions on the surface of the target are determined by introducing these coefficients back into the basis function expansion. Scattered or radiated field of the target including its radar cross-sections is then found by computing the radiation of the known surface currents and charges at required points in space. The iterative approach to determining a matrix inverse is a well-studied and established field of applied linear algebra. Among the variety of iterative solvers that exist, the generalized minimum residual (GMRES) method is a well-known technique. Antenna Toolbox uses this iterative solver.
The direct solver implemented in the Antenna Toolbox is based on EFIE. EFIE uses the electric field relationships on the surface of a metal and at any point in free space to set up the system of equations.
{E}_{t}^{s}=-{E}_{t}^{i}
{E}^{s}\left(r\right)=-j\omega A-\nabla \phi
The index t in the first of the two equations is used to describe the tangential component of the electric field on a metal surface, index s describes the scattered field, and index i denotes the incident field. In the second equation the relationship of the scattered field is shown in terms of the electric scalar potential φ and magnetic vector potential A.
Applying the Galerkin approach, where the test using the basis functions leads to the following key equation:
j\omega \left\{\frac{{l}_{m}}{2}{p}_{m}^{+}\left({r}_{m}^{+}\right)·A\left({r}_{m}^{+}\right)+\frac{{l}_{m}}{2}{p}_{m}^{-}\left({r}_{m}^{-}\right)·A\left({r}_{m}^{-}\right)\right\}-\left\{{l}_{m}\phi \left({r}_{m}^{+}\right)-{l}_{m}\phi \left({r}_{m}^{-}\right)\right\}={V}_{m}
{V}_{m}=\frac{{l}_{m}}{2}{p}_{m}^{+}\left({r}_{m}^{+}\right)·{E}^{i}\left({r}_{m}^{+}\right)+\frac{{l}_{m}}{2}{p}_{m}^{-}\left({r}_{m}^{-}\right)·{E}^{i}\left({r}_{m}^{-}\right)
MFIE equation expresses the surface current density J(r) developed on the body of a metallic object in response to a magnetic field excitation. An important observation here is that the second term of MFIE is the exact physical optics (PO) approximation. This equation captures the first order solution as the PO approximation, while the second term involving the integral captures the full-wave effects, thus providing a complete solution.
MFIE can be applied only to closed structures such as boxes, spheres, closed shells of aircraft, and so on. It cannot be applied, for example, to a strip dipole or monopole antenna.
J\left(r\right)=2n\left(r\right)×\underset{s}{\int }J\left(r\text{'}\right)×{\nabla }_{r\text{'}}\frac{\mathrm{exp}\left(-jk|r-r\text{'}|\right)}{4\pi |r-r\text{'}|}dr\text{'}+2n\left(r\right)×{H}^{i}\left(r\right)
Using the collocation approach leads to the equation for the MFIE implementation:
{c}_{m}-\left\{{I}_{m}·\sum _{n=1}^{{N}_{facets}}\left(\begin{array}{c}{M}_{1}·\nabla \\ {M}_{2}·\nabla \\ {M}_{3}·\nabla \end{array}\right)\frac{\mathrm{exp}\left(-jk|{R}_{m}-{r}_{n}|\right)}{4\pi |{R}_{m}-{r}_{n}|}\right\}={I}_{m}^{PO}
{M}_{1}=\left(\begin{array}{c}0\\ -{m}_{z}\\ +{m}_{y}\end{array}\right),{M}_{2}=\left(\begin{array}{c}+{m}_{z}\\ 0\\ -{m}_{x}\end{array}\right),{M}_{3}=\left(\begin{array}{c}-{m}_{y}\\ +{m}_{x}\\ 0\end{array}\right),m={I}_{n}{r}_{n}
CFIE uses the two equations shown for EFIE and MFIE. The term α is chosen to be 0.5 and η = 376.3Ω is the free space impedance.
\alpha LHS{E}_{m}+\left(1-\alpha \right)\eta LHS{H}_{m}=\alpha {V}_{m}+\left(1-\alpha \right)\eta {I}_{m}^{PO}
The FMM solver is applied to compute the left side of this equation. LHSEm represents the left side of EFIE and LHSHm represents the left side of MFIE.
[1] Flatironinstitute/FMM3D. Fortran. 2018. Reprint, Flatiron Institute, 2021. https://github.com/flatironinstitute/FMM3D.
[2] Greengard, L, and V Rokhlin. “A Fast Algorithm for Particle Simulations.” Journal of Computational Physics 73, no. 2 (December 1987): 325–48. https://doi.org/10.1016/0021-9991(87)90140-9.
[3] Rius JM, Úbeda E, Parrón J. On The Testing of the Magnetic Field Integral Equation With RWG Basis Functions in Method of Moments. IEEE Trans. Antennas and Propagation, vol. AP-49, no. 11, pp. 1550-1553.
[4] Rao SM, Wilton DR, Glisson AW. Electromagnetic Scattering by Surfaces of Arbitrary Shape. IEEE Trans. on Antennas and Propagation. 1982 May;30(3):409-418. doi: 001 8-926X/82/0500-O409.
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f(x)=x^2−6x+5
Is the vertex the lowest or highest point on the graph?
Remember that the domain is the set of all inputs for which there is an output, and the range is the set of all possible outputs.
Look at the graph. Does the graph extend all the way left and right (domain)?
Does the graph extend all the way up and down (range)?
Does the vertex represent the maximum or minimum value of the function?
The vertex is the lowest/highest point on the graph.
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Short-Answer Questions. Questions 1-4 are short-answer questions. Put your answers in the boxes provided. Simplify your answers as much as possible, and show your work. Each question is worth 3 marks, but not all questions are of equal difficulty.
{\displaystyle \displaystyle y=(\sin x)^{\sin x}}
{\displaystyle y'}
With variables in both the base and the exponent, the usual rules (like chain rule, product rule etc.) won't be effective. What can you do to get the variable out of the exponent?
Try taking the logarithm of both sides and applying implicit differentiation.
{\displaystyle \displaystyle \ln(a^{b})=b\ln(a)}
Following the hints, we have
{\displaystyle {\begin{aligned}\ln y&=\ln \left((\sin x)^{\sin x}\right)=\sin x\ln(\sin x)\end{aligned}}}
Differentiating both sides yields
{\displaystyle {\begin{aligned}{\frac {y'}{y}}&=(\cos x)\ln(\sin x)+(\sin x){\tfrac {d}{dx}}\ln(\sin x)\quad {\text{(product rule)}}\\{\frac {y'}{y}}&=(\cos x)\ln(\sin x)+(\sin x)(\cos x)\cdot {\frac {1}{\sin x}}\quad {\text{(chain rule)}}\\{\frac {y'}{y}}&=(\cos x)\ln(\sin x)+\cos x\\y'&=y\cos(x)(\ln(\sin x)+1)\end{aligned}}}
Finally, plugging in y yields the final answer
{\displaystyle \displaystyle y'=(\sin x)^{\sin x}\cos(x)(\ln(\sin x)+1)}
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Implicit differentiation, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag
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Receiver operating characteristic curves by SNR - MATLAB rocsnr - MathWorks France
SNRdB
MaxPfa
MinPfa
ROC Curves for Different SNRs
Receiver operating characteristic curves by SNR
[Pd,Pfa] = rocsnr(SNRdB)
[Pd,Pfa] = rocsnr(SNRdB,Name,Value)
rocsnr(...)
[Pd,Pfa] = rocsnr(SNRdB) returns the single-pulse detection probabilities, Pd, and false-alarm probabilities, Pfa, for the SNRs in the vector SNRdB. By default, for each SNR, the detection probabilities are computed for 101 false-alarm probabilities between 1e–10 and 1. The false-alarm probabilities are logarithmically equally spaced. The ROC curve is constructed assuming a coherent receiver with a nonfluctuating target.
[Pd,Pfa] = rocsnr(SNRdB,Name,Value) returns detection probabilities and false-alarm probabilities with additional options specified by one or more Name,Value pair arguments.
rocsnr(...) plots the ROC curves.
Signal-to-noise ratios in decibels, in a row or column vector.
Maximum false-alarm probability to include in the ROC calculation.
Minimum false-alarm probability to include in the ROC calculation.
Number of false-alarm probabilities to use when calculating the ROC curves. The actual probability values are logarithmically equally spaced between MinPfa and MaxPfa.
{P}_{D}=\frac{1}{2}\text{erfc}\left({\text{erfc}}^{-1}\left(2{P}_{FA}\right)-\sqrt{\chi }\right)
Detection probabilities corresponding to the false-alarm probabilities. For each SNR in SNRdB, Pd contains one column of detection probabilities.
False-alarm probabilities in a column vector. By default, the false-alarm probabilities are 101 logarithmically equally spaced values between 1e–10 and 1. To change the range of probabilities, use the optional MinPfa or MaxPfa input argument. To change the number of probabilities, use the optional NumPoints input argument.
Plot ROC curves for different SNR's for a single pulse.
SNRdB = [3 6 9 12];
[Pd,Pfa] = rocsnr(SNRdB,'SignalType','NonfluctuatingCoherent');
semilogx(Pfa,Pd)
xlabel('P_{fa}')
legend('SNR 3 dB','SNR 6 dB','SNR 9 dB','SNR 12 dB', 'location','northwest')
title('Receiver Operating Characteristic (ROC) Curves')
npwgnthresh | rocpfa | shnidman
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LMIs in Control/Matrix and LMI Properties and Tools/Schur Stabilizability - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Schur Stabilizability
LMI for Schur Stabilizability
Schur Stabilization is one method of ensuring that a controller can be made to stabilize a system. The following LMI is one that determines whether or not a system is indeed Schur Stabilizable, or having the property of being able to be Schur Stabilized.
4 The LMI: LMI for Schur stabilizability
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax+Bu\end{aligned}}}
or the matrix pair (A,B). In both cases, the matrices
{\displaystyle A\in \mathbb {R} ^{n\times n}}
{\displaystyle B\in \mathbb {R} ^{n\times r}}
{\displaystyle x\in \mathbb {R} ^{n}}
{\displaystyle u\in \mathbb {R} ^{r}}
The data required is both the matrices A and B as seen in the form above.
The goal of the optimization is to find a valid symmetric P such that the following LMI is satisfied.
The LMI: LMI for Schur stabilizabilityEdit
The LMI problem is to find a symmetric matrix P and a matrix W satisfying:
{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&AP+BW\\(AP+BW)^{T}&-P\end{bmatrix}}<0\\\end{aligned}}}
Another LMI with the same result of finding Schur Stabilizability is to find a symmetric matrix P such that:
{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&PA^{T}\\AP&-P-\gamma BB^{T}\end{bmatrix}}<0,\gamma \leq 1\\\end{aligned}}}
If the one of the above LMIs is found to be feasible, then the system is Schur Stabilizable and the Schur Stabilization LMI will always give a feasible result as well, in addition to a controller K that will Schur Stabilize the system.
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Schur_Stabilizability&oldid=4011181"
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Pedestrian and Bicyclist Classification Using Deep Learning - MATLAB & Simulink - MathWorks India
Synthetic Data Generation by Simulation
Classify Signatures Without Car Noise
Classify Signatures with Car Noise
Retrain CNN by Adding Car Noise to Training Data Set
This example shows how to classify pedestrians and bicyclists based on their micro-Doppler characteristics using a deep learning network and time-frequency analysis.
The movements of different parts of an object placed in front of a radar produce micro-Doppler signatures that can be used to identify the object. This example uses a convolutional neural network (CNN) to identify pedestrians and bicyclists based on their signatures.
This example trains the deep learning network using simulated data and then examines how the network performs at classifying two cases of overlapping signatures.
The data used to train the network is generated using backscatterPedestrian and backscatterBicyclist from Radar Toolbox™. These functions simulate the radar backscattering of signals reflected from pedestrians and bicyclists, respectively.
The helper function helperBackScatterSignals generates a specified number of pedestrian, bicyclist, and car radar returns. Because the purpose of the example is to classify pedestrians and bicyclists, this example considers car signatures as noise sources only. To get an idea of the classification problem to solve, examine one realization of a micro-Doppler signature from a pedestrian, a bicyclist, and a car. (For each realization, the return signals have dimensions
{\mathit{N}}_{\mathrm{fast}}
{\mathit{N}}_{\mathrm{slow}}
{\mathit{N}}_{\mathrm{fast}}
is the number of fast-time samples and
{\mathit{N}}_{\mathrm{slow}}
is the number of slow-time samples. See Radar Data Cube for more information.)
Plot the time-frequency maps for the pedestrian, bicyclist, and car realizations.
% Plot the first realization of objects
imagesc(T,F,SPed(:,:,1))
title('Pedestrian')
axis square xy
imagesc(T,F,SBic(:,:,1))
title('Bicyclist')
imagesc(T,F,SCar(:,:,1))
title('Car')
The normalized spectrograms (STFT absolute values) show that the three objects have quite distinct signatures. Specifically, the spectrograms of the pedestrian and the bicyclist have rich micro-Doppler signatures caused by the swing of arms and legs and the rotation of wheels, respectively. By contrast, in this example, the car is modeled as a point target with rigid body, so the spectrogram of the car shows that the short-term Doppler frequency shift varies little, indicating little micro-Doppler effect.
Classifying a single realization as a pedestrian or bicyclist is relatively simple because the pedestrian and bicyclist micro-Doppler signatures are dissimilar. However, classifying multiple overlapping pedestrians or bicyclists, with the addition of Gaussian noise or car noise, is much more difficult.
If multiple objects exist in the detection region of the radar at the same time, the received radar signal is a summation of the detection signals from all the objects. As an example, generate the received radar signal for a pedestrian and bicyclist with Gaussian background noise.
% Configure Gaussian noise level at the receiver
xRadarRec = complex(zeros(size(xPedRec)));
for ii = 1:size(xPedRec,3)
xRadarRec(:,:,ii) = rx(xPedRec(:,:,ii) + xBicRec(:,:,ii));
Then obtain micro-Doppler signatures of the received signal by using the STFT.
[S,~,~] = helperDopplerSignatures(xRadarRec,Tsamp);
imagesc(T,F,S(:,:,1)) % Plot the first realization
title('Spectrogram of a Pedestrian and a Bicyclist')
Because the pedestrian and bicyclist signatures overlap in time and frequency, differentiating between the two objects is difficult.
