text
stringlengths 256
16.4k
|
|---|
For the following problem, define a variable and write an equation (use the 5-D Process if needed). Then solve the equation to solve the problem. Write your solution as a sentence.
84
18
Remember that the 5-D Process has the following steps: Describe, Define, Do, Decide, and Declare.
Perform the Do and Decide steps. For the Do step, note that one piece is
18
m longer than the other piece.
If you are not sure how to do the 5-D process, choose among the videos at: 5-D Process Videos.
|
EUDML | The Hermite polynomials and the Bessel functions from a general point of view. EuDML | The Hermite polynomials and the Bessel functions from a general point of view.
The Hermite polynomials and the Bessel functions from a general point of view.
Dattoli, G.; Srivastava, H. M.; Sacchetti, D.
Dattoli, G., Srivastava, H. M., and Sacchetti, D.. "The Hermite polynomials and the Bessel functions from a general point of view.." International Journal of Mathematics and Mathematical Sciences 2003.57 (2003): 3633-3642. <http://eudml.org/doc/52141>.
author = {Dattoli, G., Srivastava, H. M., Sacchetti, D.},
title = {The Hermite polynomials and the Bessel functions from a general point of view.},
AU - Srivastava, H. M.
AU - Sacchetti, D.
TI - The Hermite polynomials and the Bessel functions from a general point of view.
{}_{0}{F}_{1}
Articles by Dattoli
Articles by Sacchetti
|
Experimental Study of Sub-Grid Scale Physics in Stratified Flows | FEDSM | ASME Digital Collection
Xu, D, & Chen, J. "Experimental Study of Sub-Grid Scale Physics in Stratified Flows." Proceedings of the ASME 2013 Fluids Engineering Division Summer Meeting. Volume 1C, Symposia: Gas-Liquid Two-Phase Flows; Industrial and Environmental Applications of Fluid Mechanics; Issues and Perspectives in Automotive Flows; Liquid-Solids Flows; Multiscale Methods for Multiphase Flow; Noninvasive Measurements in Single and Multiphase Flows; Numerical Methods for Multiphase Flow; Transport Phenomena in Energy Conversion From Clean and Sustainable Resources; Transport Phenomena in Materials Processing and Manufacturing Processes; Transport Phenomena in Mixing; Turbulent Flows: Issues and Perspectives. Incline Village, Nevada, USA. July 7–11, 2013. V01CT28A002. ASME. https://doi.org/10.1115/FEDSM2013-16561
The accuracy of large-eddy simulation (LES) of stratified flows is significantly influenced by sub-grid scale (SGS) stress and scalar flux models. In this study, two-dimensional high-resolution velocity and scalar (density) data (simultaneously obtained using a combined Particle Image Velocimetry and Planar Laser Induced Fluorescence technique) in a horizontal turbulent stratified jet are used to examine the SGS parameters and the performance of SGS models. The profiles of SGS dissipation of kinetic energy indicate that the flow has more capability to sustain its structure in the stable region (upper mixing layer) of stratified jet. The backscatter is observed from the components of the SGS dissipation of kinetic energy and SGS dissipation of scalar variance in the stable stratification region of jet in the high-Ri case. The SGS dissipation of kinetic energy and of scalar variance are shown strong dependence on the stability status of local flow field, which experience the ascending and descending as the stability parameter increases. In the SGS model tests, the scale-invariant dynamic model shows better performance of predicting
Cs2
than classic Smagorinsky model and scale-dependent dynamic model. From the current study, the SGS turbulent Prandtl number is suggested as constant (e.g., Pr ≃ 0.46) to a achieve a good simulation of scalar field in engineering applications to economize the computational cost.
Physics, Stratified flow, Energy dissipation, Scalars, Kinetic energy, Dynamic models, Flow (Dynamics), Stability, Turbulence, Backscattering, Density, Engineering systems and industry applications, Fluorescence, Large eddy simulation, Lasers, Particulate matter, Prandtl number, Resolution (Optics), Scalar field theory, Simulation, Stress
High Resolution Scalar and Velocity Measurements in an IC Engine
|
An experimental capability using an in-ground spin-pit facility specifically designed to investigate aeromechanic phenomena for gas turbine engine hardware rotating at engine speed is demonstrated herein to obtain specific information related to prediction and modeling of blade-casing interactions. Experiments are designed to allow insertion of a segment of engine casing into the path of single-bladed or multiple-bladed disks. In the current facility configuration, a
90deg
sector of a representative engine casing is forced to rub the tip of a single-bladed compressor disk for a selected number of rubs with predetermined blade incursion into the casing at rotational speeds in the vicinity of
20,000rpm
|
offset - Simple English Wiktionary
Offset is on the Academic Word List.
{\displaystyle x}
{\displaystyle y}
, the loss (because) of
{\displaystyle y}
is balanced by
{\displaystyle x}
The school will provide limited scholarships to offset the cost of tuition.
Increases in efficiency partially offset the increased costs.
The few problems are more than offset by the relatively large number of successes.
(transitive) If you offset
{\displaystyle x}
{\displaystyle y}
, you compare of contrast them.
All this solid colour is offset by the tiny yellow green flowers.
The past tense and past participle of offset.
(countable) An offset is something that balances (the loss of) something else.
(uncountable) (technical) A particular way of printing where the ink moves from surface A to B and then from B to the final C.
(countable); (technical) The image produced by this kind of printing.
(countable & uncountable); (technical) An offset is the distance that something moves away from where it is supposed to be or where it was.
Retrieved from "https://simple.wiktionary.org/w/index.php?title=offset&oldid=416877"
|
It is possible to multiply matrices by a constant.
\left[ \begin{array} { l l } { 1 } & { 2 } \\ { 3 } & { 4 } \end{array} \right]
\left[ \begin{array} { l l } { 2 } & { 4 } \\ { 6 } & { 8 } \end{array} \right]
, why is it natural to write
N = 2M
2 \left[ \begin{array} { l l } { 1 } & { 2 } \\ { 3 } & { 4 } \end{array} \right]
10\textbf{v}
\textbf{v} = \langle - 2,3,1 \rangle
10 \langle - 2,3,1 \rangle = \langle ? , ? , ? \rangle
When vectors are in component form, matrix operations can be performed. With the addition of matrices, many problems with vectors can be solved. The same rules apply when multiplying a vector by a matrix as multiplying two matrices together.
|
Inverse short-time Fourier transform - MATLAB istft - MathWorks Deutschland
ISTFT of Multichannel Signals
Phase Vocoder with Different Synthesis and Analysis Windows
ISTFT of Zero-Padded Complex Signal
ISTFT of Real Signal Using COLA Compliant Window and Overlap
Constant Overlap-Add (COLA) Constraint
x = istft(s) returns the Inverse Short-Time Fourier Transform (ISTFT) of s.
x = istft(s,fs) returns the ISTFT of s using sample rate fs.
x = istft(s,ts) returns the ISTFT using sample time ts.
x = istft(___,Name,Value) specifies additional options using name-value pair arguments. Options include the FFT window length and number of overlapped samples. These arguments can be added to any of the previous input syntaxes.
[x,t] = istft(___) returns the signal times at which the ISTFT is evaluated.
Generate a three-channel signal consisting of three different chirps sampled at 1 kHz for 1 second.
Plot the original and reconstructed versions of the first and second channels.
The phase vocoder performs time stretching and pitch scaling by transforming the audio into the frequency domain. This diagram shows the operations involved in the phase vocoder implementation.
The phase vocoder takes the STFT of a signal with an analysis window of hop size
{\mathit{R}}_{1}
and then performs an ISTFT with a synthesis window of hop size
{\mathit{R}}_{2}
. The vocoder thus takes advantage of the WOLA method. To time stretch a signal, the analysis window uses a larger number of overlap samples than the synthesis. As a result, there are more samples at the output than at the input (
{\mathit{N}}_{\mathit{S},\mathrm{Out}}>{\mathit{N}}_{\mathit{S},\mathrm{In}}
), although the frequency content remains the same. Now, you can pitch scale this signal by playing it back at a higher sample rate, which produces a signal with the original duration but a higher pitch.
Design a root-Hann window of length 512. Set analysis overlap length as 192 and synthesis overlap length as 166.
Implement the phase vocoder by using an analysis window of overlap 192 and a synthesis window of overlap 166.
If the analysis and synthesis windows are the same but the overlap length is changed, there will be an additional gain/loss that you will need to adjust. This is a common approach to implementing a phase vocoder.
Calculate the hop ratio and use it to adjust the gain of the reconstructed signal. Also calculate frequency of pitch-shifted data using the hop ratio.
Plot the original signal and the time stretched signal with fixed gain.
Compare the time-stretched signal and the pitch shifted signal on the same plot.
To better understand the effect of pitch shifting data, consider the following sinusoid of frequency Fs over 2 seconds.
Calculate the short-time Fourier transform and the inverse short-time Fourier transform with overlap lengths 192 and 166 respectively.
Plot the original signal on one plot and the time-stretched and pitch shifted signal on another.
Generate a complex sinusoid of frequency 1 kHz and duration 2 seconds.
Design a periodic Hann window of length 100 and set the number of overlap samples to 75. Check the window and overlap length for COLA compliance.
Zero-pad the signal to remove edge-effects. To avoid truncation, pad the input signal with zeros such that
is an integer. Set the FFT length to 128. Compute the short-time Fourier transform of the complex signal.
Calculate the inverse short-time Fourier transform and remove the zeros for perfect reconstruction.
Plot the real parts of the original and reconstructed signals. The imaginary part of the signal is also reconstructed perfectly.
Generate a sinusoid sampled at 2 kHz for 1 second.
Design a periodic Hamming window of length 120. Check the COLA constraint for the window with an overlap of 80 samples. The window-overlap combination is COLA compliant.
Set the FFT length to 512. Compute the short-time Fourier transform.
Calculate the inverse short-time Fourier transform.
Short-time Fourier transform, specified as a matrix or a 3-D array. For single-channel signals, specify s as a matrix with time increasing across the columns and frequency increasing down the rows. For multichannel signals, specify s as a 3-D array with the third dimension corresponding to the channels. The frequency and time vectors are obtained as outputs of stft.
Sample rate in hertz, specified as a positive scalar.
Sample time, specified as a duration scalar.
Example: istft(s,'Window',win,'OverlapLength',50,'FFTLength',128) windows the data using the window win, with 50 samples overlap between adjoining segments and 128 DFT points.
Window — Windowing function
Windowing function, specified as the comma-separated pair consisting of 'Window' and a vector. If you do not specify the window or specify it as empty, the function uses a periodic Hann window of length 128. The length of Window must be greater than or equal to 2.
Number of overlapped samples, specified as the comma-separated pair consisting of 'OverlapLength' and a positive integer smaller than the length of window. If you omit 'OverlapLength' or specify it as empty, it is set to the largest integer less than 75% of the window length, which turns out to be 96 samples for the default Hann window.
Number of DFT points, specified as the comma-separated pair consisting of 'FFTLength' and a positive integer. To achieve perfect time-domain reconstruction, you should set the number of DFT points to match that used in stft.
Method — Method of overlap-add
'wola' (default) | 'ola'
Method of overlap-add, specified as the comma-separated pair consisting of 'Method' and one of these:
'wola' — Weighted overlap-add
'ola' — Overlap-add
ConjugateSymmetric — Conjugate symmetry of original signal
Conjugate symmetry of the original signal, specified as the comma-separated pair consisting of 'ConjugateSymmetric' and true or false. If this option is set to true, istft assumes that the input s is symmetric, otherwise no symmetric assumption is made. When s is not exactly conjugate symmetric due to round-off error, setting the name-value pair to true ensures that the STFT is treated as if it were conjugate symmetric. If s is conjugate symmetric, then the inverse transform computation is faster, and the output is real.
STFT frequency range, specified as the comma-separated pair consisting of 'FrequencyRange' and 'centered', 'twosided', or 'onesided'.
'centered' — Treat s as a two-sided, centered STFT. If nfft is even, then s is considered to be computed over the interval (–π, π] rad/sample. If nfft is odd, then s is considered to be computed over the interval (–π, π) rad/sample. If you specify time information, then the intervals are (–fs, fs/2] cycles/unit time and (–fs, fs/2) cycles/unit time, respectively, where fs is the sample rate.
'twosided' — Treat s as a two-sided STFT computed over the interval [0, 2π) rad/sample. If you specify time information, then the interval is [0, fs) cycles/unit time.
'onesided' — Treat s as a one-sided STFT. If nfft is even, then s is considered to be computed over the interval [0, π] rad/sample. If nfft is odd, then s is considered to be computed over the interval [0, π) rad/sample. If you specify time information, then the intervals are [0, fs/2] cycles/unit time and [0, fs/2) cycles/unit time, respectively, where fs is the sample rate.
When this argument is set to 'onesided', istft assumes the values in the positive Nyquist range were computed without conserving the total power.
Input time dimension, specified as the comma-separated pair consisting of 'InputTimeDimension' and 'acrosscolumns' or 'downrows'. If this value is set to 'downrows', istft assumes that the time dimension of s is down the rows and the frequency is across the columns. If this value is set to 'acrosscolumns', the function istft assumes that the time dimension of s is across the columns and frequency dimension is down the rows.
Reconstructed signal in the time domain, returned as a vector or a matrix.
If a sample rate fs is provided, then t contains time values in seconds.
If a duration ts is provided, then t has the same time format as the input duration and is a duration array.
If no time information is provided, then t contains sample numbers.
The inverse short-time Fourier transform is computed by taking the IFFT of each DFT vector of the STFT and overlap-adding the inverted signals. The ISTFT is calculated as follows:
\begin{array}{c}x\left(n\right)=\underset{-1/2}{\overset{1/2}{\int }}\sum _{m=-\infty }^{\infty }{X}_{m}\left(f\right){e}^{j2\pi fn}df\\ =\sum _{m=-\infty }^{\infty }\underset{-1/2}{\overset{1/2}{\int }}{X}_{m}\left(f\right){e}^{j2\pi fn}df\\ =\sum _{m=-\infty }^{\infty }{x}_{m}\left(n\right)\end{array}
R
is the hop size between successive DFTs,
{X}_{m}
is the DFT of the windowed data centered about time
mR
{x}_{m}\left(n\right)=x\left(n\right)\text{ }\text{\hspace{0.17em}}g\left(n-mR\right)
. The inverse STFT is a perfect reconstruction of the original signal as long as
\sum _{m=-\infty }^{\infty }{g}^{a+1}\left(n-mR\right)=c\text{\hspace{0.17em}}\forall n\in ℤ
where the analysis window
g\left(n\right)
was used to window the original signal and
c
is a constant. The following figure depicts the steps followed in reconstructing the original signal.
To ensure successful reconstruction of nonmodified spectra, the analysis window must satisfy the COLA constraint. In general, if the analysis window satisfies the condition
\sum _{m=-\infty }^{\infty }{g}^{a+1}\left(n-mR\right)=c\text{\hspace{0.17em}}\forall n\in ℤ
, the window is considered to be COLA-compliant. Additionally, COLA compliance can be described as either weak or strong.
Weak COLA compliance implies that the Fourier transform of the analysis window has zeros at frame-rate harmonics such that
G\left({f}_{k}\right)=0,\text{ }\text{ }k=1,2,\dots ,R-1,\text{ }\text{ }{f}_{k}\triangleq \frac{k}{R}.
Alias cancellation is disturbed by spectral modifications. Weak COLA relies on alias cancellation in the frequency domain. Therefore, perfect reconstruction is possible using weakly COLA-compliant windows as long as the signal has not undergone any spectral modifications.
For strong COLA compliance, the Fourier transform of the window must be bandlimited consistently with downsampling by the frame rate such that
G\left(f\right)=0,\text{ }\text{ }f\ge \frac{1}{2R}.
This equation shows that no aliasing is allowed by the strong COLA constraint. Additionally, for strong COLA compliance, the value of the constant
c
must equal 1. In general, if the short-time spectrum is modified in any way, a stronger COLA compliant window is preferred.
You can use the iscola function to check for weak COLA compliance. The number of summations used to check COLA compliance is dictated by the window length and hop size. In general, it is common to use
a=1
\sum _{m=-\infty }^{\infty }{g}^{a+1}\left(n-mR\right)=c\text{\hspace{0.17em}}\forall n\in ℤ
for weighted overlap-add (WOLA), and
a=0
for overlap-add (OLA). By default, istft uses the WOLA method, by applying a synthesis window before performing the overlap-add method.
In general, the synthesis window is the same as the analysis window. You can construct useful WOLA windows by taking the square root of a strong OLA window. You can use this method for all nonnegative OLA windows. For example, the root-Hann window is a good example of a WOLA window.
Input size — If you invert the output of stft using istft and want the result to be the same length as the input signal x, the value of
k\text{ }=\text{ }\frac{\left(length\left(x\right)-noverlap\right)}{\left(length\left(window\right)-noverlap\right)}
must be an integer.
'InputTimeDimension' must be always specified and set to 'downrows'.
The 'ConjugateSymmetric' argument is not supported for code generation.
Unless 'ConjugateSymmetric' is set to true, the output x is always complex even if all the imaginary parts are zero.
|
The average distance from the earth to the moon is
3.844\times10^8
meters. If the length of the average pencil is
1.8 × 10^{−1}
meters, approximately how many pencils would need to be connected together to reach the moon? Use appropriate precision in your answer.
This would require dividing these two numbers in scientific notation.
Review the Math Notes box in Lessons 1.3.1 and 1.3.2.
\frac{3.844 \times 10^{8}}{1.8 \times 10^{-1}}
|
Distances/Angular momenta - Wikiversity
Distances/Angular momenta
This diagram describes the relationship between force (F), torque (τ), momentum (p), and angular momentum (L) vectors in a rotating system. 'r' is the radius. Credit: Yawe.
Angular momenta, or angular momentum, is a lecture from the radiation astronomy department. Usually, such a classical field would be a department of physics lecture.
It is a lecture in a series about distances. Large distances are significant in astronomy.
Referring to the diagram on the right, an angular momentum L of a particle about an origin is given by
{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }
where r is the radius vector of the particle relative to the origin, p is the linear momentum of the particle, and × denotes the cross product (r · p sin θ). Theta is the angle between r and p.
The radius vector consists of two parts or concepts: a distance, or displacement, and a direction, indicated by the arrow.
4 Angular velocity
7 Earth system
Def. the "quantity of matter which a body contains, irrespective of its bulk or volume"[5] is called a mass.
Kinetics[edit | edit source]
The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector v. Credit: Krishnavedala.
The angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. The direction of the angular velocity pseudovector is along the axis of rotation; in this case (counter-clockwise rotation) the vector points up. Credit: DnetSvg.
Def. a "quantity [...] cohering together so as to make one body, or an aggregation of particles or things which collectively make one body or quantity"[5] is called a mass.
Mass is an idea.
Def. "objects in motion, but not with the forces involved"[9] is called kinematics, or the science of kinematics.
Def. a "property of a body that resists any change to its uniform motion"[10] is called inertia.
Mass and inertia are generally considered equivalent.
Isaac Newton's laws of motion contain the idea of inertia.
Newton's First law: "Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed."[11]
While motion in a straight line, or rectilinear motion can be produced over limited distances in a laboratory, it may not occur naturally.
Def. "the product of [a body's] mass and velocity"[12] is called momentum.
Def. "the rotary inertia of a system [such as] an isolated rigid body [...] is a measure of the extent to which an object will continue to rotate in the absence of an applied torque"[13] is called angular momentum.
Def. a "rotational or twisting effect of a force"[14] is called a torque.
Def. a "turning effect of a force applied to a rotational system at a distance from the axis of rotation"[15] is called a moment of force.
"The moment is equal to the magnitude of the force multiplied by the perpendicular distance between its line of action and the axis of rotation."[15]
A torque and a moment of force are the same. Each is a "unit of work done, or energy expended".[16]
Def. "the effects of forces on moving bodies"[17] is called kinetics, or the science of kinetics.
Moment of inertia[edit | edit source]
The moment of inertia and angular momenta are different for every possible configuration of mass and axis of rotation. Credit: PanCiasteczko.
"For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:
{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}
The moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment of inertia depends on the shape of the body and may be different around different axes of rotation. A larger moment of inertia around a given axis requires more torque to increase the rotation, or to stop the rotation, of a body about that axis. Moment of inertia depends on the amount and distribution of its mass, and can be found through the sum of moments of inertia of the masses making up the whole object, under the same conditions.
Angular velocity[edit | edit source]
In two dimensions the angular velocity ω is given by
{\displaystyle \omega ={\frac {d\phi }{dt}}}
This is related to the cross-radial (tangential) velocity by:[18]
{\displaystyle \mathrm {v} _{\perp }=r\,{\frac {d\phi }{dt}}}
An explicit formula for v⊥ in terms of v and θ is:
{\displaystyle \mathrm {v} _{\perp }=|\mathrm {\mathbf {v} } |\,\sin(\theta )}
Combining the above equations gives a formula for ω:
{\displaystyle \omega ={\frac {|\mathrm {\mathbf {v} } |\sin(\theta )}{|\mathrm {\mathbf {r} } |}}}
Conservation of angular momentum[edit | edit source]
A figure skater conserves angular momentum – her angular rotational speed increases as her moment of inertia decreases by drawing in her arms and legs. Credit: Deerstop.
"In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque."[19]
Angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved).[20]
"A change in angular momentum is proportional to the applied torque and occurs about the same axis as that torque."[19]
Requiring the system to be closed is equivalent to requiring that no external influence, in the form of a torque, acts upon it.[20]
"A body continues in a state of rest or of uniform rotation unless compelled by a torque to change its state."[19]
With no external influence to act upon it, the original angular momentum of the system is conserved.[20]
Orbital mechanics[edit | edit source]
A massless (or per unit mass) angular momentum is defined by[21]
{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ,}
called specific angular momentum, where
{\displaystyle \mathbf {L} =m\mathbf {h} .}
Earth system[edit | edit source]
Main articles: Planets/Earth/System, Earth system, and Earth and Moon
This is a photograph of a retroreflector array placed by the crew of Apollo 14 on the lunar surface. Credit: Alan B. Shepard, Jr., and Edgar D. Mitchell, during EVA 1 of Apollo 14 on the Moon.
Plotted are the geographical distribution of the retroreflector arrays on the lunar surface. Credit: J. O. Dickey, P. L. Bender, J. E. Faller, X X Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veilet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder.
"On 21 July 1969, during the first manned lunar mission, Apollo 11, the first retroreflector was placed on the moon, enabling highly accurate measurements of the Earth - moon separation by means of laser ranging."[22]
"The locations of the three Apollo [A-11, A-14, and A-15] arrays plus one French-built array still operating on the Soviet roving vehicle Lunakhod 2 [L-1 and L-2 are shown in the image on the left and] provide a favorable geometry for studying the rotations of the moon and for separating these rotations from lunar orbital motion and geodynamic effects [...]."[22]
"Lunar laser ranging consists of measuring the round-trip travel time and thus the separation between transmitter and reflector."[22]
"Retroreflector arrays provide optical points on the moon toward which one can fire a laser pulse and receive back a localized and recognizable signal. Ranging accuracies on the order of a centimeter are immediately possible if one has sufficiently short laser pulse lengths with high power."[22]
Although an "order-of-magnitude improvement in accuracy [has occurred since the Apollo program], the early data are still important in the separation of effects with long characteristic timescales, notably precession, nutation, relativistic geodetic precession, tidal acceleration, the primary lunar oblateness term (J2), and the relative orientation of the planes of the Earth's equator, the lunar orbit, and the ecliptic."[22]
"The data set considered here consists of over 8300 normal-point ranges (8) spanning the period between August 1969 and December 1993; the observatories and the lunar reflectors included in the analysis are listed in Table 1. The data are analyzed with a model that calculates the light travel time between the observatory and the reflector, accounting for the orientation of the Earth and moon, the distance between the centers of the two bodies, solid tides on both bodies, plate motion, atmospheric delay, and relativity (13). The fitted parameters include the geocentric locations of the observatories; corrections to the variation of latitude (that is, polar motion); the orbit of the moon about the Earth; the Earth's obliquity, precession, and nutation; plus lunar parameters including the selenocentric reflector coordinates, fractional moment-of-inertia differences, gravitational third-degree harmonics, a lunar Love number, and rotational dissipation."[22]
"The mean Earth-moon distance is 385,000 km; the radii of the Earth and moon are 6371 and 1738 km, respectively."[22]
"The moon's orbit is strongly distorted from a simple elliptical path by the solar attraction-the instantaneous eccentricity varies by a factor of 2 (0.03 to 0.07)."[22]
"[A]ccuracies are degraded when extrapolated outside the span of observations."[22]
"The two largest solar perturbations in distance r [the distance between the centers of the Earth and moon] are 3699 km (monthly) and 2956 km (semimonthly)."[22]
To use angular momentum or energy a mass must be assigned.
Angular momenta quiz
↑ 5.0 5.1 Eclecticology (12 September 2003). "mass". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2013-08-12. Cite error: Invalid <ref> tag; name "MassWikt" defined multiple times with different content
↑ "kinematics". San Francisco, California: Wikimedia Foundation, Inc. 22 July 2016. Retrieved 2016-09-08.
↑ "inertia". San Francisco, California: Wikimedia Foundation, Inc. February 20, 2014. Retrieved 2014-02-28.
↑ "momentum". San Francisco, California: Wikimedia Foundation, Inc. January 30, 2014. Retrieved 2014-02-28.
↑ "angular momentum". San Francisco, California: Wikimedia Foundation, Inc. October 9, 2013. Retrieved 2014-02-28.
↑ "torque". San Francisco, California: Wikimedia Foundation, Inc. January 10, 2014. Retrieved 2014-02-28.
↑ 15.0 15.1 "moment of force". San Francisco, California: Wikimedia Foundation, Inc. December 10, 2013. Retrieved 2014-02-28.
↑ "foot-pound". San Francisco, California: Wikimedia Foundation, Inc. June 20, 2013. Retrieved 2014-02-28.
↑ Jazzy Prinker (21 May 2016). "kinetics". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2016-09-08.
↑ Russell C. Hibbeler (2009). "Engineering Mechanics". Upper Saddle River, New Jersey: Pearson Prentice Hall. pp. 314, 153. ISBN 978-0-13-607791-6.
↑ 19.0 19.1 19.2 Henry Crew (1908). The Principles of Mechanics: For Students of Physics and Engineering. Longmans, Green, and Company, New York. p. 88. https://books.google.com/books?id=sv6fAAAAMAAJ.
↑ 20.0 20.1 20.2 Arthur M. Worthington (1906). "Dynamics of Rotation". Longmans, Green and Co., London. p. 82.
↑ Richard H. Battin (1999). An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. American Institute of Aeronautics and Astronautics, Inc.. p. 115. ISBN 1-56347-342-9.
↑ 22.0 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 J. O. Dickey, P. L. Bender, J. E. Faller, X X Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veilet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder (22 July 1994). "Lunar Laser Ranging: A Continuing Legacy of the Apollo Program". Science 265 (5171): 482-90. doi:10.1126/science.265.5171.482. http://science.sciencemag.org/content/265/5171/482. Retrieved 2016-09-09.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Distances/Angular_momenta&oldid=2120137"
|
Linear prediction filter coefficients - MATLAB lpc - MathWorks France
Estimate Series Using Forward Predictor
Linear prediction filter coefficients
[a,g] = lpc(x,p)
[a,g] = lpc(x,p) finds the coefficients of a pth-order linear predictor, an FIR filter that predicts the current value of the real-valued time series x based on past samples. The function also returns g, the variance of the prediction error. If x is a matrix, the function treats each column as an independent channel.
Estimate a data series using a third-order forward predictor. Compare the estimate to the original signal.
First, create the signal data as the output of an autoregressive (AR) process driven by normalized white Gaussian noise. Use the last 4096 samples of the AR process output to avoid startup transients.
noise = randn(50000,1);
x = filter(1,[1 1/2 1/3 1/4],noise);
x = x(end-4096+1:end);
Compute the predictor coefficients and the estimated signal.
a = lpc(x,3);
est_x = filter([0 -a(2:end)],1,x);
Compare the predicted signal to the original signal by plotting the last 100 samples of each.
plot(1:100,x(end-100+1:end),1:100,est_x(end-100+1:end),'--')
legend('Original signal','LPC estimate')
Compute the prediction error and the autocorrelation sequence of the prediction error. Plot the autocorrelation. The prediction error is approximately white Gaussian noise, as expected for a third-order AR input process.
e = x-est_x;
[acs,lags] = xcorr(e,'coeff');
plot(lags,acs)
ylabel('Normalized Autocorrelation')
Input array, specified as a vector or matrix. If x is a matrix, then the function treats each column as an independent channel.
p — Prediction filter polynomial order
length(x)-1 (default) | positive integer
Prediction filter polynomial order, specified as a positive integer. p must be less than or equal to the length of x.
a — Linear predictor coefficients
Linear predictor coefficients, returned as a row vector or a matrix. The coefficients relate the past p samples of x to the current value:
\stackrel{^}{x}\left(n\right)=-a\left(2\right)x\left(n-1\right)-a\left(3\right)x\left(n-2\right)-\cdots -a\left(p+1\right)x\left(n-p\right).
g — Prediction error variance
Prediction error variance, returned as a scalar or vector.
The prediction error, e(n), can be viewed as the output of the prediction error filter A(z), where
H(z) is the optimal linear predictor.
x(n) is the input signal.
\stackrel{^}{x}\left(n\right)
is the predicted signal.
lpc determines the coefficients of a forward linear predictor by minimizing the prediction error in the least squares sense. It has applications in filter design and speech coding.
lpc uses the autocorrelation method of autoregressive (AR) modeling to find the filter coefficients. The generated filter might not model the process exactly, even if the data sequence is truly an AR process of the correct order, because the autocorrelation method implicitly windows the data. In other words, the method assumes that signal samples beyond the length of x are 0.
lpc computes the least-squares solution to Xa = b, where
\begin{array}{ccc}X=\left[\begin{array}{cccc}x\left(1\right)& 0& \cdots & 0\\ x\left(2\right)& x\left(1\right)& \cdots & ⋮\\ ⋮& x\left(2\right)& \cdots & 0\\ x\left(m\right)& ⋮& ⋮& x\left(1\right)\\ 0& x\left(m\right)& \cdots & x\left(2\right)\\ ⋮& ⋮& ⋮& ⋮\\ 0& \cdots & 0& x\left(m\right)\end{array}\right],& a=\left[\begin{array}{c}1\\ a\left(2\right)\\ ⋮\\ a\left(p+1\right)\end{array}\right],& b=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right]\end{array},
and m is the length of x. Solving the least-squares problem using the normal equations
{X}^{H}Xa={X}^{H}b
leads to the Yule-Walker equations
\left[\begin{array}{cccc}r\left(1\right)& r{\left(2\right)}^{\ast }& \cdots & r{\left(p\right)}^{\ast }\\ r\left(2\right)& r\left(1\right)& \cdots & ⋮\\ ⋮& ⋮& \ddots & r{\left(2\right)}^{\ast }\\ r\left(p\right)& \cdots & r\left(2\right)& r\left(1\right)\end{array}\right]\left[\begin{array}{c}a\left(2\right)\\ a\left(3\right)\\ ⋮\\ a\left(p+1\right)\end{array}\right]=\left[\begin{array}{c}-r\left(2\right)\\ -r\left(3\right)\\ ⋮\\ -r\left(p+1\right)\end{array}\right],
where r = [r(1) r(2) ... r(p+1)] is an autocorrelation estimate for x computed using xcorr. The Levinson-Durbin algorithm (see levinson) solves the Yule-Walker equations in O(p2) flops.
[1] Jackson, L. B. Digital Filters and Signal Processing. 2nd Edition. Boston: Kluwer Academic Publishers, 1989, pp. 255–257.
aryule | levinson | prony | pyulear | stmcb
|
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
Then increase Ar accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir (i.e., permeable pavement) area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging, assuming Ar = Ap.
|
Disturbance rejection requirement for control system tuning - MATLAB - MathWorks América Latina
TuningGoal.Rejection class
distloc
attfact
MinAttenuation
Constant Minimum Attenuation in Frequency Band
Frequency-Dependent Attenuation Profile
Disturbance rejection requirement for control system tuning
Use TuningGoal.Rejection to specify the minimum attenuation of a disturbance injected at a specified location in a control system. This tuning goal helps you tune control systems with tuning commands such as systune or looptune.
When you use TuningGoal.Rejection, the software attempts to tune the system so that the attenuation of a disturbance at the specified location exceeds the minimum attenuation factor you specify. This attenuation factor is the ratio between the open- and closed-loop sensitivities to the disturbance and is a function of frequency. You can achieve disturbance attenuation only inside the control bandwidth. The loop gain must be larger than one for the disturbance to be attenuated (attenuation factor > 1).
Req = TuningGoal.Rejection(distloc,attfact) creates a tuning goal for rejecting a disturbance entering at distloc. This tuning goal constrains the minimum disturbance attenuation factor to the frequency-dependent value, attfact.
Disturbance location, specified as a character vector or, for multiple-input tuning goals, a cell array of character vectors.
If you are using the tuning goal to tune a Simulink® model of a control system, then distloc can include any signal identified as an analysis point in an slTuner (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points to the slTuner interface. Use getPoints (Simulink Control Design) to get the list of analysis points available in an slTuner interface to your model.
For example, suppose that the slTuner interface contains analysis points u1 and u2. Use 'u1' to designate that point as the disturbance input when creating tuning goals. Use {'u1','u2'} to designate a two-channel disturbance input.
If you are using the tuning goal to tune a generalized state-space model (genss) of a control system, then inputname can include any AnalysisPoint channel in the model. For example, if you are tuning a control system model, T, which contains an AnalysisPoint block with a location named AP_u, then distloc can include 'AP_u'. (Use getPoints to get a list of analysis points available in a genss model.) The constrained disturbance location is injected at the implied input associated with the analysis point, and measured at the implied output:
Attenuation factor as a function of frequency, specified as a numeric LTI model.
TuningGoal.Rejection constrains the minimum disturbance attenuation to the frequency-dependent value attfact. You can specify attfact as a smooth transfer function (tf , zpk, or ss model). Alternatively, you can specify a piecewise gain profile using a frd model. For example, the following code specifies an attenuation factor of 100 (40 dB) below 1 rad/s, gradually dropping to 1 (0 dB) past 10 rad/s, for a disturbance injected at u.
attfact = frd([100 100 1 1],[0 1 10 100]);
Req = TuningGoal.Rejection('u',attfact);
bodemag(attfact)
When you use an frd model to specify attfact, the gain profile is automatically mapped onto a zpk model. The magnitude of this zpk model approximates the desired gain profile. Use viewGoal(Req) to visualize the resulting attenuation profile.
If you are tuning in discrete time (that is, using a genss model or slTuner interface with nonzero Ts), you can specify attfact as a discrete-time model with the same Ts. If you specify attfact in continuous time, the tuning software discretizes it. Specifying the attenuation profile in discrete time gives you more control over the profile near the Nyquist frequency.
Minimum disturbance attenuation as a function of frequency, expressed as a SISO zpk model.
The software automatically maps the attfact input argument to a zpk model. The magnitude of this zpk model approximates the desired attenuation factor and is stored in the MinAttenuation property. Use viewGoal(Req) to plot the magnitude of MinAttenuation.
For multiloop or MIMO disturbance rejection tuning goals, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the open-loop transfer function. Set LoopScaling to 'off' to disable such scaling and shape the unscaled open-loop response.
Location of disturbance, specified as a cell array of character vectors that identify one or more analysis points in the control system to tune. For example, if Location = {'u'}, the tuning goal evaluates disturbance rejection at an analysis point 'u'. If Location = {'u1','u2'}, the tuning goal evaluates the rejection at based on the MIMO open-loop response measured at analysis points 'u1' and 'u2'.
The initial value of the Location property is set by the distloc input argument when you create the tuning goal.
Create a tuning goal that enforces a attenuation of at least a factor of 10 between 0 and 5 rad/s. The tuning goal applies to a disturbance entering a control system at a point identified as 'u'.
Req.Name = 'Rejection spec';
Req.Focus = [0 5]
Create a tuning goal that enforces an attenuation factor of at least 100 (40 dB) below 1 rad/s, gradually dropping to 1 (0 dB) past 10 rad/s. The tuning goal applies to a disturbance entering a control system at a point identified as 'u'.
These commands use a frd model to specify the minimum attenuation profile as a function of frequency. The minimum attenuation of 100 below 1 rad/s, together with the minimum attenuation of 1 at the frequencies of 10 and 100 rad/s, specifies the desired rolloff.
attfact is converted into a smooth function of frequency that approximates the piecewise specified profile. Display the gain profile using viewGoal.
The shaded region indicates where the tuning goal is violated.
When you tune a control system using a TuningGoal, the software converts the tuning goal into a normalized scalar value f(x). In this case, x is the vector of free (tunable) parameters in the control system. The parameter values are adjusted automatically to minimize f(x) or drive f(x) below 1 if the tuning goal is a hard constraint.
For TuningGoal.Rejection, f(x) is given by:
f\left(x\right)=\underset{\omega \in \text{\hspace{0.17em}}\Omega }{\mathrm{max}}{‖{W}_{S}\left(j\omega \right)S\left(j\omega ,x\right)‖}_{\infty },
or its discrete-time equivalent. Here, S(jω,x) is the closed-loop sensitivity function measured at the disturbance location. Ω is the frequency interval over which the tuning goal is enforced, specified in the Focus property. WS is a frequency weighting function derived from the specified attenuation profile. The gains of WS and MinAttenuation roughly match for gain values ranging from –20 dB to 60 dB. For numerical reasons, the weighting function levels off outside this range, unless the specified attenuation profile changes slope outside this range. This adjustment is called regularization. Because poles of WS close to s = 0 or s = Inf might lead to poor numeric conditioning of the systune optimization problem, it is not recommended to specify attenuation profiles with very low-frequency or very high-frequency dynamics.
looptune | viewGoal | systune | systune (for slTuner) (Simulink Control Design) | looptune (for slTuner) (Simulink Control Design) | TuningGoal.Tracking | TuningGoal.LoopShape | slTuner (Simulink Control Design)
|
\mathrm{LerchPhi}\left(z,a,v\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{z}^{n}}{{\left(v+n\right)}^{a}}
|z|<1
|z|=1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}1<\mathrm{ℜ}\left(a\right)
z
and
v
v
are positive integers, LerchPhi(z, a, v) has a branch cut in the
z
z=1
z=1
a
z
1-a
z=1
1<\mathrm{ℜ}\left(a\right)
\mathrm{ℜ}\left(a\right)\le 1
0\le \mathrm{ℜ}\left(a\right)
a
\mathrm{LerchPhi}\left(3,4,1\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{polylog}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)}{\textcolor[rgb]{0,0,1}{3}}
\mathrm{LerchPhi}\left(0,7,4\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{16384}}
\mathrm{LerchPhi}\left(4,0,3\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}}
\mathrm{LerchPhi}\left(z,a,1\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{polylog}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)}{\textcolor[rgb]{0,0,1}{z}}
\mathrm{LerchPhi}\left(1,z,1\right)
\textcolor[rgb]{0,0,1}{\mathrm{\zeta }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)
\mathrm{diff}\left(\mathrm{LerchPhi}\left(z,3,4\right),z\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{LerchPhi}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\right)}{\textcolor[rgb]{0,0,1}{z}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{LerchPhi}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\right)}{\textcolor[rgb]{0,0,1}{z}}
|
Continuous-time process model with identifiable parameters - MATLAB idproc - MathWorks 日本
sys=\frac{{K}_{p}}{1+{T}_{p1}s}{e}^{â{T}_{d}s}.
sys={K}_{p}\frac{1+{T}_{z}s}{\left(1+{T}_{p1}s\right)\left(1+{T}_{p2}s\right)\left(1+{T}_{p3}s\right)}{e}^{â{T}_{d}s}.
sys={K}_{p}\frac{1+{T}_{z}s}{\left(1+2\mathrm{ζ}{T}_{\mathrm{Ï}}s+{\left({T}_{\mathrm{Ï}}s\right)}^{2}\right)\left(1+{T}_{p3}s\right)}{e}^{â{T}_{d}s}.
Tω is the time constant of the complex pair of poles, and ζ is the associated damping constant.
sys={K}_{p}\frac{1}{s\left(1+2\mathrm{ζ}{T}_{\mathrm{Ï}}s+{\left({T}_{\mathrm{Ï}}s\right)}^{2}\right)}{e}^{â{T}_{d}s}.
sys=\frac{0.01}{1+2\left(0.1\right)\left(10\right)s+{\left(10s\right)}^{2}}{e}^{-5s}
Z The process model includes a zero (Tz ≠0). A type with P0 cannot include Z (a process model with no poles cannot include a zero).
D The process model includes a time delay (deadtime) (Td ≠0).
sys=\frac{{K}_{p}}{1+{T}_{p1}s}{e}^{â{T}_{d}s}.
sys=\frac{{K}_{p}}{\left(1+2\mathrm{ζ}{T}_{\mathrm{Ï}}s+{\left({T}_{\mathrm{Ï}}s\right)}^{2}\right)}.
sys={K}_{p}\frac{1+{T}_{z}s}{s\left(1+{T}_{p1}s\right)\left(1+{T}_{p2}s\right)\left(1+{T}_{p3}s\right)}{e}^{â{T}_{d}s}.
Z The process model includes a zero (Tz ≠0).
|
Status Iatrogenicus: Sunk Kidney Bias: A Lethal Form of Sunk Cost Bias
The anchoring and adjustment heuristic was first theorized by Amos Tversky and Daniel Kahneman. In one of their first studies, participants were asked to compute, within 5 seconds, the product of the numbers one through eight, either as
{\displaystyle 1\times 2\times 3\times 4\times 5\times 6\times 7\times 8}
or reversed as
{\displaystyle 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}
. Because participants did not have enough time to calculate the full answer, they had to make an estimate after their first few multiplications. When these first multiplications gave a small answer – because the sequence started with small numbers – the median estimate was 512; when the sequence started with the larger numbers, the median estimate was 2,250. (The correct answer was 40,320.) In another study by Tversky and Kahneman, participants observed a roulette wheel that was predetermined to stop on either 10 or 65. Participants were then asked to guess the percentage of the United Nations that were African nations. Participants whose wheel stopped on 10 guessed lower values (25% on average) than participants whose wheel stopped at 65 (45% on average).[5] The pattern has held in other experiments for a wide variety of different subjects of estimation.
While it is certainly possible that you can anchor to and fail to adjust away from a non-numerical thing like a diagnosis, this is conceptually strained and I am not aware of any empirical support for it. When something is as carefully and clearly defined as anchoring and adjustment, I do not think we should play loose with it.
Sunk cost bias (my erstwhile mentor and co-author Hal Arkes, pictured above, provided one of the earliest descriptions) refers to situations where decision makers base decisions about expenditures not on future expected benefits, but rather on irretrievable costs that have already been incurred. It is beyond the scope of this post to explain the proposed underlying mechanisms, but sunk cost bias has well-recognized real world examples. Wall Street traders recognize it easily and call it "throwing good money after bad." A proverbial description is "failure to cut bait." Presidents Johnson and Nixon, according to the recent PBS documentary, failed to end the Viet Nam war sooner, because doing so, they feared, would represent wasting the lives that had already been lost, while apparently failing to recognize that overall casualties would simply be greater.
I have long looked for sunk cost bias in medicine, and the only recurring example I have seen is when something is ordered but later information suggests it should not be administered - say, 2 units of blood for a hemoglobin level of 4.5, which on repeat testing is 7.5. Should we give the blood just so it is not wasted? This is a mild form of sunk cost bias. We have incurred costs, which are sunk, crossmatching and thawing it for this patient, but the patient is not expected to benefit, and may be harmed from the blood. We give it not for expected future benefit, but rather because we have sunk costs into the course of action by the time the repeat hemoglobin is received.
This year, the eureka! moment came. The first example was a kidney-pancreas transplant recipient who was desperately ill, whose kidney had failed and had required dialysis for several weeks. There were multiple infections with multi-drug-resistant bacteria, as well as poor wound healing from multiple surgical and decubitus wounds. The patient was receiving insulin. The patient was also receiving immunosuppression for the remote possibility that the transplants would recover. The main focus of recovery in these dire circumstances is for the patient, not her transplanted organs. But those organs represent massive sunk costs, especially to sub-communities. The primary team pushed to discontinue them, appropriately weighing the threats from infection and poor wound healing to be greater than the benefits of possible transplanted organ recovery. The team was told to "wean" the immunosuppressives, but elected instead to stop them cold turkey. Months later, the patient continues to suffer from poor wound healing and MDR bacterial infections. The patient's life and well-being was threatened by sunk kidney bias.
At nearly the same time, another patient had failure of a kidney transplant, was started on hemodialysis, immunosuppression was continued to preserve the pancreas and the possibility of renal recovery, and the patient died from invasive pulmonary mucormycosis. A month after starting dialysis, the patient died, essentially a result of immunosuppression ("good money") to potentially save a sunk kidney (and an afloat pancreas). The sunk kidney was "bad money" because it was likely irretrievable. Maybe it was not, you say, it may have recovered! Thus is one essence of sunk cost bias - overestimation of the probability that the good money will save the bad.
While the correct decision regarding immunosuppression in both of these cases cannot be known, and notwithstanding other caveats of retrospective analysis, these cases highlight the dangers of sunk cost, and of failure to avoid biases it may engender. Whenever sunk cost, or any other sacrosanct goal leads you to fail to accurately weigh costs and benefits of all courses of action, the results can be lethal.
Scott K. Aberegg, M.D., M.P.H. February 19, 2018 at 10:49 PM
Saw it again, twice in the past week - dead kidney transplants, or quiescent inflammatory diseases with continued immumosuppression that caused harm without benefit
I got into intensive care after nephrology and while I appreciate your example, I should mention that immunosuppression after a kidney fails is complicated, especially when you consider quality of life. Average life expectancy on dialysis is 4.5yrs. With a functional transplant, mortality closely matches that in patients without CKD. If you can save the kidney, you do more than improve the patients' quality of life, you give the patient years of life.
Transplanted organs can still reject after failing and immunosuppression is continued for this. Immunosuppression also reduces the formation of antibodies and so can improve the chance of matching to another kidney transplant.
Rejection post transplant failure may require surgical removal of the allograft. The allograft at this point is a fibrosed ball of calcium fused to surrounding tissue and surgeons hate taking them out.
We typically explain all this to patients and allow them to make their choice. A really large number opt to continue immunosuppression even post multiple infections (although immunosuppression is always held during an active infection). Being on dialysis leads to terrible quality of life and kidneys allografts can recover even after bad episodes of rejection.
Overall its a difficult decision, but their are clear benefits to continuing immunosuppresion. Informed patients can make their own choices.
Scott K. Aberegg, M.D., M.P.H. March 5, 2018 at 1:01 PM
^best comment of 2018
|
Get 3-D sensitivity matrix from SimData object - MATLAB getsensmatrix - MathWorks France
getsensmatrix
outputFactorNames
inputFactorNames
outputFactors
inputFactors
Get 3-D sensitivity matrix from SimData object
[t,r,outputFactors,inputFactors] = getsensmatrix(simdata)
[t,r,outputFactors,inputFactors] = getsensmatrix(simdata,outputFactorNames,inputFactorNames)
[t,r,outputFactors,inputFactors] = getsensmatrix(simdata) returns the time t and sensitivity data r as well as all the outputFactors and inputFactors (sensitivity outputs and inputs) from the SimData object simdata.
[t,r,outputFactors,inputFactors] = getsensmatrix(simdata,outputFactorNames,inputFactorNames) returns the sensitivity data for only the outputs and inputs specified by outputFactorNames and inputFactorNames, respectively.
sensMatrix2=\left[\begin{array}{cc}\begin{array}{c}\frac{\partial y1}{\partial c1}\\ \\ \frac{\partial y2}{\partial c1}\end{array}& \begin{array}{c}\frac{\partial y1}{\partial c2}\\ \\ \frac{\partial y2}{\partial c2}\end{array}\end{array}\right]
Simulation data, specified as a SimData object or array of SimData objects. If simdata is an array of objects, the outputs are cell arrays in which each cell contains data for the corresponding object in the SimData array.
outputFactorNames — Names of sensitivity outputs
[] (default) | character vector | string | string vector | cell array of character vectors
Names of sensitivity outputs, specified as an empty array [], character vector, string, string vector, or cell array of character vectors.
By default, the function uses an empty array [] to return sensitivity data for all output factors in simdata.
inputFactorNames — Names of sensitivity inputs
Names of sensitivity inputs, specified as an empty array [], character vector, string, string vector, or cell array of character vectors.
By default, the function uses an empty array [] to return sensitivity data on all input factors in simdata.
m-by-1 numeric vector | cell array
Simulation time points for the sensitivity data, returned as an m-by-1 numeric vector or cell array. m is the number of time points.
r — Sensitivity data
m-by-n-by-p array | cell array
Sensitivity data, returned as an m-by-n-by-p array or cell array. m is the number of time points, n is the number of sensitivity outputs, and p is the number of sensitivity inputs.
The outputFactors output argument labels the second dimension of r and inputFactors labels the third dimension of r. For example, r(:,i,j) is the time course for the sensitivity of the state outputFactors{i} to the input inputFactor{j}.
The function returns only the sensitivity data already in the SimData object. It does not calculate the sensitivities. For details on setting up and performing a sensitivity calculation, see Local Sensitivity Analysis (LSA). During setup, you can also specify how to normalize the sensitivity data.
outputFactors — Names of sensitivity outputs
Names of sensitivity outputs, returned as an n-by-1 cell array. n is the number of sensitivity outputs.
The output factors are the states for which you calculated the sensitivities. In other words, the sensitivity outputs are the numerators. For more information, see Local Sensitivity Analysis (LSA).
inputFactors — Names of sensitivity inputs
Names of sensitivity inputs, returned as an p-by-1 cell array. p is the number of input factors.
The input factors are the states with respect to which you calculated the sensitivities. In other words, the sensivity inputs are the denominators as explained in Local Sensitivity Analysis (LSA).
SimData | sbiosimulate | SimFunctionSensitivity object
|
Extract a frequency subband using a one-sided (complex) bandpass decimator - MATLAB - MathWorks Nordic
MinimizeComplexCoefficients
MixToBaseband
Specific to dsp.ComplexBandpassDecimator
Extract a frequency subband using a one-sided (complex) bandpass decimator
The dsp.ComplexBandpassDecimator System object™ extracts a specific sub-band of frequencies using a one-sided, multistage, complex bandpass decimator. The object determines the bandwidth of interest using the specified CenterFrequency, DecimationFactor and Bandwidth values.
To extract a frequency subband using a complex bandpass decimator:
Create the dsp.ComplexBandpassDecimator object and set its properties.
bpdecim = dsp.ComplexBandpassDecimator
bpdecim = dsp.ComplexBandpassDecimator(d)
bpdecim = dsp.ComplexBandpassDecimator(d,Fc)
bpdecim = dsp.ComplexBandpassDecimator(d,Fc,Fs)
bpdecim = dsp.ComplexBandpassDecimator(Name,Value)
bpdecim = dsp.ComplexBandpassDecimator creates a System object that filters each channel of the input over time using a one-sided, multistage, complex bandpass decimation filter. The object determines the bandwidth of interest using the default center frequency, decimation factor, and bandwidth values.
bpdecim = dsp.ComplexBandpassDecimator(d) creates a complex bandpass decimator object with the DecimationFactor property set to d.
bpdecim = dsp.ComplexBandpassDecimator(d,Fc) creates a complex bandpass decimator object with the DecimationFactor property set to d, and the CenterFrequency property set to Fc.
bpdecim = dsp.ComplexBandpassDecimator(d,Fc,Fs) creates a complex bandpass decimator object with the DecimationFactor property set to d, the CenterFrequency property set to Fc, and the SampleRate property set to Fs.
Example: dsp.ComplexBandpassDecimator(48e3/1e3,2e3,48e3);
bpdecim = dsp.ComplexBandpassDecimator(Name,Value) creates a complex bandpass decimator object with each specified property set to the specified value. Enclose each property name in quotes. You can use this syntax with any previous input argument combinations.
Example: dsp.ComplexBandpassDecimator(48e3/1e3,2e3,48e3,'CenterFrequency',1e3);
CenterFrequency — Center frequency in Hz
Center frequency of the desired band in Hz, specified as a real, finite numeric scalar in the range [-SampleRate/2, SampleRate/2].
'Decimation factor' (default) | 'Bandwidth' | 'Decimation factor and bandwidth'
Filter design parameters, specified as:
'Decimation factor' –– The object specifies the decimation factor through the Decimation Factor property. The bandwidth of interest (BW) is computed using the following equation:
BW=Fs/D
Fs –– Sample rate specified through SampleRate property.
D –– Decimation factor.
'Bandwidth' –– The object specifies the bandwidth through the Bandwidth property. The decimation factor (D) is computed using the following equation:
D=\text{floor}\left(\frac{Fs}{BW+TW}\right)
BW –– Bandwidth of interest.
TW –– Transition width specified through the TransitionWidth property.
'Decimation factor and bandwidth' –– The decimation factor and the bandwidth of interest are specified through the DecimationFactor and Bandwidth properties.
Factor by which to reduce the bandwidth of the input signal, specified as a positive integer. The frame size (number of rows) of the input signal must be a multiple of the decimation factor.
This property applies when you set Specification to either 'Decimation factor' or 'Decimation factor and bandwidth'.
StopbandAttenuation — Stopband attenuation in dB
Stopband attenuation of the filter in dB, specified as finite positive scalar.
TransitionWidth — Transition width in Hz
Transition width of the filter in Hz, specified as a positive scalar.
Bandwidth — Bandwidth (Hz)
Width of the frequency band of interest, specified as a real positive scalar in Hz.
This property applies when you set Specification to either 'Bandwidth' or 'Decimation factor and bandwidth'.
PassbandRipple — Passband ripple (dB)
Passband ripple of the filter, specified as a positive scalar in dB.
MinimizeComplexCoefficients — Flag to minimize number of complex coefficients
Flag to minimize the number of complex filter coefficients, specified as:
true –– The first stage of the multistage filter is bandpass (complex coefficients) centered at the specified center frequency. The first stage is followed by a mixing stage that heterodynes the signal to DC. The remaining filter stages, all with real coefficients, follow.
false –– The input signal is first passed through the different stages of the multistage filter. All stages are bandpass (complex coefficients). The signal is then heterodyned to DC if MixToBaseband is true, and the frequency offset resulting from the decimation is nonzero.
MixToBaseband — Flag to mix signal to baseband
Flag to mix the signal to baseband, specified as:
true –– The object heterodynes the filtered, decimated signal to DC. This mixing stage runs at the output sample rate of the filter.
false –– The object skips the mixing stage.
This property applies when you set MinimizeComplexCoefficients to false.
SampleRate — Input sample rate in Hz
Sampling rate of the input signal in Hz, specified as a real positive scalar.
y = bpdecim(x)
y = bpdecim(x) filters the real or complex input signal, x, to produce the output, y. The output contains the subband of frequencies specified by the System object properties. The System object filters each channel of the input signal independently over time. The frame size (first dimension) of x must be a multiple of the decimation factor.
Data input, specified as a vector or a matrix. The number of rows in the input must be a multiple of the decimation factor.
Output of the complex bandpass decimator, returned as a vector or a matrix. The output contains the subband of frequencies specified by the System object properties. The number of rows (frame size) in the output signal is 1/D times the number of rows in the input signal, where D is the decimation factor. The number of channels (columns) does not change.
The data type of the output is same as the data type of the input. The output signal is always complex.
cost Implementation cost of the complex bandpass decimator
freqz Frequency response of the multirate multistage filter
visualizeFilterStages Visualize filter stages
The complex bandpass decimator is designed by applying a complex frequency shift transformation on a lowpass prototype filter. The lowpass prototype in this case is a multirate, multistage finite impulse response (FIR) filter. The desired frequency shift applies only to the first stage. Subsequent stages scale the desired frequency shift by their respective cumulative decimation factors. For details, see Complex Bandpass Filter Design and Zoom FFT.
cost | freqz | info | visualizeFilterStages
|
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
Where the total contributing drainage area of the pavement (Ac) and total depth of clear stone aggregate needed for load bearing capacity are known (i.e., storage reservoir depth is fixed) or if available space is constrained in the vertical dimension due to water table or bedrock elevation, the minimum footprint area of the water storage reservoir, Ar can be calculated as follows:
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
Then increase Ar accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir (i.e., permeable pavement) area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging, assuming Ar = Ap/sub>.
|
A fuzzy approach to option pricing in a Levy process setting
Piotr Nowak, Maciej Romaniuk (2013)
In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets. We assume that some parameters of the financial instrument cannot be precisely described and therefore they are...
Jean-Pierre Fouque, Chuan-Hsiang Han (2007)
A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility...
r
-balayages of matrix-exponential Lev́y processes.
Sheu, Yuan-Chung, Chen, Yu-Ting (2009)
A priori error estimates for reduced order models in finance
Ekkehard W. Sachs, Matthias Schu (2013)
Mathematical models for option pricing often result in partial differential equations. Recent enhancements are models driven by Lévy processes, which lead to a partial differential equation with an additional integral term. In the context of model calibration, these partial integro differential equations need to be solved quite frequently. To reduce the computational cost the implementation of a reduced order model has shown to be very successful numerically. In this paper we give a priori error...
A reduced modelling approach to the pricing of mortgage backed securities.
Parshad, Rana D. (2010)
Actuarial Approach to Option Pricing in a Fractional Black-Scholes Model with Time-Dependent Volatility
Adrian Falkowski (2013)
We study actuarial methods of option pricing in a fractional Black-Scholes model with time-dependent volatility. We interpret the option as a potential loss and we show that the fair premium needed to insure this loss coincides with the expectation of the discounted claim payoff under the average risk neutral measure.
An American convert close to maturity.
Alobaidi, G, Mallier, R. (2009)
An analytic solution for a Vasicek interest rate convertible bond model.
Deakin, A.S., Davison, Matt (2010)
An approximation formula for the price of credit default swaps under the fast-mean reversion volatility model
Xin-Jiang He, Wenting Chen (2019)
We consider the pricing of credit default swaps (CDSs) with the reference asset assumed to follow a geometric Brownian motion with a fast mean-reverting stochastic volatility, which is often observed in the financial market. To establish the pricing mechanics of the CDS, we set up a default model, under which the fair price of the CDS containing the unknown “no default” probability is derived first. It is shown that the “no default” probability is equivalent to the price of a down-and-out binary...
Analytical approximation of the transition density in a local volatility model
Stefano Pagliarani, Andrea Pascucci (2012)
Applications of simulation methods to barrier options driven by Lévy processes.
Roşca, Alin V., Roşca, Natalia C. (2010)
A. V. Nagaev, S. A. Nagaev (2003)
Callable Russian options and their optimal boundaries.
Suzuki, Atsuo, Sawaki, Katsushige (2009)
Rafael Company, Lucas Jódar, José-Ramón Pintos (2009)
This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical...
Cost-efficiency in multivariate Lévy models
Ludger Rüschendorf, Viktor Wolf (2015)
In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence...
Defaultable bonds with an infinite number of Lévy factors
Jacek Jakubowski, Mariusz Niewęgłowski (2010)
A market with defaultable bonds where the bond dynamics is in a Heath-Jarrow-Morton setting and the forward rates are driven by an infinite number of Lévy factors is considered. The setting includes rating migrations driven by a Markov chain. All basic types of recovery are investigated. We formulate necessary and sufficient conditions (generalized HJM conditions) under which the market is arbitrage-free. Connections with consistency conditions are discussed.