In this example, you train a CNN by using data consisting of simulated realizations of objects with varying properties—for example, bicyclists pedaling at different speeds and pedestrians with different heights walking at different speeds. Assuming the radar is fixed at the origin, in one realization, one object or multiple objects are uniformly distributed in a rectangular area of [5, 45] and [–10, 10] meters along the X and Y axes, respectively.
The other properties of the three objects that are randomly tuned are as follows:
Height — Uniformly distributed in the interval of [1.5, 2] meters
Heading — Uniformly distributed in the interval of [–180, 180] degrees
Speed — Uniformly distributed in the interval of [0, 1.4h] meters/second, where h is the height value
2) Bicyclists
Speed — Uniformly distributed in the interval of [1, 10] meters/second
Gear transmission ratio — Uniformly distributed in the interval of [0.5, 6]
Pedaling or coasting — 50% probability of pedaling (coasting means that the cyclist is moving without pedaling)
Velocity — Uniformly distributed in the interval of [0, 10] meters/second along the X and Y directions
The input to the convolutional network is micro-Doppler signatures consisting of spectrograms expressed in decibels and normalized to [0, 1], as shown in this figure:
Radar returns originate from different objects and different parts of objects. Depending on the configuration, some returns are much stronger than others. Stronger returns tend to obscure weaker ones. Logarithmic scaling augments the features by making return strengths comparable. Amplitude normalization helps the CNN converge faster.
The data set contains realizations of the following scenes:
One pedestrian present in the scene
One bicyclist present in the scene
One pedestrian and one bicyclist present in the scene
Two pedestrians present in the scene
Two bicyclists present in the scene
The data for this example consists of 20,000 pedestrian, 20,000 bicyclist, and 12,500 car signals generated by using the helper functions helperBackScatterSignals and helperDopplerSignatures. The signals are divided into two data sets: one without car noise samples and one with car noise samples.
For the first data set (without car noise), the pedestrian and bicyclist signals were combined, Gaussian noise was added, and micro-Doppler signatures were computed to generate 5000 signatures for each of the five scenes to be classified.
In each category, 80% of the signatures (that is, 4000 signatures) are reserved for the training data set while 20% of the signatures (that is, 1000 signatures) are reserved for the test data set.
To generate the second data set (with car noise), the procedure for the first data set was followed, except that car noise was added to 50% of the signatures. The proportion of signatures with and without car noise is the same in the training and test data sets.
Download and unzip the data in your temporary directory, whose location is specified by MATLAB®'s tempdir command. The data has a size of 21 GB and the download process may take some time. If you have the data in a folder different from tempdir, change the directory name in the subsequent instructions.
dataURL = 'https://ssd.mathworks.com/supportfiles/SPT/data/PedBicCarData.zip';
saveFolder = fullfile(tempdir,'PedBicCarData');
zipFile = fullfile(tempdir,'PedBicCarData.zip');
if ~exist(zipFile,'file')
elseif ~exist(saveFolder,'dir')
unzip(zipFile,tempdir)
trainDataNoCar.mat contains the training data set trainDataNoCar and its label set trainLabelNoCar.
testDataNoCar.mat contains the test data set testDataNoCar and its label set testLabelNoCar.
trainDataCarNoise.mat contains the training data set trainDataCarNoise and its label set trainLabelCarNoise.
testDataCarNoise.mat contains the test data set testDataCarNoise and its label set testLabelCarNoise.
TF.mat contains the time and frequency information for the micro-Doppler signatures.
Create a CNN with five convolution layers and one fully connected layer. The first four convolution layers are followed by a batch normalization layer, a rectified linear unit (ReLU) activation layer, and a max pooling layer. In the last convolution layer, the max pooling layer is replaced by an average pooling layer. The output layer is a classification layer after softmax activation. For network design guidance, see Deep Learning Tips and Tricks (Deep Learning Toolbox).
imageInputLayer([size(S,1),size(S,2),1],'Normalization','none')
convolution2dLayer(10,16,'Padding','same')
maxPooling2dLayer(10,'Stride',2)
1 '' Image Input 400x144x1 images
2 '' Convolution 16 10x10 convolutions with stride [1 1] and padding 'same'
5 '' Max Pooling 10x10 max pooling with stride [2 2] and padding [0 0 0 0]
6 '' Convolution 32 5x5 convolutions with stride [1 1] and padding 'same'
10 '' Convolution 32 5x5 convolutions with stride [1 1] and padding 'same'
11 '' Batch Normalization Batch normalization
13 '' Max Pooling 10x10 max pooling with stride [2 2] and padding [0 0 0 0]
21 '' Average Pooling 2x2 average pooling with stride [2 2] and padding [0 0 0 0]
22 '' Fully Connected 5 fully connected layer
Specify the optimization solver and the hyperparameters to train the CNN using trainingOptions. This example uses the ADAM optimizer and a mini-batch size of 128. Train the network using either a CPU or GPU. Using a GPU requires Parallel Computing Toolbox™. To see which GPUs are supported, see GPU Support by Release (Parallel Computing Toolbox). For information on other parameters, see trainingOptions (Deep Learning Toolbox). This example uses a GPU for training.
Load the data set without car noise and use the helper function helperPlotTrainData to plot one example of each of the five categories in the training data set,
load(fullfile(tempdir,'PedBicCarData','trainDataNoCar.mat')) % load training data set
load(fullfile(tempdir,'PedBicCarData','testDataNoCar.mat')) % load test data set
load(fullfile(tempdir,'PedBicCarData','TF.mat')) % load time and frequency information
helperPlotTrainData(trainDataNoCar,trainLabelNoCar,T,F)
Train the CNN that you created. You can view the accuracy and loss during the training process. In 30 epochs, the training process achieves almost 95% accuracy.
trainedNetNoCar = trainNetwork(trainDataNoCar,trainLabelNoCar,layers,options);
Use the trained network and the classify function to obtain the predicted labels for the test data set testDataNoCar. The variable predTestLabel contains the network predictions. The network achieves about 95% accuracy for the test data set without the car noise.
predTestLabel = classify(trainedNetNoCar,testDataNoCar);
testAccuracy = mean(predTestLabel == testLabelNoCar)
Use a confusion matrix to view detailed information about prediction performance for each category. The confusion matrix for the trained network shows that, in each category, the network predicts the labels of the signals in the test data set with a high degree of accuracy.
confusionchart(testLabelNoCar,predTestLabel);
To analyze the effects of car noise, classify data containing car noise with the trainedNetNoCar network, which was trained without car noise.
Load the car-noise-corrupted test data set testDataCarNoise.mat.
load(fullfile(tempdir,'PedBicCarData','testDataCarNoise.mat'))
Input the car-noise-corrupted test data set to the network. The prediction accuracy for the test data set with the car noise drops significantly, to around 70%, because the network never saw training samples containing car noise.
predTestLabel = classify(trainedNetNoCar,testDataCarNoise);
testAccuracy = mean(predTestLabel == testLabelCarNoise)
The confusion matrix shows that most prediction errors occur when the network takes in scenes from the "pedestrian," "pedestrian+pedestrian," or "pedestrian+bicyclist" classes and classifies them as "bicyclist."
confusionchart(testLabelCarNoise,predTestLabel);
Car noise significantly impedes the performance of the classifier. To solve this problem, train the CNN using data that contains car noise.
Load the car-noise-corrupted training data set trainDataCarNoise.mat.
load(fullfile(tempdir,'PedBicCarData','trainDataCarNoise.mat'))
Retrain the network by using the car-noise-corrupted training data set. In 30 epochs, the training process achieves almost 90% accuracy.
trainedNetCarNoise = trainNetwork(trainDataCarNoise,trainLabelCarNoise,layers,options);
Input the car-noise-corrupted test data set to the network trainedNetCarNoise. The prediction accuracy is about 87%, which is approximately 15% higher than the performance of the network trained without car noise samples.
predTestLabel = classify(trainedNetCarNoise,testDataCarNoise);
The confusion matrix shows that the network trainedNetCarNoise performs much better at predicting scenes with one pedestrian and scenes with two pedestrians.
To better understand the performance of the network, examine its performance in classifying overlapping signatures. This section is just for illustration. Due to the non-deterministic behavior of GPU training, you may not get the same classification results in this section when you rerun this example.
For example, signature #4 of the car-noise-corrupted test data, which does not have car noise, has two bicyclists with overlapping micro-Doppler signatures. The network correctly predicts that the scene has two bicyclists.
imagesc(T,F,testDataCarNoise(:,:,:,k))
title('Ground Truth: '+string(testLabelCarNoise(k))+', Prediction: '+string(predTestLabel(k)))
From the plot, the signature appears to be from only one bicyclist. Load the data CaseStudyData.mat of the two objects in the scene. The data contains return signals summed along the fast time. Apply the STFT to each signal.
load CaseStudyData.mat
M = 200; % FFT window length
beta = 6; % window parameter
w = kaiser(M,beta); % kaiser window
R = floor(1.7*(M-1)/(beta+1)); % ROUGH estimate
noverlap = M-R; % overlap length
[Sc,F,T] = stft(x,1/Tsamp,'Window',w,'FFTLength',M*2,'OverlapLength',noverlap);
imagesc(T,F,10*log10(abs(Sc(:,:,ii))))
title(['Bicyclist ' num2str(ii)])
c.Label.String = 'dB';
The amplitudes of the Bicyclist 2 signature are much weaker than those of Bicyclist 1, and the signatures of the two bicyclists overlap. When they overlap, the two signatures cannot be visually distinguished. However, the neural network classifies the scene correctly.
Another case of interest is when the network confuses car noise with a bicyclist, as in signature #267 of the car-noise-corrupted test data:
The signature of the bicyclist is weak compared to that of the car, and the signature has spikes from the car noise. Because the signature of the car closely resembles that of a bicyclist pedaling or a pedestrian walking at a low speed, and has little micro-Doppler effect, there is a high possibility that the network will classify the scene incorrectly.
[1] Chen, V. C. The Micro-Doppler Effect in Radar. London: Artech House, 2011.
[2] Gurbuz, S. Z., and Amin, M. G. "Radar-Based Human-Motion Recognition with Deep Learning: Promising Applications for Indoor Monitoring." IEEE Signal Processing Magazine. Vol. 36, Issue 4, 2019, pp. 16–28.
[3] Belgiovane, D., and C. C. Chen. "Micro-Doppler Characteristics of Pedestrians and Bicycles for Automotive Radar Sensors at 77 GHz." In 11th European Conference on Antennas and Propagation (EuCAP), 2912–2916. Paris: European Association on Antennas and Propagation, 2017.
[4] Angelov, A., A. Robertson, R. Murray-Smith, and F. Fioranelli. "Practical Classification of Different Moving Targets Using Automotive Radar and Deep Neural Networks." IET Radar, Sonar & Navigation. Vol. 12, Number 10, 2017, pp. 1082–1089.
[5] Parashar, K. N., M. C. Oveneke, M. Rykunov, H. Sahli, and A. Bourdoux. "Micro-Doppler Feature Extraction Using Convolutional Auto-Encoders for Low Latency Target Classification." In 2017 IEEE Radar Conference (RadarConf), 1739–1744. Seattle: IEEE, 2017.
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Ghost Leg - Wikipedia
Find sources: "Ghost Leg" – news · newspapers · books · scholar · JSTOR (June 2013) (Learn how and when to remove this template message)
An example of how an amidakuji can be used.
Ghost Leg (Chinese: 畫鬼腳), known in Japan as Amidakuji (阿弥陀籤, "Amida lottery", so named because the paper was folded into a fan shape resembling Amida's halo[1]) or in Korea as Sadaritagi (사다리타기, literally "ladder climbing"), is a method of lottery designed to create random pairings between two sets of any number of things, as long as the number of elements in each set is the same. This is often used to distribute things among people, where the number of things distributed is the same as the number of people. For instance, chores or prizes could be assigned fairly and randomly this way.
It consists of vertical lines with horizontal lines connecting two adjacent vertical lines scattered randomly along their length; the horizontal lines are called "legs". The number of vertical lines equals the number of people playing, and at the bottom of each line there is an item - a thing that will be paired with a player. The general rule for playing this game is: choose a line on the top, and follow this line downwards. When a horizontal line is encountered, follow it to get to another vertical line and continue downwards. Repeat this procedure until reaching the end of the vertical line. Then the player is given the thing written at the bottom of the line.
If the elements written above the Ghost Leg are treated as a sequence, and after the Ghost Leg is used, the same elements are written at the bottom, then the starting sequence has been transformed to another permutation. Hence, Ghost Leg can be treated as a kind of permuting operator.
3.4 Odd/Even property of permutation
3.5 Infinite Ghost Legs with same permutation
4.1 Bubble sort and highest simplicity
4.2 Maximum number of legs of prime
4.3 Bubblization
As an example, consider assigning roles in a play to actors.
To start with, the two sets are enumerated horizontally across a board. The actors' names would go on top, and the roles on the bottom. Then, vertical lines are drawn connecting each actor with the role directly below it.
The names of the actors and/or roles are then concealed so that people do not know which actor is on which line, or which role is on which line.
Next, each actor adds a leg to the board. Each leg must connect two adjacent vertical lines, and must not touch any other horizontal line.
Once this is done, a path is traced from top of each vertical line to the bottom. As you follow the line down, if you come across a leg, you must follow it to the adjacent vertical line on the left or right, then resume tracing down. You continue until you reach the bottom of a vertical line, and the top item you started from is now paired with the bottom item you ended on.
Another process involves creating the ladder beforehand, then concealing it. Then people take turns choosing a path to start from at the top. If no part of the amidakuji is concealed, then it is possible to fix the system so that you are guaranteed to get a certain pairing, thus defeating the idea of random chance.
Part of the appeal for this game is that, unlike random chance games like rock, paper, scissors, amidakuji will always create a 1:1 correspondence, and can handle arbitrary numbers of pairings. It is guaranteed that two items at the top will never have the same corresponding item at the bottom, nor will any item on the bottom ever lack a corresponding item at the top.
It also works regardless of how many horizontal lines are added. Each person could add one, two, three, or any number of lines, and the 1:1 correspondence would remain.