Defaultable game options in a hazard process model.
Bielecki, Tomasz R., Crépey, Stéphane, Jeanblanc, Monique, Rutkowski, Marek (2009)
DG method for numerical pricing of multi-asset Asian options—the case of options with floating strike
Jiří Hozman, Tomáš Tichý (2017)
Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two...
Diffusion approximations of the geometric Markov renewal processes and option price formulas.
Swishchuk, Anatoliy, Islam, M.Shafiqul (2010)
|
Cohen Courses:Learning Indian Classical Using Sequential Models - Cohen Courses
2.1.1 Pakad
4 Grand Idea
Indian Classical music is a very structured when it comes to melody. A composition is (generally) within a constraints of a raag. It has a specific grammar, which lends the emotions to the composition. This aspect of music lends an interesting application of sequential models for note prediction, and raga classification.
Pakad is a string of notes characteristic to a Raga to which a musician frequently returns while improvising in a performance. A pakad has the potential to illustrate the grammar and aesthetics of a raga. For example consider raga Bageshree. The pakad is F G A F D# D C. It can be rendered in various ways as -
F G A F D# F DC
F G A F G D# F D C
F G A , D# F D C
The following are valid sequences in Bageshree, but they are not pakads -
F G A G F D# D C
F A G F D# F D C
Since pakad enforces a raga, the objective would be to identify a pakad in a sequence of notes.
Questions from William:
Don't you need duration and stress as well as the notes?
> Yes. These are additional features. Since I'm using midi files, I do have the stress (velocity) and the duration of the notes (which will be preserved in the annotation). But the baseline doesn't need it.
> It means a stop.
How do you plan to encode this? as a BIO labeling for notes?
> Right now, I'm planning with attr/non-attr type labels for each notes. I haven't figured out what difference would BIO make.
How hard is it to go from midi to a sequence of nodes (maybe with stress and duration, if you need that)?
> I have the code in place for that. It identifies the note; the duration and the stress.
Sorry for all the questions - it's partly my unfamiliarity with the domain... --Wcohen 14:47, 11 October 2011 (UTC)
In [2], the pakad matching was done using
{\displaystyle \delta }
-Occurence with
{\displaystyle \alpha }
-Bounded Gaps. This however, fails for the two sequences displayed above.
The grand idea is to view this task as a sequence alignment problem. There has been considerable work in machine translation. The challenge would be to adapt this work.
Question from William: ie, you would be learning a similarity metric? or constructing alignments between a midi file and some designated prototypes? please explain in more detail what the inputs and outputs of the system would be. --Wcohen 14:43, 11 October 2011 (UTC) > I'm constucting alignments between a midi file and designated prototypes.
There are midi files available at http://www.cse.iitk.ac.in/users/tvp/music/. These will be manually annotated for pakads.
how long will it take to do the annotation (do you have a clear idea yet)? It seems like this might be a hard annotation task, since you're labeling subsequences of the song rather than just adding labels to a complete song.
> I plan to complete the annotations by this weekend. I'm not doing inter-annotator agreement to start with.
What will be your baseline method? I see the related work, but I don't know if that is a difficult thing to re-implement or not. Is there some sort of off-the-shelf learning method that can be used?
> I want to compare it with the existing technique to demonstrate that using sequence alignment makes sense.
--Wcohen 14:42, 11 October 2011 (UTC)
1. http://www.slideshare.net/butest/music-and-machine-learning
2. TANSEN : A SYSTEM FOR AUTOMATIC RAGA IDENTIFICATION
3. C. S. Iliopoulos and M. Kurokawa: "StringMatching with Gaps for Musical Melodic Recognition": Proc. Prague Stringology Conference, pp. 55-64: 2002.
Retrieved from "http://curtis.ml.cmu.edu/w/courses/index.php?title=Cohen_Courses:Learning_Indian_Classical_Using_Sequential_Models&oldid=9067"
|
Watt's Law (Power Law) | James's Knowledge Graph
Watt's Law, sometimes called the Power Law, describes the relationship between power (
P
), current (
I
), and voltage (
V
), so that power is defined as:
P=IV
Given tha values for any two variables, we solve for the third so that voltage, electrical potential energy, is defined as:
V=P/I
And current, which is the flow of electricity, is defined as:
I=P/V
Power and Watts
Power is measured in watts. One watt (
W
) is equal to one joule (
J
) per second (
s
W = \frac{J}{s}
. Another way to put it is, a watt is the amount of electricity required to accelerate 1kg by 1 meter per second squared (a joule) over 1 second. In other words, a watt is "how much work can be done per second".
Video: Power, Work, and Energy
Full course on Khan Academy: Power)
The terms watt and Watt's Law are named after James Watt, best known for his work to improve the steam engine.
Broader Topics Related to Watt's Law (Power Law)
Watt's Law (Power Law) Knowledge Graph
|
CoCalc – Features
Overview of CoCalc features
These pages are an overview of what CoCalc is able to do. You can also learn about our mission, developers and features...
We provide a CoCalc specific version of Jupyter notebooks with real-time collaboration, chat, and high precision edit history. Explore in more detail in the documentation.
Huge installed Python stack
Use Python in CoCalc for data science, statistics, mathematics, physics, machine learning. Many packages are included in CoCalc!
CoCalc's collaborative whiteboard fully supports writing mathematics using LaTeX and doing computation using Jupyter code cells on an infinite canvas.
Use Jupyter notebooks with the R kernel, the R command line, X11 graphics,
\LaTeX
with Knitr and RMarkdown, and more.
\LaTeX
CoCalc's
\LaTeX
editor can help you be a more productive author online. Check out its documentation.
SageMath Online
SageMath is very well supported in CoCalc, because William Stein, who started SageMath, also started CoCalc. Many versions of Sage are preinstalled and there is excellent integration with
\LaTeX
Run GNU Octave on CoCalc – the syntax is largely compatible with MATLAB®. Use Jupyter notebooks, write programs, and display X11 graphics.
Use Julia on CoCalc with Pluto and Jupyter notebooks. Edit Julia code and run it in a terminal or notebook. Teach classes using nbgrader with the Julia kernel.
Linux graphical X11 desktop
Run graphical applications in CoCalc's remote virtual desktop environment. Read more in the X11 documentation.
Online Linux environment
Use a collaborative online Linux terminal, edit and run Bash scripts, or work in a Jupyter Notebooks running the Bash kernel.
Work in a collaborative remote Linux shell. Read more in our documentation.
Organize and teach a course and automatically grade Jupyter notebooks. Read more in the instructor guide.
Programmatically control CoCalc from your own server. Embed CoCalc within other products with a customized external look and feel.
Use Sage Worksheets, Course management, Task management, Chat, and more...
|
An Innovative Ultradeepwater Subsea Blowout Preventer Control System Using Shape-Memory Alloy Actuators | J. Energy Resour. Technol. | ASME Digital Collection
, 4800 Calhoun Road, Houston, TX 77004
Dr. Gangbing Song is an Associate Professor of Mechanical Engineering and the director of the Smart Materials and Structures Laboratory at the University of Houston. He is a NSF CAREER award recipient of 2001. Dr. Song received his Ph.D. and MS degrees from the Department of Mechanical Engineering at Columbia University in the City of New York in 1995 and 1991, respectively. Dr. Song received his BS degree in 1989 from Zhejing University, P.R.C. He has research interests in smart materials and structures, structural vibration control, and advance control methods. He has developed two new courses in smart materials and published more than 100 journal and conference papers. Dr. Song is also a coinventor of a US patent.
Ziping Hu,
Ziping Hu
, 14990 Yorktown Plaza Drive, Houston, TX 77040
Mr. Ziping Hu is a Design Engineer at Baker Hughes Inc. in Houston. With
14years
of experience in Mechanical Engineering, Hu currently works on well completion projects. Hu holds a MS degree in petroleum engineering from the University of Houston and he received his BME degree in 1987 from Hefei University of Technology, Hefei, China. Hu’s interest is in the application of innovative technology and material in petroleum industry.
WellDynamics Inc.
, 445 Woodline Drive, Spring, TX 77386
Mr. Kai Sun is a Petroleum Engineer who currently works in Baker Oil Tools in Houston, focusing on the application/optimization of the Intelligent Well System. With ten years of experience, Mr. Sun previously worked in WellDynamics as a Reservoir Engineer and CNPC as an Oil/Gas Gathering/Transportation Engineer. Sun holds MS degrees in Petroleum Engineering and Engineering and Technology Management from University of Houston and University of Louisiana and a BS degree in Mechanical Engineering from Harbin Engineering University. Mr. Sun has published 14 SPE papers with 4 of them in SPE journals.
OptiSolar Inc.
, 31302 Huntwood Avenue, Hayward, CA 94544
Dr. Ning Ma is a Research Associate in the Smart Materials and Structures Laboratory at the University of Houston. Dr. Ma received his Ph.D. in the Mechanical Engineering at University of Houston, Houston TX in 2005, an MS from the Department of Mechanical Engineering at the University of Akron, Akron, OH in 2002, and a BS degree in 1993 from Shenyang Institute of Aeronautic Engineering, P.R.C. His research interests are modeling and applications of smart materials and structures, especially the SMAS, for motion control, structural vibration control, and so on. He has published five journal papers and holds one US patent.
Michael J. Economides,
Dr. Michael J. Economides is a Professor at the Cullen College of Engineering, University of Houston, and the Managing Partner of a petroleum engineering and petroleum strategy consulting firm. Publications include authoring or coauthoring of 11 professional textbooks and books, including “The Color Of Oil” and almost 200 journal papers and articles. Economides does a wide range of industrial consulting, including major retainers by national oil companies at the country level and by Fortune 500 companies. He has had professional activities in over 70 countries.
Samuel G. Robello,
Samuel G. Robello
Dr. Christine Ehlig–Economides is a Professor of Petroleum Engineering and Albert B. Stevens Endowed Chair at Texas A&M University in College Station, TX. She received a BA degree in Math Science from Rice University in 1971, MS in Chemical Engineering from the University of Kansas in 1976, and a Ph.D. in Petroleum Engineering from Stanford University in 1979. Along with Dr. Michael J. Economides, she established the BS and MS programs at the University of Alaska, Fairbanks. She then worked in various capacities for
20years
in more than 30 countries with Schlumberger before returning to the academia. She has published more than 50 papers and 2 patents, and was inducted into the US National Academy of Engineering in 2003.
Gangbing Song Associate Professor
Ziping Hu Design Engineer Sr.
14years
Kai Sun Petroleum Engineer
Michael J. Economides Professor
Samuel G. Robello Principal Technical Advisor
Christine Ehlig-Economides Professor
20years
Song, G., Hu, Z., Sun, K., Ma, N., Economides, M. J., Robello, S. G., and Ehlig-Economides, C. (August 11, 2008). "An Innovative Ultradeepwater Subsea Blowout Preventer Control System Using Shape-Memory Alloy Actuators." ASME. J. Energy Resour. Technol. September 2008; 130(3): 033101. https://doi.org/10.1115/1.2955558
This paper presents an innovative undersea blowout preventer (BOP) using shape-memory alloy (SMA). The new device using SMA actuators could easily be implemented into existing conventional subsea control system so that they can work solely or as a backup of other methods. Most important, the innovative all-electric BOP will provide much faster response than its hydraulic counterpart and will improve safety for subsea drilling. To demonstrate the feasibility of such a device, a proof-of-concept prototype of a pipe RAM type BOP with SMA actuation has been designed, fabricated, and tested at the University of Houston. The BOP actuator uses strands of SMA wires to achieve large force and large displacement in a remarkably small space. Experimental results demonstrate that the BOP can be activated and fully closed in less than
10s
. The concept of this innovative device is illustrated, and detailed comparisons of the response time for hydraulic and nitinol SMA actuation mechanisms are included. This preliminary research reveals the potential of smart material technology in subsea drilling systems.
drilling, electric actuators, offshore installations, shape memory effects, underwater equipment
Actuators, Control systems, Engineering prototypes, Ocean engineering, Reliability, Shape memory alloys, Wire, Drilling, Nickel titanium alloys, Temperature
API Spec. 16D, 2004, API RP16E: Recommended Practice for Design of Control Systems for Drilling Well Control Equipment, Section 16E.3.1 2nd ed.
BOP Diverter & Drilling Riser Sustem: System No. 12–30, pp.
Drilling in Brazil in 2887m Water Depth Using a Surface BOP System and a DP Vessel
,” Paper No. IADC/SPE 87113.
A Deepwater Well Construction Alternative: Surface BOP Drilling Concept Using Environmental Safe Guard
IADC/SPE 87108 Drilling Conference
, Dallas, TX, Mar. 2–4.
Subsea Accumulators—Are They a False Reliance?
Electro-Hydraulic Control Systems for Subsea Applications
,” Paper No. SPE 3762.
Design In Sight, Torben Lenau, 1994–2003; http://www.designinsite.dk/htmsider/m1310.htmhttp://www.designinsite.dk/htmsider/m1310.htm, accessed October 23, 2005.
Actuator Design Using Shape Memory Alloys
T. C. Waram
A Robust Co-Sputtering Fabrication Procedure for NiTi Shape Memory Alloys for MEMS
SINTEF Petroleum Research, Detailed Study of Shape Memory Alloys in Oil Well Applications, 32.0924.00∕01∕99. Oslo, Norway.
SINTEF Petroleum Research, Feasibility Study of Shape Memory Alloys in Oil Well Applications, 32.0896.00∕01∕97. Oslo, Norway.
, 2006, Design and Performance Evaluation of an Ultradeepwater Subsea Blowout Preventor Control System Using Shaping Memory Alloy Actuators, Paper No. SPE 101080.
Deepwater Bop Control Systems—A Look At Reliability Issues
” OTC 15194,
|
To G. H. K. Thwaites 20 June [1862]1
My dear Mr Thwaites
By an odd chance, two days before receiving your letter of May 15th I wrote to you on Primula.—2
I am particularly glad to hear of Sethia. Menyanthes is said to be dimorphic like Primula; so I am not surprised at Limnanthemum;3 it will be a curious point to compare Villarsia (I have been blundering, I fancied Villarsia was diœcious.) with Menyanthes, if I can make out any difference in fertility in the two of Menyanthes.4 Have you any Malpighiaceæ? if so, I very much wish you would mark the imperfect flowers & see if they set seed.— Also whether they are closed, & whether the pollen-tubes are emitted from the pollen-grains within the anthers & then penetrate the stigma.— This is the case in the imperfect flowers of Viola & Oxalis.—5
Many thanks for your Governor’s letter: you do not say whether I am to return it, so I will keep it till I hear.—6
In Haste, pray believe me | yours very sincerely | Ch. Darwin—
I suppose it would be too troublesome for you to mark
\frac{1}{2}
a dozen plants of the two forms Limnanthemum & count the capsules, & compare the produce of seed by weighing or counting.— I suspect the dimorphism of Primula is often, (though not at all necessarily) the high-road to diœciousness.7
The year is established by the relationship to the letter from G. H. K. Thwaites, 15 May 1862.
Letter from G. H. K. Thwaites, 15 May 1862, and letter to G. H. K. Thwaites, 15 June [1862].
In his letter of 15 May 1862, Thwaites reported that he had just read CD’s paper, ‘Dimorphic condition in Primula’, and that he had noticed the same phenomenon in the genera Sethia and Limnanthemum.
In Forms of flowers, p. 116, CD noted that the genera Menyanthes, Limnanthemum, and Villarsia constituted ‘a well-marked sub-tribe of the Gentianeæ’ and that all the species, as far as was then known, were ‘heterostyled’. CD had been anxious to see specimens of Menyanthes since he had learned earlier in the year that it was dimorphic (see letter to C. C. Babington, 20 January [1862], and letter from C. W. Crocker, 13 March 1862); he had recently acquired a short-styled specimen from the Royal Botanic Gardens, Kew (see letter to J. D. Hooker, 9 [April 1862] and n. 3). See also letter from G. H. K. Thwaites, 15 May 1862, CD annotations.
For CD’s interest in Viola and Oxalis, see also the letters to Daniel Oliver, 12 [April 1862] and 15 April [1862], the letter to J. D. Hooker, 30 May [1862], and the letter to Alphonse de Candolle, 17 June [1862].
See the enclosure to the letter from G. H. K. Thwaites, 15 May 1862. Charles Justin MacCarthy was the governor of Ceylon.
See ‘Dimorphic condition in Primula’, p. 95 (Collected papers 2: 61–2).
Asks for information concerning heterostyled and dioecious plants.
Thwaites, G. H. K.
|
Lines of minima in outer space
15 March 2014 Lines of minima in outer space
We define lines of minima in the thick part of outer space for the free group
{F}_{n}
n\ge 3
generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of
{F}_{n}
defines such a line of minima. Now let
\Gamma
be a subgroup of the outer automorphism group of
{F}_{n}
which is not virtually abelian. We obtain that if
\Gamma
contains at least one fully irreducible element, then for every
p\in \left(1,\infty \right)
the second bounded cohomology group
{H}_{b}^{2}\left(\Gamma ,{\ell }^{p}\left(\Gamma \right)\right)
is infinite-dimensional.
Ursula Hamenstädt. "Lines of minima in outer space." Duke Math. J. 163 (4) 733 - 776, 15 March 2014. https://doi.org/10.1215/00127094-2429807
Ursula Hamenstädt "Lines of minima in outer space," Duke Mathematical Journal, Duke Math. J. 163(4), 733-776, (15 March 2014)
|
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
NB: The ratio of impervious contributing drainage area (Ai) to permeable pavement area (Ap) R = Ai/Ap should not exceed 2 and Ai should not contain pervious areas.
|
Möbius Function | Brilliant Math & Science Wiki
Patrick Corn, Guillermo Templado, Satyabrata Dash, and
The Möbius function
μ(n)
is a multiplicative function which is important in the study of Dirichlet convolution. It is an important multiplicative function in number theory and combinatorics. While the values of the function itself are not difficult to calculate, the function is the Dirichlet inverse of the unit function
{\bf 1}(n)=1
. This fact, called Möbius inversion, gives rise to formulas involving
\mu
for many sums and identities involving arithmetic functions.
Applications of Möbius Inversion
Interesting Formulas involving the Möbius Function
n,
μ(n)
as the sum of the primitive
n^\text{th}
It has values in
\{-1, 0, 1\}
depending on the factorization of
n
into prime factors:
\mu(n) = 1
n
is a square-free positive integer with an even number of prime factors.
\mu(n) = -1
n
is a square-free positive integer with an odd number of prime factors.
\mu(n) = 0
n
has a squared prime factor.
\mu(n) = \begin{cases} 1 & \text{ if } n=1, \\ 0 & \text{ if } a^2 \mid n \text{ for some } a > 1 \text{ (i.e., } n \text{ has a squared prime factor)}, \\ (-1)^k & \text { if } n \text{ is the product of } k \text{ distinct primes.} \end{cases}
\mu(p) = -1
p
From the above definition, it is straightforward to check that
\mu(n)
is a multiplicative function.
Many arithmetic functions can be expressed as sums of other functions over the positive divisors of their argument:
f(n) = \sum_{d|n} g(d)
. In this situation, it is possible to solve for
g(n)
in terms of values of
; the general solution turns out to be furnished by the Möbius function.
The following lemma explains why the Möbius function is so fundamental:
\sum_{d|n}\mu(d) = \begin{cases} 1 & \text{ if } n=1, \\ 0 & \text{ if } n>1. \end{cases}
n > 1,
n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_r^{\alpha_r}
\alpha_i \ge 1 \text{ for all }i =1,\ldots,r).
d|n
d = p_1^{\beta_1} p_2^{\beta_2} \cdots p_r^{\beta_r}
0 \leq \beta_i \leq \alpha_i
i = 1,\ldots, r.
\beta_i \ge 2
i = 1, \ldots , r,
\mu(d) = 0
\begin{aligned}\large \sum_{d|n} \mu(d) &= \sum_{\overset{(\beta_1, \ldots , \beta_r)}{\beta_i = 0 \text{ or } 1}} \mu\left(p_1^{\beta_1} \cdots p_r^{\beta_r}\right) \\ &=1 - \binom{r}{1} + \binom{r}{2} - \cdots + (-1)^{r} \binom {r}{r} \\ &= (1 - 1)^{r} \\ &= 0. \end{aligned}
The next-to-last equality follows from the binomial theorem.
\big(
\binom{r}{k}
in the above equation count the factors which are products of
k
distinct primes, which have a
\mu
(-1)^k. \big)
n = 1,
\sum_{d|n} \mu(d) = \mu(1) = 1. \ _\square
e(n)
as the function on the right side of the equality in the lemma, and defining
\mathbf{1}(n) = 1
, the lemma can be written more compactly in the language of Dirichlet convolution:
\mu * \mathbf{1} = e.
So if
g
are arithmetic functions such that
f(n) = \sum_{d|n} g(d)
f = g * \mathbf{1}
f * \mu = (g * \mathbf{1}) * \mu = g * (\mathbf{1} * \mu) = g * e = g.
This is referred to as Möbius inversion.
f
g
be arithmetic functions such that
f(n) = \sum_{d|n} g(d)
n
g(n) = \sum_{d|n} \mu(d) f\left(\frac{n}{d}\right) = \sum_{d|n} \mu\left(\frac{n}{d}\right) f(d).
Here is an explicit proof that does not use the language of Dirichlet convolution; the work done in this proof is essentially a special case of the proof that Dirichlet convolution is associative. Consider
\begin{aligned} \sum_{d|n} f(n/d) \mu(d) &= \sum_{d|n} \sum_{r|n/d} g(r) \mu(d) \\ &= \sum_{\overset{r,d}{rd|n}} g(r) \mu(d) \\ &= \sum_{r|n} \left( \sum_{d|n/r} \mu(d) \right) g(r), \end{aligned}
but the sum in parentheses is
0
r \ne n
1
r = n
, by the lemma, so this equals
g(n)
_\square
\sum_{d|n}\mu\left(\frac{n}{d}\right)f(d)=n
f(d)
n
f(2015)
\mu
There is also a multiplicative version of Möbius inversion, proved exactly the same way:
f
g
f(n) = \prod_{d|n} g(d)
g(n) = \prod_{d|n} f(n/d)^{\mu(d)} = \prod_{d|n} f(d)^{\mu(n/d)}.
\phi(n)
be Euler's totient function. It is a standard fact that
\sum_{d|n} \phi(d) = n
. Möbius inversion immediately gives
\phi(n) = \sum_{d|n} \mu(d) \frac nd \implies \frac{\phi(n)}{n} =\sum_{d|n} \frac{\mu(d)}{d}.
f(n)
be the sum of the primitive
n^\text{th}
f(1) = 1
n \ge 2
\zeta_n = \text{exp}\left(\frac{2\pi i}n\right).
Since the powers of
\zeta_n
are all the primitive
d^\text{th}
d
runs over the positive divisors of
n
\begin{aligned} \sum_{d|n} f(d) &= 1+\zeta_n+\zeta_n^2+\cdots+\zeta_n^{n-1} \\ &= \frac{\zeta_n^n-1}{\zeta_n-1} = 0, \end{aligned}
\sum_{d|n} f(d) = e(n)
f
\mu
have the same summation, so Möbius inversion implies that they are equal to each other:
f(n) = \sum_{d|n} \mu\left(\frac nd\right) e(d) = \mu(n).
So the Möbius function is the sum of the primitive
n^\text{th}
\Phi_n(x)
n^\text{th}
cyclotomic polynomial, the polynomial whose roots are equal to the primitive
n^\text{th}
x^n-1 = \prod_{d|n} \Phi_d(x),
and now multiplicative Möbius inversion gives
\Phi_n(x) = \prod_{d|n} (x^d-1)^{\mu(n/d)}.
\Phi_{48}(x) = \frac{(x^{48}-1)(x^8-1)}{(x^{24}-1)(x^{16}-1)} = \frac{x^{24}+1}{x^8+1} = x^{16}-x^8+1.
A common strategy to prove facts about multiplicative functions is to first restrict attention to their values on prime powers. That is, if two multiplicative functions agree on prime powers, they must agree everywhere. The proofs of the first two identities below use this idea.
It is a fact that
\sum_{d|n} \frac{\mu^2(d)}{\phi(d)} = \frac{n}{\phi(n)}.
To see this, note that both sides are multiplicative, so we can restrict our attention to
n = p^k
p
1 + \frac1{p-1}
\frac{p}{p-1}
\omega(n)
denote the number of distinct prime factors of
n
\sum_{d|n} \mu(d)^2 = 2^{\omega(n)}.
2^{\omega(n)}
is multiplicative
\big(
\omega(ab) = \omega(a)+\omega(b)
if gcd
(a,b)=1\big),
so both sides are multiplicative. For
n = p^k
, the left side is
1+1
2
s
be a complex number with Re
(s) > 1
\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} = \frac1{\zeta(s)},
\zeta
is the Riemann zeta function. This is an example of a Dirichlet series.
The average value of the Möbius function is
0 .
a(x) = \frac1{x} \sum_{1 \le n \le x} \mu(n),
\lim_{x\to\infty} a(x) = 0.
This statement turns out to be equivalent to the famous prime number theorem, which gives an asymptotic estimate of the number of primes less than
x
. The point here is that answers to simple questions about the Möbius function are related to quite deep facts about prime numbers.
Cite as: Möbius Function. Brilliant.org. Retrieved from https://brilliant.org/wiki/mobius-function/
|
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
R = Ai/Ap; the ratio of impervious contributing drainage area (Ai) to permeable pavement area (Ap). Note that the contributing drainage area should not contain pervious areas. R should not exceed 2.
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
Gauss's law | Brilliant Math & Science Wiki
Agnishom Chattopadhyay, Abhijeet Vats, Satyabrata Dash, and
Gauss's law states that any charge
q
can be thought to give rise to a definite quantity of flux through any enclosing surface. Physically, we might think of any source of light, such as a lightbulb, or the Sun, which has a definite rating of power which it emits in all directions. No matter what shape of enclosing surface we trap it in, and no matter how near or far the light source is from the surface, the enclosing surface will receive the same amount of energy per unit time
P_0
In a similar way, a distribution of total charge
\sum_i q_i
will give rise to an invariant amount of flux through any enclosing surface, which is defined to be the sum over all infinitesimal patches of surface
dA
, of the electric field component perpendicular to the surface. Although not always a practical tool, in situations where geometrical symmetries can be exploited, Gauss's law is an incredibly powerful tool to quickly calculate electric fields. Analogous laws hold for other inverse square laws, e.g. Newtonian gravity.
Gauss's Law for Electric Field
Gauss Law for other Important Fields
Equivalence with Coulomb's Law
Gauss's law of electric flux states that the net electric flux through any closed surface is directly proportional to the charge enclosed by the surface:
\Phi_E = \frac{q_\text{tot}}{\varepsilon_0} .
It can be shown that this statement is equivalent to Coulomb's inverse square law, i.e. any force for which the inverse square law holds will also satisfy Gauss's law, and any force which satisfies Gauss's law will behave like an inverse square. Gauss's law is one of the four Maxwell equations for electrodynamics and describes an important property of electric fields. If one day magnetic monopoles are shown to exist, then Maxwell's equations would require slight modification, for one to show that magnetic fields can have divergence, i.e.
\nabla \cdot B \sim \rho_m
. Cosmological theories do, however, predict that magnetic monopoles did exist at the beginning of the universe but collapsed due to their high instabilities.
A closed surface is a surface that is compact and without boundary. In other words, a closed surface is one that divides space (excluding itself) into two disjoint parts, an exterior and an interior. Some simple examples of closed surfaces include intact bubbles, Dyson spheres, or the enclosure one would be inside of if they were to get into a sleeping bag and sew the opening shut.
A sheet of paper An empty soda bottle A bowl An inflated swimming tube
Gauss's law is a very powerful method to determine the electric field due to a distribution of charges. The mathematical expression for Gauss's law is
\int_{S} \vec{E} \cdot \vec{dA}=\frac{Q_{enc}}{\epsilon_0} ,
S
is a surface,
\vec{E}
is the electric field vector,
\vec{dA}
is the infinitesimal area element,
Q_{enc}
is the charge enclosed by
S,
\epsilon_0
In order to apply Gauss's law, we need to understand what each of the parts of this expression means. This set of problems will help you understand each of the components. Let's start with
S
. You may be more familiar with integrals as the limit of a sum of a function over a line interval, which gives the "area under the curve." An integral over a surface of a function is just the sum of that function over all the points on the surface.
The surface in Gauss's law is a closed two-dimensional surface, such as the surface of a sphere or the surface of a cube. A closed surface is a surface that divides space into an inside and an outside, whereby dividing we mean there is no path that goes from inside to outside that does not penetrate the surface. Consider the surface
S
of the objects below. For which of the objects is
S
a closed surface?
Loosely spoken, the flux of a field through a surface is the net flow through it. We develop this intuition in the example below.
Assume you fit a cotton membrane in the middle of a pipe through which water is flowing. What is the flow of water through the membrane?
Of course, the answer would be the average normal component of the velocity times the area of the membrane.
This is what we call the flux through the membrane!
Why do we take the normal component? Because the alignment of the membrane with the direction of the flow matters. What if both the membrane and the flow are aligned horizontally?
\Phi = \oint_{\mathcal{S}} \overrightarrow{E} \cdot d \overrightarrow{A}.
Divergence of a vector field at a point is the magnitude of the field's source or sink at that point. Of course, this is the same as stating that the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
Formally stated, the above translates to
The divergence of a vector field
\overrightarrow{E}
p
is defined as the limit of the net flow of
\overrightarrow{E}
across the smooth boundary of a three-dimensional region
V
divided by the volume of
V
V
shrinks to
p
\operatorname{div}\,\overrightarrow{E}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} {\overrightarrow{E}\cdot\widehat{n} \over |V| } \; dS,
|V|
is the volume of
V
S(V)
V
, and the integral is a surface integral with
\widehat{n}
being the outward unit normal to that surface.
Application in Cartesian coordinates:
If there is a vector field such that
\overrightarrow{E}(x \widehat{i} + y \widehat{j} + z \widehat{k}) = u \widehat{i} + v \widehat{j} + w \widehat{k},
\operatorname{div}\,\overrightarrow{E} = \nabla \cdot \overrightarrow{E} = \frac{\partial{u}}{\partial{x}}+\frac{\partial{v}}{\partial{y}}+\frac{\partial{w}}{\partial{z}} .
S
is a closed surface in which a charge
Q
is enclosed, then the flux
\Phi_E
S
\Phi_E = \oint_{\mathcal{S}} \overrightarrow{E} \cdot d \overrightarrow{A} =\frac{Q}{\varepsilon_0}.
\rho \left(\overrightarrow{r}\right)
is the volume density of the charge at
\overrightarrow{r}
, then the divergence of the electric field
\overrightarrow{E}
\overrightarrow{r}
\nabla \cdot \overrightarrow{E}\left(\overrightarrow{r}\right) = \frac{\rho \left(\overrightarrow{r}\right) }{\varepsilon_0} .
The above discussion on flux and divergence should make it clear why these two forms are equivalent. Nevertheless, this equivalence comes from Gauss's theorem or the divergence theorem.
A similar statement such as the electric Gauss law could be made for several other fields. Here is a table of such expressions where symbols have their usual meanings.
Field Integral Form Differential Form
\hspace{20mm}
\oint \overrightarrow{g} \cdot d \overrightarrow{A} = -4 \pi G M
\hspace{20mm}
\nabla \cdot \overrightarrow{g} =-4 \pi G \rho
\oint \overrightarrow{B} \cdot d \overrightarrow{A} = 0
\nabla \cdot \overrightarrow{B} = 0
Deriving Coulomb's from Gauss:
Consider a charge
Q
and a sphere of radius
r
By Gauss's law the flux of the field from the sphere is
\frac{Q}{\varepsilon},
4 \pi r^2 |E|,
4 \pi r^2 |E| = \frac{Q}{\varepsilon} \Rightarrow |E| = \frac{Q}{4 \pi r^2 \varepsilon}.
So, placing a test charge
q
in the field which does not disturb it results in a force of
|F| = q |E| = q \frac{Q}{4 \pi r^2 \varepsilon},
which is the Coulomb's law.
Please read John Muradeli's note for a better explanation of the above proof.
Deriving Gauss from Coulomb's:
We start with Coulomb's law for a single point charge
E = \frac{\gamma}{r^2}
\gamma = \frac{q}{4\pi \varepsilon_0}
Now, we integrate the electric field over a closed spherical surface encasing the charge:
\begin{aligned} \oint E \cdot \hat{n}\, dA &= \oint \frac{\gamma}{r^2} \, dA \\ &= \frac{\gamma}{r^2}\oint\, dA \\ &= \frac{\gamma}{r^2} 4\pi r^2 \\ &= 4\pi\gamma \\ &= \frac{q}{\varepsilon_0}, \end{aligned}
which is Gauss's law, as desired.
_\square
Note: The proof offered only works for spherical surfaces; a more general proof ends up using vector calculus and Green's/Delta functions. Feel free to take a shot at the general case.
Gauss's law is a powerful statement about inverse square fields. Not just in problem-solving, it has also found its place in the four Maxwell equations as well as in gravity.
\frac{\lambda}{2\epsilon_0}
\frac{\lambda}{2\pi \epsilon_0 x}
\frac{x}{\epsilon_0 \lambda}
\frac{\lambda}{4\pi \epsilon_0 x^2}
Suppose that an infinitely long straight current-carrying wire has a uniform linear charge density
\lambda.
Let this wire be on the
y
-axis of the
xy
-plane, and let
x > 0
be the distance between the
y
-axis and a point
P
xy
-plane. What is the strength of the electric field at the point
P?
\epsilon_0
in the choices below denotes the electric constant.
The gravity train is a hypothetical idea proposed by Robert Hooke (he of Hooke's law fame) to Isaac Newton in the 1600's. It consists of a simple idea that's hard to implement in practice. Dig a tunnel that runs straight through Earth between two points on the surface. If you can figure out how to remove friction and lower air resistance, you now have a mechanism for extremely efficient and rapid travel between widely separated points. Simply drop something into the tunnel at one end. Gravity will initially pull it downwards through the tunnel, eventually reaching high speeds. Once the object is halfway through the tunnel, gravity will now slow it back down, so you can retrieve the object easily on the other side.
While there are obvious engineering impracticalities, it winds up being pretty amazing how fast a gravity train can get things from point to point without needing fuel. So, let's say we wanted to go from Beijing to Paris using such a gravity train. How long does the trip take assuming a frictionless and drag-free train to the nearest minute?
Earth can be modeled as a sphere of uniform density, with total mass
6 \times 10^{24}~\mbox{kg}
6370~\mbox{km}
. The shortest distance from Beijing to Paris on the surface of Earth is approximately
8200~\mbox{km}
6.67 \times 10^{-11}~\mbox{N m}^2/\mbox{kg}^2
6 \times 10^{24}~\mbox{kg}
See David's Set for a tour of Gauss's theorem and its applications.
Cite as: Gauss's law. Brilliant.org. Retrieved from https://brilliant.org/wiki/gauss-law/
|
Complementarity (physics) - Wikipedia
(Redirected from Complementarity principle)
{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }
In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory.[1][2] The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously. An example of such a pair is position and momentum. Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an electron, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.[3]
Niels Bohr apparently conceived of the principle of complementarity during a skiing vacation in Norway in February and March 1927, during which he received a letter from Werner Heisenberg regarding an as-yet-unpublished result, a thought experiment about a microscope using gamma rays. This thought experiment implied a tradeoff between uncertainties that would later be formalized as the uncertainty principle. To Bohr, Heisenberg's paper did not make clear the distinction between a position measurement merely disturbing the momentum value that a particle carried and the more radical idea that momentum was meaningless or undefinable in a context where position was measured instead. Upon returning from his vacation, by which time Heisenberg had already submitted his paper for publication, Bohr convinced Heisenberg that the uncertainty tradeoff was a manifestation of the deeper concept of complementarity.[4] Heisenberg duly appended a note to this effect to his paper, before its publication, stating:
Bohr publicly introduced the principle of complementarity in a lecture he delivered on 16 September 1927 at the International Physics Congress held in Como, Italy, attended by most of the leading physicists of the era, with the notable exceptions of Einstein, Schrödinger, and Dirac. However, these three were in attendance one month later when Bohr again presented the principle at the Fifth Solvay Congress in Brussels, Belgium. The lecture was published in the proceedings of both of these conferences, and was republished the following year in Naturwissenschaften (in German) and in Nature (in English).[5]
In his original lecture on the topic, Bohr pointed out that just as the finitude of the speed of light implies the impossibility of a sharp separation between space and time (relativity), the finitude of the quantum of action implies the impossibility of a sharp separation between the behavior of a system and its interaction with the measuring instruments and leads to the well-known difficulties with the concept of 'state' in quantum theory; the notion of complementarity is intended to capture this new situation in epistemology created by quantum theory. Physicists F.A.M. Frescura and Basil Hiley have summarized the reasons for the introduction of the principle of complementarity in physics as follows:[6]
In the traditional view, it is assumed that there exists a reality in space-time and that this reality is a given thing, all of whose aspects can be viewed or articulated at any given moment. Bohr was the first to point out that quantum mechanics called this traditional outlook into question. To him the "indivisibility of the quantum of action" [...] implied that not all aspects of a system can be viewed simultaneously. By using one particular piece of apparatus only certain features could be made manifest at the expense of others, while with a different piece of apparatus another complementary aspect could be made manifest in such a way that the original set became non-manifest, that is, the original attributes were no longer well defined. For Bohr, this was an indication that the principle of complementarity, a principle that he had previously known to appear extensively in other intellectual disciplines but which did not appear in classical physics, should be adopted as a universal principle.
Complementarity was a central feature of Bohr's reply to the EPR paradox, an attempt by Albert Einstein, Boris Podolsky and Nathan Rosen to argue that quantum particles must have position and momentum even without being measured and so quantum mechanics must be an incomplete theory.[7] The thought experiment proposed by Einstein, Podolsky and Rosen involved producing two particles and sending them far apart. The experimenter could choose to measure either the position or the momentum of one particle. Given that result, they could in principle make a precise prediction of what the corresponding measurement on the other, faraway particle would find. To Einstein, Podolsky and Rosen, this implied that the faraway particle must have precise values of both quantities whether or not that particle is measured in any way. Bohr argued in response that the deduction of a position value could not be transferred over to the situation where a momentum value is measured, and vice versa.[8]
Later expositions of complementarity by Bohr include a 1938 lecture in Warsaw[9][10] and a 1949 article written for a festschrift honoring Albert Einstein.[11][12] It was also covered in a 1953 essay by Bohr's collaborator Léon Rosenfeld.[13]
Complementarity is mathematically expressed by the operators that represent the observable quantities being measured failing to commute:
{\displaystyle \left[{\hat {A}},{\hat {B}}\right]:={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}\neq {\hat {0}}.}
Observables corresponding to non-commuting operators are called incompatible observables. Incompatible observables cannot have a complete set of common eigenstates. Note that there can be some simultaneous eigenstates of
{\displaystyle {\hat {A}}}
{\displaystyle {\hat {B}}}
, but not enough in number to constitute a complete basis.[14][15] The canonical commutation relation
{\displaystyle \left[{\hat {x}},{\hat {p}}\right]=i\hbar }
implies that this applies to position and momentum. Likewise, an analogous relationship holds for any two of the spin observables defined by the Pauli matrices; measurements of spin along perpendicular axes are complementary.[7] This has been generalized to discrete observables with more than two possible outcomes using mutually unbiased bases, which provide complementary observables defined on finite-dimensional Hilbert spaces.[16][17]
^ Wheeler, John A. (January 1963). ""No Fugitive and Cloistered Virtue"—A tribute to Niels Bohr". Physics Today. Vol. 16, no. 1. p. 30. Bibcode:1963PhT....16a..30W. doi:10.1063/1.3050711.
^ Howard, Don (2004). "Who invented the Copenhagen Interpretation? A study in mythology" (PDF). Philosophy of Science. 71 (5): 669–682. CiteSeerX 10.1.1.164.9141. doi:10.1086/425941. JSTOR 10.1086/425941. S2CID 9454552.
^ Bohr, Niels; Rosenfeld, Léon (1996). "Complementarity: Bedrock of the Quantal Description". Foundations of Quantum Physics II (1933–1958). Niels Bohr Collected Works. Vol. 7. Elsevier. pp. 284–285. ISBN 978-0-444-89892-0.
^ Baggott, Jim (2011). The Quantum Story: A History in 40 moments. Oxford Landmark Science. Oxford: Oxford University Press. p. 97. ISBN 978-0-19-956684-6.
^ Bohr, N. (1928). "The Quantum Postulate and the Recent Development of Atomic Theory". Nature. 121 (3050): 580–590. Bibcode:1928Natur.121..580B. doi:10.1038/121580a0. Available in the collection of Bohr's early writings, Atomic Theory and the Description of Nature (1934).
^ Frescura, F. A. M.; Hiley, B. J. (July 1984). "Algebras, quantum theory and pre-space" (PDF). Revista Brasileira de Física. Special volume "Os 70 anos de Mario Schonberg": 49–86, 2.
^ a b Fuchs, Christopher A. (2017). "Notwithstanding Bohr: The Reasons for QBism". Mind and Matter. 15: 245–300. arXiv:1705.03483. Bibcode:2017arXiv170503483F.
^ Jammer, Max (1974). The Philosophy of Quantum Mechanics. John Wiley and Sons. ISBN 0-471-43958-4.
^ Bohr, Niels (1939). "The causality problem in atomic physics". New theories in physics. Paris: International Institute of Intellectual Co-operation. pp. 11–38.
^ Chevalley, Catherine (1999). "Why Do We Find Bohr Obscure?". In Greenberger, Daniel; Reiter, Wolfgang L.; Zeilinger, Anton (eds.). Epistemological and Experimental Perspectives on Quantum Physics. Springer Science+Business Media. pp. 59–74. doi:10.1007/978-94-017-1454-9. ISBN 978-9-04815-354-1.
^ Bohr, Niels (1949). "Discussions with Einstein on Epistemological Problems in Atomic Physics". In Schilpp, Paul Arthur (ed.). Albert Einstein: Philosopher-Scientist. Open Court.
^ Saunders, Simon (2005). "Complementarity and Scientific Rationality". Foundations of Physics. 35 (3): 417–447. arXiv:quant-ph/0412195. Bibcode:2005FoPh...35..417S. doi:10.1007/s10701-004-1982-x. S2CID 17301341.
^ Griffiths, David J. (2017). Introduction to Quantum Mechanics. Cambridge University Press. p. 111. ISBN 978-1-107-17986-8.
^ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019-12-04). Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications. Wiley. p. 232. ISBN 978-3-527-34553-3.
^ Bengtsson, Ingemar; Ericsson, Åsa (June 2005). "Mutually Unbiased Bases and the Complementarity Polytope". Open Systems & Information Dynamics. 12 (2): 107–120. arXiv:quant-ph/0410120. Bibcode:2004quant.ph.10120B. doi:10.1007/s11080-005-5721-3. ISSN 1230-1612. S2CID 37108528.
^ Blanchfield, Kate (2014-04-04). "Orbits of mutually unbiased bases". Journal of Physics A: Mathematical and Theoretical. 47 (13): 135303. arXiv:1310.4684. Bibcode:2014JPhA...47m5303B. doi:10.1088/1751-8113/47/13/135303. ISSN 1751-8113. S2CID 118340150.
Berthold-Georg Englert, Marlan O. Scully & Herbert Walther, Quantum Optical Tests of Complementarity, Nature, Vol 351, pp 111–116 (9 May 1991) and (same authors) The Duality in Matter and Light Scientific American, pg 56–61, (December 1994).
Rhodes, Richard (1986). The Making of the Atomic Bomb. Simon & Schuster. ISBN 0-671-44133-7. OCLC 231117096.
Wikiquote has quotations related to Complementarity (physics).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Complementarity_(physics)&oldid=1072181593"
|
Gaussian kernel regression model using random feature expansion - MATLAB - MathWorks í•œêµ
Kernel Regression Properties
Gaussian kernel regression model using random feature expansion
RegressionKernel is a trained model object for Gaussian kernel regression using random feature expansion. RegressionKernel is more practical for big data applications that have large training sets but can also be applied to smaller data sets that fit in memory.
Unlike other regression models, and for economical memory usage, RegressionKernel model objects do not store the training data. However, they do store information such as the dimension of the expanded space, the kernel scale parameter, and the regularization strength.
You can use trained RegressionKernel models to continue training using the training data, predict responses for new data, and compute the mean squared error or epsilon-insensitive loss. For details, see resume, predict, and loss.
Create a RegressionKernel object using the fitrkernel function. This function maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space by minimizing the regularized objective function. Obtaining the linear model in the high-dimensional space is equivalent to applying the Gaussian kernel to the model in the low-dimensional space. Available linear regression models include regularized support vector machines (SVM) and least-squares regression models.
Half the width of the epsilon-insensitive band, specified as a nonnegative scalar.
f\left(x\right)=T\left(x\right)\mathrm{β}+b.
T\left(·\right)
{\mathrm{â}}^{p}
{\mathrm{â}}^{m}
β is a vector of coefficients.
\mathrm{â}\left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|yâf\left(x\right)|â\mathrm{ε}\right]
\mathrm{â}\left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yâf\left(x\right)\right]}^{2}
\mathrm{â}\left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|yâf\left(x\right)|â\mathrm{ε}\right]
\mathrm{â}\left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yâf\left(x\right)\right]}^{2}
\mathrm{λ}\underset{j=1}{\overset{p}{â}}|{\mathrm{β}}_{j}|
\frac{\mathrm{λ}}{2}\underset{j=1}{\overset{p}{â}}{\mathrm{β}}_{j}^{2}
Parameters used for training the RegressionKernel model, specified as a structure.
ResponseTransform — Response transformation function to apply to predicted responses
Response transformation function to apply to predicted responses, specified as 'none' or a function handle.
For kernel regression models and before the response transformation, the predicted response for the observation x (row vector) is
f\left(x\right)=T\left(x\right)\mathrm{β}+b.
T\left(·\right)
β corresponds to Mdl.Beta.
b corresponds to Mdl.Bias.
fitrkernel | fitrlinear | RegressionLinear
|
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
It is important to note that R = Ai/Ap; the ratio of impervious contributing drainage area (Ai) to permeable pavement area (Ap) should not exceed 2 and that the contributing drainage area should not contain pervious areas that are sources of sediment that can lead to premature clogging.
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
Complementary Error Function for Floating-Point and Symbolic Numbers
Complementary Error Function for Vectors and Matrices
Special Values of Complementary Error Function
Handling Expressions That Contain Complementary Error Function
Plot Complementary Error Function
Iterated Integral of Complementary Error Function
erfc(K,X)
erfc(X) represents the complementary error function of X, that is,erfc(X) = 1 - erf(X).
erfc(K,X) represents the iterated integral of the complementary error function of X, that is, erfc(K, X) = int(erfc(K - 1, y), y, X, inf).
Depending on its arguments, erfc can return floating-point or exact symbolic results.
Compute the complementary error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:
A = [erfc(1/2), erfc(1.41), erfc(sqrt(2))]
Compute the complementary error function for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, erfc returns unresolved symbolic calls:
symA = [erfc(sym(1/2)), erfc(sym(1.41)), erfc(sqrt(sym(2)))]
[ erfc(1/2), erfc(141/100), erfc(2^(1/2))]
[ 0.4795001222, 0.04614756064, 0.0455002639]
For most symbolic variables and expressions, erfc returns unresolved symbolic calls.
Compute the complementary error function for x and sin(x) + x*exp(x):
erfc(f)
erfc(sin(x) + x*exp(x))
If the input argument is a vector or a matrix, erfc returns the complementary error function for each element of that vector or matrix.
Compute the complementary error function for elements of matrix M and vector V:
[ 1, 0]
[ erfc(1/3), 2]
1 + Inf*1i
Compute the iterated integral of the complementary error function for the elements of V and M, and the integer -1:
erfc(-1, M)
erfc(-1, V)
[ 2/pi^(1/2), 0]
[ (2*exp(-1/9))/pi^(1/2), 0]
(2*exp(-1))/pi^(1/2)
erfc returns special values for particular parameters.
Compute the complementary error function for x = 0, x = ∞, and x = –∞. The complementary error function has special values for these parameters:
[erfc(0), erfc(Inf), erfc(-Inf)]
Compute the complementary error function for complex infinities. Use sym to convert complex infinities to symbolic objects:
[erfc(sym(i*Inf)), erfc(sym(-i*Inf))]
[ 1 - Inf*1i, 1 + Inf*1i]
Many functions, such as diff and int, can handle expressions containing erfc.
Compute the first and second derivatives of the complementary error function:
diff(erfc(x), x)
diff(erfc(x), x, 2)
-(2*exp(-x^2))/pi^(1/2)
(4*x*exp(-x^2))/pi^(1/2)
int(erfc(-1, x), x)
int(erfc(x), x)
x*erfc(x) - exp(-x^2)/pi^(1/2)
int(erfc(2, x), x)
(x^3*erfc(x))/6 - exp(-x^2)/(6*pi^(1/2)) +...
(x*erfc(x))/4 - (x^2*exp(-x^2))/(6*pi^(1/2))
Plot the complementary error function on the interval from -5 to 5.
fplot(erfc(x),[-5 5])
K — Input representing an integer larger than -2
number | symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix
Input representing an integer larger than -2, specified as a number, symbolic number, variable, expression, or function. This arguments can also be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.
The following integral defines the complementary error function:
erfc\left(x\right)=\frac{2}{\sqrt{\pi }}\underset{x}{\overset{\infty }{\int }}{e}^{-{t}^{2}}dt=1-erf\left(x\right)
Here erf(x) is the error function.
The following integral is the iterated integral of the complementary error function:
erfc\left(k,x\right)=\underset{x}{\overset{\infty }{\int }}erfc\left(k-1,y\right)dy
erfc\left(0,x\right)=erfc\left(x\right)
Calling erfc for a number that is not a symbolic object invokes the MATLAB® erfc function. This function accepts real arguments only. If you want to compute the complementary error function for a complex number, use sym to convert that number to a symbolic object, and then call erfc for that symbolic object.
For most symbolic (exact) numbers, erfc returns unresolved symbolic calls. You can approximate such results with floating-point numbers using vpa.
At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then erfc expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.
erf | erfcinv | erfi | erfinv
|
Effective_stress Knowpia
The effective stress can be defined as the stress, depending on the applied tension
{\displaystyle {\boldsymbol {\sigma }}_{ij}}
and pore pressure
{\displaystyle p}
, which controls the strain or strength behaviour of soil and rock (or a generic porous body) for whatever pore pressure value or, in other terms, the stress which applied over a dry porous body (i.e. at
{\displaystyle p=0}
) provides the same strain or strength behaviour which is observed at
{\displaystyle p}
≠ 0.[1] In the case of granular media it can be viewed as a force that keeps a collection of particles rigid. Usually this applies to sand, soil, or gravel, as well as every kind of rock and several other porous materials such as concrete, metal powders, biological tissues etc.[1] The usefulness of an appropriate ESP formulation consists in allowing to assess the behaviour of a porous body for whatever pore pressure value on the basis of experiments involving dry samples (i.e. carried out at zero pore pressure).
Karl von Terzaghi first proposed the relationship for effective stress in 1925.[2][3][4] For him, the term "effective" meant the calculated stress that was effective in moving soil, or causing displacements. It has been often interpreted as the average stress carried by the soil skeleton.[citation needed] Afterwards, different formulations have been proposed for the effective stress. Maurice Biot fully developed the three-dimensional soil consolidation theory, extending the one-dimensional model previously developed by Terzaghi to more general hypotheses and introducing the set of basic equations of Poroelasticity. Alec Skempton in his work in 1960, has carried out an extensive review of available formulations and experimental data in literature about effective stress valid in soil, concrete and rock, in order to reject some of these expressions, as well as clarify what expression was appropriate according to several work hypotheses, such as stress–strain or strength behaviour, saturated or nonsaturated media, rock/concrete or soil behaviour, etc.
Effective stress (σ') acting on a soil is calculated from two parameters, total stress (σ) and pore water pressure (u) according to:
{\displaystyle \sigma '=\sigma -u\,}
Typically, for simple examples
{\displaystyle {\begin{aligned}\sigma &=H_{\mathrm {soil} }\,\gamma _{\mathrm {soil} }\\u&=H_{\mathrm {w} }\,\gamma _{\mathrm {w} }\end{aligned}}}
Much like the concept of stress itself, the formula is a construct, for the easier visualization of forces acting on a soil mass, especially simple analysis models for slope stability, involving a slip plane.[5] With these models, it is important to know the total weight of the soil above (including water), and the pore water pressure within the slip plane, assuming it is acting as a confined layer.[citation needed]
However, the formula becomes confusing when considering the true behaviour of the soil particles under different measurable conditions, since none of the parameters are actually independent actors on the particles.[citation needed]
Arrangement of Spheres showing contacts
Consider a grouping of round quartz sand grains, piled loosely, in a classic "cannonball" arrangement. As can be seen, there is a contact stress where the spheres actually touch. Pile on more spheres and the contact stresses increase, to the point of causing frictional instability (dynamic friction), and perhaps failure. The independent parameter affecting the contacts (both normal and shear) is the force of the spheres above. This can be calculated by using the overall average density of the spheres and the height of spheres above.[citation needed]
Spheres immersed in water, reducing effective stress
If we then have these spheres in a beaker and add some water, they will begin to float a little depending on their density (buoyancy). With natural soil materials, the effect can be significant, as anyone who has lifted a large rock out of a lake can attest. The contact stress on the spheres decreases as the beaker is filled to the top of the spheres, but then nothing changes if more water is added. Although the water pressure between the spheres (pore water pressure) is increasing, the effective stress remains the same, because the concept of "total stress" includes the weight of all the water above. This is where the equation can become confusing, and the effective stress can be calculated using the buoyant density of the spheres (soil), and the height of the soil above.[citation needed]
Spheres being injected with water, reducing effective stress
The concept of effective stress truly becomes interesting when dealing with non-hydrostatic pore water pressure. Under the conditions of a pore pressure gradient, the ground water flows, according to the permeability equation (Darcy's law). Using our spheres as a model, this is the same as injecting (or withdrawing) water between the spheres. If water is being injected, the seepage force acts to separate the spheres and reduces the effective stress. Thus, the soil mass becomes weaker. If water is being withdrawn, the spheres are forced together and the effective stress increases.[6]
Two extremes of this effect are quicksand, where the groundwater gradient and seepage force act against gravity; and the "sandcastle effect",[7] where the water drainage and capillary action act to strengthen the sand. As well, effective stress plays an important role in slope stability, and other geotechnical engineering and engineering geology problems, such as groundwater-related subsidence.
Terzaghi, K. (1925). Principles of Soil Mechanics. Engineering News-Record, 95(19-27).
^ a b Guerriero, V; Mazzoli, S. (2021). "Theory of Effective Stress in Soil and Rock and Implications for Fracturing Processes: A Review". Geosciences. 11 (3): 119. Bibcode:2021Geosc..11..119G. doi:10.3390/geosciences11030119.
^ Terzaghi, Karl (1925). Erdbaumechanik auf Bodenphysikalischer Grundlage. F. Deuticke.
^ Terzaghi, Karl (1936). "Relation Between Soil Mechanics and Foundation Engineering: Presidential Address". Proceedings, First International Conference on Soil Mechanics and Foundation Engineering, Boston. 3, 13–18.
^ http://fbe.uwe.ac.uk/public/geocal/SoilMech/stresses/stresses.htm Archived June 18, 2006, at the Wayback Machine
^ http://www.dur.ac.uk/~des0www4/cal/slopes/page4.htm
^ http://fbe.uwe.ac.uk/public/geocal/SoilMech/water/water.htm Archived September 2, 2006, at the Wayback Machine
^ http://home.tu-clausthal.de/~pcdj/publ/PRL96_058301.pdf Archived May 30, 2008, at the Wayback Machine
|
Find the equation of circle touching 3x-4y+1=0 at (1,1) and having radius 10 unit - Maths - Conic Sections - 10888697 | Meritnation.com
Find the equation of circle touching
3x-4y+1=0 at (1,1) and having radius 10 unit.
Hi, \phantom{\rule{0ex}{0ex}}Let the center is \left(h,k\right) so its dis\mathrm{tan}ce from \left(1,1\right) will be 10 \phantom{\rule{0ex}{0ex}}{\left(h-1\right)}^{2}+{\left(k-1\right)}^{2}= {10}^{2}\phantom{\rule{0ex}{0ex}}{h}^{2}+{k}^{2}-2h-2k+2-100=0\phantom{\rule{0ex}{0ex}}Replacing, \left(h,k\right)\to \left(x,y\right)\phantom{\rule{0ex}{0ex}}{x}^{2}+{y}^{2}-2x-2y-98=0\phantom{\rule{0ex}{0ex}}Regards
No dhruv
|
Soil_consolidation Knowpia
Soil consolidation refers to the mechanical process by which soil changes volume gradually in response to a change in pressure. This happens because soil is a two-phase material, comprising soil grains and pore fluid, usually groundwater. When soil saturated with water is subjected to an increase in pressure, the high volumetric stiffness of water compared to the soil matrix means that the water initially absorbs all the change in pressure without changing volume, creating excess pore water pressure. As water diffuses away from regions of high pressure due to seepage, the soil matrix gradually takes up the pressure change and shrinks in volume. The theoretical framework of consolidation is therefore closely related to the diffusion equation, the concept of effective stress, and hydraulic conductivity.
In the narrow sense, "consolidation" refers strictly to this delayed volumetric response to pressure change due to gradual movement of water. Some publications also use "consolidation" in the broad sense, to refer to any process by which soil changes volume due to a change in applied pressure. This broader definition encompasses the overall concept of soil compaction, subsidence, and heave. Some types of soil, mainly those rich in organic matter, show significant creep, whereby the soil changes volume slowly at constant effective stress over a longer time-scale than consolidation due to the diffusion of water. To distinguish between the two mechanisms, "primary consolidation" refers to consolidation due to dissipation of excess water pressure, while "secondary consolidation" refers to the creep process.
The effects of consolidation are most conspicuous where a building sits over a layer of soil with low stiffness and low permeability, such as marine clay, leading to large settlement over many years. Types of construction project where consolidation often poses technical risk include land reclamation, the construction of embankments, and tunnel and basement excavation in clay.
Geotechnical engineers use oedometers to quantify the effects of consolidation. In an oedometer test, a series of known pressures are applied to a thin disc of soil sample, and the change of sample thickness with time is recorded. This allows the consolidation characteristics of the soil to be quantified in terms of the coefficient of consolidation (
{\displaystyle C_{v}}
) and hydraulic conductivity (
{\displaystyle K}
Clays undergo consolidation settlement not only the action action of external loads (surcharge loads) but also under its own weight or weight of soils that exist above the clay.
Clays also undergo settlement when dewatered (groundwater pumping) because the effective stress on the clay increases.
Coarse-grained soils do not undergo consolidation settlement due to relativity high hydraulic conductivity compared to clays. Instead, Coarse-grained soils undergo the immediate settlement.
According to the "father of soil mechanics", Karl von Terzaghi, consolidation is "any process which involves a decrease in water content of saturated soil without replacement of water by air". More generally, consolidation refers to the process by which soils change volume in response to a change in pressure, encompassing both compaction and swelling.[1]
Magnitude of volume changeEdit
The experimentally determined consolidation curve (blue dots) for a saturated clay showing a procedure for computing the preconsolidation stress.