One way of realizing how this works is to consider the analogy of coins in cups. You have n coins in n cups, representing the items at the bottom of the amidakuji. Then, each leg that is added represents swapping the position of two adjacent cups. Thus, it is obvious that in the end there will still be n cups, and each cup will have one coin, regardless of how many swaps you perform.
Permutation[edit]
A Ghost Leg transforms an input sequence into an output sequence with the same number of elements with (possibly) different order. Thus, it can be regarded a permutation of n symbols, where n is the number of vertical lines in the Ghost Leg.,[2] hence it can be represented by the corresponding permutation matrix.
Applying a Ghost Leg a finite number of times to an input sequence eventually generates an output sequence identical to the original input sequence.
I.e., if M is a matrix representing a particular Ghost Leg, then Mn = I for some finite n.
Reversibility[edit]
For any Ghost Leg with matrix representation M, there exists a Ghost Leg with representation M−1, such that M M−1 = I
Odd/Even property of permutation[edit]
As each leg exchanges the two neighboring elements at its ends, the number of legs indicates the odd/even permutation property of the Ghost Leg. An odd number of legs represents an odd permutation, and an even number of legs gives an even permutation.
Infinite Ghost Legs with same permutation[edit]
It is possible to express every permutation as a Ghost Leg, but the expression is not one-to-one, i.e. a particular permutation does not correspond to a unique Ghost Leg. An infinite number of Ghost Legs represents the same permutation.
As there are an infinite number of Ghost Legs representing a particular permutation, it is obvious those Ghost Legs have a kind of equivalence. Among those equivalent Ghost Legs, the one(ones) which have smallest number of legs are called Prime.
Bubble sort and highest simplicity[edit]
A Ghost Leg can be constructed arbitrarily, but such a Ghost Leg is not necessarily prime. It can be proven that only those Ghost Legs constructed by bubble sort contains the fewest legs, and hence is prime. This is equivalent to saying that bubble sort performs the minimum number of adjacent exchanges to sort a sequence.
Maximum number of legs of prime[edit]
For a permutation with n elements, the maximum number of neighbor exchanging =
{\displaystyle {\frac {n(n-1)}{2}}}
In the same way, the maximum number of legs in a prime with n tracks =
{\displaystyle {\frac {n(n-1)}{2}}}
Bubblization[edit]
For an arbitrary Ghost Leg, it is possible to transform it into prime by a procedure called bubblization. When bubblization operates, the following two identities are repeatedly applied in order to move and eliminate "useless" legs.
When the two identities cannot be applied any more, the ghost leg is proven to be exactly the same as the Ghost Leg constructed by bubble sort, thus bubblization can reduce Ghost Legs to primes.
Randomness[edit]
Since, as mentioned above, an odd number of legs produces an odd permutation and an even number of legs produces an even permutation, a given number of legs can produce a maximum of half the total possible permutations (less than half if the number of legs is small relative to the number of tracks, reaching half as the number of legs increases beyond a certain critical number).
If the legs are drawn randomly (for reasonable definitions of "drawn randomly"), the evenness of the distribution of permutations increases with the number of legs. If the number of legs is small relative to number of tracks, the probabilities of different attainable permutations may vary greatly; for large numbers of legs the probabilities of different attainable permutations approach equality.
The 1981 arcade game Amidar, programmed by Konami and published by Stern, uses the same lattice as a maze. The game took its name from Amidakuji. and most of the enemy movement conformed to the game's rules.
An early Sega Master System game called Psycho Fox uses the mechanics of an Amidakuji board as a means to bet a bag of coins on a chance at a prize at the top of the screen. Later Sega Genesis games based on the same game concept DecapAttack and its Japanese predecessor "Magical Hat no Buttobi Tabo! Daibōken" follow the same game mechanics, including the Amidakuji bonus levels.
Super Mario Land 2: 6 Golden Coins features an Amidakuji-style bonus game that rewards the player with a power-up. New Super Mario Bros. and Wario: Master of Disguise feature an Amidakuji-style minigame in which the player uses the stylus to trace lines that will lead the character down the right path.
In Mario Party there is a mini game where one of the four players pours money into an Amidakuji made out of pipes. The goal is to try to choose the path leading to the character controlled by the player.
The BoSpider in Mega Man X and Maverick Hunter X descends onto the player via an Amidakuji path.
In Mega Man Zero 3, an Amidakuji-like mini game can be unlocked, requiring the player to guide five colored jewels to the right colored beaker.
In Super Monkey Ball 2, there is a level in the Advanced-Extra difficulty named "Amida Lot" (Advanced-EX 7) that features a floor resembling an Amidakuji board, which bumper travels around the way and may knock off the player if they happen to hit them. The goal only travels through one of the vertical lines and the player must reach the goal using the ghost legs while avoiding the bumpers to not fall out.
In Digimon World the player must travel through the Amida Forest, the Forest being a Amidakuji. Travelling the correct paths will eventually see them recruit Centarumon. Travelling the incorrect paths however will cause their Digimon damage (multiple times for each wrong path taken).
In WarioWare, Inc.: Mega Microgames!, the microgame "Noodle Cup" features Amidakuji-style gameplay.
Azalea Gym in Pokémon HeartGold and SoulSilver was redesigned with an Amidakuji-based system of carts to cross. The correct choices lead to the gym leader; the wrong ones lead to other trainers to fight.
Phantasy Star Online 2 uses the principle of Amidakuji for a randomly appearing bomb-defusing minigame. One must trace an Amidakuji path around each bomb to determine which button defuses it; incorrect selections knock players away for a few seconds, wasting time.
In the manga Liar Game (vol 17), an Amidakuji is used for determinate the rank of each participant to the penultimate stage of the game.
In the Japanese drama Don Quixote (episode 10), the character Shirota (Shota Matsuda) uses Amidakuji to help decide between candidate families for an adoption.
In Raging Loop, a "ghost leg lottery" is described as an analogy for the selection of roles across a village for a ceremony that is central to the game's plot.
^ Frédéric, Louis (2002). Japan Encyclopedia. ISBN 9780674017535.
^ Ho 2012, p.31
Wikimedia Commons has media related to Ghost leg.
https://www.webcitation.org/query?url=http://www.geocities.com/Athens/Acropolis/7247/amidakuji.html&date=2009-10-25+05:45:20
Ladders: A Research Paper by David Senft (PDF)
Man-Kit Ho, Hoi-Kwan Lau, Ting-Fai Man, Shek Yeung (2012). "Ghost Leg", Hang Lung Mathematics Awards Collection of Winning Papers, 2004. International Press. ISBN 978-1-57146-254-1.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Ghost_Leg&oldid=1084820781"
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EUDML | Geometrically constructed bases for homology of partition lattices of types , and . EuDML | Geometrically constructed bases for homology of partition lattices of types , and .
Geometrically constructed bases for homology of partition lattices of types
A
B
D
Björner, Anders, and Wachs, Michelle L.. "Geometrically constructed bases for homology of partition lattices of types , and .." The Electronic Journal of Combinatorics [electronic only] 11.2 (2004): Research paper R3, 26 p.-Research paper R3, 26 p.. <http://eudml.org/doc/124691>.
@article{Björner2004,
author = {Björner, Anders, Wachs, Michelle L.},
keywords = {hyperplane arrangement; Coxeter arrangements; interpolating arrangements},
title = {Geometrically constructed bases for homology of partition lattices of types , and .},
AU - Björner, Anders
TI - Geometrically constructed bases for homology of partition lattices of types , and .
KW - hyperplane arrangement; Coxeter arrangements; interpolating arrangements
hyperplane arrangement, Coxeter arrangements, interpolating arrangements
Arrangements of points, flats, hyperplanes
Articles by Björner
Articles by Wachs
|
EUDML | Local moduli for plane curve singularities, the dimension of the ...-constant stratum. EuDML | Local moduli for plane curve singularities, the dimension of the ...-constant stratum.
Local moduli for plane curve singularities, the dimension of the ...-constant stratum.
Hans Olav Heröy
Heröy, Hans Olav. "Local moduli for plane curve singularities, the dimension of the ...-constant stratum.." Mathematica Scandinavica 65.1 (1989): 33-40. <http://eudml.org/doc/167063>.
@article{Heröy1989,
author = {Heröy, Hans Olav},
keywords = {Milnor number; -constant deformations; minimal Tjurina-number; plane curve singularity},
title = {Local moduli for plane curve singularities, the dimension of the ...-constant stratum.},
AU - Heröy, Hans Olav
TI - Local moduli for plane curve singularities, the dimension of the ...-constant stratum.
KW - Milnor number; -constant deformations; minimal Tjurina-number; plane curve singularity
Milnor number,
\mu
-constant deformations, minimal Tjurina-number, plane curve singularity
Local deformation theory, Artin approximation, etc.
Articles by Hans Olav Heröy
|
Definition 15.11.1 (09XE)—The Stacks project
Definition 15.11.1. A henselian pair is a pair $(A, I)$ satisfying
$I$ is contained in the Jacobson radical of $A$, and
for any monic polynomial $f \in A[T]$ and factorization $\overline{f} = g_0h_0$ with $g_0, h_0 \in A/I[T]$ monic generating the unit ideal in $A/I[T]$, there exists a factorization $f = gh$ in $A[T]$ with $g, h$ monic and $g_0 = \overline{g}$ and $h_0 = \overline{h}$.
Comment #457 by Kestutis Cesnavicius on March 10, 2014 at 23:43
I would replace 'radical ideal' by 'Jacobson radical' for clarity.
Well, I changed it in this section, because it is clearer as you say. But there are other locations where we use the language "the radical of
A
" which I did not change. For example in Nakayama's lemma 10.20.1.
You can find the change here.
Condition (1) implies uniqueness of the decomposition in (2). It would be nice to mention it; in fact I would not be suprised if this were used somewhere later.
Going to leave as is for now. Some uniqueness is mentined in Lemma 15.11.6.
|
Lie_superalgebra Knowpia
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or superalgebra, over a commutative ring (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading):
Super skew-symmetry:
{\displaystyle [x,y]=-(-1)^{|x||y|}[y,x].\ }
The super Jacobi identity:[1]
{\displaystyle (-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,}
where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1). The degree of [x,y] is the sum of degree of x and y modulo 2.
One also sometimes adds the axioms
{\displaystyle [x,x]=0}
for |x| = 0 (if 2 is invertible this follows automatically) and
{\displaystyle [[x,x],x]=0}
for |x| = 1 (if 3 is invertible this follows automatically). When the ground ring is the integers or the Lie superalgebra is a free module, these conditions are equivalent to the condition that the Poincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).
Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.
A graded Lie algebra (say, graded by Z or N) that is anticommutative and Jacobi in the graded sense also has a
{\displaystyle Z_{2}}
grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See note at graded Lie algebra for discussion.
{\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\oplus {\mathfrak {g}}_{1}}
be a Lie superalgebra. By inspecting the Jacobi identity, one sees that there are eight cases depending on whether arguments are even or odd. These fall into four classes, indexed by the number of odd elements:[2]
No odd elements. The statement is just that
{\displaystyle {\mathfrak {g}}_{0}}
is an ordinary Lie algebra.
One odd element. Then
{\displaystyle {\mathfrak {g}}_{1}}
{\displaystyle {\mathfrak {g}}_{0}}
-module for the action
{\displaystyle \mathrm {ad} _{a}:b\rightarrow [a,b],\quad a\in {\mathfrak {g}}_{0},\quad b,[a,b]\in {\mathfrak {g}}_{1}}
Two odd elements. The Jacobi identity says that the bracket
{\displaystyle {\mathfrak {g}}_{1}\otimes {\mathfrak {g}}_{1}\rightarrow {\mathfrak {g}}_{0}}
is a symmetric
{\displaystyle {\mathfrak {g}}_{1}}
Three odd elements. For all
{\displaystyle b\in {\mathfrak {g}}_{1}}
{\displaystyle [b,[b,b]]=0}
Thus the even subalgebra
{\displaystyle {\mathfrak {g}}_{0}}
of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while
{\displaystyle {\mathfrak {g}}_{1}}
is a linear representation of
{\displaystyle {\mathfrak {g}}_{0}}
, and there exists a symmetric
{\displaystyle {\mathfrak {g}}_{0}}
-equivariant linear map
{\displaystyle \{\cdot ,\cdot \}:{\mathfrak {g}}_{1}\otimes {\mathfrak {g}}_{1}\rightarrow {\mathfrak {g}}_{0}}
{\displaystyle [\left\{x,y\right\},z]+[\left\{y,z\right\},x]+[\left\{z,x\right\},y]=0,\quad x,y,z\in {\mathfrak {g}}_{1}.}
Conditions (1)–(3) are linear and can all be understood in terms of ordinary Lie algebras. Condition (4) is nonlinear, and is the most difficult one to verify when constructing a Lie superalgebra starting from an ordinary Lie algebra (
{\displaystyle {\mathfrak {g}}_{0}}
) and a representation (
{\displaystyle {\mathfrak {g}}_{1}}
A ∗ Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z2 grading and satisfies [x,y]* = [y*,x*] for all x and y in the Lie superalgebra. (Some authors prefer the convention [x,y]* = (−1)|x||y|[y*,x*]; changing * to −* switches between the two conventions.) Its universal enveloping algebra would be an ordinary *-algebra.
Given any associative superalgebra
{\displaystyle A}
one can define the supercommutator on homogeneous elements by
{\displaystyle [x,y]=xy-(-1)^{|x||y|}yx\ }
and then extending by linearity to all elements. The algebra
{\displaystyle A}
together with the supercommutator then becomes a Lie superalgebra. The simplest example of this procedure is perhaps when
{\displaystyle A}
is the space of all linear functions
{\displaystyle \mathbf {End} (V)}
of a super vector space
{\displaystyle V}
to itself. When
{\displaystyle V=\mathbb {K} ^{p|q}}
, this space is denoted by
{\displaystyle M^{p|q}}
{\displaystyle M(p|q)}
.[3] With the Lie bracket per above, the space is denoted
{\displaystyle {\mathfrak {gl}}(p|q)}
The Whitehead product on homotopy groups gives many examples of Lie superalgebras over the integers.
The simple complex finite-dimensional Lie superalgebras were classified by Victor Kac.