Construction of compression and recompressiom curve.The curve, generally referred to as the virgin compression curve, approximately intersects the laboratory curve at a void ratio of 0.42
{\displaystyle e_{0}}
(Terzaghi and Peck, 1967). Note that
{\displaystyle e_{0}}
is the void ratio of the clay in the field. Knowing the values of
{\displaystyle e_{0}}
{\displaystyle \sigma _{c}^{'}}
you can easily construct the virgin curve and calculate its compression index by using Eq.
{\displaystyle C_{C}={\frac {e_{1}-e_{2}}{log({\frac {\sigma _{2}^{'}}{\sigma _{1}^{'}}})}}}
Consolidation is the process in which reduction in volume takes place by the gradual expulsion or absorption of water under long-term static loads.[2]
When stress is applied to a soil, it causes the soil particles to pack together more tightly. When this occurs in a soil that is saturated with water, water will be squeezed out of the soil. The magnitude of consolidation can be predicted by many different methods. In the classical method developed by Terzaghi, soils are tested with an oedometer test to determine their compressibility. In most theoretical formulations, a logarithmic relationship is assumed between the volume of the soil sample and the effective stress carried by the soil particles. The constant of proportionality (change in void ratio per order of magnitude change in effective stress) is known as the compression index, given the symbol
{\displaystyle \lambda }
when calculated in natural logarithm and
{\displaystyle C_{C}}
when calculated in base-10 logarithm.[2][3]
This can be expressed in the following equation, which is used to estimate the volume change of a soil layer:
{\displaystyle \delta _{c}={\frac {C_{c}}{1+e_{0}}}H\log \left({\frac {\sigma _{zf}'}{\sigma _{z0}'}}\right)\ }
δc is the settlement due to consolidation.
Cc is the compression index.
e0 is the initial void ratio.
H is the height of the compressible soil.
σzf is the final vertical stress.
σz0 is the initial vertical stress.
When stress is removed from a consolidated soil, the soil will rebound, regaining some of the volume it had lost in the consolidation process. If the stress is reapplied, the soil will consolidate again along a recompression curve, defined by the recompression index. The gradient of the swelling and recompression lines on a plot of void ratio against the logarithm of effective stress often idealised to take the same value, known as the "swelling index" (given the symbol
{\displaystyle \kappa }
{\displaystyle C_{S}}
when calculated in base-10 logarithm).
Cc can be replaced by Cr (the recompression index) for use in overconsolidated soils where the final effective stress is less than the preconsolidation stress. When the final effective stress is greater than the preconsolidation stress, the two equations must be used in combination to model both the recompression portion and the virgin compression portion of the consolidation processes, as follows,
{\displaystyle \delta _{c}={\frac {C_{r}}{1+e_{0}}}H\log \left({\frac {\sigma _{zc}'}{\sigma _{z0}'}}\right)+{\frac {C_{c}}{1+e_{0}}}H\log \left({\frac {\sigma _{zf}'}{\sigma _{zc}'}}\right)\ }
where σzc is the preconsolidation stress of the soil.
This method assumes consolidation occurs in only one-dimension. Laboratory data is used to construct a plot of strain or void ratio versus effective stress where the effective stress axis is on a logarithmic scale. The plot's slope is the compression index or recompression index. The equation for consolidation settlement of a normally consolidated soil can then be determined to be:
The soil which had its load removed is considered to be "overconsolidated". This is the case for soils that have previously had glaciers on them. The highest stress that it has been subjected to is termed the "preconsolidation stress". The "over-consolidation ratio" (OCR) is defined as the highest stress experienced divided by the current stress. A soil that is currently experiencing its highest stress is said to be "normally consolidated" and has an OCR of one. A soil could be considered "underconsolidated" or "unconsolidated" immediately after a new load is applied but before the excess pore water pressure has dissipated. Occasionally, soil strata form by natural deposition in rivers and seas may exist in an exceptionally low density that is impossible to achieve in an oedometer; this process is known as "intrinsic consolidation".[4]
Time dependencyEdit
Spring analogyEdit
The process of consolidation is often explained with an idealized system composed of a spring, a container with a hole in its cover, and water. In this system, the spring represents the compressibility or the structure of the soil itself, and the water which fills the container represents the pore water in the soil.
Schematic diagram of spring analogy
The container is completely filled with water, and the hole is closed. (Fully saturated soil)
A load is applied onto the cover, while the hole is still unopened. At this stage, only the water resists the applied load. (Development of excess pore water pressure)
As soon as the hole is opened, water starts to drain out through the hole and the spring shortens. (Drainage of excess pore water pressure)
After some time, the drainage of water no longer occurs. Now, the spring alone resists the applied load. (Full dissipation of excess pore water pressure. End of consolidation)
Analytical formulation of consolidation rateEdit
The time for consolidation to occur can be predicted. Sometimes consolidation can take years. This is especially true in saturated clays because their hydraulic conductivity is extremely low, and this causes the water to take an exceptionally long time to drain out of the soil. While drainage is occurring, the pore water pressure is greater than normal because it is carrying part of the applied stress (as opposed to the soil particles).
{\displaystyle T_{v}={\frac {c_{v}*t}{(H_{dr})^{2}}}\ }
Where Tv is the time factor.
Hdr is the average longest drain path during consolidation.
t is the time at measurement
Cv is defined as the coefficient of consolidation found using the log method with
{\displaystyle C_{v}={\frac {T_{50}H_{dr}^{2}}{t_{50}}}}
or the root method with
{\displaystyle C_{v}={\frac {T_{95}H_{dr}^{2}}{t_{95}}}}
t50 time to 50% deformation (consolidation) and t95 is 95%
Where T95=1.129 T50=0.197
CreepEdit
The theoretical formulation above assumes that time-dependent volume change of a soil unit only depends on changes in effective stress due to the gradual restoration of steady-state pore water pressure. This is the case for most types of sand and clay with low amounts of organic material. However, in soils with a high amount of organic material such as peat, the phenomenon of creep also occurs, whereby the soil changes volume gradually at constant effective stress. Soil creep is typically caused by viscous behavior of the clay-water system and compression of organic matter.
This process of creep is sometimes known as "secondary consolidation" or "secondary compression" because it also involves gradual change of soil volume in response to an application of load; the designation "secondary" distinguishes it from "primary consolidation", which refers to volume change due to dissipation of excess pore water pressure. Creep typically takes place over a longer time-scale than (primary) consolidation, such that even after the restoration of hydrostatic pressure some compression of soil takes place at slow rate.
Analytically, the rate of creep is assumed to decay exponentially with time since application of load, giving the formula:
{\displaystyle S_{s}={\frac {H_{0}}{1+e_{0}}}C_{a}\log \left({\frac {t}{t_{95}}}\right)\ }
Where H0 is the height of the consolidating medium
e0 is the initial void ratio
Ca is the secondary compression index
t is the length of time after consolidation considered
t95 is the length of time for achieving 95% consolidation
Wikimedia Commons has media related to Category:Consolidation.
^ Schofield, Andrew Noel; Wroth, Peter (1968). Critical State Soil Mechanics. McGraw-Hill. ISBN 9780641940484.
^ a b Lambe, T. William; Whitman, Robert V. (1969). Soil mechanics. Wiley. ISBN 9780471511922.
^ Chan, Deryck Y.K. (2016). Base slab heave in over-consolidated clay (MRes thesis). University of Cambridge.
^ Burland, J. B. (1990-09-01). "On the compressibility and shear strength of natural clays". Géotechnique. 40 (3): 329–378. doi:10.1680/geot.1990.40.3.329. ISSN 0016-8505.
Coduto, Donald (2001), Foundation Design, Prentice-Hall, ISBN 0-13-589706-8
Kim, Myung-mo (2000), Soil Mechanics (in Korean) (4 ed.), Seoul: Munundang, ISBN 89-7393-053-2
Terzaghi, Karl (1943), Theoretical soil mechanics, John Wiley&Sons, Inc., p. 265
|
Bond Equity Earnings Yield Ratio (BEER) Definition
Bond Equity Earnings Yield Ratio
Understanding the BEER
BEER vs. Fed Model
Limitations of the BEER
What Is the Bond Equity Earnings Yield Ratio (BEER)?
The bond equity earnings yield ratio (BEER) is a metric used to evaluate the relationship between bond yields and earnings yields in the stock market.
The bond equity earnings yield ratio may also go by the gilt-equity yield ratio (GEYR).
The bond equity earnings yield ratio (BEER) is a way investors can use bond yields to estimate the direction of the stock market.
The ratio is determined by dividing the yield of a government bond by the current earnings yield of a stock or stock benchmark.
A ratio greater than 1.0 indicates the stock market is overvalued, while a rating of under 1.0 suggests stocks are undervalued.
A particular example of a BEER that uses the S&P 500 and 10-year Treasuries is the so-called Fed model.
Understanding the Bond Equity Earnings Yield Ratio (BEER)
BEER has two parts—the numerator is represented by a benchmark bond yield, such as a five- or 10-year Treasury, while the denominator is the current earnings yield of a stock benchmark, such as the S&P 500.
A comparison of the yield on long-term government debt and the average yield on an equity market benchmark can be used as a form of indicator on when to buy stocks. If the ratio is above 1.0 the stock market is said to be overvalued; a reading of less than 1.0 indicates the stock market is undervalued.
The theory behind the ratio is that if stocks are yielding more than bonds, that is, BEER < 1, then stocks are cheap given that more value is being created by investing in equities. As investors increase their demand for stocks, the prices increase, causing P/E ratios to increase. As P/E ratios increase, earnings yield decreases, bringing it more in line with bond yields.
Conversely, if the earnings yield on stocks is less than the yield on Treasury bonds (BEER > 1), the proceeds from the sale of stocks are reinvested in bonds. This results in a decreased P/E ratio and increased earnings yield. Theoretically, a BEER of 1 would indicate equal levels of perceived risk in the bond market and the stock market.
Analysts often feel that BEER ratios greater than 1 imply that equity markets are overvalued, while numbers less than 1 mean they are undervalued, or that prevailing bond yields are not adequately pricing risk. If the BEER is above normal levels, the assumption is that the price of stocks will decrease, thus lowering the BEER.
The formula for BEER:
BEER is calculated by dividing the yield of a government bond by the current earnings yield of a stock benchmark in the same market. The current earnings yield of the stock market (or simply an individual stock) is just the inverse of the price-to-earnings (P/E) ratio. The earnings yield is quoted as a percentage, which measures the percentage of each dollar invested that was earned by a company, sector, or the whole market during the past twelve months.
For example, if the P/E ratio of the S&P 500 is 25, then the earnings yield is 1/25 = 0.04 or 4%. It is easier to compare the earnings yield to bond yields than to compare the P/E ratio to bond yields.
The idea behind the BEER ratio is that if stocks are yielding more than bonds, then they are undervalued; inversely, if bonds are yielding more than stocks, then stocks are overvalued.
Consider a 10-year Treasury bond with a yield of 2.8% and the earnings yield on the S&P 500 at 4% (indicative of a P/E of 25x). The BEER ratio can thus be calculated as:
\text{BEER}=\text{Bond Yield}\left(0.028\right)/\text{Earnings Yield}\left(0.04\right)=0.7
Using the results above, an investor can conclude that the stock market is undervalued as the ratio is calculated to be below 1.0.
The Fed model is a particular case of a bond equity earnings yield ratio. A BEER ratio can be calculated using any benchmark bond yield and any benchmark stock market's earnings yield. The Fed model is a tool for determining whether the U.S. stock market is fairly valued at a given time. The model is based on an equation that compares the earnings yield specifically of the S&P 500 with the yield on 10-year U.S. Treasury bonds.
Economist Ed Yardeni created the Fed model. He gave it this name saying it was the "Fed's stock valuation model, though no one at the Fed ever officially endorsed it." The Fed model dictates that if the S&P’s earnings yield is higher than the U.S. 10-year bonds yield, the market is "bullish."
A bullish market assumes stock prices are going to rise and a good time to buy shares. If the earnings yield dips below the yield of the 10-year bond, the market is considered "bearish." A bearish market assumes stock prices will decline. The Fed model did not seem to work during and following the 2008 financial crisis. The widely used and accepted model still has many investing experts questioning its utility in recent years.
The bond equity earnings yield ratio helps investors understand the value created by investing one dollar in bonds versus investing that dollar in stocks. However, critics have pointed out that the BEER ratio has zero predictive value, based on research that was carried out on historical yields in the Treasury and stock markets.
In addition, creating a correlation between stocks and bonds is said to be flawed as both investments are different in a number of ways—while government bonds are contractually guaranteed to pay back the principal, stocks promise nothing. Similarly, unlike the interest on a bond, a stock’s earnings and dividends are unpredictable and its value is not contractually guaranteed.
Yardeni Research. ”Stock Valuation Models - Topical Study #56,” Pages 2-3. Accessed Sept. 8, 2020.
The Wall Street Journal. ”When Is It Time to Buy Stocks Again?” Accessed Sept. 8, 2020.
ScienceDirect. ”The fed model: The bad, the worse, and the ugly.” Accessed Sept. 8, 2020.
How the Fed Model Works
The Fed model is a tool used to determine whether the U.S. stock market is bullish or bearish at a given time.
|
Ramanujan–Nagell equation - Wikipedia
In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.
1 Equation and solution
2 Triangular Mersenne numbers
3 Equations of Ramanujan–Nagell type
4 Equations of Lebesgue–Nagell type
Equation and solutionEdit
{\displaystyle 2^{n}-7=x^{2}\,}
and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15 (sequence A060728 in the OEIS).
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of n correspond to the values of x as:-
x = 1, 3, 5, 11 and 181 (sequence A038198 in the OEIS).[1]
Triangular Mersenne numbersEdit
The problem of finding all numbers of the form 2b − 1 (Mersenne numbers) which are triangular is equivalent:
{\displaystyle {\begin{aligned}&\ 2^{b}-1={\frac {y(y+1)}{2}}\\[2pt]\Longleftrightarrow &\ 8(2^{b}-1)=4y(y+1)\\\Longleftrightarrow &\ 2^{b+3}-8=4y^{2}+4y\\\Longleftrightarrow &\ 2^{b+3}-7=4y^{2}+4y+1\\\Longleftrightarrow &\ 2^{b+3}-7=(2y+1)^{2}\end{aligned}}}
The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan–Nagell numbers) are:
{\displaystyle {\frac {y(y+1)}{2}}={\frac {(x-1)(x+1)}{8}}}
for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more (sequence A076046 in the OEIS).
Equations of Ramanujan–Nagell typeEdit
{\displaystyle x^{2}+D=AB^{n}}
for fixed D, A , B and variable x, n is said to be of Ramanujan–Nagell type. The result of Siegel[2] implies that the number of solutions in each case is finite.[3] By representing
{\displaystyle n=3m+r}
{\displaystyle r\in \{0,1,2\}}
{\displaystyle B^{n}=B^{r}y^{3}}
{\displaystyle y=B^{m}}
, the equation of Ramanujan–Nagell type is reduced to three Mordell curves (indexed by
{\displaystyle r}
), each of which has a finite number of integer solutions:
{\displaystyle r=0:\qquad (Ax)^{2}=(Ay)^{3}-A^{2}D}
{\displaystyle r=1:\qquad (ABx)^{2}=(ABy)^{3}-A^{2}B^{2}D}
{\displaystyle r=2:\qquad (AB^{2}x)^{2}=(AB^{2}y)^{3}-A^{2}B^{4}D}
The equation with
{\displaystyle A=1,\ B=2}
has at most two solutions, except in the case
{\displaystyle D=7}
corresponding to the Ramanujan–Nagell equation. There are infinitely many values of D for which there are two solutions, including
{\displaystyle D=2^{m}-1}
Equations of Lebesgue–Nagell typeEdit
{\displaystyle x^{2}+D=Ay^{n}}
for fixed D, A and variable x, y, n is said to be of Lebesgue–Nagell type. This is named after Victor-Amédée Lebesgue, who proved that the equation
{\displaystyle x^{2}+1=y^{n}}
has no nontrivial solutions.[4]
Results of Shorey and Tijdeman[5] imply that the number of solutions in each case is finite.[6] Bugeaud, Mignotte and Siksek[7] solved equations of this type with A = 1 and 1 ≤ D ≤ 100. In particular, the following generalization of the Ramanujan-Nagell equation:
{\displaystyle y^{n}-7=x^{2}\,}
has positive integer solutions only when x = 1, 3, 5, 11, or 181.
Scientific equations named after people
^ a b Saradha & Srinivasan 2008, p. 208.
^ Saradha & Srinivasan 2008, p. 207.
^ Shorey & Tijdeman 1986.
^ Bugeaud, Mignotte & Siksek 2006.
Bugeaud, Y.; Mignotte, M.; Siksek, S. (2006). "Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation". Compositio Mathematica. 142: 31–62. arXiv:math/0405220. doi:10.1112/S0010437X05001739. S2CID 18534268.
Lebesgue (1850). "Sur l'impossibilité, en nombres entiers, de l'équation xm = y2 + 1". Nouv. Ann. Math. Série 1. 9: 178–181.
Ljunggren, W. (1943). "Oppgave nr 2". Norsk Mat. Tidsskr. 25: 29.
Nagell, T. (1948). "Løsning till oppgave nr 2". Norsk Mat. Tidsskr. 30: 62–64.
Nagell, T. (1961). "The Diophantine equation x2 + 7 = 2n". Ark. Mat. 30 (2–3): 185–187. Bibcode:1961ArM.....4..185N. doi:10.1007/BF02592006.
Ramanujan, S. (1913). "Question 464". J. Indian Math. Soc. 5: 130.
Saradha, N.; Srinivasan, Anitha (2008). "Generalized Lebesgue–Ramanujan–Nagell equations". In Saradha, N. (ed.). Diophantine Equations. Narosa. pp. 207–223. ISBN 978-81-7319-898-4.
Shorey, T. N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. Vol. 87. Cambridge University Press. pp. 137–138. ISBN 0-521-26826-5. Zbl 0606.10011.
Siegel, C. L. (1929). "Uber einige Anwendungen Diophantischer Approximationen". Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1: 41–69.
"Values of X corresponding to N in the Ramanujan–Nagell Equation". Wolfram MathWorld. Retrieved 2012-05-08.
Can N2 + N + 2 Be A Power Of 2?, Math Forum discussion
Retrieved from "https://en.wikipedia.org/w/index.php?title=Ramanujan–Nagell_equation&oldid=1071660150"
|
Modal logic - Routledge Encyclopedia of Philosophy
Propositional S5
Philosophical questions about S5
Quantified S5
Weaker systems
Retrieved May 22, 2022, from https://www.rep.routledge.com/articles/thematic/modal-logic/v-1
Modal logic, narrowly conceived, is the study of principles of reasoning involving necessity and possibility. More broadly, it encompasses a number of structurally similar inferential systems. In this sense, deontic logic (which concerns obligation, permission and related notions) and epistemic logic (which concerns knowledge and related notions) are branches of modal logic. Still more broadly, modal logic is the study of the class of all possible formal systems of this nature.
It is customary to take the language of modal logic to be that obtained by adding one-place operators ‘□’ for necessity and ‘◇’ for possibility to the language of classical propositional or predicate logic. Necessity and possibility are interdefinable in the presence of negation:
\begin{array}{l}\square A↔\mathrm{¬}◊\mathrm{¬}A\text{ and }\\ ◊A↔\mathrm{¬}\square \mathrm{¬}A\end{array}
hold. A modal logic is a set of formulas of this language that contains these biconditionals and meets three additional conditions: it contains all instances of theorems of classical logic; it is closed under modus ponens (that is, if it contains A and A→B it also contains B); and it is closed under substitution (that is, if it contains A then it contains any substitution instance of A; any result of uniformly substituting formulas for sentence letters in A). To obtain a logic that adequately characterizes metaphysical necessity and possibility requires certain additional axiom and rule schemas:
\begin{array}{ll}\hfill K& \square \left(A\to B\right)\to \left(\square A\to \square B\right)\hfill \\ \hfill T& \square A\to A\hfill \\ \hfill 5& ◊A\to \square ◊A\hfill \\ \hfill Necessitation\text{ }A/\square A.& \hfill \end{array}
By adding these and one of the □–◇ biconditionals to a standard axiomatization of classical propositional logic one obtains an axiomatization of the most important modal logic, S5, so named because it is the logic generated by the fifth of the systems in Lewis and Langford’s Symbolic Logic (1932). S5 can be characterized more directly by possible-worlds models. Each such model specifies a set of possible worlds and assigns truth-values to atomic sentences relative to these worlds. Truth-values of classical compounds at a world w depend in the usual way on truth-values of their components. □A is true at w if A is true at all worlds of the model; ◇A, if A is true at some world of the model. S5 comprises the formulas true at all worlds in all such models. Many modal logics weaker than S5 can be characterized by models which specify, besides a set of possible worlds, a relation of ‘accessibility’ or relative possibility on this set. □A is true at a world w if A is true at all worlds accessible from w, that is, at all worlds that would be possible if w were actual. Of the schemas listed above, only K is true in all these models, but each of the others is true when accessibility meets an appropriate constraint.
The addition of modal operators to predicate logic poses additional conceptual and mathematical difficulties. On one conception a model for quantified modal logic specifies, besides a set of worlds, the set Dw of individuals that exist in w, for each world w. For example, ∃x□A is true at w if there is some element of Dw that satisfies A in every possible world. If A is satisfied only by existent individuals in any given world ∃x□A thus implies that there are necessary individuals; individuals that exist in every accessible possible world. If A is satisfied by non-existents there can be models and assignments that satisfy A, but not ∃xA. Consequently, on this conception modal predicate logic is not an extension of its classical counterpart.
The modern development of modal logic has been criticized on several grounds, and some philosophers have expressed scepticism about the intelligibility of the notion of necessity that it is supposed to describe.
Kuhn, Steven T.. Modal logic, 1998, doi:10.4324/9780415249126-Y039-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/modal-logic/v-1.
|
Filter order - MATLAB filtord - MathWorks í•œêµ
Verify Order of FIR Filter
Determine the Order Difference Between FIR and IIR Designs
n = filtord(b,a)
n = filtord(sos)
n = filtord(b,a) returns the filter order, n, for the causal rational system function specified by the numerator coefficients, b, and denominator coefficients, a.
n = filtord(sos) returns the filter order for the filter specified by the second-order sections matrix, sos. sos is a K-by-6 matrix. The number of sections, K, must be greater than or equal to 2. Each row of sos corresponds to the coefficients of a second-order filter. The ith row of the second-order section matrix corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].
n = filtord(d) returns the filter order, n, for the digital filter, d. Use the function designfilt to generate d.
Design a 20th-order FIR filter with normalized cutoff frequency
0.5\mathrm{Ï}
rad/sample using the window method. Verify the filter order.
b = fir1(20,0.5);
n = filtord(b)
Design the same filter using designfilt and verify its order.
di = designfilt('lowpassfir','FilterOrder',20,'CutoffFrequency',0.5);
ni = filtord(di)
Design FIR equiripple and IIR Butterworth filters from the same set of specifications. Determine the difference in filter order between the two designs.
fir = designfilt('lowpassfir','DesignMethod','equiripple','SampleRate',1e3, ...
'PassbandFrequency',100,'StopbandFrequency',120, ...
iir = designfilt('lowpassiir','DesignMethod','butter','SampleRate',1e3, ...
FIR = filtord(fir)
FIR = 114
IIR = filtord(iir)
IIR = 41
Numerator coefficients, specified as a scalar or a vector. If the filter is an allpole filter, b is a scalar. Otherwise, b is a row or column vector.
Example: b = fir1(20,0.25)
Denominator coefficients, specified as a scalar or a vector. If the filter is an FIR filter, a is a scalar. Otherwise, a is a row or column vector.
Example: [b,a] = butter(20,0.25)
sos — Matrix of second-order sections
Matrix of second order-sections, specified as a K-by-6 matrix. The system function of the Kth biquad filter has the rational Z-transform
{H}_{k}\left(z\right)=\frac{{B}_{k}\left(1\right)+{B}_{k}\left(2\right){z}^{â1}+{B}_{k}\left(3\right){z}^{â2}}{{A}_{k}\left(1\right)+{A}_{k}\left(2\right){z}^{â1}+{A}_{k}\left(3\right){z}^{â2}}.
The coefficients in the Kth row of the matrix, sos, are ordered as follows.
\left[\begin{array}{cccccc}{B}_{k}\left(1\right)& {B}_{k}\left(2\right)& {B}_{k}\left(3\right)\text{â}& {A}_{k}\left(1\right)& {A}_{k}\left(2\right)& {A}_{k}\left(3\right)\end{array}\right].
The frequency response of the filter is the system function evaluated on the unit circle with
z={e}^{j2\mathrm{Ï}f}.
Example: d = designfilt('lowpassiir','FilterOrder',3,'HalfPowerFrequency',0.5) specifies a third-order Butterworth filter with normalized 3 dB frequency 0.5Ï€ rad/sample.
designfilt | digitalFilter | isallpass | isminphase | ismaxphase | isstable
|
The walleye (Sander vitreus, synonym Stizostedion vitreum), also called the yellow pike or yellow pickerel,[3] is a freshwater perciform fish native to most of Canada and to the Northern United States. It is a North American close relative of the European zander, also known as the pikeperch. The walleye is sometimes called the yellow walleye to distinguish it from the blue walleye, which is a subspecies that was once found in the southern Ontario and Quebec regions, but is now presumed extinct.[4] However, recent genetic analysis of a preserved (frozen) 'blue walleye' sample suggests that the blue and yellow walleye were simply phenotypes within the same species and do not merit separate taxonomic classification.[5]
In parts of its range in English-speaking Canada, the walleye is known as a pickerel, though the fish is not related to the true pickerels, which are a member of the family Esocidae.[6]
Walleyes show a fair amount of variation across watersheds. In general, fish within a watershed are quite similar and are genetically distinct from those of nearby watersheds. The species has been artificially propagated for over a century and has been planted on top of existing populations or introduced into waters naturally devoid of the species, sometimes reducing the overall genetic distinctiveness of populations.
The name "walleye" comes from its pearlescent eyes caused by the reflective tapetum lucidum which, in addition to allowing the fish to see well in low-light conditions, gives its eyes an opaque appearance. Their vision affects their behavior. They avoid bright light and feed in low light on fish that cannot see as well as they do.[7] Many anglers look for walleyes at night since this is when major feeding efforts occur. The fish's eyes also allow them to see well in turbid waters (stained or rough, breaking waters), which gives them an advantage over their prey. Thus, walleye anglers commonly look for locations where a good "walleye chop" (i.e., rough water) occurs. Their vision also allows the fish to populate the deeper regions in a lake, and they can often be found in deeper water, particularly during the warmest part of the summer and at night.[8]
Walleyes are largely olive and gold in color (hence the French common name: doré—golden). The dorsal side of a walleye is olive, grading into a golden hue on the flanks. The olive/gold pattern is broken up by five darker saddles that extend to the upper sides. The color shades to white on the belly. The mouth of a walleye is large and is armed with many sharp teeth. The first dorsal and anal fins are spinous, as is the operculum. Walleyes are distinguished from their close relative the sauger by the white coloration on the lower lobe of the caudal fin, which is absent on the sauger. In addition, the two dorsals and the caudal fin of the sauger are marked with distinctive rows of black dots which are absent from or indistinct on the same fins of walleyes.[9]
Weight and length of walleyes
Walleyes grow to about 80 cm (31 in) in length, and weigh up to about 9 kg (20 lb). The maximum recorded size for the fish is 107 cm (42 in) in length and 13 kilograms (29 lb) in weight. The rate depends partly on where in their range they occur, with southern populations often growing faster and larger. In general, females grow larger than males. Walleyes may live for decades; the maximum recorded age is 29 years. In heavily fished populations, however, few walleye older than five or six years of age are encountered. In North America, where they are highly prized, their typical size when caught is on the order of 30 to 50 cm (12 to 20 in), substantially below their potential size.
As walleye grow longer, they increase in weight. The relationship between total length (L) and total weight (W) for nearly all species of fish can be expressed by an equation of the form
{\displaystyle W=cL^{b}\,}
Invariably, b is close to 3.0 for all species, and c is a constant that varies among species. For walleye, b = 3.180 and c = 0.000228 (with units in inches and pounds) or b = 3.180 and c = 0.000005337 (with units in cm and kg).[10]
This relationship suggests a 50 cm (20 in) walleye will weigh about 1.5 kg (3.3 lb), while a 60 cm (24 in) walleye will likely weigh about 2.5 kg (5.5 lb).
The Garrison Dam National Fish Hatchery at Garrison Dam, North Dakota is the largest walleye hatchery in the world. Although they are in high demand for fishing and consumption in North Dakota, elsewhere they are considered a nuisance. For that reason GDNFH is also researching hormonal population control to provide control options to other areas.[11]
In most of the species' range, male walleyes mature sexually between three and four years of age. Females normally mature about a year later. Adults migrate to tributary streams in late winter or early spring to lay eggs over gravel and rock, although open-water reef or shoal-spawning strains are seen, as well. Some populations are known to spawn on sand or vegetation. Spawning occurs at water temperatures of 6 to 10 °C (43 to 50 °F). A large female can lay up to 500,000 eggs, and no care is given by the parents to the eggs or fry. The eggs are slightly adhesive and fall into spaces between rocks. The incubation period for the embryos is temperature-dependent, but generally lasts from 12 to 30 days. After hatching, the free-swimming embryos spend about a week absorbing a relatively small amount of yolk. Once the yolk has been fully absorbed, the young walleyes begin to feed on invertebrates, such as fly larvæ and zooplankton. After 40 to 60 days, juvenile walleyes become piscivorous. Thenceforth, both juvenile and adult walleyes eat fish almost exclusively, frequently yellow perch or ciscoes, moving onto bars and shoals at night to feed. Walleye also feed heavily on crayfish, minnows, and leeches.
The walleye is part of the North American clade within the genus Sander, alongside the sauger (S. canadensis). Hubbs described a taxon called the blue walleye (S. glaucus) from the Great Lakes but subsequent taxonomic work showed no consistent differences between this form and the "yellow" walleye and the blue walleye is now considered to be a synonym and color variant of the walleye.[12] The walleye was first formally described by the American naturalist Samuel Latham Mitchill (1764-1831) with the type locality given as Cayuga Lake near Ithaca, New York.[13]
Fresh walleye being cooked over a fire
The walleye is considered to be a quite palatable freshwater fish, and consequently, is fished recreationally and commercially for food.[14] Because of its nocturnal feeding habits, it is most easily caught at night using live minnows or lures that mimic small fish. Most commercial fisheries for walleye are situated in the Canadian waters of the Great Lakes,[15] and fried walleye is considered a staple of Canadian cuisine.[16][17] In Minnesota, the walleye is often fished for in the late afternoon on windy days (known as a "walleye chop") or at night. Often served as a sandwich in Minnesota's pubs where the fish is very popular, deep fried walleye on a stick is a Minnesota State Fair food.[18]
Main article: Walleye fishing
Because walleyes are popular with anglers, fishing for walleyes is regulated by most natural resource agencies. Management may include the use of quotas and length limits to ensure that populations are not overexploited. For example, in Michigan, walleyes shorter than 15 in (38 cm) may not be legally kept, except in Lake St. Clair, the St. Clair River, and Saginaw Bay where fish as short as 13 in (33 cm) may be taken.
Since walleyes have excellent visual acuity under low illumination levels, they tend to feed more extensively at dawn and dusk, on cloudy or overcast days, and under choppy conditions when light penetration into the water column is disrupted. Although anglers interpret this as light avoidance, it is merely an expression of the walleyes' competitive advantage over their prey under those conditions. Similarly, in darkly stained or turbid waters, walleyes tend to feed throughout the day. In the spring and fall, walleyes are located near the shallower areas due to the spawning grounds, and they are most often located in shallower areas during higher winds due to the murkier, higher oxygenated water at around six feet deep.[19] On calm spring days, walleyes are more often located at the deep side of the shoreline drop-off and around shore slopes around or deeper than 10 feet.[20]
As a result of their widespread presence in Canada and the northern United States, walleyes are frequently caught while ice fishing, a popular winter pastime throughout those regions.
"Walleye chop" is a term used by walleye anglers for rough water typically with winds of 10 to 25 km/h (6 to 16 mph), and is one of the indicators for good walleye fishing due to the walleyes' increased feeding activity during such conditions. In addition to fishing this chop, night fishing with live bait can be very effective.
The current all-tackle world record for a walleye is held by Mabry Harper, who caught an 11.34-kg (25-lb) walleye in Old Hickory Lake in Tennessee on 2 August 1960.[21]
Large walleye statue at Lake Mille Lacs in Garrison, Minnesota
The walleye is the state fish of Minnesota, Vermont, and South Dakota, and the official provincial fish of Manitoba[22] and Saskatchewan.[23]
It is very popular with Minnesota residents; more walleye is eaten in Minnesota than in any other jurisdiction of the United States. Both Garrison and Baudette, Minnesota, claim to be the "Walleye Capital of the World", each with a large statue of the fish.[24]
Winnipeg, Manitoba, considers the walleye (referred to locally as "pickerel") its most important local fish.[25]: 76 Icelandic fishermen in Lake Winnipeg traditionally supplied the Winnipeg market.[25]: 23–26
Wisconsin Walleye War
^ NatureServe (2013). "Sander vitreus". IUCN Red List of Threatened Species. 2013: e.T202605A18229159. doi:10.2305/IUCN.UK.2013-1.RLTS.T202605A18229159.en. Retrieved 19 November 2021.
^ Froese, Rainer; Pauly, Daniel (eds.) (2019). Sander &speciesname= vitreum" Sander vitreum " in FishBase. December 2019 version.
^ "Ontario Freshwater Fishes Life History Database Species Detail".
^ "Le doré bleu existe!". lapresse.ca. 16 August 2008. Retrieved 24 March 2018.
^ Haponski, Amanda E.; Stepien, Carol A. (2014). "A population genetic window into the past and future of the walleye Sander vitreus: relation to historic walleye and the extinct "blue pike" S. v. "glaucus"". Giornale della Libreria. 14 (1): 133. doi:10.1186/1471-2148-14-133. PMC 4229939. PMID 24941945. Retrieved 10 July 2015.
^ Crossman, E.J. "Walleye - The Canadian Encyclopedia". Retrieved 29 April 2017.
^ "Walleye biology and identification". Minnesota Department of Natural Resources. Retrieved 29 May 2021.
^ Northern Wisconsin All-Outdoors Atlas & Field Guide. Sportsman's Connection. 2012. p. 5.
^ "In-Fisherman - The World's Foremost Authority On Freshwater Fishing". In-Fisherman. Retrieved 24 March 2018.
^ Anderson, R. O.; Neumann, R. M. (1996). "Length, Weight, and Associated Structural Indices". In Murphy, B. E.; Willis, D. W. (eds.). Fisheries Techniques (Second ed.). Bethesda, MD: American Fisheries Society. ISBN 1-888569-00-X.
^ Wilson, Malik (12 February 2021). "Garrison Dam National Fish Hatchery spearheading walleye population control project". KX NEWS. Retrieved 14 February 2021.
^ Carol A. Stepien & Amanda Haponski (2015). "Taxonomy, Distribution, and Evolution of the Percidae". In Patrick Kestemont; Konrad Dabrowski & Robert C. Summerfelt (eds.). Biology and Culture of Percid Fishes. Springer, Dordrecht. pp. 3–60. doi:10.1007/978-94-017-7227-3_1. ISBN 978-94-017-7227-3.
^ Eschmeyer, William N.; Fricke, Ron & van der Laan, Richard (eds.). "Perca vitrea". Catalog of Fishes. California Academy of Sciences. Retrieved 16 September 2020.
^ "Walleye, Sander vitreus". Department of Natural Resources (DNR), State of Michigan. Retrieved 15 March 2013.
^ "Walleye". Retrieved 25 April 2022. Seafood Source
^ "Best Fried Walleye". Retrieved 25 April 2022. Cook Me
^ "Best Fried Walleye". Retrieved 25 April 2022. All Recipes
^ "Field and Stream July 2005". July 2005.
^ Joe Fellegy, Jr., Walleyes and Walleye Fishing (Dillon Press, 1974), 57, 58
^ Fellegy, 60
^ International Game and Fish Association (1960). "IGFA All-Tackle World Record". IGFA. Retrieved 20 April 2014.
^ "OFFICIAL EMBLEMS OF MANITOBA|Fish Emblem|Pickerel" (PDF). gov.mb.ca. Retrieved 3 June 2021. ((cite web)): CS1 maint: url-status (link)
^ "Saskatchewan". Canadian Heritage. Retrieved 15 March 2013.
^ "Walleyed War of the Walleye Capitals". RoadsideAmerica.com. Doug Kirby, Ken Smith, Mike Wilkins.
^ a b Nicholson, Karen (May 2007). "A History of Manitoba's Commercial Fishery 1872-2005" (PDF). Manitoba Historic Resources Branch. Retrieved 8 May 2017.
"Sander vitreus". Integrated Taxonomic Information System. Retrieved 19 March 2006.
"Sander vitreus, Walleye". Fishbase. Retrieved 15 March 2013.
Cena, Christopher J; George E. Morgan; Michael D. Malette; Daniel D. Heath (2006). "Inbreeding, Outbreeding and Environmental Effects on Genetic Diversity in 46 Walleye (Sander Vitreus) Populations". Molecular Ecology. 15 (2): 303–20. doi:10.1111/j.1365-294x.2005.02637.x. PMID 16448402. S2CID 22802903.
Grant, Gerold C; Paul Radomski; Charles S. Anderson (2004). "Using Underwater Video to Directly Estimate Gear Selectivity: the Retention Probability for Walleye (Sander Vitreus) in Gill Nets". Canadian Journal of Fisheries and Aquatic Sciences. 61 (2): 168–74. doi:10.1139/f03-166.
Huppert, Boyd (7 December 2004). "Walleye or Zander? What Are You Really Eating?". Kare 11. Retrieved 15 March 2013.
Kaufman, Scott D.; John M. Gunn; George E. Morgan; Patrice Couture (2006). "Muscle Enzymes Reveal Walleye (Sander Vitreus) Are Less Active When Larger Prey (cisco, Coregonus Artedi) Are Present". Canadian Journal of Fisheries and Aquatic Sciences. 63 (5): 970–79. doi:10.1139/f06-004.
Simoneau, M (2005). "Fish Growth Rates Modulate Mercury Concentrations in Walleye (Sander Vitreus) from Eastern Canadian Lakes". Environmental Research. 98 (1): 73–82. doi:10.1016/j.envres.2004.08.002. PMID 15721886.
Suski, C. D.; S. J. Cooke; S. S. Killen; D. H. Wahl; B. L. Tufts (2005). "Behaviour of Walleye, Sander Vitreus, and Largemouth Bass, Micropterus Salmoides, Exposed to Different Wave Intensities and Boat Operating Conditions during Livewell Confinement". Fisheries Management and Ecology. 12 (1): 119–26. doi:10.1111/j.1365-2400.2004.00415.x.
Wikimedia Commons has media related to Sander vitreus.
Wikispecies: Sander vitreus
Fish of the United States
Cuisine of Manitoba
|
Balance Sheet | Hifi Docs
Hifi DocsProtocolSDKSmart Contracts
Chainlink Operator
Fintroller
The BalanceSheet is the global debt registry for Hifi. It tracks the collateral deposits and the debt taken on by all users.
When a user borrow hTokens, a vault is opened for them in the BalanceSheet. All vaults are recorded and managed in this contract.
This is the only upgradeable contract in the Hifi protocol. The most up to date version is the BalanceSheetV2.
getBondList
function getBondList(
) external returns (contract IHToken[])
Returns the list of bond markets the given account entered.
It is not an error to provide an invalid address.
account address The borrower account to make the query against.
getCollateralAmount
function getCollateralAmount(
contract IErc20 collateral
) external returns (uint256 collateralAmount)
Returns the amount of collateral deposited by the given account for the given collateral type.
collateral contract IErc20 The collateral to make the query against.
function getCollateralList(
) external returns (contract IErc20[])
Returns the list of collaterals the given account deposited.
getCurrentAccountLiquidity
function getCurrentAccountLiquidity(
) external returns (uint256 excessLiquidity, uint256 shortfallLiquidity)
Calculates the current account liquidity.
account address The account to make the query against.
excessLiquidity uint256 account liquidity in excess of collateral requirements.
shortfallLiquidity uint256 account shortfall below collateral requirements
getDebtAmount
function getDebtAmount(
contract IHToken bond
) external returns (uint256 debtAmount)
Returns the amount of debt accrued by the given account in the given bond market.
bond contract IHToken The bond to make the query against.
getHypotheticalAccountLiquidity
contract IErc20 collateralModify,
uint256 collateralAmountModify,
contract IHToken bondModify,
uint256 debtAmountModify
Calculates the account liquidity given a modified collateral, collateral amount, bond and debt amount, using the current prices provided by the oracle.
Works by summing up each collateral amount multiplied by the USD value of each unit and divided by its respective collateral ratio, then dividing the sum by the total amount of debt drawn by the user.
This function expects that the "collateralList" and the "bondList" are each modified in advance to include the collateral and bond due to be modified.
collateralModify contract IErc20 The collateral to make the check against.
collateralAmountModify uint256 The hypothetical normalized amount of collateral.
bondModify contract IHToken The bond to make the check against.
debtAmountModify uint256 The hypothetical amount of debt.
excessLiquidity uint256 hypothetical account liquidity in excess of collateral requirements.
shortfallLiquidity uint256 hypothetical account shortfall below collateral requirements
getRepayAmount
function getRepayAmount(
contract IErc20 collateral,
uint256 seizableCollateralAmount,
) external returns (uint256 repayAmount)
Calculates the amount of hTokens that should be repaid in order to seize a given amount of collateral. Note that this is for informational purposes only, it doesn't say anything about whether the user can be liquidated.
The formula applied:
repayAmount = \frac{seizableCollateralAmount * collateralPrice}{liquidationIncentive * underlyingPrice}
seizableCollateralAmount uint256 The amount of collateral to seize.
repayAmount uint256 The amount of hTokens that should be repaid.
getSeizableCollateralAmount
function getSeizableCollateralAmount(
contract IHToken bond,
) external returns (uint256 seizableCollateralAmount)
Calculates the amount of collateral that can be seized when liquidating a borrow. Note that this is for informational purposes only, it doesn't say anything about whether the user can be liquidated.
seizableCollateralAmount = \frac{repayAmount * liquidationIncentive * underlyingPrice}{collateralPrice}
repayAmount uint256 The amount of hTokens to repay.
seizableCollateralAmount uint256 The amount of seizable collateral.
Non-Constant Functions
Increases the debt of the caller and mints new hTokens.
Emits a {Borrow} event.
The Fintroller must allow this action to be performed.
The maturity of the bond must be in the future.
The amount to borrow cannot be zero.
The new length of the bond list must be below the max bonds limit.
The new total amount of debt cannot exceed the debt ceiling.
The caller must not end up having a shortfall of liquidity.
bond contract IHToken The address of the bond contract.
borrowAmount uint256 The amount of hTokens to borrow and print into existence.
function depositCollateral(
uint256 depositAmount
Deposits collateral in the caller's account.
Emits a {DepositCollateral} event.
The amount to deposit cannot be zero.
The caller must have allowed this contract to spend collateralAmount tokens.
The new collateral amount cannot exceed the collateral ceiling.
collateral contract IErc20 The address of the collateral contract.
depositAmount uint256 The amount of collateral to deposit.
liquidateBorrow
Repays the debt of the borrower and rewards the caller with a surplus of collateral.
Emits a {LiquidateBorrow} event.
All from "repayBorrow".
The caller cannot be the same with the borrower.
The borrower must have a shortfall of liquidity if the bond didn't mature.
The amount of seized collateral cannot be more than what the borrower has in the vault.
borrower address The account to liquidate.
repayBorrow
function repayBorrow(
Erases the borrower's debt and takes the hTokens out of circulation.
Emits a {RepayBorrow} event.
The amount to repay cannot be zero.
The caller must have at least repayAmount hTokens.
The caller must have at least repayAmount debt.
function repayBorrowBehalf(
Same as the repayBorrow function, but here borrower is the account that must have at least repayAmount hTokens to repay the borrow.
borrower address The borrower account for which to repay the borrow.
bond contract IHToken The address of the bond contract
setFintroller
function setFintroller(
contract IFintroller newFintroller
Updates the Fintroller contract this BalanceSheet is connected to.
Emits a {SetFintroller} event.
The caller must be the owner.
The new address cannot be the zero address.
newFintroller contract IFintroller The new Fintroller contract.
contract IChainlinkOperator newOracle
Updates the oracle contract.
Emits a {SetOracle} event.
newOracle contract IChainlinkOperator The new oracle contract.
function withdrawCollateral(
uint256 withdrawAmount
Withdraws a portion or all of the collateral.
Emits a {WithdrawCollateral} event.
The amount to withdraw cannot be zero.
There must be enough collateral in the vault.
The caller's account cannot fall below the collateral ratio.
withdrawAmount uint256 The amount of collateral to withdraw.
Emitted when a borrow is made.
account address The address of the borrower.
borrowAmount uint256 The amount of hTokens borrowed.
event DepositCollateral(
Emitted when collateral is deposited.
collateral contract IErc20 The related collateral.
collateralAmount uint256 The amount of deposited collateral.
uint256 seizedCollateralAmount
Emitted when a borrow is liquidated.
liquidator address The address of the liquidator.
borrower address The address of the borrower.
repayAmount uint256 The amount of repaid funds.
seizedCollateralAmount uint256 The amount of seized collateral.
uint256 newDebtAmount
Emitted when a borrow is repaid.
payer address The address of the payer.
newDebtAmount uint256 The amount of the new debt.
event SetFintroller(
address oldFintroller,
address newFintroller
Emitted when a new Fintroller contract is set.
owner address The address of the owner.
oldFintroller address The address of the old Fintroller contract.
newFintroller address The address of the new Fintroller contract.
event SetOracle(
address oldOracle,
address newOracle
Emitted when a new oracle contract is set.
oldOracle address The address of the old oracle contract.
newOracle address The address of the new oracle contract.
event WithdrawCollateral(
Emitted when collateral is withdrawn.
collateralAmount uint256 The amount of withdrawn collateral.
BalanceSheet__BondMatured
error BalanceSheet__BondMatured(contract IHToken bond)
Emitted when the bond matured.
BalanceSheet__BorrowMaxBonds
error BalanceSheet__BorrowMaxBonds(contract IHToken bond, uint256 newBondListLength, uint256 maxBonds)
Emitted when the account exceeds the maximum numbers of bonds permitted.
BalanceSheet__BorrowNotAllowed
error BalanceSheet__BorrowNotAllowed(contract IHToken bond)
Emitted when borrows are not allowed by the Fintroller contract.
BalanceSheet__BorrowZero
error BalanceSheet__BorrowZero()
Emitted when borrowing a zero amount of hTokens.
BalanceSheet__CollateralCeilingOverflow
error BalanceSheet__CollateralCeilingOverflow(uint256 newTotalSupply, uint256 debtCeiling)
Emitted when the new collateral amount exceeds the collateral ceiling.
BalanceSheet__DebtCeilingOverflow
error BalanceSheet__DebtCeilingOverflow(uint256 newCollateralAmount, uint256 debtCeiling)
Emitted when the new total amount of debt exceeds the debt ceiling.
BalanceSheet__DepositCollateralNotAllowed
error BalanceSheet__DepositCollateralNotAllowed(contract IErc20 collateral)
Emitted when collateral deposits are not allowed by the Fintroller contract.
BalanceSheet__DepositCollateralZero
error BalanceSheet__DepositCollateralZero()
Emitted when depositing a zero amount of collateral.
BalanceSheet__FintrollerZeroAddress
error BalanceSheet__FintrollerZeroAddress()
Emitted when setting the Fintroller contract to the zero address.
BalanceSheet__LiquidateBorrowInsufficientCollateral
error BalanceSheet__LiquidateBorrowInsufficientCollateral(address account, uint256 vaultCollateralAmount, uint256 seizableAmount)
Emitted when there is not enough collateral to seize.
BalanceSheet__LiquidateBorrowNotAllowed
error BalanceSheet__LiquidateBorrowNotAllowed(contract IHToken bond)
Emitted when borrow liquidations are not allowed by the Fintroller contract.
BalanceSheet__LiquidateBorrowSelf
error BalanceSheet__LiquidateBorrowSelf(address account)
Emitted when the borrower is liquidating themselves.
BalanceSheet__LiquidityShortfall
error BalanceSheet__LiquidityShortfall(address account, uint256 shortfallLiquidity)
Emitted when there is a liquidity shortfall.
BalanceSheet__NoLiquidityShortfall
error BalanceSheet__NoLiquidityShortfall(address account)
Emitted when there is no liquidity shortfall.
BalanceSheet__OracleZeroAddress
error BalanceSheet__OracleZeroAddress()
Emitted when setting the oracle contract to the zero address.
BalanceSheet__RepayBorrowInsufficientBalance
error BalanceSheet__RepayBorrowInsufficientBalance(contract IHToken bond, uint256 repayAmount, uint256 hTokenBalance)
Emitted when the repayer does not have enough hTokens to repay the debt.
BalanceSheet__RepayBorrowInsufficientDebt
error BalanceSheet__RepayBorrowInsufficientDebt(contract IHToken bond, uint256 repayAmount, uint256 debtAmount)
Emitted when repaying more debt than the borrower owes.
BalanceSheet__RepayBorrowNotAllowed
error BalanceSheet__RepayBorrowNotAllowed(contract IHToken bond)
Emitted when borrow repays are not allowed by the Fintroller contract.
BalanceSheet__RepayBorrowZero
error BalanceSheet__RepayBorrowZero()
Emitted when repaying a borrow with a zero amount of hTokens.
BalanceSheet__WithdrawCollateralUnderflow
error BalanceSheet__WithdrawCollateralUnderflow(address account, uint256 vaultCollateralAmount, uint256 withdrawAmount)
Emitted when withdrawing more collateral than there is in the vault.
BalanceSheet__WithdrawCollateralZero
error BalanceSheet__WithdrawCollateralZero()
Emitted when withdrawing a zero amount of collateral.
Chainlink Operator »
Copyright © 2022 Mainframe Group Inc.
|
Home : Support : Online Help : Mathematics : Discrete Mathematics : Combinatorics : conversions : GAMMA
convert factorials and binomials to GAMMAs
convert(expr, GAMMA, indets)
\mathrm{convert}\left(\mathrm{expr},\mathrm{\Gamma },\mathrm{indets}\right)
The convert/GAMMA function converts factorials, binomials and multinomial coefficients in an expression to the GAMMA function.
You can enter the command convert/GAMMA using either the 1-D or 2-D calling sequence. For example, convert(x!, GAMMA) is equivalent to
\mathrm{convert}\left(x!,\mathrm{\Gamma }\right)
If an indeterminate or set of indeterminates is specified, then only factorials and binomials involving a specified indeterminate will be converted to the GAMMA function.
\mathrm{convert}\left(x!,\mathrm{\Gamma }\right)
\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{convert}\left(\mathrm{binomial}\left(m,3\right),\mathrm{\Gamma }\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}{\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}
\mathrm{convert}\left(x!y!z!,\mathrm{\Gamma },{x,y}\right)
\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{!}
\mathrm{convert}\left(\left(x+y\right)!,\mathrm{\Gamma },{x,y}\right)
\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)
|
Revision as of 21:15, 10 December 2021 by Dean Young (talk | contribs) (→For full infiltration design, to calculate the total depth of clear stone aggregate layers needed for the water storage reservoir)
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
(Adding QRL, quantum finite automata.)
m (alphabetize, oops)
According to [[zooref#bg94|[BG94]]], Beigel and Feigenbaum and (independently) Krawczyk showed that QPSPACE is not contained in [[Complexity Zoo:C#check|Check]].
===== <span id="qrl" style="color:red">QRL</span>: Quantum Regular Languages =====
The class of problems nondeterministically recognized by ''Moore-Crutchfield Quantum Finite Automata''. These are automata that undergo a unitary transformation for each symbol consumed, and at the end are observed to be in an accepting state or not. The "nondeterminism" refers to the fact that the automaton will accept with positive probability if in the language, or with exactly zero probability otherwise.
Has the same caveats as [[Complexity Zoo:N#nqp|NQP]] and [[Complexity Zoo:E#eqp|EQP]] because of the exact acceptance probabilities.
Also called NMCL, in <ref>Abuzer Yakaryilmaz, A. C. Cem Say. Languages recognized by nondeterministic quantum finite automata. https://arxiv.org/abs/0902.2081</ref>, in order to disambiguate from [[Complexity Zoo:N#nql_2|NQL]].
Contrast with [[Complexity Zoo:N#nql_2|NQL]], which is an analogous class for a different model of quantum finite automaton: there the automaton is measured after each step for acceptance or rejection.
QRG(1) is trivially contained in [[Complexity Zoo:Q#qrg2|QRG(2)]] (and hence [[Complexity Zoo:P#pspace|PSPACE]]).
{\displaystyle k\geq 4}
{\displaystyle k\geq 2}
{\displaystyle k=1}
|
How to Discount Cash Flow: 11 Steps (with Pictures) - wikiHow
How to Discount Cash Flow
2 Discounting Cash Flows
3 Using Discounted Cash Flows
Everyone knows that a dollar today is worth more than a dollar tomorrow — this is because of inflation and the opportunity cost of what you miss out on by not having the dollar today. But how much less is that dollar received tomorrow actually worth? Discounting is a technique designed to answer this question by reducing the value of future cash flows to their present-day values. Once calculated, discounted future cash flows can be used to analyze investments and value companies.
Gathering Your Variables Download Article
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/2b\/Discount-Cash-Flow-Step-1-Version-2.jpg\/v4-460px-Discount-Cash-Flow-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/2\/2b\/Discount-Cash-Flow-Step-1-Version-2.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Identify a situation in which you would need to discount cash flows. Discounted cash flow (DCF) calculations are used to adjust the value of money received in the future. In order to calculate DCFs, you will need to identify a situation in which money will be received at a later date or dates in one or more installments. DCFs are commonly used for things like investments in securities or companies that will provide cash flows over a number of years. Alternately, a business might use DCFs to estimate the return from an investment in production equipment, for example.
In order to calculate DCFs, you will need a definable set of future cash flows and know the date(s) that you will receive those cash flows.[1] X Research source
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/4\/49\/Discount-Cash-Flow-Step-2-Version-2.jpg\/v4-460px-Discount-Cash-Flow-Step-2-Version-2.jpg","bigUrl":"\/images\/thumb\/4\/49\/Discount-Cash-Flow-Step-2-Version-2.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-2-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Determine the value of future cash flows. To calculate the present value of future cash flows, you will first need to know their future values. With fixed payments like annuities or bond coupon payments, these cash flows are set in stone; however, with cash flows from company operations or project returns you will need to estimate future cash flows, which is an entire calculation in itself. While it may seem that you could just project current growth trends over the next set of years, the proper calculation of future cash flows will involve much more.
For example, you might include industry trends, market conditions, and operational developments in cash flow projections for a company. Even then it may not be even close to accurate when the cash flows actually arrive.[2] X Research source
For simplicity, though, let's say you are considering an investment that will return you a set amount at the end each year for three years. Specifically, you will receive $1,000 the first year, $2,000 the second year, and $3,000 the third year.
The investment costs $5,000 to buy, and you want to know if it is a good investment based on the present value of the money you will receive.
Calculate your discount rate. The discount rate is used to "discount" the future cash flow value back to its present value. The discount rate, sometimes also called the personal rate of return, represents the amount that is "lost" each year due to inflation and missed investment opportunities. You might choose to use the return on a safe investment, plus a risk premium.[3] X Research source
For example, imagine that instead of investing in the investment providing future cash flows, you could invest your money in treasuries earning a guaranteed return of 2 percent per year.
In addition, you expect to be compensated for taking the risk of loss of your money, say a risk premium of 7 percent.
Your discount rate would be the sum of these two figures, which is 9 percent. This represents the rate of return you would earn by investing your money elsewhere, such as in the stock market.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5c\/Discount-Cash-Flow-Step-4.jpg\/v4-460px-Discount-Cash-Flow-Step-4.jpg","bigUrl":"\/images\/thumb\/5\/5c\/Discount-Cash-Flow-Step-4.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Figure out the number of compounding periods. The only other variable you'll need once you have the discount rate and cash flow future values is the dates at which those cash flows will be received. This should be pretty self-explanatory if you've purchased an investment, have a set of structured payouts, or have created a model for a company's future cash flows; however, make sure to clearly record the cash flows with their associated years. Creating a chart may help you organize your ideas. For example, you might organize the example payouts as follows:[4] X Research source
Discounting Cash Flows Download Article
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/f\/fd\/Discount-Cash-Flow-Step-5.jpg\/v4-460px-Discount-Cash-Flow-Step-5.jpg","bigUrl":"\/images\/thumb\/f\/fd\/Discount-Cash-Flow-Step-5.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-5.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Set up your equation. In its simplest form, the DCF formula is
{\displaystyle {\text{DCF}}={\frac {CF_{n}}{(1+r)^{n}}}}
. In the formula,
{\displaystyle CF_{n}}
refers to the future value of the cash flow for year n and r represents the discount rate. For example, using the first year of the example investment from the part "Gathering Your Variables," the present value of that cash flow for $1,000 after one year, using the discount rate of 9 percent, would be represented as:
{\displaystyle {\text{DCF}}={\frac {\$1,000}{(1+0.09)^{1}}}}
The discount rate must be represent as a decimal rather than by a percentage. This is done by dividing the discount rate by 100. Therefore, the 9 percent rate from above is shown as 0.09 (
{\displaystyle 9\div 100}
) in the equation.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/53\/Discount-Cash-Flow-Step-6.jpg\/v4-460px-Discount-Cash-Flow-Step-6.jpg","bigUrl":"\/images\/thumb\/5\/53\/Discount-Cash-Flow-Step-6.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-6.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Add up all discounted cash flows. The total value of discounted cash flows for an investment is calculated as the present values of each cash flow. So, the other cash flows must be added to the calculation in the same method as the first one. For the previous example, we would add the $2,000 and $3,000 payments at the end of the second and third years to the equation.[6] X Research source
In total, this gives:
{\displaystyle {\text{DCF}}={\frac {\$1,000}{(1+0.09)^{1}}}+{\frac {\$2,000}{(1+0.09)^{2}}}+{\frac {\$3,000}{(1+0.09)^{3}}}}
Arrive at the discounted value. Solve your equation to get your total discounted value. The result will be the present value of your future cash flows. Start by adding the discounted rate to the 1 within parentheses:[7] X Research source
{\displaystyle {\text{DCF}}={\frac {\$1,000}{(1.09)^{1}}}+{\frac {\$2,000}{(1.09)^{2}}}+{\frac {\$3,000}{(1.09)^{3}}}}
From there, calculate the exponent. This is done by raising the "1.09" in parentheses to the power above it (1,2, or 3). Solve this by either typing "[lower value]^[exponent]" into Google or using the exponent button,
{\displaystyle x^{y}}
on a calculator.
After solving the exponent, the equation will be:
{\displaystyle {\text{DCF}}={\frac {\$1,000}{1.09}}+{\frac {\$2,000}{1.1881}}+{\frac {\$3,000}{1.295029}}}
Next, divide each cash flow by the number underneath it. This yields:
{\displaystyle {\text{DCF}}=\$917.43+\$1,683.36+\$2,316.55}
Finally, add up the present values to get the total, which is
{\displaystyle {\text{DCF}}=\$4,917.34}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b8\/Discount-Cash-Flow-Step-8.jpg\/v4-460px-Discount-Cash-Flow-Step-8.jpg","bigUrl":"\/images\/thumb\/b\/b8\/Discount-Cash-Flow-Step-8.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-8.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Adjust your discount rate. In some cases it may be necessary to change the discount rate used to account for changes to expectations, risk, or taxes. For example, businesses analyzing a project might add a risk premium on to the discount rate used to discount the cash flows from a risky project. This artificially lowers the returns to account for risk. The same might be done for a very long time window between the present and the future cash flows to account for uncertainty.