The basic classical compact Lie superalgebras (that are not Lie algebras) are: [1]
SU(m/n) These are the superunitary Lie algebras which have invariants:
{\displaystyle z.{\overline {z}}+iw.{\overline {w}}}
This gives two orthosymplectic (see below) invariants if we take the m z variables and n w variables to be non-commutative and we take the real and imaginary parts. Therefore, we have
{\displaystyle SU(m/n)=OSp(2m/2n)\cap OSp(2n/2m)}
SU(n/n)/U(1) A special case of the superunitary Lie algebras where we remove one U(1) generator to make the algebra simple.
OSp(m/2n) These are the orthosymplectic groups. They have invariants given by:
{\displaystyle x.x+y.z-z.y}
for m commutative variables (x) and n pairs of anti-commutative variables (y,z). They are important symmetries in supergravity theories.
D(2/1;
{\displaystyle \alpha }
) This is a set of superalgebras parameterised by the variable
{\displaystyle \alpha }
. It has dimension 17 and is a sub-algebra of OSp(9|8). The even part of the group is O(3)×O(3)×O(3). So the invariants are:
{\displaystyle A_{\mu }A_{\mu }+B_{\mu }B_{\mu }+C_{\mu }C_{\mu }+\psi ^{\alpha \beta \gamma }\psi ^{\alpha '\beta '\gamma '}\varepsilon _{\alpha \alpha '}\varepsilon _{\beta \beta '}\varepsilon _{\gamma \gamma '}}
{\displaystyle A_{\{1}A_{2}A_{3\}}+B_{\{1}B_{2}B_{3\}}+C_{\{1}C_{2}C_{3\}}+A_{\mu }\Gamma _{\mu }^{\alpha \alpha '}\psi \psi +B_{\mu }\Gamma _{\mu }^{\beta \beta '}\psi \psi +C_{\mu }\Gamma _{\mu }^{\gamma \gamma '}\psi \psi }
for particular constants
{\displaystyle \gamma }
F(4) This exceptional Lie superalgebra has dimension 40 and is a sub-algebra of OSp(24|16). The even part of the group is O(3)xSO(7) so three invariants are:
{\displaystyle B_{\mu \nu }+B_{\nu \mu }=0}
{\displaystyle A_{\mu }A_{\mu }+B_{\mu \nu }B_{\mu \nu }+\psi _{\{1}^{\alpha }\psi _{2\}}^{\alpha }}
{\displaystyle A_{\{1}A_{2}A_{3\}}+B_{\{\mu \nu }B_{\nu \tau }B_{\tau \mu \}}+B_{\mu \nu }\sigma _{\mu \nu }^{\alpha \beta }\psi _{k}^{\alpha }\psi _{k}^{\beta }+A_{\mu }\Gamma _{\mu }^{\alpha \beta }\psi _{\alpha }^{k}\psi _{\beta }^{k}+({\text{sym.}})}
This group is related to the octonions by considering the 16 component spinors as two component octonion spinors and the gamma matrices acting on the upper indices as unit octonions. We then have
{\displaystyle f^{\mu \nu \tau }\sigma _{\nu \tau }\equiv \gamma _{\mu }}
where f is the structure constants of octonion multiplication.
G(3) This exceptional Lie superalgebra has dimension 31 and is a sub-algebra of OSp(17|14). The even part of the group is O(3)×G2. The invariants are similar to the above (it being a subalgebra of the F(4)?) so the first invariant is:
{\displaystyle A_{\mu }A_{\mu }+C_{\alpha }^{\mu }C_{\alpha }^{\mu }+\psi _{\{1}^{\mu }\psi _{2\}}^{\nu }}
There are also two so-called strange series called p(n) and q(n).
Classification of infinite-dimensional simple linearly compact Lie superalgebrasEdit
The classification consists of the 10 series W(m, n), S(m, n) ((m, n) ≠ (1, 1)), H(2m, n), K(2m + 1, n), HO(m, m) (m ≥ 2), SHO(m, m) (m ≥ 3), KO(m, m + 1), SKO(m, m + 1; β) (m ≥ 2), SHO ∼ (2m, 2m), SKO ∼ (2m + 1, 2m + 3) and the five exceptional algebras:
E(1, 6), E(5, 10), E(4, 4), E(3, 6), E(3, 8)
The last two are particularly interesting (according to Kac) because they have the standard model gauge group SU(3)×SU(2)×U(1) as their zero level algebra. Infinite-dimensional (affine) Lie superalgebras are important symmetries in superstring theory. Specifically, the Virasoro algebras with
{\displaystyle {\mathcal {N}}}
supersymmetries are
{\displaystyle K(1,{\mathcal {N}})}
which only have central extensions up to
{\displaystyle {\mathcal {N}}=4}
In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra whose product satisfies
{\displaystyle [\cdot ,\cdot ]\circ ({\operatorname {id} }+\tau _{A,A})=0}
{\displaystyle [\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes {\operatorname {id} }\circ ({\operatorname {id} }+\sigma +\sigma ^{2})=0}
where σ is the cyclic permutation braiding
{\displaystyle ({\operatorname {id} }\otimes \tau _{A,A})\circ (\tau _{A,A}\otimes {\operatorname {id} })}
. In diagrammatic form:
^ Freund 1983, p. 8
^ Varadarajan 2004, p. 89
Cheng, S.-J.; Wang, W. (2012). Dualities and Representations of Lie Superalgebras. Graduate Studies in Mathematics. Vol. 144. pp. 302pp. ISBN 978-0-8218-9118-6.
Freund, P. G. O. (1983). Introduction to supersymmetry. Cambridge Monographs on Mathematical Physics. Cambridge University Press. doi:10.1017/CBO9780511564017. ISBN 978-0521-356-756.
Grozman, P.; Leites, D.; Shchepochkina, I. (2005). "Lie Superalgebras of String Theories". Acta Mathematica Vietnamica. 26 (2005): 27–63. arXiv:hep-th/9702120. Bibcode:1997hep.th....2120G.
Kac, V. G. (1977). "Lie superalgebras". Advances in Mathematics. 26 (1): 8–96. doi:10.1016/0001-8708(77)90017-2.
Kac, V. G. (2010). "Classification of Infinite-Dimensional Simple Groups of Supersymmetries and Quantum Field Theory". Visions in Mathematics: 162–183. arXiv:math/9912235. doi:10.1007/978-3-0346-0422-2_6. ISBN 978-3-0346-0421-5. S2CID 15597378.
Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry ((2nd ed.) ed.). Berlin: Springer. ISBN 978-3-540-61378-7.
Musson, I. M. (2012). Lie Superalgebras and Enveloping Algebras. Graduate Studies in Mathematics. Vol. 131. pp. 488 pp. ISBN 978-0-8218-6867-6.
Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics. Vol. 11. American Mathematical Society. ISBN 978-0-8218-3574-6.
Frölicher, A.; Nijenhuis, A. (1956). "Theory of vector valued differential forms. Part I". Indagationes Mathematicae. 59: 338–350. doi:10.1016/S1385-7258(56)50046-7. .
Gerstenhaber, M. (1963). "The cohomology structure of an associative ring". Annals of Mathematics. 78 (2): 267–288. doi:10.2307/1970343. JSTOR 1970343.
Gerstenhaber, M. (1964). "On the Deformation of Rings and Algebras". Annals of Mathematics. 79 (1): 59–103. doi:10.2307/1970484. JSTOR 1970484.
Milnor, J. W.; Moore, J. C. (1965). "On the structure of Hopf algebras". Annals of Mathematics. 81 (2): 211–264. doi:10.2307/1970615. JSTOR 1970615.
Irving Kaplansky + Lie Superalgebras
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EUDML | A new characterization of -bounded semigroups with application to implicit evolution equations. EuDML | A new characterization of -bounded semigroups with application to implicit evolution equations.
B
Arlotti, Luisa. "A new characterization of -bounded semigroups with application to implicit evolution equations.." Abstract and Applied Analysis 5.4 (2001): 227-244. <http://eudml.org/doc/49644>.
@article{Arlotti2001,
author = {Arlotti, Luisa},
keywords = {one-parameter family of linear operators; -bounded semigroups; implicit evolution equations; -bounded semigroups},
title = {A new characterization of -bounded semigroups with application to implicit evolution equations.},
AU - Arlotti, Luisa
TI - A new characterization of -bounded semigroups with application to implicit evolution equations.
KW - one-parameter family of linear operators; -bounded semigroups; implicit evolution equations; -bounded semigroups
one-parameter family of linear operators,
B
-bounded semigroups, implicit evolution equations,
B
Articles by Arlotti
|
Density — lesson. Science State Board, Class 7.
In a beaker filled with water, an iron ball and a cork is dropped simultaneously. Now, what will you observe?
From the picture, it is observed that the cork floats and the iron ball sinks. What may be the reason for this?
If lighter objects float and heavier objects sink in water, then how does a small iron piece sink, whereas a heavier wooden log floats in water?
To answer all these questions, let us learn the concept of density.
Density is defined as the mass of the substance contained in a unit volume of \(1\ m^3\).
The formula for density (\(D\)) is given as,
D\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{M}{V}
where \(M\) is the mass of a substance, and \(V\) is the volume of a substance.
The SI unit of density is \(kg/ m^3\) and the CGS (Centimetre Gram Second) unit is \(g/ cm^3\).
Lighter and heavier objects
1. Wooden block and iron ball
A wooden block with the same mass as an iron ball takes up more volume or space. Also, the wooden block is lighter than the iron ball of the same size. Hence, it floats on water.
The lightness or heaviness of a body depends on density. If the mass of an object is greater than the volume, then the object will have a higher density. For example, the mass of the iron ball will be greater than the mass of the wooden block of equal size. As a result, the iron ball has a higher density and sinks.
When a single drop of water is dropped into the oil, it sinks. However, if one drop of oil is dropped into water, it floats and builds a layer on the surface. Through this, we can say, some oil are denser than water.
When an oil spill occurs in the ocean, the oil rises to the water surface, creating an oil slick on the top.
Water is denser than cooking oil and castor oil, even though these oils appear to be denser than water. Castor oil has a density of \(961\ kg/m^3\), whereas water has a density of \(1000\ kg/m^3\).
https://upload.wikimedia.org/wikipedia/commons/f/f1/Water_and_oil.jpg
https://live.staticflickr.com/7434/12685861633_cf5c25abc8_b.jpg
https://pixahive.com/wp-content/uploads/2020/09/Two-Birds-Sitting-on-a-Wooden-Log-floating-on-the-lake-51745-pixahive.jpg
|
Prediction Using Discriminant Analysis Models - MATLAB & Simulink - MathWorks í•œêµ
\stackrel{^}{y}=\underset{y=1,...,K}{\mathrm{arg}\mathrm{min}}\underset{k=1}{\overset{K}{â}}\stackrel{^}{P}\left(k|x\right)C\left(y|k\right),
\stackrel{^}{y}
\stackrel{^}{P}\left(k|x\right)
C\left(y|k\right)
The posterior probability that a point x belongs to class k is the product of the prior probability and the multivariate normal density. The density function of the multivariate normal with 1-by-d mean μk and d-by-d covariance Σk at a 1-by-d point x is
P\left(x|k\right)=\frac{1}{{\left({\left(2\mathrm{Ï}\right)}^{d}|{\mathrm{Σ}}_{k}|\right)}^{1/2}}\mathrm{exp}\left(â\frac{1}{2}\left(xâ{\mathrm{μ}}_{k}\right){\mathrm{Σ}}_{k}^{â1}{\left(xâ{\mathrm{μ}}_{k}\right)}^{T}\right),
|{\mathrm{Σ}}_{k}|
is the determinant of Σk, and
{\mathrm{Σ}}_{k}^{â1}
\stackrel{^}{P}\left(k|x\right)=\frac{P\left(x|k\right)P\left(k\right)}{P\left(x\right)},
\underset{i=1}{\overset{K}{â}}\stackrel{^}{P}\left(i|X\left(n\right)\right)C\left(k|i\right),
\stackrel{^}{P}\left(i|X\left(n\right)\right)
C\left(k|i\right)
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Fit binary decision tree for regression - MATLAB fitrtree - MathWorks Switzerland
\epsilon =\sum _{i=1}^{n}{w}_{i}{\left({y}_{i}-\overline{y}\right)}^{2}.
\sum _{i=1}^{n}{w}_{i}=1
\overline{y}=\sum _{i=1}^{n}{w}_{i}{y}_{i}
{\stackrel{^}{\pi }}_{jk}=\sum _{i=1}^{n}I\left\{{y}_{i}=k\right\}{w}_{i}.
\sum {w}_{i}=1
{\stackrel{^}{\pi }}_{jk}=\frac{{n}_{jk}}{n}
t=n\sum _{k=1}^{K}\sum _{j=1}^{J}\frac{{\left({\stackrel{^}{\pi }}_{jk}-{\stackrel{^}{\pi }}_{j+}{\stackrel{^}{\pi }}_{+k}\right)}^{2}}{{\stackrel{^}{\pi }}_{j+}{\stackrel{^}{\pi }}_{+k}}
{\stackrel{^}{\pi }}_{j+}=\sum _{k}{\stackrel{^}{\pi }}_{jk}
{\stackrel{^}{\pi }}_{+k}=\sum _{j}{\stackrel{^}{\pi }}_{jk}
{\lambda }_{jk}=\frac{\text{min}\left({P}_{L},{P}_{R}\right)-\left(1-{P}_{{L}_{j}{L}_{k}}-{P}_{{R}_{j}{R}_{k}}\right)}{\text{min}\left({P}_{L},{P}_{R}\right)}.
{P}_{{L}_{j}{L}_{k}}
{P}_{{R}_{j}{R}_{k}}
{\epsilon }_{t}=\sum _{j\in T}{w}_{j}{\left({y}_{j}-{\overline{y}}_{t}\right)}^{2}.
P\left(T\right)=\sum _{j\in T}{w}_{j}.
\Delta I=P\left(T\right){\epsilon }_{t}-P\left({T}_{L}\right){\epsilon }_{{t}_{L}}-P\left({T}_{R}\right){\epsilon }_{{t}_{R}}.
\Delta {I}_{U}=P\left(T-{T}_{U}\right){\epsilon }_{t}-P\left({T}_{L}\right){\epsilon }_{{t}_{L}}-P\left({T}_{R}\right){\epsilon }_{{t}_{R}}.
{r}_{ti}={y}_{ti}-{\overline{y}}_{t}
{\overline{y}}_{t}=\frac{1}{{\sum }_{i}{w}_{i}}{\sum }_{i}{w}_{i}{y}_{ti}
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Entropy | Free Full-Text | Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations
Halun, J.
Karbowniczek, P.
Kuterba, P.
Danel, Z.
Correction published on 16 March 2022, see Entropy 2022, 24(3), 413.
Institute of Nuclear Physics, Polish Academy of Sciences, 31-342 Cracow, Poland
Institute of Physics, Cracow University of Technology, 30-084 Cracow, Poland
Faculty of Physics, Astronomy and Applied Computer Sciences, Jagiellonian University in Cracow, 30-348 Cracow, Poland
Former name: Zoryana Usatenko.
Academic Editor: Zoltán Néda
Received: 15 December 2020 / Revised: 6 February 2021 / Accepted: 8 February 2021 / Published: 19 February 2021 / Corrected: 16 March 2022
f=4
\Theta
N=300
, 300 (360), and 1201 (4 × 300 + 1-star polymer with four arms) beads accordingly. The obtained analytical and numerical results for phantom ring and star polymers are compared with the results for linear polymer chains in confined geometries. View Full-Text
Keywords: critical phenomena; surface effects; renormalization group; polymers critical phenomena; surface effects; renormalization group; polymers
Halun, J.; Karbowniczek, P.; Kuterba, P.; Danel, Z. Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations. Entropy 2021, 23, 242. https://doi.org/10.3390/e23020242
Halun J, Karbowniczek P, Kuterba P, Danel Z. Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations. Entropy. 2021; 23(2):242. https://doi.org/10.3390/e23020242
Halun, Joanna, Pawel Karbowniczek, Piotr Kuterba, and Zoriana Danel. 2021. "Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations" Entropy 23, no. 2: 242. https://doi.org/10.3390/e23020242
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TopologicSort - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : TopologicSort
compute topological order
TopologicSort(G)
output = one of default or permutation
When output=permutation, returns a list of integers representing the permutation to Vertices(G) to obtain the topological order. When output=default, a list of the actual vertices is returned.
The TopologicSort command returns a linear ordering of vertices of an acyclic digraph that is consistent with the arcs of the digraph. This means a vertex u precedes a vertex v if there is an arc from u to v. The output is a list.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{DG}≔\mathrm{Digraph}\left({[a,b],[a,d],[b,d],[c,a],[c,b],[c,d]}\right):
\mathrm{IsAcyclic}\left(\mathrm{DG}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{TopologicSort}\left(\mathrm{DG}\right)
[\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]
The GraphTheory[TopologicSort] command was updated in Maple 2021.
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Clifford torus - Wikipedia
Find sources: "Clifford torus" – news · newspapers · books · scholar · JSTOR (November 2019) (Learn how and when to remove this template message)
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1
b (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
b each exists in its own independent embedding space R2
b, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
A stereographic projection of a Clifford torus performing a simple rotation
Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together.
Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
b each has a radius of
{\displaystyle \textstyle {\sqrt {1/2}}}
, their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.
The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry[clarification needed] as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.[1]
1.1 Alternative derivation using complex numbers
2 More general definition of Clifford tori
3 Still more general definition of Clifford tori in higher dimensions
5 Uses in mathematics
The unit circle S1 in R2 can be parameterized by an angle coordinate:
{\displaystyle S^{1}=\{(\cos \theta ,\sin \theta )\mid 0\leq \theta <2\pi \}.}
In another copy of R2, take another copy of the unit circle
{\displaystyle S^{1}=\{(\cos \varphi ,\sin \varphi )\mid 0\leq \varphi <2\pi \}.}
Then the Clifford torus is
{\displaystyle {\frac {1}{\sqrt {2}}}S^{1}\times {\frac {1}{\sqrt {2}}}S^{1}=\left\{{\frac {1}{\sqrt {2}}}(\cos \theta ,\sin \theta ,\cos \varphi ,\sin \varphi )\mid 0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.}
Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in R2 × R2 = R4.
If R4 is given by coordinates (x1, y1, x2, y2), then the Clifford torus is given by
{\displaystyle x_{1}^{2}+y_{1}^{2}={\frac {1}{2}}=x_{2}^{2}+y_{2}^{2}.}
This shows that in R4 the Clifford torus is a submanifold of the unit 3-sphere S3.
It is easy to verify that the Clifford torus is a minimal surface in S3.
Alternative derivation using complex numbersEdit
It is also common to consider the Clifford torus as an embedded torus in C2. In two copies of C, we have the following unit circles (still parametrized by an angle coordinate):
{\displaystyle S^{1}=\left\{e^{i\theta }\mid 0\leq \theta <2\pi \right\}}
{\displaystyle S^{1}=\left\{e^{i\varphi }\mid 0\leq \varphi <2\pi \right\}.}
Now the Clifford torus appears as
{\displaystyle {\frac {1}{\sqrt {2}}}S^{1}\times {\frac {1}{\sqrt {2}}}S^{1}=\left\{{\frac {1}{\sqrt {2}}}\left(e^{i\theta },e^{i\varphi }\right)\,|\,0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.}
As before, this is an embedded submanifold, in the unit sphere S3 in C2.
If C2 is given by coordinates (z1, z2), then the Clifford torus is given by
{\displaystyle \left|z_{1}\right|^{2}={\frac {1}{2}}=\left|z_{2}\right|^{2}.}
In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of C2 is
{\displaystyle {\sqrt {{\frac {1}{2}}\left|e^{i\theta }\right|^{2}+{\frac {1}{2}}\left|e^{i\varphi }\right|^{2}}}=1.}
The set of all points at a distance of 1 from the origin of C2 is the unit 3-sphere, and so the Clifford torus sits inside this 3-sphere. In fact, the Clifford torus divides this 3-sphere into two congruent solid tori (see Heegaard splitting[2]).
Since O(4) acts on R4 by orthogonal transformations, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations. These are all called "Clifford tori". The six-dimensional group O(4) acts transitively on the space of all such Clifford tori sitting inside the 3-sphere. However, this action has a two-dimensional stabilizer (see group action) since rotation in the meridional and longitudinal directions of a torus preserves the torus (as opposed to moving it to a different torus). Hence, there is actually a four-dimensional space of Clifford tori.[2] In fact, there is a one-to-one correspondence between Clifford tori in the unit 3-sphere and pairs of polar great circles (i.e., great circles that are maximally separated). Given a Clifford torus, the associated polar great circles are the core circles of each of the two complementary regions. Conversely, given any pair of polar great circles, the associated Clifford torus is the locus of points of the 3-sphere that are equidistant from the two circles.
More general definition of Clifford toriEdit
The flat tori in the unit 3-sphere S3 that are the product of circles of radius r in one 2-plane R2 and radius √1 − r2 in another 2-plane R2 are sometimes also called "Clifford tori".
The same circles may be thought of as having radii that are cos(θ) and sin(θ) for some angle θ in the range 0 ≤ θ ≤ π/2 (where we include the degenerate cases θ = 0 and θ = π/2).
The union for 0 ≤ θ ≤ π/2 of all of these tori of form
{\displaystyle T_{\theta }=S(\cos \theta )\times S(\sin \theta )}
(where S(r) denotes the circle in the plane R2 defined by having center (0, 0) and radius r) is the 3-sphere S3. (Note that we must include the two degenerate cases θ = 0 and θ = π/2, each of which corresponds to a great circle of S3, and which together constitute a pair of polar great circles.)
This torus Tθ is readily seen to have area
{\displaystyle \operatorname {area} (T_{\theta })=4\pi ^{2}\cos \theta \sin \theta =2\pi ^{2}\sin 2\theta ,}
so only the torus Tπ/4 has the maximum possible area of 2π2. This torus Tπ/4 is the torus Tθ that is most commonly called the "Clifford torus" – and it is also the only one of the Tθ that is a minimal surface in S3.
Still more general definition of Clifford tori in higher dimensionsEdit
Any unit sphere S2n−1 in an even-dimensional euclidean space R2n = Cn may be expressed in terms of the complex coordinates as follows:
{\displaystyle S^{2n-1}=\left\{(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{1}|^{2}+\cdots +|z_{n}|^{2}=1\right\}.}
Then, for any non-negative numbers r1, ..., rn such that r12 + ... + rn2 = 1, we may define a generalized Clifford torus as follows:
{\displaystyle T_{r_{1},\ldots ,r_{n}}=\left\{(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{k}|=r_{k},~1\leqslant k\leqslant n\right\}.}
These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori Tr1, ..., rn is the unit (2n − 1)-sphere S2n−1 (where we must again include the degenerate cases where at least one of the radii rk = 0).
The Clifford torus is "flat"; it can be flattened out to a plane without stretching, unlike the standard torus of revolution.
The Clifford torus divides the 3-sphere into two congruent solid tori. (In a stereographic projection, the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior, which is not easily visualized).
Uses in mathematicsEdit
In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of C2 with the standard symplectic structure. (Of course, any product of embedded circles in C gives a Lagrangian torus of C2, so these need not be Clifford tori.)
The Lawson conjecture states that every minimally embedded torus in the 3-sphere with the round metric must be a Clifford torus. This conjecture was proved by Simon Brendle in 2012.
Clifford tori and their images under conformal transformations are the global minimizers of the Willmore functional.
Clifford parallel and Clifford surface
William Kingdom Clifford
^ Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238 .
^ a b Norbs, P (September 2005). "The 12th problem" (PDF). The Australian Mathematical Society Gazette. 32 (4): 244–246.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Clifford_torus&oldid=1088307052"
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Inverse transform sampling - Wikipedia
Basic method for pseudo-random number sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden rule[1]) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function.
Inverse transformation sampling takes uniform samples of a number
{\displaystyle u}
between 0 and 1, interpreted as a probability, and then returns the largest number
{\displaystyle x}
from the domain of the distribution
{\displaystyle P(X)}
{\displaystyle P(-\infty <X<x)\leq u}
. For example, imagine that
{\displaystyle P(X)}
is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution.
Transformation from uniform sample to normal
{\displaystyle u}
{\displaystyle F^{-1}(u)}
1-2−52 8.12589
Inverse transform sampling for normal distribution
We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs to the left of that number. Intuitively, we are unlikely to choose a number in the far end of tails because there is very little area in them which would require choosing a number very close to zero or one.
Computationally, this method involves computing the quantile function of the distribution — in other words, computing the cumulative distribution function (CDF) of the distribution (which maps a number in the domain to a probability between 0 and 1) and then inverting that function. This is the source of the term "inverse" or "inversion" in most of the names for this method. Note that for a discrete distribution, computing the CDF is not in general too difficult: we simply add up the individual probabilities for the various points of the distribution. For a continuous distribution, however, we need to integrate the probability density function (PDF) of the distribution, which is impossible to do analytically for most distributions (including the normal distribution). As a result, this method may be computationally inefficient for many distributions and other methods are preferred; however, it is a useful method for building more generally applicable samplers such as those based on rejection sampling.
For the normal distribution, the lack of an analytical expression for the corresponding quantile function means that other methods (e.g. the Box–Muller transform) may be preferred computationally. It is often the case that, even for simple distributions, the inverse transform sampling method can be improved on:[2] see, for example, the ziggurat algorithm and rejection sampling. On the other hand, it is possible to approximate the quantile function of the normal distribution extremely accurately using moderate-degree polynomials, and in fact the method of doing this is fast enough that inversion sampling is now the default method for sampling from a normal distribution in the statistical package R.[3]
6 Truncated distribution
7 Reduction of the number of inversions
The probability integral transform states that if
{\displaystyle X}
is a continuous random variable with cumulative distribution function
{\displaystyle F_{X}}
, then the random variable
{\displaystyle Y=F_{X}(X)}
has a uniform distribution on [0, 1]. The inverse probability integral transform is just the inverse of this: specifically, if
{\displaystyle Y}
has a uniform distribution on [0, 1] and if
{\displaystyle X}
has a cumulative distribution
{\displaystyle F_{X}}
{\displaystyle F_{X}^{-1}(Y)}
{\displaystyle X}
Graph of the inversion technique from
{\displaystyle x}
{\displaystyle F(x)}
. On the bottom right we see the regular function and in the top left its inversion.
{\displaystyle U\sim \mathrm {Unif} [0,1]}
, we want to generate
{\displaystyle X}
with CDF
{\displaystyle F_{X}(x).}
{\displaystyle F_{X}(x)}
to be a strictly increasing function, which provides good intuition.
We want to see if we can find some strictly monotone transformation
{\displaystyle T:[0,1]\mapsto \mathbb {R} }
{\displaystyle T(U){\overset {d}{=}}X}
. We will have
{\displaystyle F_{X}(x)=\Pr(X\leq x)=\Pr(T(U)\leq x)=\Pr(U\leq T^{-1}(x))=T^{-1}(x),{\text{ for }}x\in \mathbb {R} ,}
where the last step used that
{\displaystyle \Pr(U\leq y)=y}
{\displaystyle U}
is uniform on
{\displaystyle (0,1)}
So we got
{\displaystyle F_{X}}
to be the inverse function of
{\displaystyle T}
, or, equivalently
{\displaystyle T(u)=F_{X}^{-1}(u),u\in [0,1].}
Therefore, we can generate
{\displaystyle X}
{\displaystyle F_{X}^{-1}(U).}
Schematic of the inverse transform sampling. The inverse function of
{\displaystyle y=F_{X}(x)}
{\displaystyle F_{X}^{-1}(y)=\mathrm {inf} \{x|F_{X}(x)\geq y\}}
An animation of how inverse transform sampling generates normally distributed random values from uniformly distributed random values
The problem that the inverse transform sampling method solves is as follows:
{\displaystyle X}
be a random variable whose distribution can be described by the cumulative distribution function
{\displaystyle F_{X}}
We want to generate values of
{\displaystyle X}
which are distributed according to this distribution.
The inverse transform sampling method works as follows:
Generate a random number
{\displaystyle u}
from the standard uniform distribution in the interval
{\displaystyle [0,1]}
, e.g. from
{\displaystyle U\sim \mathrm {Unif} [0,1].}
Find the inverse of the desired CDF, e.g.