Discount rates may be converted to real rates (rather than nominal rates) by removing inflation from the discount rate.
A spreadsheet program, such as Excel, has functions that can help with these calculations.
Using Discounted Cash Flows Download Article
Analyze your result. To use your DCF result, you will need to understand what your figures represent. Your total DCF is the sum of the present values of future payments. That is, if you received an amount equivalent to your future payments today it would be the total DCF value; therefore, you can now compare future amounts of money directly to the present cost of investing to get that money.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/bd\/Discount-Cash-Flow-Step-10.jpg\/v4-460px-Discount-Cash-Flow-Step-10.jpg","bigUrl":"\/images\/thumb\/b\/bd\/Discount-Cash-Flow-Step-10.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-10.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Evaluate an investment. In general, DCF calculations are used to discount cash flows from an investment to see if that investment is worthwhile. This is done by comparing the value of buying into the investment to the present value of its future cash flows. If the present value of the future cash flows is higher than the cost of investing, it may be a good investment. If they are lower, you will be effectively losing money.
For example, in the example used in the other two parts, you had the option of buying an investment that would pay $6,000 total over three years ($1,000 + $2,000 +$3,000) at an initial investment cost of only $5,000.
While this may seem like a good deal, you can see that, using a discount rate of 9 percent, you are better off investing your money elsewhere.
This is because the present value of the cash flows, $4,917.34 is lower than the cost of the investment, $5,000.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5b\/Discount-Cash-Flow-Step-11.jpg\/v4-460px-Discount-Cash-Flow-Step-11.jpg","bigUrl":"\/images\/thumb\/5\/5b\/Discount-Cash-Flow-Step-11.jpg\/aid1800540-v4-728px-Discount-Cash-Flow-Step-11.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Use discounted cash flows for company valuation. In finance, DCF calculations are used for DCF analysis, which is a method used to assess the value of a company. In this method, the company's free cash flows are as estimated for the next five or ten years and a "terminal value" are discounted back to the present. The present value of these amounts is then used as the "enterprise value" of company. Then, debt is removed from the enterprise value to arrive at a valuation for the company.[8] X Research source
↑ http://www.investopedia.com/terms/d/dcf.asp
↑ http://www.morningstar.co.uk/uk/news/65385/the-discounted-cash-flow-method.aspx
↑ https://www.thestreet.com/story/10385275/1/getting-started-with-discounted-cash-flows.html
↑ https://www.business-case-analysis.com/discounted-cash-flow.html
↑ http://macabacus.com/valuation/dcf/overview
Español:descontar los flujos de efectivo
Italiano:Calcolare il Discounted Cash Flow
"Thanks, this was a very good and very clear explanation. You broke it down into simple steps, and you kept the language simple."..." more
Caroline Twesigye
"The simplicity and clarity with which the DCF concept was explained stood out for me."
|
Solar radius is a unit of distance used to express the size of stars in astronomy relative to the Sun. The solar radius is usually defined as the radius to the layer in the Sun's photosphere where the optical depth equals 2/3[citation needed]:
{\displaystyle 1\,R_{\odot }=6.957\times 10^{8}{\hbox{ m}}}
695,700 kilometres (432,300 miles) is approximately 10 times the average radius of Jupiter, about 109 times the radius of the Earth, and 1/215th of an astronomical unit, the distance of the Earth from the Sun. It varies slightly from pole to equator due to its rotation, which induces an oblateness in the order of 10 parts per million.[1]
2 Nominal solar radius
Evolution of the solar luminosity, radius and effective temperature compared to the present-day Sun. After Ribas (2009)[2]
The unmanned SOHO spacecraft was used to measure the radius of the Sun by timing transits of Mercury across the surface during 2003 and 2006. The result was a measured radius of 696,342 ± 65 kilometres (432,687 ± 40 miles).[3]
Haberreiter, Schmutz & Kosovichev (2008)[4] determined the radius corresponding to the solar photosphere to be 695,660 ± 140 kilometres (432,263 ± 87 miles). This new value is consistent with helioseismic estimates; the same study showed that previous estimates using inflection point methods had been overestimated by approximately 300 km (190 mi).
Nominal solar radius[edit]
In 2015, the International Astronomical Union passed Resolution B3, which defined a set of nominal conversion constants for stellar and planetary astronomy. Resolution B3 defined the nominal solar radius (symbol
{\displaystyle R_{\odot }^{N}}
) to be equal to exactly 695700 km.[5] The nominal value, which is the rounded value, within the uncertainty, given by Haberreiter, Schmutz & Kosovichev (2008), was adopted to help astronomers avoid confusion when quoting stellar radii in units of the Sun's radius, even when future observations will likely refine the Sun's actual photospheric radius (which is currently[6] only known to about an accuracy of ±100–200 km).
Solar radii as a unit are common when describing spacecraft moving close to the sun. Two spacecraft in the 2010s include:
Solar Orbiter (as close as 45 R☉)
Parker Solar Probe (as close as 9 R☉)
^ NASA RHESSI oblateness measurements 2012
^ Ribas, Ignasi (August 2009). "The Sun and Stars as the Primary Energy Input in Planetary Atmospheres" (PDF). Proceedings of the International Astronomical Union. 5 (S264 [Solar and Stellar Variability: Impact on Earth and Planets]): 3–18. arXiv:0911.4872. Bibcode:2010IAUS..264....3R. doi:10.1017/S1743921309992298. S2CID 119107400.
^ Emilio, Marcelo; Kuhn, Jeff R.; Bush, Rock I.; Scholl, Isabelle F. (2012), "Measuring the Solar Radius from Space during the 2003 and 2006 Mercury Transits", The Astrophysical Journal, 750 (2): 135, arXiv:1203.4898, Bibcode:2012ApJ...750..135E, doi:10.1088/0004-637X/750/2/135, S2CID 119255559
^ Haberreiter, M; Schmutz, W; Kosovichev, A.G. (2008), "Solving the Discrepancy between the Seismic and Photospheric Solar Radius", Astrophysical Journal, 675 (1): L53–L56, arXiv:0711.2392, Bibcode:2008ApJ...675L..53H, doi:10.1086/529492, S2CID 14584860
^ Mamajek, E.E.; Prsa, A.; Torres, G.; et, al. (2015), IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties, arXiv:1510.07674, Bibcode:2015arXiv151007674M
^ Meftah, M; Corbard, T; Hauchecorne, A.; Morand, F.; Ikhlef, R.; Chauvineau, B.; Renaud, C.; Sarkissian, A.; Damé, L. (2018), "Solar radius determined from PICARD/SODISM observationsand extremely weak wavelength dependence in the visibleand the near-infrared", Astronomy & Astrophysics, 616: A64, Bibcode:2018A&A...616A..64M, doi:10.1051/0004-6361/201732159
S. C. Tripathy; H. M. Antia (1999). "Influence of surface layers on the seismic estimate of the solar radius". Solar Physics. 186 (1/2): 1–11. Bibcode:1999SoPh..186....1T. doi:10.1023/A:1005116830445. S2CID 118037693.
T. M. Brown; J. Christensen-Dalsgaard (1998). "Accurate Determination of the Solar Photospheric Radius". Astrophysical Journal Letters. 500 (2): L195. arXiv:astro-ph/9803131. Bibcode:1998ApJ...500L.195B. doi:10.1086/311416. S2CID 13875360.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Solar_radius&oldid=1046468449"
|
(→P-RLOCAL: Polylogarithmic rounds in the Randomized LOCAL model)
Drwadu (talk | contribs)
<li>Π<sub>i</sub>P = [[Complexity Zoo:C#conp|coNP]] with Σ<sub>i-1</sub>P oracle.</li>
Then PH is the union of these classes for all nonnegative constants i.
PH can also be defined using alternating quantifiers: it's the class of problems of the form, "given an input x, does there exist a y such that for all z, there exists a w ... such that φ(x,y,z,w,...)," where y,z,w,... are polynomial-size strings and φ is a polynomial-time computable predicate. It's not totally obvious that this is equivalent to the first definition, since the first one involves adaptive [[Complexity Zoo:N#np|NP]] oracle queries and the second one doesn't, but it is.
{\displaystyle k}
{\displaystyle n^{O(1)}}
It was later shown that, if NP is contained in P/poly, then PH collapses to ZPPNP [KW98] and indeed to O2P [CR06] (which is unconditionally included in P/poly). This seems close to optimal, since there exists an oracle relative to which the collapse cannot be improved to Δ2P [Wil85].
{\displaystyle O(n)}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle F:X_{1}\times X_{2}\times \cdots \times X_{k}\to \{0,1\}}
{\displaystyle i\in [1..k]}
{\displaystyle X_{k}\in \{0,1\}^{n}}
{\displaystyle F}
{\displaystyle k}
{\displaystyle X_{i}}
{\displaystyle O\left(\mathrm {poly} (\log n)\right)}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle {\mathsf {P}}^{\mathrm {SAT} [1]}={\mathsf {P}}^{\mathrm {SAT} [2]}\Rightarrow {\mathsf {PH}}\subseteq {\mathsf {NP}}}
P-LOCAL: Polylogarithmic rounds in the LOCAL model
The family of locally checkable problems that can be solved by deterministic algorithms in poly(log n) rounds of the LOCAL model of computation on an n node graph.
In the local model of computation, there is a graph with n vertices, each with its own, unique Θ(log(n)) bit ID. Each vertex starts with the state of knowing only its ID and has unbounded, deterministic, computational ability. At each round, every vertex synchronously sends and receives one unbounded message from each neighbor. At the end of the algorithm, each vertex gives an output satisfying the problem. For instance, the vertices may output a valid coloring.
Proven in [RG20] that P-LOCAL = P-RLOCAL.
P-RLOCAL: Polylogarithmic rounds in the Randomized LOCAL model
The family of locally checkable problems that can be solved by randomized algorithms in poly(log n) rounds of the LOCAL model of computation on an n node graph.
The same as P-LOCAL, except now the vertices may use randomness in the messages they send and their final output, and must give a valid output with probability at least 1 - 1/n.
{\displaystyle c}
{\displaystyle 2^{c}}
{\displaystyle 2^{n^{O(1)}}}
{\displaystyle n^{O(1)}}
|
The Goal: A Business Graphic Novel by Eliyahu Goldratt | James's Knowledge Graph
The Goal: A Business Graphic Novel is an adaptation of the book The Goal: A Process of Ongoing Improvement by Dwight Jon Zimmerman and Dean Motter.
Key Lessons from The Goal
The Goal is to make money (this is a business book, after all). To remain competitive, businesses must continuously and systematically improve their operations. The theory of constraints provides a model to continuously identify and improve systems as they relate to The Goal.
The theory of constraints defines a model of how cash flows through a system:
Throughput is the rate at which a system generates money through sales.
Inventory is all of the money a system has invested in purchasing things it intends to sell but hasn't sold; this includes anything from raw materials to finished products.
Operational expenses are all of the money a system spends in order to convert inventory to throughput.
In short: Throughput is money in, inventory is money out, and operational expenses are money that is "stuck" in the system.
The process of converting inventory to throughput is constrained by one or more bottlenecks. A bottleneck is a resource with less capacity than the demand placed upon it. Bottlenecks aren't good or bad, they're reality.
"The Goal" (to make money) therefore translates to increasing throughput while reducing inventory and operational expenses simultaneously. Numbers not based on the constraints of the system are meaningless; producing work and profiting from it are two very different things.
A critical mistake many companies make is to optimize throughput, inventory, or operational expenses in isolation which can harm the system as a whole. For example, a reduction in operational expenses may look like a success "on paper" but if it leads to quality issues and an increase in returns, the system is not improved.
Continuous Improvement with the Theory of Constraints
Time lost on a bottleneck directly equates to reduced throughput and includes defects produced both by the bottleneck and prior to the bottleneck because defective throughput will have to pass through the bottleneck again.
Bottleneck cost (
c
) is the system's total expenses (
\$
), divided by the bottleneck's production hours (
h
c = \$/h
. For example, if a system's total expenses are $1,000,000/year and a bottleneck is operational for 7,200 hours/year, the bottleneck cost is
\$1,000,000/7,200 hours = ~\$139/year
. Note that as utilization of the bottleneck decreases, the cost to the system increases.
Ways to Get the Most from a Bottleneck
Reduce defects processed by the bottleneck
Reduce idle time of the bottleneck (but not at the expense of maintenance!)
Prioritize only what contributes to throughput, not to inventory
Distribute or reduce the load through alternative processes
It's okay to slow down steps that proceed the bottleneck to reduce defects sent to the bottleneck or to "get around" the bottleneck, as long as those steps do not become slower than the bottleneck (thus becoming the bottleneck).
When a bottleneck's predecessor exceeds throughput, inventory is created; inventory increases tend to increase operational expenses as well. Running a non-bottleneck at maximum capacity is therefore a waste. Only bottlenecks should be utilized to full capacity. To "subordinate non-bottleneck resources to bottlenecks" is to run non-bottleneck resources at the rate of throughput as constrained by the bottleneck.
Theory of Constraints: Focus Steps
graph TB Identify[1. Identify
Constraints] --> Decide[2. Decide how to
fully utilize the
constraints] Decide --> Subordinate[3. Subordinate
everything else to
that decision] Subordinate --> Elevate[4. Elevate the
constraints] Elevate -->|Step five: Go back
to step one| Identify
Broader Topics Related to The Goal: A Business Graphic Novel by Eliyahu Goldratt
Methods of minimizing costs and maximizing profit
The Goal: A Business Graphic Novel by Eliyahu Goldratt Knowledge Graph
|
Fluorescence - New World Encyclopedia
Previous (Fluorescein)
Next (Fluorescent lamp)
Fluorescence is a luminescence that is mostly found as an optical phenomenon in cold bodies, in which the molecular absorption of a photon at a certain wavelength triggers the emission of another photon with a longer wavelength. The substance that fluoresces is called a fluorophore. The energy difference between the absorbed and emitted photons ends up as molecular vibrations or heat. Usually the absorbed photon is in the ultraviolet range and the emitted light is in the visible range, but this depends on the fluorophore used and other factors.
1 Examples of fluorescent materials
2.3 Biochemistry and medicine
4.2 Fluorescence quantum yield
4.3 Fluorescence lifetime
Fluorescence is named after the mineral fluorite, composed of calcium fluoride, which often exhibits this phenomenon. A variety of other minerals and organic materials also fluoresce, and they are used for a number of different applications. For example, fluorescence is useful for lighting and tagging molecules in analytical chemistry and biochemistry. Fluorophores have been used to label cells, antibodies, and other biological structures, and to determine their structures and modes of action.
Examples of fluorescent materials
Gemstones, minerals, fibers, and many other materials which that be encountered in forensics or with a relationship to various collectibles may have a distinctive fluorescence or may fluoresce differently under short-wave ultraviolet, long-wave ultraviolet, or X-rays.
Many types of calcite and amber will fluoresce under shortwave UV. Rubies, emeralds, and the Hope Diamond exhibit red fluorescence under short-wave UV light; diamonds also emit light under X-ray radiation.
Crude oil (petroleum) fluoresces in a range of colors, from dull brown for heavy oils and tars through to bright yellowish and bluish white for very light oils and condensates. This phenomenon is used in oil exploration drilling to identify very small amounts of oil in drill cuttings and core sample.
Organic liquids such as mixtures of anthracene in benzene or toluene, or stilbene in the same solvents, fluoresce with ultraviolet or gamma ray irradiation. The decay times of this fluorescence is of the order of nanoseconds since the duration of the light depends on the lifetime of the excited states of the fluorescent material, in this case anthracene or stilbene.
There are many natural and synthetic compounds that exhibit fluorescence, and they have a number of applications. Some deep-sea animals, such as the Greeneye, use fluorescence.
The common fluorescent tube relies on fluorescence. Inside the glass tube is a partial vacuum and a small amount of mercury. An electric discharge in the tube causes the mercury atoms to emit light. The emitted light is in the ultraviolet (UV) range, is invisible, and is harmful to most living organisms. The tube is lined with a coating of a fluorescent material, called the phosphor, which absorbs the ultraviolet and re-emits visible light. Fluorescent lighting is very energy efficient compared to incandescent technology, but the spectra produced may cause certain colors to appear unnatural.
In the mid 1990s, white light-emitting diodes (LEDs) became available, which work through a similar process. Typically, the actual light-emitting semiconductor produces light in the blue part of the spectrum, which strikes a phosphor compound deposited on the chip; the phosphor fluoresces from the green to red part of the spectrum. The combination of the blue light that goes through the phosphor and the light emitted by the phosphor produce a net emission of white light.
The modern mercury vapor streetlight is said to have been evolved from the fluorescent lamp.
Glow sticks oxidize phenyl oxalate ester to produce light.
Compact fluorescent lighting (CFL) is the same as any typical fluorescent lamp with advantages. It is self-ballasted and used to replace incandescents in most applications. They produce a quarter of the heat per lumen as incandescent bulbs and last about five times as long. These bulbs contain mercury and must be handled and disposed with care.
Fluorescence in several wavelengths can be detected by an array detector, to detect compounds from HPLC flow. Also, thin layer chromatography (TLC) plates can be visualized if the compounds or a coloring reagent is fluorescent.
Fingerprints can be visualized with fluorescent compounds such as ninhydrin.
Biological molecules can be tagged with a fluorescent chemical group (fluorophore) by a simple chemical reaction, and the fluorescence of the tag enables sensitive and quantitative detection of the molecule. Examples include:
Fluorescence microscopy of tissues, cells or subcellular structures is accomplished by labeling an antibody with a fluorophore and allowing the antibody to find its target antigen within the sample. Labeling multiple antibodies with different fluorophores allows visualization of multiple targets within a single image.
Automated sequencing of DNA by the chain termination method; each of four different chain terminating bases has its own specific fluorescent tag. As the labeled DNA molecules are separated, the fluorescent label is excited by a UV source, and the identity of the base terminating the molecule is identified by the wavelength of the emitted light.
DNA detection: the compound ethidium bromide, when free to change its conformation in solution, has very little fluorescence. Ethidium bromide's fluorescence is greatly enhanced when it binds to DNA, so this compound is very useful in visualizing the location of DNA fragments in agarose gel electrophoresis. Ethidium bromide can be toxic; a safer alternative is the dye SYBR Green.
The DNA microarray
Immunology: An antibody has a fluorescent chemical group attached, and the sites (e.g., on a microscopic specimen) where the antibody has bound can be seen, and even quantified, by the fluorescence.
FACS (fluorescent-activated cell sorting)
Fluorescence has been used to study the structure and conformations of DNA and proteins with techniques such as Fluorescence resonance energy transfer, which measures distance at the angstrom level. This is especially important in complexes of multiple biomolecules.
Aequorin, from the jellyfish Aequorea victoria, produces a blue glow in the presence of Ca2+ ions (by a chemical reaction). It has been used to image calcium flow in cells in real time. The success with aequorin spurred further investigation of A. victoria and led to the discovery of Green Fluorescent Protein (GFP), which has become an extremely important research tool. GFP and related proteins are used as reporters for any number of biological events including such things as sub-cellular localization. Levels of gene expression are sometimes measured by linking a gene for GFP production to another gene.
Also, many biological molecules have an intrinsic fluorescence that can sometimes be used without the need to attach a chemical tag. Sometimes this intrinsic fluorescence changes when the molecule is in a specific environment, so the distribution or binding of the molecule can be measured. Bilirubin, for instance, is highly fluorescent when bound to a specific site on serum albumin. Zinc protoporphyrin, formed in developing red blood cells instead of hemoglobin when iron is unavailable or lead is present, has a bright fluorescence and can be used to detect these problems.
As of 2006, the number of fluorescence applications is growing in the biomedical biological and related sciences. Methods of analysis in these fields are also growing, albeit with increasingly unfortunate nomenclature in the form of acronyms such as: FLIM, FLI, FLIP, CALI, FLIE, FRET, FRAP, FCS, PFRAP, smFRET, FIONA, FRIPS, SHREK, SHRIMP, TIRF. Most of these techniques rely on fluorescence microscopes. These microscopes use high intensity light sources, usually mercury or xenon lamps, LEDs, or lasers, to excite fluorescence in the samples under observation. Optical filters then separate excitation light from emitted fluorescence, to be detected by eye, or with a (CCD) camera or other light detectors (photomultiplier tubes, spectrographs, etc). Much research is underway to improve the capabilities of such microscopes, the fluorescent probes used, and the applications they are applied to. Of particular note are confocal microscopes, which use a pinhole to achieve optical sectioning—affording a quantitative, 3D view of the sample.
Fluorescent bulbs create far less waste heat than incandescent and halogen bulbs. Halogen bulbs are implicated in a large number of fires, and incandescent bulbs also carry a higher risk of fire than fluorescent bulbs, due to waste heat. Lamps may topple accidentally, or sometimes by events such as earthquakes. Using fluorescent bulbs can thus be a means of preventing accidental fires. However, fluorescent bulbs may contain mercury, and breakage of such a bulb could result in a costly mercury spill.
Fluorescence occurs when a molecule or quantum dot relaxes to its ground state after being electronically excited.
Excitation:
{\displaystyle S_{0}+h\nu \to S_{1}}
Fluorescence (emission):
{\displaystyle S_{1}\to S_{0}+h\nu }
{\displaystyle h\nu }
is a generic term for photon energy where: h = Planck's constant and
{\displaystyle \nu }
= frequency of light. (The specific frequencies of exciting and emitted light are dependent on the particular system.)
State S0 is called the ground state of the fluorophore (fluorescent molecule) and S1 is its first (electronically) excited state.
A molecule in its excited state, S1, can relax by various competing pathways. It can undergo 'non-radiative relaxation' in which the excitation energy is dissipated as heat (vibrations) to the solvent. Excited organic molecules can also relax via conversion to a triplet state which may subsequently relax via phosphorescence or by a secondary non-radiative relaxation step.
Relaxation of an S1 state can also occur through interaction with a second molecule through fluorescence quenching. Molecular oxygen (O2) is an extremely efficient quencher of fluorescence because of its unusual triplet ground state.
Molecules that are excited through light absorption or via a different process (e.g. as the product of a reaction) can transfer energy to a second 'sensitized' molecule, which is converted to its excited state and can then fluoresce. This process is used in lightsticks.
The fluorescence quantum yield gives the efficiency of the fluorescence process. It is defined as the ratio of the number of photons emitted to the number of photons absorbed.
{\displaystyle \Phi ={\frac {\rm {\#\ photons\ emitted}}{\rm {\#\ photons\ absorbed}}}}
The maximum fluorescence quantum yield is 1.0 (100 percent); every photon absorbed results in a photon emitted. Compounds with quantum yields of 0.10 are still considered quite fluorescent. Another way to define the quantum yield of fluorescence, is by the rates excited state decay:
{\displaystyle {\frac {{k}_{f}}{\sum _{i}{k}_{i}}}}
{\displaystyle {k}_{f}}
is the rate of spontaneous emission of radiation and
{\displaystyle \sum _{i}{k}_{i}}
is the sum of all rates of excited state decay. Other rates of excited state decay are caused by mechanisms other than photon emission and are therefore often called "non-radiative rates," which can include: dynamic collisional quenching, near-field dipole-dipole interaction (or resonance energy transfer), internal conversion and intersystem crossing. Thus, if the rate of any pathway changes, this will affect both the excited state lifetime and the fluorescence quantum yield.
Fluorescence quantum yield are measured by comparison to a standard with known quantology; the quinine salt, quinine sulfate, in a sulfuric acid solution is a common fluorescence standard.
The fluorescence lifetime refers to the average time the molecule stays in its excited state before emitting a photon. Fluorescence typically follows first-order kinetics:
{\displaystyle \left[S1\right]=\left[S1\right]_{0}e^{-\Gamma t},}
{\displaystyle \left[S1\right]}
is the concentration of excited state molecules at time
{\displaystyle t}
{\displaystyle \left[S1\right]_{0}}
is the initial concentration and
{\displaystyle \Gamma }
is the decay rate or the inverse of the fluorescence lifetime. This is an instance of exponential decay. Various radiative and non-radiative processes can de-populate the excted state. In such case the total decay rate is the sum over all rates:
{\displaystyle \Gamma _{tot}=\Gamma _{rad}+\Gamma _{nrad}}
{\displaystyle \Gamma _{tot}}
is the total decay rate,
{\displaystyle \Gamma _{rad}}
the radiative decay rate and
{\displaystyle \Gamma _{nrad}}
the non-radiative decay rate. It is similar to a first-order chemical reaction in which the first-order rate constant is the sum of all of the rates (a parallel kinetic model). If the rate of spontaneous emission, or any of the other rates are fast, the lifetime is short. For commonly used fluorescent compounds typical excited state decay times for fluorescent compounds that emit photons with energies from the UV to near infrared are within the range of 0.5 to 20 nanoseconds. The fluorescence lifetime is an important parameter for practical applications of fluorescence such as fluorescence resonance energy transfer.
There are several rules that deal with fluorescence. The Kasha–Vavilov rule dictates that the quantum yield of luminescence is independent of the wavelength of exciting radiation.
This rule is not always valid and is violated severely in many simple molecules. A somewhat more reliable statement, although still with exceptions, is that the fluorescence spectrum shows very little dependence on the wavelength of the exciting radiation.
Lakowicz, Joseph R. 2006. Principles of Fluorescence Spectroscopy, 3rd ed. New York: Springer. ISBN 978-0387312781
Turro, Nicholas J. 1991. Modern Molecular Photochemistry. Mill Valley, CA: University Science Books. ISBN 0935702717
Valeur, Bernard. 2002. Molecular Fluorescence: Principles and Applications. Weinheim: Wiley-VCH. ISBN 352729919X
Fluorophores.org The database of fluorescent dyes
Fluorescence on Scienceworld
History of "Fluorescence"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Fluorescence&oldid=1004322
|
Home : Support : Online Help : Mathematics : Algebra : Magma : DirectProduct
compute the direct product of two magmas
DirectProduct( A, B )
The direct product of two magmas A and B is the set of pairs (a,b), with a in A and b in B, and with binary operation defined componentwise.
The DirectProduct( A, B ) command returns the Cayley table of the direct product of the magmas A and B.
\mathrm{with}\left(\mathrm{Magma}\right):
A≔〈〈〈1|2〉,〈2|1〉〉〉
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
B≔〈〈〈1|1|2〉,〈2|3|2〉,〈3|2|1〉〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\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}{3}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{DirectProduct}\left(A,B\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{AreIsomorphic}\left(\mathrm{DirectProduct}\left(A,B\right),\mathrm{DirectProduct}\left(B,A\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
C≔〈〈〈1|2|3〉,〈2|3|1〉,〈3|1|2〉〉〉
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{AreIsomorphic}\left(\mathrm{DirectProduct}\left(\mathrm{DirectProduct}\left(A,B\right),C\right),\mathrm{DirectProduct}\left(A,\mathrm{DirectProduct}\left(B,C\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The Magma[DirectProduct] command was introduced in Maple 2016.
|
Revision as of 10:04, 6 February 2021 by Timeroot (talk | contribs) (Adding QRL, quantum finite automata.)
The class of decision problems for which a "yes" answer can be verified by a quantum computer with access to a classical proof. Also known as the subclass of of QMA with classical witnesses. Defined in [AN02].
In [BT09], the authored presented a QMAlog(2) (that is, the length of the certificates is logarithmic in the size of the problem -- see, e.g., QMAlog) protocol for 3-Coloring, with perfect completeness and 1-1/poly(n) soundness. Note that a similar construction for QMA is highly unlikely, since it would imply NP is contained in QMAlog=BQP. An analogous result, with constant soundness but with quadratic proof length was shown in [ABD+08]. It was shown shown there that a conjecture they call the Strong Amplification Conjecture implies that QMA(2) is contained in PSPACE. The authors also show that there exists no perfectly disentangler that can be used to simulate QMA(2) in QMA, though other approaches to showing QMA = QMA(2) may still exist.
{\displaystyle k\geq 4}
QRL: Quantum Regular Languages
The class of problems nondeterministically recognized by Moore-Crutchfield Quantum Finite Automata. These are automata that undergo a unitary transformation for each symbol consumed, and at the end are observed to be in an accepting state or not. The "nondeterminism" refers to the fact that the automaton will accept with positive probability if in the language, or with exactly zero probability otherwise.
Initially studied in <ref>Cristopher Moore and James P. Crutchfield. Quantum automata and quantum grammars.Theoretical ComputerScience, 237(1-2):275–306, 2000.</ref>, although they described QRL as probabilities over words, rather a language in the strict sense. Several important properties established by <ref>Alberto Bertoni and Marco Carpentieri. Analogies and differences between quantum and stochastic automata.Theoretical Computer Science, 262(1-2):69–81, 2001</ref>, where it was shown that QRL has no finite-size languages, but also contains several non-regular languages.
Also called NMCL, in <ref>Abuzer Yakaryilmaz, A. C. Cem Say. Languages recognized by nondeterministic quantum finite automata. https://arxiv.org/abs/0902.2081</ref>, in order to disambiguate from NQL.
Contrast with NQL, which is an analogous class for a different model of quantum finite automaton: there the automaton is measured after each step for acceptance or rejection.
{\displaystyle k\geq 2}
{\displaystyle k=1}
|
Birthday Paradox Practice Problems Online | Brilliant
Three friends find out their birthdays are all within the same week.
Supposing their birthdays are otherwise random, what is the probability they all have their birthday on the same day?
\frac{1}{7}
\frac{3}{49}
\frac{3}{7^3}
\frac{1}{49}
Thirty people all have their birthdays in November (which has 30 days).