{\displaystyle F_{X}^{-1}(x)}
{\displaystyle X=F_{X}^{-1}(u)}
. The computed random variable
{\displaystyle X}
{\displaystyle F_{X}(x)}
Expressed differently, given a continuous uniform variable
{\displaystyle U}
{\displaystyle [0,1]}
and an invertible cumulative distribution function
{\displaystyle F_{X}}
{\displaystyle X=F_{X}^{-1}(U)}
{\displaystyle F_{X}}
{\displaystyle X}
is distributed
{\displaystyle F_{X}}
A treatment of such inverse functions as objects satisfying differential equations can be given.[4] Some such differential equations admit explicit power series solutions, despite their non-linearity.[citation needed]
As an example, suppose we have a random variable
{\displaystyle U\sim \mathrm {Unif} (0,1)}
and a cumulative distribution function
{\displaystyle {\begin{aligned}F(x)=1-\exp(-{\sqrt {x}})\end{aligned}}}
In order to perform an inversion we want to solve for
{\displaystyle F(F^{-1}(u))=u}
{\displaystyle {\begin{aligned}F(F^{-1}(u))&=u\\1-\exp \left(-{\sqrt {F^{-1}(u)}}\right)&=u\\F^{-1}(u)&=(-\log(1-u))^{2}\\&=(\log(1-u))^{2}\end{aligned}}}
From here we would perform steps one, two and three.
As another example, we use the exponential distribution with
{\displaystyle F_{X}(x)=1-e^{-\lambda x}}
for x ≥ 0 (and 0 otherwise). By solving y=F(x) we obtain the inverse function
{\displaystyle x=F^{-1}(y)=-{\frac {1}{\lambda }}\ln(1-y).}
It means that if we draw some
{\displaystyle y_{0}}
{\displaystyle U\sim \mathrm {Unif} (0,1)}
{\displaystyle x_{0}=F_{X}^{-1}(y_{0})=-{\frac {1}{\lambda }}\ln(1-y_{0}),}
{\displaystyle x_{0}}
has exponential distribution.
The idea is illustrated in the following graph:
Random numbers yi are generated from a uniform distribution between 0 and 1, i.e. Y ~ U(0, 1). They are sketched as colored points on the y-axis. Each of the points is mapped according to x=F−1(y), which is shown with gray arrows for two example points. In this example, we have used an exponential distribution. Hence, for x ≥ 0, the probability density is
{\displaystyle \varrho _{X}(x)=\lambda e^{-\lambda \,x}}
and the cumulative distribution function is
{\displaystyle F(x)=1-e^{-\lambda \,x}}
{\displaystyle x=F^{-1}(y)=-{\frac {\ln(1-y)}{\lambda }}}
. We can see that using this method, many points end up close to 0 and only few points end up having high x-values - just as it is expected for an exponential distribution.
Note that the distribution does not change if we start with 1-y instead of y. For computational purposes, it therefore suffices to generate random numbers y in [0, 1] and then simply calculate
{\displaystyle x=F^{-1}(y)=-{\frac {1}{\lambda }}\ln(y).}
Let F be a continuous cumulative distribution function, and let F−1 be its inverse function (using the infimum because CDFs are weakly monotonic and right-continuous):[5]
{\displaystyle F^{-1}(u)=\inf \;\{x\mid F(x)\geq u\}\qquad (0<u<1).}
Claim: If U is a uniform random variable on (0, 1) then
{\displaystyle F^{-1}(U)}
has F as its CDF.
{\displaystyle {\begin{aligned}&\Pr(F^{-1}(U)\leq x)\\&{}=\Pr(U\leq F(x))\quad &({\text{applying }}F,{\text{ to both sides}})\\&{}=F(x)\quad &({\text{because }}\Pr(U\leq y)=y,{\text{ when U is uniform on}}(0,1))\\\end{aligned}}}
Truncated distribution[edit]
Inverse transform sampling can be simply extended to cases of truncated distributions on the interval
{\displaystyle (a,b]}
without the cost of rejection sampling: the same algorithm can be followed, but instead of generating a random number
{\displaystyle u}niformly distributed between 0 and 1, generate
{\displaystyle u}niformly distributed between
{\displaystyle F(a)}
{\displaystyle F(b)}
, and then again take
{\displaystyle F^{-1}(u)}
Reduction of the number of inversions[edit]
In order to obtain a large number of samples, one needs to perform the same number of inversions of the distribution. One possible way to reduce the number of inversions while obtaining a large number of samples is the application of the so-called Stochastic Collocation Monte Carlo sampler (SCMC sampler) within a polynomial chaos expansion framework. This allows us to generate any number of Monte Carlo samples with only a few inversions of the original distribution with independent samples of a variable for which the inversions are analytically available, for example the standard normal variable.[6]
Copula, defined by means of probability integral transform.
Quantile function, for the explicit construction of inverse CDFs.
Inverse distribution function for a precise mathematical definition for distributions with discrete components.
^ Aalto University, N. Hyvönen, Computational methods in inverse problems. Twelfth lecture https://noppa.tkk.fi/noppa/kurssi/mat-1.3626/luennot/Mat-1_3626_lecture12.pdf[permanent dead link]
^ Luc Devroye (1986). Non-Uniform Random Variate Generation (PDF). New York: Springer-Verlag.
^ "R: Random Number Generation".
^ Steinbrecher, G., Shaw, W.T. (2008). Quantile mechanics. European Journal of Applied Mathematics 19 (2): 87–112.
^ Luc Devroye (1986). "Section 2.2. Inversion by numerical solution of F(X) = U" (PDF). Non-Uniform Random Variate Generation. New York: Springer-Verlag.
^ L.A. Grzelak, J.A.S. Witteveen, M. Suarez, and C.W. Oosterlee. The stochastic collocation Monte Carlo sampler: Highly efficient sampling from “expensive” distributions. https://ssrn.com/abstract=2529691
Retrieved from "https://en.wikipedia.org/w/index.php?title=Inverse_transform_sampling&oldid=1087831605"
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Writing functions - Nice R Code
At some point, you will want to write a function, and it will probably be sooner than you think. Functions are core to the way that R works, and the sooner that you get comfortable writing them, the sooner you’ll be able to leverage R’s power, and start having fun with it.
The first function many people seem to need to write is to compute the standard error of the mean for some variable, because curiusly this function does not come with R’s base package. This is defined as $\sqrt{\mathrm{var}(x)/n}$ (that is the square root of the variance divided by the sample size.
Start by reloading our data set again.
We can already easily compute the mean
mean(data$Height)
var(data$Height)
and the sample size
length(data$Height)
so it seems easy to compute the standard error:
sqrt(var(data$Height)/length(data$Height))
notice how data$Height is repeated there — not desirable.
Suppose we now want the standard error of the dry weight too:
sqrt(var(data$Weight)/length(data$Weight))
This is basically identical to the height case above. We’ve copied and pasted the definition and replaced the variable that we are interested in. This sort of substitution is tedious and error prone, and the sort of things that computers are a lot better at doing reliably than humans are.
It is also just not that clear from what is written what the point of these lines is. Later on, you’ll be wondering what those lines are doing.
Look more carefully at the two statements and see the similarity in form, and what is changing between them. This pattern is the key to writing functions.
Here is the syntax for defining a function, used to make a standard error function:
The result of the last line is “returned” from the function.
standard.error(data$Height)
standard.error(data$Weight)
Note that x has a special meaning within the curly braces. If we do this:
we get the same answer. Because x appears in the “argument list”, it will be treated specially. Note also that it is completely unrelated to the name of what is provided as value to the function.
You can define variables within functions
This can often help you structure your function and your thoughts.
These are also treated specially — they do not affect the main workspace (the “global environment”) and are destroyed when the function ends. If you had some value v in the global environment, it would be ignored in this function as soon as the local v was defined, with the local definition used instead.
We used the variance function above, but let’s rewrite it. The sample variance is defined as
\frac{1}{n-1}\left(\sum_{i=1}^n (x_i - \bar x)^2 \right)
This case is more compliated, so we’ll do it in pieces.
We’re going to use x for the argument, so name our first input data x so we can use it.
x <- data$Height
(1/(n - 1))
The second term is harder. We want the difference between all the x values and the mean.
## [1] -24.5444 -14.5444 -13.5444 8.4556 -8.5444 -3.5444 1.4556
## [8] -28.5444 -15.5444 -22.5444 -4.5444 -14.5444 -17.5444 5.4556
## [15] -9.5444 -21.5444 -5.5444 -4.5444 -22.5444 -14.5444 11.4556
## [22] -25.5444 -7.5444 -18.5444 5.4556 -5.5444 -11.5444 -5.5444
## [29] -10.5444 10.4556 20.4556 21.4556 4.4556 26.4556 5.4556
## [36] 16.4556 -18.5444 -6.5444 16.4556 9.4556 -4.5444 21.4556
## [43] 3.4556 11.4556 -13.5444 -22.5444 -16.5444 -12.5444 -14.5444
## [50] 2.4556 -8.5444 -27.5444 -0.5444 -15.5444 -39.5444 -1.5444
## [57] 1.4556 -9.5444 -25.5444 -19.5444 3.4556 -10.5444 -28.5444
## [64] -2.5444 16.4556 -11.5444 -1.5444 13.4556 28.4556 -15.5444
## [71] 3.4556 -14.5444 33.4556 -13.5444 -0.5444 11.4556 -5.5444
## [78] -8.5444 -7.5444 -17.5444 -5.5444 3.4556 -9.5444 -23.5444
## [85] -12.5444 -18.5444 -17.5444 15.4556 18.4556 1.4556 -9.5444
## [92] 6.4556 -4.5444 -9.5444 -0.5444 14.4556 34.4556 19.4556
## [99] 0.4556 14.4556 5.4556 1.4556 3.4556 7.4556 3.4556
## [106] -13.5444 -32.5444 -8.5444 -23.5444 -2.5444 24.4556 24.4556
## [113] -10.5444 8.4556 28.4556 4.4556 -12.5444 -19.5444 -4.5444
## [120] -11.5444 -3.5444 0.4556 17.4556 8.4556 -14.5444 -10.5444
## [127] 38.4556 17.4556 5.4556 11.4556 21.4556 -5.5444 11.4556
## [134] 41.4556 11.4556 6.4556 20.4556 12.4556 32.4556 8.4556
## [141] 1.4556 -4.5444 -5.5444 9.4556 -9.5444 7.4556 24.4556
## [148] -1.5444 4.4556 -1.5444 -7.5444 0.4556 4.4556 -5.5444
## [155] 24.4556 15.4556 18.4556 -9.5444 -15.5444 8.4556 -10.5444
## [162] 12.4556 13.4556 27.4556 41.4556 23.4556 15.4556 -7.5444
## [169] 18.4556
Then we want to square those differences:
(x - m)^2
## [1] 602.4265 211.5390 183.4502 71.4975 73.0064 12.5626 2.1188
## [8] 814.7816 241.6277 508.2490 20.6514 211.5390 307.8052 29.7638
## [15] 91.0952 464.1603 30.7401 20.6514 508.2490 211.5390 131.2313
## [22] 652.5153 56.9176 343.8940 29.7638 30.7401 133.2727 30.7401
## [29] 111.1839 109.3200 418.4324 460.3437 19.8526 699.8999 29.7638
## [36] 270.7875 343.8940 42.8289 270.7875 89.4088 20.6514 460.3437
## [43] 11.9413 131.2313 183.4502 508.2490 273.7165 157.3614 211.5390
## [50] 6.0301 73.0064 758.6928 0.2963 241.6277 1563.7579 2.3851
## [57] 2.1188 91.0952 652.5153 381.9827 11.9413 111.1839 814.7816
## [64] 6.4739 270.7875 133.2727 2.3851 181.0537 809.7224 241.6277
## [71] 11.9413 211.5390 1119.2786 183.4502 0.2963 131.2313 30.7401
## [78] 73.0064 56.9176 307.8052 30.7401 11.9413 91.0952 554.3378
## [85] 157.3614 343.8940 307.8052 238.8762 340.6100 2.1188 91.0952
## [92] 41.6750 20.6514 91.0952 0.2963 208.9650 1187.1898 378.5212
## [99] 0.2076 208.9650 29.7638 2.1188 11.9413 55.5863 11.9413
## [106] 183.4502 1059.1366 73.0064 554.3378 6.4739 598.0774 598.0774
## [113] 111.1839 71.4975 809.7224 19.8526 157.3614 381.9827 20.6514
## [120] 133.2727 12.5626 0.2076 304.6987 71.4975 211.5390 111.1839
## [127] 1478.8348 304.6987 29.7638 131.2313 460.3437 30.7401 131.2313
## [134] 1718.5685 131.2313 41.6750 418.4324 155.1425 1053.3674 71.4975
## [141] 2.1188 20.6514 30.7401 89.4088 91.0952 55.5863 598.0774
## [148] 2.3851 19.8526 2.3851 56.9176 0.2076 19.8526 30.7401
## [155] 598.0774 238.8762 340.6100 91.0952 241.6277 71.4975 111.1839
## [162] 155.1425 181.0537 753.8111 1718.5685 550.1662 238.8762 56.9176
## [169] 340.6100
and compute the sum:
sum((x - m)^2)
Watch that you don’t do this, which is quite different
sum(x - m)^2
(this follows from the definition of the mean)
Putting both halves together, the variance is
(1/(n - 1)) * sum((x - m)^2)
Which agrees with R’s variance function
The rm function cleans up:
rm(n, x, m)
We can then define our function, using the pieces that we wrote above.
variance(data$Weight)
var(data$Weight)
An aside on floating point comparisons:
Our function does not exactly agree with R’s function
variance(data$Height) == var(data$Height)
This is not because one is more accurate than the other, it is because “machine precision” is finite (that is, the number of decimal places being kept).
variance(data$Height) - var(data$Height)
This affects all sorts of things:
sqrt(2) * sqrt(2) # looks like 2
sqrt(2) * sqrt(2) - 2 # but not quite
So be careful with == for floating point comparisons. Usually you have do something like:
abs(x - y) < eps
For some small value eps. The all.equal function can be very helpful here.
Exercise: define a function to compute skew
Skewness is a measure of asymmetry of a probability distribution.
Write a function that computes the skewness.
Don’t try to do this in one step, but use intermediate variables like the second version of standard.error, or like our variance function.
The term on the top of the fraction is a lot like the variance function.
Remember that parentheses can help with order-of-operation control.
Get the pieces of your function working before putting it all together.
skewness <- function(x) {
third.moment <- (1/(n - 2)) * sum((x - m)^3)
third.moment/(var(x)^(3/2))
skewness(data$Height) # should be 0.301
skewness(data$Weight) # should be 1.954
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Multiple Studies - Open Targets Genetics Documentation
From the Study page, multiple studies can quickly be compared to identify overlapping signals. To navigate to the study comparison view, click through the
button, in the header of the study page of the first study for comparison.
The first study will be loaded into the comparison view as the root, with reported loci at genome-wide significance plotted by position. Studies for comparison can be loaded into the view using the drop-down menu. Only studies with at least one overlapping locus will be displayed as an option for comparison. Studies in the drop-down are ordered decreasing on number of overlaps with the loaded root. When >1 study is loaded, intersecting loci across all of the loaded studies are displayed in red both on the intersection bar at the top of the view, and within each study. Loci within each study which overlap with the root study are displayed black. Non-overlapping loci are plotted grey.
Below the plot view, each overlapping locus across all loaded studies is summarised in a table. Accessing the locus view from the links in this table will pre-select all of the currently-loaded studies in the generated plot. The table can be ordered by column and downloaded in a range of flat or nested formats. Lead variants and genes can be clicked through to view the corresponding entity page.
Note that the top-ranked genes displayed are the top genes implicated directly by the lead variant shown, and do not take account of genes assigned to any tag variants of the lead. There may, therefore, be an element of mismatch between the gene shown here and the expected functional gene at the locus. The locus plot should be used to explore the full range of genes assigned to the locus via the lead variants and their proxies.
In the summary table, only the lead variant from the root study is displayed for each overlapping signal. For example, if there were one overlapping signal across four loaded studies, the table would include one variant: the lead from the root study.
How is Overlap Defined?
The comparison view displays pre-calculated locus overlap between all studies currently available in Open Targets Genetics. To define overlap for a given lead variant
x
, the LD-defined tag variants of
x
are cross-referenced to the tag variants of all lead variants within 5mB of
x
. In any case where a tag variant of
x
is shared with another lead variant, that lead is considered part of the same signal as
x
. Each shared locus, therefore, can be considered as a set of signals occupying a common haplotype.
Next - How To Use Open Targets Genetics starting with
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RTableAssign - Maple Help
Home : Support : Online Help : Connectivity : Calling External Routines : ExternalCalling : C Application Programming Interface : RTableAssign
retrieve an element of an rtable in external code
assign into an rtable in external code
RTableSelect(kv, rt, index)
RTableAssign(kv, rt, index, val)
integer array denoting the element index
RTableData union value
RTableSelect extracts the element at the index from the rtable rt. RTableAssign sets the element at the index in the rtable rt to val.
These functions are especially useful for extracting elements from rtables with indexing functions or unusual storage. For example, assigning directly to the data-block of a dense symmetric Matrix may violate the symmetric property of the Matrix unless you are careful to ensure the element reflected along the diagonal is also updated. This is automatically handled when using RTableSelect and RTableAssign.
The value val set by RTableAssign, and returned by RTableSelect is one of the datatypes in the RTableData union. The type must exactly correspond to the rtable data_type. The following union members match with these rtable_types.
complexf64
CXDAG
ALGEB M_DECL MySwapRow( MKernelVector kv, ALGEB *args )
M_INT argc, i, index1[2], index2[2];
RTableData rtd1, rtd2;
MapleRaiseError(kv,"three argument expected");
index1[0] = MapleToM_INT(kv,args[2]);
for( i=RTableLowerBound(kv,rt,2); i<=RTableUpperBound(kv,rt,2); ++i ) {
index1[1] = i;
rtd1 = RTableSelect(kv,rt,index1);
if( rts.data_type == RTABLE_INTEGER32 ) {
MaplePrintf(kv,"Swapping rt[%d,%d] = %d and rt[%d,%d] = %dn",
index1[0],index1[1],rtd1.int32,
index2[0],index2[1],rtd2.int32 );
RTableAssign(kv,rt,index1,rtd2);
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{dll}≔\mathrm{ExternalLibraryName}\left("HelpExamples"\right):
\mathrm{swaprow}≔\mathrm{DefineExternal}\left("MySwapRow",\mathrm{dll}\right):
M≔\mathrm{Matrix}\left(4,4,\mathrm{storage}=\mathrm{sparse}\right):
M[1,1]≔1:
M[1,3]≔2:
M[3,2]≔3:
M[3,4]≔4:
M
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{swaprow}\left(M,1,3\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
M≔\mathrm{Matrix}\left(4,4,\left(i,j\right)↦i,\mathrm{shape}=\mathrm{symmetric},\mathrm{datatype}=\mathrm{integer}[4]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
\mathrm{swaprow}\left(M,2,3\right)
Swapping rt[2,1] = 1 and rt[3,1] = 1
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4}\end{array}]
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Simulate linear models with uncertainty using Monte Carlo method - MATLAB simsd - MathWorks France
Simulate State-Space Model Using Monte Carlo Method
Simulate Estimated Model Using Monte Carlo Method
Simulate Time Series Model Using Monte Carlo Method
Study Effect of Initial Condition Uncertainty on Model Response
Study Effect of Additive Disturbance on Response Uncertainty
Generating Perturbations of Identified Model
Simulate linear models with uncertainty using Monte Carlo method
simsd(sys,udata)
simsd(sys,udata,N)
simsd(sys,udata,N,opt)
y = simsd(___)
[y,y_sd] = simsd(___)
simsd simulates linear models using the Monte Carlo method. The command performs multiple simulations using different values of the uncertain parameters of the model, and different realizations of additive noise and simulation initial conditions. simsd uses Monte Carlo techniques to generate response uncertainty, whereas sim generates the uncertainty using the Gauss Approximation Formula.
simsd(sys,udata) simulates and plots the response of 10 perturbed realizations of the identified model sys. Simulation input data udata is used to compute the simulated response.
The parameters of the perturbed realizations of sys are consistent with the parameter covariance of the original model, sys. If sys does not contain parameter covariance information, the 10 simulated responses are identical. For information about how the parameter covariance information is used to generate the perturbed models, see Generating Perturbations of Identified Model.
simsd(sys,udata,N) simulates and plots the response of N perturbed realizations of the identified model sys.
simsd(sys,udata,N,opt) simulates the system response using the simulation behavior specified in the option set opt. Use opt to specify uncertainties in the initial conditions and include the effect of additive disturbances.
The simulated responses are all identical if sys does not contain parameter covariance information, and you do not specify additive noise or covariance values for initial states. You specify these values in the AddNoise and X0Covariance options of opt.
y = simsd(___) returns the N simulation results in y as a cell array. No simulated response plot is produced. Use with any of the input argument combinations in the previous syntaxes.
[y,y_sd] = simsd(___) also returns the estimated standard deviation y_sd for the simulated response.
z1 is an iddata object that stores the input-output estimation data.
Simulate the response of the estimated model using the Monte Carlo method and input estimation data, and plot the response.
simsd(sys,z1);
The blue line plots the simulated response of the original nominal model sys. The green lines plot the simulated response of 10 perturbed realizations of sys.
Simulate an estimated model using the Monte Carlo method for a specified number of model perturbations.
Estimate a second-order state-space model using estimation data. Obtain sys in the observability canonical form.
sys = ssest(z3,2,'Form','canonical');
Compute the simulated response of the estimated model using the Monte Carlo method, and plot the responses. Specify the number of random model perturbations as 20.
simsd(sys,z3,N)
The blue line plots the simulated response of the original nominal model sys. The green lines plot the simulated response of the 20 perturbed realizations of sys.
You can also obtain the simulated response for each perturbation of sys. No plot is generated when you use this syntax.
y = simsd(sys,z3,N);
y is the simulated response, returned as a cell array of N+1 elements. y{1} contains the nominal response for sys. The remaining elements contain the simulated response for the N perturbed realizations.
z9 is an iddata object with 200 output data samples and no inputs.
Estimate a sixth-order AR model using the least-squares algorithm.
For time series data, specify the desired simulation length, Ns = 200 using an Ns-by-0 input data set.
Set the initial conditions to use the initial samples of the time series as historical output samples. The past data is mapped to the initial states of each perturbed system individually.
opt = simsdOptions('InitialCondition',IC);
Simulate the model using Monte Carlo method and specified initial conditions. Specify the number of random model perturbations as 20.
simsd(sys,data,20,opt)
Load data, and split it into estimation and simulation data.
zsim = z3(201:256);
Estimate a second-order state-space model sys using estimation data. Specify that no parameter covariance data is generated. Obtain sys in the observability canonical form.
opt = ssestOptions('EstimateCovariance',false);
sys = ssest(ze,2,'Form','canonical',opt);
Set the initial conditions for simulating the estimated model. Specify initial state values x0 for the two states and also the covariance of initial state values x0Cov. The covariance is specified as a 2-by-2 matrix because there are two states.
x0 = [1.2; -2.4];
x0Cov = [0.86 -0.39; -0.39 1.42];
opt = simsdOptions('InitialCondition',x0,'X0Covariance',x0Cov);
Simulate the model using Monte Carlo method and specified initial conditions. Specify the number of random model perturbations as 100.
simsd(sys,zsim,100,opt)
The blue line plots the simulated response of the original nominal model sys. The green lines plot the simulated response of the 100 perturbed realizations of sys. The software uses a different realization of the initial states to simulate each perturbed model. Initial states are drawn from a Gaussian distribution with mean InitialCondition and covariance X0Covariance.
z1 is an idddata object that stores 300 input-output estimation data samples.
Estimate a second-order state-space model using the estimation data.
Create a default option set for simsd, and modify the option set to add noise.
Compute the simulated response of the estimated model using the Monte Carlo method. Specify the number of random model perturbations as 20, and simulate the model using the specified option set.
[y,y_sd] = simsd(sys,z1,20,opt);
y is the simulated response, returned as a cell array of 21 elements. y{1} contains the nominal, noise-free response for sys. The remaining elements contain the simulated response for the 20 perturbed realizations of sys with additive disturbances added to each response.
y_sd is the estimated standard deviation of simulated response, returned as an iddata object with no inputs. The standard deviations are computed from the 21 simulated outputs. To access the standard deviation, use y_sd.OutputData.
sys — Model to be simulated
parametric linear identified model
Model to be simulated, specified as one of the following parametric linear identified models: idtf, idproc, idpoly, idss, or idgrey.
To generate the set of simulated responses, the software perturbs the parameters of sys in a way that is consistent with the parameter covariance information. Use getcov to examine the parameter uncertainty for sys. For information about how the perturbed models are generated from sys, see rsample.
The simulated responses are all identical if sys does not contain parameter covariance information and you do not specify additive noise or covariance values for initial states. You specify these values in the AddNoise and X0Covariance options of opt.
Simulation input data, specified as one of the following:
iddata object — Input data can be either time-domain or frequency-domain. The software uses only the input channels of the iddata object.
If sys is a time series model, that is, a model with no inputs, specify udata as an Ns-by-0 signal, where Ns is the wanted number of simulation output samples for each of the N perturbed realizations of sys. For example, to simulate 100 output samples, specify udata as follows.
For an example, see Simulate Time Series Model Using Monte Carlo Method.
matrix — For simulation of discrete-time systems using time-domain data only. Columns of the matrix correspond to each input channel.
N — Number of perturbed realizations
Number of perturbed realizations of sys to be simulated, specified as a positive integer.
Simulation options for simulating models using Monte Carlo methods, specified as a simsdOptions option set. You can use this option set to specify:
Input and output signal offsets — Specify an offset to remove from the input signal and an offset to add to the response of sys.
Initial condition handling — Specify initial conditions for simulation and their covariance. For state-space and linear grey-box models (idss and idgrey), if you want to simulate the effect of uncertainty in initial states, set the InitialCondition option to a double vector, and specify its covariance using the X0Covariance option. For an example, see Study Effect of Initial Condition Uncertainty on Model Response.
Addition of noise to simulated data — If you want to include the influence of additive disturbances, specify the AddNoise option as true. For an example, see Study Effect of Additive Disturbance on Response Uncertainty.
Simulated response, returned as a cell array of N+1 elements. y{1} contains the nominal response for sys. The remaining elements contain the simulated response for the N perturbed realizations.
The command performs multiple simulations using different values of the uncertain parameters of the model, and different realizations of additive noise and simulation initial conditions. Thus, the simulated responses are all identical if sys does not contain parameter covariance information and you do not specify additive noise and covariance values for initial states in opt.
y_sd — Estimated standard deviation of simulated response
Estimated standard deviation of simulated response, returned as an iddata object. The standard deviation is computed as the sample standard deviation of the y ensemble:
y_sd=\sqrt{\frac{1}{N}\sum _{i=2}^{N+1}{\left(y\left\{1\right\}-y\left\{i\right\}\right)}^{2}}
Here y{1} is the nominal response for sys, and y{i} (i = 2:N+1) are the simulated responses for the N perturbed realizations of sys.
The software generates N perturbations of the identified model sys and then simulates the response of each of these perturbations. The parameters of the perturbed realizations of sys are consistent with the parameter covariance of the original model sys. The parameter covariance of sys gives information about the distribution of the parameters. However, for some parameter values, the resulting perturbed systems can be unstable. To reduce the probability of generation of unrealistic systems, the software prescales the parameter covariance.
If Δp is the parameter covariance for the parameters p of sys, then the simulated output f(p+Δp) of a perturbed model as a first-order approximation is:
f\left(p+\Delta p\right)=f\left(p\right)+\frac{\partial f}{\partial p}\Delta p
The simsd command first scales Δp by a scaling factor s (approximately 0.1%) to generate perturbed systems with parameters (p+sΔp). The command then computes f(p+sΔp), the simulated response of these perturbed systems. Where,
f\left(p+s\Delta p\right)=f\left(p\right)+s\frac{\partial f}{\partial p}\Delta p
The command then computes the simulated response f(p+Δp) as:
f\left(p+\Delta p\right)=f\left(p\right)+\frac{1}{s}\left(f\left(p+s\Delta p\right)-f\left(p\right)\right)
This scaling is not applied to the free delays of idproc or idtf models.
If you specify the AddNoise option of simsdOptions as true, the software adds different realizations of the noise sequence to the noise-free responses of the perturbed system. The realizations of the noise sequence are consistent with the noise component of the model.
For state-space models, if you specify the covariance of initial state values in X0Covariance option of simsdOptions, different realizations of the initial states are used to simulate each perturbed model. Initial states are drawn from a Gaussian distribution with mean InitialCondition and covariance X0Covariance.
simsdOptions | getcov | sim | rsample | showConfidence
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Canonical growth conditions associated to ample line bundles
15 February 2018 Canonical growth conditions associated to ample line bundles
We propose a new construction which associates to any ample (or big) line bundle
L
on a projective manifold
X
a canonical growth condition (i.e., a choice of a plurisubharmonic (psh) function well defined up to a bounded term) on the tangent space
{T}_{p}X
of any given point
p
. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows one to recover all the infinitesimal Okounkov bodies of
L
p
. The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case, the growth condition is “equivalent” to the moment polytope. As in the toric case, the growth condition says a lot about the Kähler geometry of the manifold. We prove a theorem about Kähler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.
David Witt Nyström. "Canonical growth conditions associated to ample line bundles." Duke Math. J. 167 (3) 449 - 495, 15 February 2018. https://doi.org/10.1215/00127094-2017-0031
Received: 7 October 2015; Revised: 24 June 2017; Published: 15 February 2018
Keywords: ample line bundle , growth condition , Kähler embeddings , Okounkov body , Seshadri constant , toric geometry
David Witt Nyström "Canonical growth conditions associated to ample line bundles," Duke Mathematical Journal, Duke Math. J. 167(3), 449-495, (15 February 2018)
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Speed of a sound — lesson. Science State Board, Class 9.
The distance travelled by a sound wave per unit time as it propagates through an elastic medium is known as the speed of sound.
\mathit{Speed}\left(v\right)=\frac{\mathit{Distance}}{\mathit{Time}}
If one wavelength () represents the distance travelled by one wave, and one time period (\(T\)) represents the time taken for this propagation, then
And, we know
T=\frac{1}{v}
By applying this in speed formula, we get
Under the same physical conditions, the speed of sound in a given medium remains nearly constant for all frequencies.
Let us solve following example for better understanding.
A sound wave has a frequency of \(2\) \(kHz\) and a wavelength of \(15\) \(cm\). How much time will it take to travel \(1.5\) \(km\)?
Frequency \(=\) \(2\) \(kHz\) \(=\) \(2000\) \(Hz\)
Wavelength \(=\) \(15\) \(cm\) \(=\) \(0.15\) \(m\)
Distance \(=\) \(1.5\) \(km\) \(=\) \(1500\) \(m\)
To find: Time period
\mathit{Time}=\frac{\mathit{Distance}}{\mathit{Speed}}
We don't know the value of speed,
Now, apply in the value of speed in time formula
\begin{array}{l}\mathit{Time}=\frac{1500}{300}\\ =5\phantom{\rule{0.147em}{0ex}}s\end{array}
The sound will take \(5\) \(s\) to travel a distance of \(1.5\) \(km\).
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Algebra/Binomial Theorem - Wikibooks, open books for an open world
Algebra/Binomial Theorem
← Quadratic Equation Binomial Theorem Complex Numbers →
Wikipedia has related information at Binomial theorem
The notation '
{\displaystyle n!}
' is defined as n factorial.
{\displaystyle n!=n\times (n-1)\times (n-2)\times (n-3)\times \dots \times 3\times 2\times 1}
0 factorial is equal to 1.
{\displaystyle 0!=1}
Proof of 0 factorial = 1
{\displaystyle n!=n\times (n-1)!}
{\displaystyle 1!=1\times (1-1)!}
{\displaystyle 1=1\times 0!}
{\displaystyle 0!=1}
The binomial thereom gives the coefficients of the polynomial
{\displaystyle (x+y)^{n}}
We may consider without loss of generality the polynomial, of order n, of a single variable z. Assuming
{\displaystyle x\neq 0}
set z = y / x
{\displaystyle (x+y)^{n}=x^{n}(1+z)^{n}}
The expansion coefficients of
{\displaystyle (1+z)^{n}}
are known as the binomial coefficients, and are denoted
{\displaystyle (1+z)^{n}=\sum _{k=0}^{n}{n \choose k}z^{k}}
{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}
is symmetric in x and y, the identity
{\displaystyle {n \choose n-k}={n \choose k}}
may be shown by replacing k by n - k and reversing the order of summation.
A recursive relationship between the
{\displaystyle {n \choose k}}
may be established by considering
{\displaystyle (1+z)^{n+1}=(1+z)(1+z)^{n}=\sum _{k=0}^{n+1}{n+1 \choose k}z^{k}=(1+z)\sum _{k=0}^{n}{n \choose k}z^{k}}
{\displaystyle \sum _{k=0}^{n+1}{n+1 \choose k}z^{k}=\sum _{k=0}^{n}{n \choose k}z^{k}+\sum _{k=0}^{n}{n \choose k}z^{k+1}=\sum _{k=0}^{n}{n \choose k}z^{k}+\sum _{k=1}^{n+1}{n \choose k-1}z^{k}}
Since this must hold for all values of z, the coefficients of
{\displaystyle z^{k}}
on both sides of the equation must be equal
{\displaystyle {n+1 \choose k}={n \choose k}+{n \choose k-1}}
for k ranging from 1 through n, and
{\displaystyle {n+1 \choose n+1}={n \choose n}={\frac {n!}{(n-n)!n!}}={\frac {n!}{n!}}=1}
{\displaystyle {n+1 \choose 0}={n \choose 0}={\frac {n!}{(n-0)!0!}}={\frac {n!}{n!}}=1}
Pascal's Triangle is a schematic representation of the above recursion relation ...
{\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}}
(proof by induction on n).
A useful identity results by setting
{\displaystyle z=1}
{\displaystyle \sum _{k=0}^{n}{n \choose k}=2^{n}}
The visual way to do the binomial theoremEdit
Wikipedia has related information at Pascal's triangle
(this section is from difference triangles)
Lets look at the results for (x+1)n where n ranges from 0 to 3.
(x+1)0 = 1x0 = 1
(x+1)1 = 1x1+1x0 = 1 1
(x+1)2 = 1x2+2x1+1x0 = 1 2 1
(x+1)3 = 1x3+3x2+3x1+1x0 = 1 3 3 1
This new triangle is Pascal’s Triangle.
It follows a counting method different from difference triangles.
The sum of the x-th number in the n-th difference and
the (x+1)-th number in the n-th difference yields the
(x+1)-th number in the (n-1)-th difference.
It would take a lot of adding if we were to use the difference triangles in the X-gon to compute (x+1)10. However, using the Pascal’s Triangle which we have derived from it, the task becomes much simpler. Let’s expand Pascal’s Triangle.
(x+1)0 1
(x+1)1 1 1
(x+1)2 1 2 1
(x+1)3 1 3 3 1
(x+1)4 1 4 6 4 1
(x+1)5 1 5 10 10 5 1
(x+1)6 1 6 15 20 15 6 1
(x+1)7 1 7 21 35 35 21 7 1
(x+1)8 1 8 28 56 70 56 28 8 1
(x+1)9 1 9 36 84 126 126 84 36 9 1
(x+1)10 1 10 45 120 210 252 210 120 45 10 1
The final line of the triangle tells us that
(x+1)10 = 1x10 + 10x9 + 45x8 + 120x7 + 210x6 + 252x5 + 210x4 + 120x3 + 45x2 + 10x1 + 1x0.
Example ProblemsEdit
Retrieved from "https://en.wikibooks.org/w/index.php?title=Algebra/Binomial_Theorem&oldid=3545052"
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Erratum to: Sports participation, perceived neighborhood safety, and individual cognitions: how do they interact? | International Journal of Behavioral Nutrition and Physical Activity | Full Text
Erratum to: Sports participation, perceived neighborhood safety, and individual cognitions: how do they interact?
Mariëlle A Beenackers1,
Carlijn BM Kamphuis1,
Johan P Mackenbach1 &
After publication of this work [Beenackers et al: Int J Behav Nutr Phys Act 2011, 8:76] it was realized that formula 3 and formula 4 in the Statistical Analysis section of the Methods were incorrectly listed. Since the formulas were correctly used in the analysis, this correction does not affect the results or conclusions of the paper.
After publication of this work [1] it was realized that formula 3 and formula 4 in the Statistical Analysis section of the Methods were incorrectly listed. Since the formulas were correctly used in the analysis, this correction does not affect the results or conclusions of the paper. The formulas should be:
{\beta }_{1\text{_}conditional\text{_}on\text{_}{Z}_{medium}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={\beta }_{1}+{\beta }_{6}
{\beta }_{1\text{_}conditional\text{_}on\text{_}{Z}_{low}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={\beta }_{1}+{\beta }_{7}
So, to obtain the coefficient of the individual cognition (X) for the second category of perceived neighborhood safety (Zmedium), the coefficient of X (β1) should be added to the coefficient of the interaction term XZmedium (β6) (equation 3). Because of the logarithmic scale, the odds ratio of an interaction term can be interpreted as a multiplicative factor. To obtain the odds ratio of the individual cognition (X) for the second category of perceived neighborhood safety (Zmedium), the odds ratio of X (EXPβ1) should be multiplied by to the odds ratio of the interaction term XZmedium (EXPβ6).
To obtain the coefficient of the individual cognition (X) for the last category of perceived neighborhood safety (Zlow), the coefficient of X (β1) should be added to the coefficient of the interaction term XZlow (β7) (equation 4). Again, to obtain the odds ratio of the individual cognition (X) for the last category of perceived neighborhood safety (Zlow), the odds ratio of X (EXPβ1) should be multiplied by to the odds ratio of the interaction term XZlow (EXPβ7).
We regret any inconvenience that this may have caused.
Beenackers MA, Kamphuis CBM, Burdorf A, Mackenbach JP, van Lenthe FJ: Sports participation, perceived neighborhood safety, and individual cognitions: how do they interact?. Int J Behav Nutr Phys Act. 2011, 8: 76-10.1186/1479-5868-8-76.
Department of Public Health, Erasmus University Medical Center, PO Box 2040, 3000, Rotterdam, CA, the Netherlands
Mariëlle A Beenackers, Carlijn BM Kamphuis, Alex Burdorf, Johan P Mackenbach & Frank J van Lenthe
Mariëlle A Beenackers
Carlijn BM Kamphuis
Correspondence to Mariëlle A Beenackers.
Beenackers, M.A., Kamphuis, C.B., Burdorf, A. et al. Erratum to: Sports participation, perceived neighborhood safety, and individual cognitions: how do they interact?. Int J Behav Nutr Phys Act 8, 114 (2011). https://doi.org/10.1186/1479-5868-8-114
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LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation (Completion of the Squares) - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation (Completion of the Squares)
This method is used to solve quadratic equations that can't be factorized.
1 Matrix inequality
3 Reformulation of Young’s Relation
Matrix inequalityEdit
{\displaystyle X,Y\in \mathbb {R} ^{n\times m}}
{\displaystyle S\in \mathbb {S} ^{n\times n}}
{\displaystyle S}
>0, The matrix inequality given by
{\displaystyle {\begin{aligned}\ X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\\\end{aligned}}}
which is named Young’s relation or Young’s inequality.
Young’s relation can be derived from a completion of the squares as follows.
{\displaystyle {\begin{aligned}0\leq (X-SY)^{T}S^{-1}(X-SY)\\0\leq X^{T}S^{-1}X+Y^{T}SY-X^{T}Y-Y^{T}X\\X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\end{aligned}}}
which is named Young’s relation.
Reformulation of Young’s RelationEdit
{\displaystyle X,Y\in \mathbb {R} ^{n\times m}}
{\displaystyle S\in \mathbb {S} ^{n\times n}}
{\displaystyle S}
{\displaystyle {\begin{aligned}\ X^{T}Y+Y^{T}X\leq {\frac {1}{2}}(X+SY)^{T}S^{-1}(X+SY),\\\end{aligned}}}
is a reformulation of Young’s relation.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (2.4.1 page 23)
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Young’s_Relation_(Completion_of_the_Squares)&oldid=4006754"
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Classify observations using support vector machine (SVM) classifier for one-class and binary classification - Simulink - MathWorks Australia
{D}^{2}={‖x-s‖}^{2}
G\left(x,s\right)=\mathrm{exp}\left(-{D}^{2}\right)
G\left(x,s\right)=xs\text{'}
G\left(x,s\right)={\left(1+xs\text{'}\right)}^{p}
f\left(x\right)
f\left(x\right)
f\left(x\right)=\sum _{j=1}^{n}{\alpha }_{j}{y}_{j}G\left({x}_{j},x\right)+b,
\left({\alpha }_{1},...,{\alpha }_{n},b\right)
G\left({x}_{j},x\right)
f\left(x\right)=\left(x/s\right)\prime \beta +b.
P\left({s}_{j}\right)=\left\{\begin{array}{l}\begin{array}{cc}0;& s<\underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\end{array}\\ \begin{array}{cc}\pi ;& \underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\le {s}_{j}\le \underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\\ \begin{array}{cc}1;& {s}_{j}>\underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\end{array},
P\left({s}_{j}\right)=\frac{1}{1+\mathrm{exp}\left(A{s}_{j}+B\right)},
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m (→Deriving Physical Quantities: added Band-Averaged Solar Spectral Irradiance as constants)
PanTDI="10" # evaluate below...
# 1st column: K Conversion Factors for 16-Bit products
K16_Pan10=0.08381880; Pan10_Width=0.398
K16_Green=0.01438470; Green_Width=0.099
# set band parameters as variables -- using K16 for 16-bit data!
# calculate ToAR -- ${BAND}_Radiance is already 32-bit -- see above!
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L(\lambda ,Band)={\frac {K*DN\lambda }{Bandwidth\lambda }}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
The conversion process can be scripted to avoid repeating the same steps for each band separately. Note, however, in this example script constants, band parameters and acquisition related metadata are hard-coded! Review the code and alter appropriately, i.e. check for the parameters ESD, SEA, PanTDI, K_BAND.
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