If their birthdays are otherwise random, what is the probability that none of them share a birthday?
\frac{1}{30^{30}}
\frac{30!}{30^{30}}
\frac{29}{30}
\frac{1}{30!}
Suppose a computer system locked via a 32-bit password (so there are
2^{32}
possible numbers). The passwords are generated randomly but stored in such a way that the system is vulnerable if a password is a duplicate of another one.
What's the probability of a password duplicate with 50,000 users?
1 - \frac{(2^{32})!}{((2^{32}))^{50000}(50000)!}
1 - \frac{(2^{32})!}{((2^{32}))(2^{32} - 50000)!}
1 - \frac{(2^{32})!}{((2^{32}))(2^{32})!}
1 - \frac{(2^{32})!}{((2^{32}))^{50000}(2^{32} - 50000)!}
Suppose 5 people meet whose birthdays all fall within a span of 7 days. Supposing their birthdays are otherwise randomly distributed, is the probability greater or less than 50% that at least two people will share a birthday?
Minah is trying to solve the problem below:
2 people, Angel and Bob, have birthdays in September (which has 30 days). Assuming their birthdays are otherwise random, what is the probability they have the same birthday?
Minah's argument goes like this: Angel's birthday can be any of the days. We want to know the chances that Bob's birthday matches Angel's. Each day has a
\frac{1}{30}
chance of being Bob's birthday. So Bob's chance of sharing a birthday with Angel is
\frac{1}{30} .
Is this argument correct?
|
Call and Put Spreads | Brilliant Math & Science Wiki
A call spread is an option strategy in which a call option is bought, and another less expensive call option is sold. A put spread is an option strategy in which a put option is bought, and another less expensive put option is sold. As the call and put options share similar characteristics, this trade is less risky than an outright purchase, though it also offers less of a reward.
These strategies are useful to pursue if you believe that the underlying price would move in a particular direction, and want to reduce your initial outlay if the prediction is incorrect.
Call spreads and Put spreads
Conditions for position
A call spread refers to buying a call on a strike, and selling another call on a higher strike of the same expiry.
A put spread refers to buying a put on a strike, and selling another put on a lower strike of the same expiry.
Most often, the strikes of the spread are on the same side of the underlying (i.e. both higher, or both lower).
An investor buys the 30-35 call spread for $2. At what price must the stock expire at for him to have made money on this trade?
If the expiration price was below $30, then both of the call would have expired worthless, and the investor would be down by the $2 premium, hence would have lost money.
If the expiration price was below $32 and above $30, then the $30 call would have expired and be worth less than $2, and the $35 call would have expired worthless, and the investor has to pay the $2 premium, hence it would have been a loss.
If the expiration price was below $35 and above $32, then the $30 call would have expired and be worth more than $2, and the $35 call would have expired worthless, and the investor has to pay the $2 premium, hence it would have been a gain.
If the expiration price was above $35, then the $30 call would be worth $5 more than the $30 call, less the $2 premium, hence the investor would have gained
\$5 - \$2 = \$3
Thus, if the stock expires at $32 and above, the investor would have made money.
An investor bought the $50 call for $4, and sold the $53 call for $2. What is the minimum price of the stock on expiration, in order for the investor to have not lost money on this trade?
Assume that no other trades were made. Ignore transaction costs and interest rates.
Most often, this strategy involves an equal number of options on each strike. Though, sometimes people sell proportionally more of the further strike. These are known as 1 by 2, or even 2 by 3, call/put spreads.
If you own a (1 by 1) call spread on a stock, your potential profit is unlimited.
Note: Compare with Calls.
A long call spread is
1. Always long delta
2. Gamma, Vega, Theta depends on the position of the underlying in relation to the strikes.
3. (Typically) Long skew risk
4. Limited profit potential
A long put spread is
1. Always short delta
3. (Typically) Short skew risk
Theta neutral, net option position is 0. Paying Theta, OTM options have less theta to collect. It depends on the time to expiry Collecting Theta, OTM options have more theta to collect
The stock is currently trading at 20. If you are long the 20-22 call spread (same expiry), what is your theta position?
Spreads are good to trade when you want to minimize risk, since these options often have complementary greeks. Arbitragers trade spreads with close strikes for edge on the trade, and then manage the position. Position takes like to trade spreads because the short option premium helps to offset the long option cost.
Consider buying call spreads in the following situations:
1. You believe that the underlying is going to move up.
2. You believe that the underlying is going to move up and after which volatility will come off (e.g. a news event)
3. You believe that the underlying is going to move up in a limited range.
4. You believe that the underlying will drop sharply in price, hence sold the underlying. In this case, the call spread will offer you protection against a small move up.
Consider buying put spreads in the following situations:
1. You believe that the underlying is going to move down and after which volatility will come off (e.g. a news event)
2. You believe that the underlying is going to move down in a limited range.
3. You believe that the underling is going to drop down sharply.
Note that if you believe that the underlying is going to drop down sharply, buying put spreads could be dangerous due to the sharp increase in volatility and slope as the future ticks down to your short put. In such a case, you should not be hedging your deltas, but instead let the future settle down.
Cite as: Call and Put Spreads. Brilliant.org. Retrieved from https://brilliant.org/wiki/call-and-put-spreads/
|
WIKIPEDIA TALK:WIKIPROJECT MATHEMATICS - Encyclopedia Information
Wikipedia talk:WikiProject Mathematics Information
7 Deletion of "Greatest integer" proposed
He is listed in Category:Icelandic mathematicians, but I was wondering whether he satisifes WP:PROF.-- SilverMatsu ( talk) 15:55, 2 May 2022 (UTC)[ reply]
He may not pass WP:NPROF, but as a member of the Icelandic Parliament should be notable by WP:NPOLITICIAN. Felix QW ( talk) 17:10, 2 May 2022 (UTC)[ reply]
Thank you for your comment. I agree that this article meets WP: NPOLITICIAN, but I wondering whether it is possible to add Category: Icelandic mathematicians to this article.-- SilverMatsu ( talk) 16:52, 3 May 2022 (UTC)[ reply]
@ SilverMatsu Having a doctorate in mathematics does not make one a mathematician. Voting to not add to your proposed category unless you can document that he has produced substantial mathematical research output. PatrickR2 ( talk) 08:28, 5 May 2022 (UTC)[ reply]
Thank you for your comment. Removed Category:Icelandic mathematicians from article.-- SilverMatsu ( talk) 15:21, 5 May 2022 (UTC)[ reply]
Getting a PhD requires writing a PhD thesis, which requires producing substantial mathematical research output. Separately, I doubt you will find consensus for the idea that "producing substantial mathematical research output" is a necessary criterion to be classified as a mathematician. -- JBL ( talk) 17:10, 5 May 2022 (UTC)[ reply]
True about the need to write a PhD thesis, although I have personally witnessed a few cases where the corresponding research was not "substantial" (without mentioning any names, e.g., a case where the advisor moved to another school, the student was passed to another professor not really familiar with the area, and the student ended up putting in his thesis results that were not even original research but things that the first professor had mentioned and explained in one of his classes. The advising committee said it was pretty weak, but let him pass anyway, knowing he was going to go to a teaching school.) [Note I am not claiming this is the case here for Blondal, just mentioning in support of the point that a PhD does not a mathematician make.]. More of a case in point, looking at Math Genealogy project for example, you can see lots of new PhD's being granted every year. Quite a few of these don't stay in academics, move to industry, become programmers, work in finance, etc, either right after the PhD or after just a few years, realizing academics is not for them. I don't think anyone can say these people are mathematicians (not to diminish anything to what they might have done for their thesis). PatrickR2 ( talk) 04:44, 6 May 2022 (UTC)[ reply]
@ JayBeeEll and PatrickR2: Thank you for your(s) comments. Both agree to comments. I'd like WP: PROF will give an explicit explanation for this case, but as already pointed out, there seems to be no consensus, so I think it is better to decide each case individually.-- SilverMatsu ( talk) 03:25, 7 May 2022 (UTC)[ reply]
So the question is whether our mathematician-turned-politician was notable as a mathematician. And this notability may come from WP:NPROF, but in our case the subject is almost certainly not notable by WP:NPROF standards. So on that basis I would remove him from the category. Felix QW ( talk) 15:19, 7 May 2022 (UTC)[ reply]
WP:COP is just what I was looking for. Thank you so much !-- SilverMatsu ( talk) 01:09, 8 May 2022 (UTC)[ reply]
The article Hole (topology) seems to revolve around an idiosyncratic definition which is better subsumed in the homology (mathematics) and homotopy groups articles, the term hole is often used in a colloquial sense to give an idea of what these notions mean but presenting it as a formal notion as in done in the article seems to be counterproductive to me (and it is not supported by the given reference). I think the article should be deleted or made a redirect (probably to the article on homotopy groups or homotopical connectivity ; it may also make sense as a disambiguation page). jraimbau ( talk) 07:40, 3 May 2022 (UTC)[ reply]
Just to clarify: Did you check the offline reference (If not, I could do so at some point this week in our library)? Felix QW ( talk) 07:57, 3 May 2022 (UTC)[ reply]
Never mind, I managed to check it and you are right. It doesn't support the formal definition. Felix QW ( talk) 08:15, 3 May 2022 (UTC)[ reply]
I blanked and redirected. I'll notify the page creator, if they disagree i'll make an afd. jraimbau ( talk) 12:02, 8 May 2022 (UTC)[ reply]
I feel like the original redirect target was better. — JBL ( talk) 20:32, 8 May 2022 (UTC)[ reply]
The reference [1] defines "a hole in dimension
{\displaystyle \ell }
{\displaystyle S^{\ell }}
{\displaystyle f:S^{\ell }\to X}
{\displaystyle {\bar {f}}:B^{\ell }\to X}
{\displaystyle f:S^{\ell }\to X}
If you still think that the concept of a "hole" does not deserve a page on its own, then I think it is better to merge it into homotopical connectivity. -- Erel Segal ( talk) 04:48, 9 May 2022 (UTC)[ reply]
Since there was no objection, I changed the redirect to homotopical connectivity. -- Erel Segal ( talk) 11:24, 15 May 2022 (UTC)[ reply]
I do support the suggestion of putting the original redirect, to Hole#In_mathematics, back (i chose "homotopy groups" as a target because i was not aware of this original redirect). jraimbau ( talk) 16:03, 15 May 2022 (UTC)[ reply]
I agree wholeheartedly with Jean Raimbault and Felix QW -- the presentation of this as a formal definition of "hole" is deeply misleading at best, close to source falsification (even if not intentionally so). JBL ( talk) 17:16, 15 May 2022 (UTC)[ reply]
I have gone ahead and restored the earlier redirect. -- JBL ( talk) 17:17, 15 May 2022 (UTC)[ reply]
Question: the "main article" link is to Homotopy group, but the text at Hole#In_mathematics is about homology. (The latter usage is what I have always heard!) Should there be multiple main article links, and/or should the text under Hole in mathematics be adjusted? Russ Woodroofe ( talk) 17:38, 15 May 2022 (UTC)[ reply]
I added some stuff about homotopy in the hole article (i think the Matoušek quote given by Erel Segal actually fits perfectly there). jraimbau ( talk) 18:13, 15 May 2022 (UTC)[ reply]
Vertical alignment of \overrightarrow
{\displaystyle {\overrightarrow {PQ}}}
{\displaystyle {\overrightarrow {P}},}
{\displaystyle \left\|{\overrightarrow {PQ}}\right\|.}
{\displaystyle {\overrightarrow {PQ}}.}
Does anybody know some work-around, and/or asking for fixing the bug? D.Lazard ( talk) 16:51, 4 May 2022 (UTC)[ reply]
{\displaystyle {\vec {PQ}}.}
{\displaystyle \left\|{\vec {PQ}}\right\|.}
-- SilverMatsu ( talk) 15:11, 5 May 2022 (UTC)[ reply]
The image below is what I see when I look at the posting by D.Lazard above. I suspect others looking at that, including D.Lazaard, see something different because of different settings. @ D.Lazard: Is something "awfully aligned" about the arrows as they appear in this screenshot, or do you see something different when you look at the articles and at your own posting above? Michael Hardy ( talk) 17:44, 5 May 2022 (UTC)[ reply]
{\displaystyle {\overrightarrow {PQ}}}
, the bottom of P is aligned with the middle of text characters such as "n". For the displayed formula with brackets the bottom of PQ and the period are aligned with the middle of the brackets, when the formula should be centered with respect to the brackets. In both displayed formulas the upper part of the arrowhead is lacking. I ignore which sort of setting can produce this sort of display errors. D.Lazard ( talk) 20:01, 5 May 2022 (UTC)[ reply]
Apparently, this is a bug in "MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools)", as, when I change my math preferences to "PNG images", I get the same rendering as you. D.Lazard ( talk) 20:11, 5 May 2022 (UTC)[ reply]
I think the svg fallback is likely to look more or less the same as Michael Hardy's screenshot, so my guess is that Wikimedia thinks your browser can properly render mathml and is not using the fallback, but your browser's rendering of mathml is bad (as is most browsers' renderings of mathml). — David Eppstein ( talk) 21:40, 5 May 2022 (UTC)[ reply]
@ D.Lazard: Why don't you post a screenshot, as I did, so that we can tell what you're trying to say? In your comment saying "the upper part of the arrow is not displayed, as in", I see something displaying the entire arrow normally. You require us to take on faith that you see something of which you offer this verbal description but no image matching the discription, while at the same time you appear to intend to show us an image. It's not working at all. Michael Hardy ( talk) 18:09, 11 May 2022 (UTC)[ reply]
. Note that the connection between arrow and extension line is also misaligned. It looks bad. --{{u| Mark viking}} { Talk} 19:22, 11 May 2022 (UTC)[ reply]
This is a problem with the SVG rendering, you can tell this by looking at the code in the developer console. It seems like somewhere in the process, the vertical-align style attribute is incorrect. I've added a Phabricator bug T308188. -- Salix alba ( talk): 21:19, 11 May 2022 (UTC)[ reply]
Labeling "press recognition" and "a proof" based on Quanta & press releases
See Talk:Jinyoung Park (mathematician)#"Widespread recognition" for a content dispute about a recent arXiv preprint that has been discussed in a Quanta article ( link) and two department/institute press releases ( link 1, link 2).
Any input is much appreciated. Thanks! — MarkH21 talk 05:50, 9 May 2022 (UTC)[ reply]
Agree with MarkH21's comments at the link, particularly that the IAS and Stanford press releases are not good sources. The Quanta article is solid verification that the paper is of interest. But it is probably too early to say unambiguously that the result is proved, although (just making educated guess as non-expert in combinatorics) it seems very likely. Gumshoe2 ( talk) 06:39, 9 May 2022 (UTC)[ reply]
Thank you for the input here! Resolved. — Preceding unsigned comment added by Caleb Stanford ( talk • contribs)
Chinese Postman Problem and other arc routing Variants
I am reading a lot about the Chinese Postman problem, which is NP hard for mixed graphs that contain undirected edges and directed arcs. These arcs and edges can be weighted, and solving the mixed Chinese Postman is something I've been working on a lot recently its possible to fit everything about the Chinese Postman Problem in to the article Route Inspection. I would like to suggest a series or template on operations research and arc routing problems. I would like to make the documentation of the Chinese postman problem more comprehensive. ScientistBuilder ( talk) 23:24, 9 May 2022 (UTC)[ reply]
Need to rephrase Commutative property
I think this article needs to rephrase, for example, the section in Commutative property#Example made the reader, probably, confused to read what it means. It's not also be made confused to read, but it also didn't well written. I'm afraid that this GA will be delisted due to didn't meet one of the criteria of GA. Dedhert.Jr ( talk) 13:15, 11 May 2022 (UTC)[ reply]
Your concern is not written in coherent English, but it lacks references and does seem to contain original research. – LaundryPizza03 ( d c̄) 20:52, 19 May 2022 (UTC)[ reply]
EDIT: Partially redacted per WP:BITE rule. – LaundryPizza03 ( d c̄) 01:26, 21 May 2022 (UTC) 03:37, 21 May 2022 (UTC)[ reply]
Deletion of "Greatest integer" proposed
Greatest integer redirects to Floor and ceiling functions. Deletion of the redirect is proposed. Post opinions at Wikipedia:Redirects_for_discussion/Log/2022_May_19#Greatest_integer. Michael Hardy ( talk) 23:47, 20 May 2022 (UTC)[ reply]
Retrieved from " https://en.wikipedia.org/?title=Wikipedia_talk:WikiProject_Mathematics&oldid=1088971599"
Project-Class mathematics articles
Wikipedia Talk:wikiproject Mathematics Videos
Wikipedia Talk:wikiproject Mathematics Websites
Wikipedia Talk:wikiproject Mathematics Encyclopedia Articles
|
what is sheet metal prototype/bending parts | China Cnc Machining
what is sheet metal prototype/bending parts
fymicohuang 1 year ago 2020-12-20 cnc machining
2020-12-20 2020-12-20 cnc machining
Stainless steel[edit]
Decambering[edit]
Deep drawing[edit]
Expanding[edit]
Hemming and seaming[edit]
Hydroforming[edit]
Incremental sheet forming[edit]
Ironing[edit]
Laser cutting[edit]
Photochemical machining[edit]
Perforating[edit]
Press brake forming[edit]
Roll forming[edit]
Stamping[edit]
Water jet cutting[edit]
Wheeling[edit]
Sheet metal is metal formed by an industrial process into thin, flat pieces. Sheet metal is one of the fundamental forms used inmetalworking and it can be cut and bent into a variety of shapes. Countless everyday objects are fabricated from sheet metal. Thicknesses can vary significantly; extremely thin sheets are considered foil or leaf, and pieces thicker than 6 mm (0.25 in) are considered plate steel or "structural steel."
In most of the world, sheet metal thickness is consistently specified in millimeters. In the US, the thickness of sheet metal is commonly specified by a traditional, non-linear measure known as its gauge. The larger the gauge number, the thinner the metal. Commonly used steel sheet metal ranges from 30 gauge to about 7 gauge. Gauge differs between ferrous (iron based) metals and nonferrous metals such as aluminum or copper; copper thickness, for example is measured in ounces, which represents the weight of copper contained in an area of one square foot. Parts manufactured from sheet metal must maintain a uniform thickness for ideal results.[1]
There are many different metals that can be made into sheet metal, such as aluminium, brass, copper, steel, tin, nickel and titanium. For decorative uses, some important sheet metals include silver, gold, and platinum (platinum sheet metal is also utilized as a catalyst.)
Sheet metal is used in automobile and truck (lorry) bodies, airplane fuselages and wings, medical tables, roofs for buildings (architecture) and many other applications. Sheet metal of iron and other materials with high magnetic permeability, also known aslaminated steel cores, has applications in transformers and electric machines. Historically, an important use of sheet metal was in plate armor worn by cavalry, and sheet metal continues to have many decorative uses, including in horse tack. Sheet metal workers are also known as "tin bashers" (or "tin knockers"), a name derived from the hammering of panel seams when installing tin roofs.[2][3]
1.3Brass
3Forming processes
3.1Bending
3.3Decambering
3.4Deep drawing
3.5Expanding
3.6Hemming and seaming
3.7Hydroforming
3.8Incremental sheet forming
3.9Ironing
3.10Laser Cutting
3.11Photochemical cncmachine.ltd/en/News/news1251.html’ target=’_blank’>machining
3.12Perforating
3.13Press brake forming
3.14Punching
3.15Roll forming
3.16Rolling
3.17Spinning
3.18Stamping
3.19Water jet cutting
3.20Wheeling
Grade 430 is popular grade, low cost alternative to series 300's grades. This is used when high corrosion resistance is not a primary criteria. Common grade for appliance products, often with a brushed finish.
Aluminum is also a popular metal used in sheet metal due to its flexibility, wide range of options, cost effectiveness, and other properties.[5] The four most common aluminiumgrades available as sheet metal are 1100-H14, 3003-H14, 5052-H32, and 6061-T6.[4][6]
Grade 3003-H14 is stronger than 1100, while maintaining the same formability and low cost. It is corrosion resistant and weldable. It is often used in stampings, spun anddrawn parts, mail boxes, cabinets, tanks, and fan blades.[4]
In sheet hydroforming, variation in incoming sheet coil properties is a common problem for forming process, especially with materials for automotive applications. Even though incoming sheet coil may meet tensile test specifications, high rejection rate is often observed in production due to inconsistent material behavior. Thus there is a strong need for a discriminating method for testing incoming sheet material formability.The hydraulic sheet bulge test emulates biaxial deformation conditions commonly seen in production operations.
For Forming Limit curves (FLCs)of materials Aluminium, Mild steel and Brass.Theoretical analysis is carried out by deriving governing equations for determining of Equivalentstress and Equivalent strain based on the bulging to be spherical andTresca’s yield criterion with the associated flow rule. For experimentation Circular Grid Analysis is used.
Investigation of Forming Limit Curves of Various Sheet Materials Using Hydraulic Bulge Testing With Analytical, Experimental and FEA Techniques. Available from:https://www.researchgate.net/publication/321168677_Investigation_of_Forming_Limit_Curves_of_Various_Sheet_Materials_Using_Hydraulic_Bulge_Testing_With_Analytical_Experimental_and_FEA_Techniques.
This article may be confusing or unclear to readers. In particular, it does not explain the difference among the various gauge standards like Manufacturers' Standard Gauge, Standard Decimal Gauge, US Standard Gauge, Birmingham Gage and British Standard Gauge and their appropriate application. Please help us clarify the article. There might be a discussion about this on the talk page. (June 2013) (Learn how and when to remove this template message)
Use of gauge numbers to designate sheet metal thickness is discouraged by numerous international standards organizations. For example, ASTM states in specification ASTM A480-10a: "The use of gauge number is discouraged as being an archaic term of limited usefulness not having general agreement on meaning." [8]
Manufacturers' Standard Gauge for Sheet Steel is based on an average weight of 41.82 lb (18.96 kg) per square foot per inch thick.[9] Gauge is defined differently for ferrous (iron-based) and non-ferrous metals (e.g. aluminium and brass).
Standard sheet metal gauges[10]
US standard[11][12]
0000000 0.5000 (12.70) …… …… …… …… ……
000000 0.4688 (11.91) …… …… …… …… ……
00000 0.4375 (11.11) …… …… …… …… ……
0000 0.4063 (10.32) …… …… …… …… ……
000 0.3750 (9.53) …… …… …… …… ……
00 0.3438 (8.73) …… …… …… …… ……
0 0.3125 (7.94) …… …… …… …… ……
3 0.2500 (6.35) 0.2391 (6.07) …… …… …… 0.006 (0.15)
6 0.2031 (5.16) 0.1943 (4.94) …… …… 0.162 (4.1) 0.012 (0.30)
7 0.1875 (4.76) 0.1793 (4.55) …… 0.1875 (4.76) 0.1443 (3.67) 0.014 (0.36)
25 0.0219 (0.56) 0.0209 (0.53) 0.0247 (0.63) 0.022 (0.56) 0.018 (0.46) ……
28 0.0156 (0.40) 0.0149 (0.38) 0.0187 (0.47) 0.016 (0.41) 0.0126 (0.32) ……
32 0.0102 (0.26) 0.0097 (0.25) …… …… …… ……
{displaystyle F_{Max}=k{frac {TLt^{2}}{W}}}
The curling process is used to form an edge on a ring. This process is used to remove sharp edges. It also increases the moment of inertia near the curled end. The flare/burr should be turned away from the die. It is used to curl a material of specific thickness. Tool steel is generally used due to the amount of wear done by operation.
It is a metal working process of removing camber, the horizontal bend, from a strip shaped material. It may be done to a finite length section or coils. It resembles flattening of leveling process, but on a deformed edge.
Drawing is a forming process in which the metal is stretched over a form or die.[15] In deep drawing the depth of the part being made is more than half its diameter. Deep drawing is used for making automotive fuel tanks, kitchen sinks, two-piece aluminum cans, etc. Deep drawing is generally done in multiple steps called draw reductions. The greater the depth, the more reductions are required. Deep drawing may also be accomplished with fewer reductions by heating the workpiece, for example in sink manufacture.
Seaming is a process of folding two sheets of metal together to form a joint.
Hydroforming is a process that is analogous to deep drawing, in that the part is formed by stretching the blank over a stationary die. The force required is generated by the direct application of extremely high hydrostatic pressure to the workpiece or to a bladder that is in contact with the workpiece, rather than by the movable part of a die in a mechanical or hydraulic press. Unlike deep drawing, hydroforming usually does not involve draw reductions—the piece is formed in a single step.
Incremental sheet forming or ISF forming process is basically sheet metal working or sheet metal forming process. In this case, sheet is formed into final shape by a series of processes in which small incremental deformation can be done in each series.
Ironing is a sheet metal working or sheet metal forming process. It uniformly thins the workpiece in a specific area. This is a very useful process. It is used to produce a uniform wall thickness part with a high height-to-diameter ratio. It is used in making aluminium beverage cans.
CNC laser involves moving a lens assembly carrying a beam of laser light over the surface of the metal. Oxygen, nitrogen or air is fed through the same nozzle from which the laser beam exits. The metal is heated and burnt by the laser beam, cutting the metal sheet. The quality of the edge can be mirror smooth and a precision of around 0.1 mm (0.0039 in) can be obtained. Cutting speeds on thin 1.2 mm (0.047 in) sheet can be as high as 25 m (82 ft) per minute. Most laser cutting systems use a CO2 based laser source with a wavelength of around 10 µm; some more recent systems use a YAG based laser with a wavelength of around 1 µm.
The picture shown is air bending. Press brake bending is a different machine. But similar.
Punching is performed by placing the sheet of metal stock between a punch and a die mounted in a press. The punch and die are made of hardened steel and are the same shape. The punch is sized to be a very close fit in the die. The press pushes the punch against and into the die with enough force to cut a hole in the stock. In some cases the punch and die "nest" together to create a depression in the stock. In progressive stamping, a coil of stock is fed into a long die/punch set with many stages. Multiple simple shaped holes may be produced in one stage, but complex holes are created in multiple stages. In the final stage, the part is punched free from the "web".
A typical CNC turret punch has a choice of up to 60 tools in a "turret" that can be rotated to bring any tool to the punching position. A simple shape (e.g. a square, circle, or hexagon) is cut directly from the sheet. A complex shape can be cut out by making many square or rounded cuts around the perimeter. A punch is less flexible than a laser for cutting compound shapes, but faster for repetitive shapes (for example, the grille of an air-conditioning unit). A CNC punch can achieve 600 strokes per minute.
Rolling is metal working or metal forming process. In this method, stock is pass through one or more pair of rolls to reduce thickness. It is used to make thickness uniform. It is classified according to its temperature of rolling.
1.Hot rolling: in this temperature is above recrystallisation temperature.
2. Cold rolling: In this temperature is below recrystallisation temperature.
3. Warm rolling: In this temperature is used is in between Hot rolling and cold rolling.
Alternatively, the related techniques repoussé and chasing have low tooling and equipment costs, but high labor costs.
The process of using an English wheel is called wheeling. It is basically a metal working or metal forming process. An English wheel is used by a craftsperson to form compound curves from a flat sheet of metal of aluminium or steel. It is costly, as highly skilled labour is required. It can produce different panels by the same method. A stamping press is used for high numbers in production.
Prev 3D printing
Next 3D Printing SLS (Selective Laser Sintering)
|
Sinestream input signal with fixed sample time - MATLAB frest.createFixedTsSinestream - MathWorks Deutschland
frest.createFixedTsSinestream
Sinestream input signal with fixed sample time
input = frest.createFixedTsSinestream(ts)
input = frest.createFixedTsSinestream(ts,{wmin wmax})
input = frest.createFixedTsSinestream(ts,w)
input = frest.createFixedTsSinestream(ts,sys)
input = frest.createFixedTsSinestream(ts,sys,{wmin wmax})
input = frest.createFixedTsSinestream(ts,sys,w)
input = frest.createFixedTsSinestream(ts) creates sinestream input signal in which each frequency has the same fixed sample time ts in seconds. The signal has 30 frequencies between 1 and ωs, where
{\omega }_{s}=\frac{2\pi }{{t}_{s}}
is the sample rate in radians per second. The software adjusts the SamplesPerPeriod option to ensure that each frequency has the same sample time. Use when your Simulink® model has linearization input I/Os on signals with discrete sample times.
input = frest.createFixedTsSinestream(ts,{wmin wmax}) creates sinestream input signal with up to 30 frequencies logarithmically spaced between wmin and wmax in radians per second.
input = frest.createFixedTsSinestream(ts,w) creates sinestream input signal with frequencies w, specified as a vector of frequency values in radians per second. The values of w must satisfy
w=\frac{2\pi }{Nts}
for integer N such that the sample rate
{\omega }_{s}=\frac{2\pi }{{t}_{s}}
is an integer multiple of each element of w.
input = frest.createFixedTsSinestream(ts,sys) creates sinestream input signal with a fixed sample time ts. The signal's frequencies, settling periods, and number of periods automatically set based on the dynamics of a linear system sys.
input = frest.createFixedTsSinestream(ts,sys,{wmin wmax}) creates sinestream input signal with up to 30 frequencies logarithmically spaced between wmin and wmax in radians per second.
input = frest.createFixedTsSinestream(ts,sys,w) creates sinestream input signal at frequencies w, specified as a vector of frequency values in radians per second. The values of w must satisfy
w=\frac{2\pi }{Nts}
for integer N such that the sample rate ts is an integer multiple of each element of w.
Sample time of 0.02 sec
Frequencies of the sinusoidal signal are between 1 rad/s and 10 rad/s
input = frest.createFixedTsSinestream(0.02,{1, 10});
frest.Sinestream | frestimate
|
PUCCH format 2 DM-RS uplink subframe timing estimate - MATLAB lteULFrameOffsetPUCCH2 - MathWorks Nordic
lteULFrameOffsetPUCCH2
Synchronize and Demodulate Using PUCCH Format 2 DM-RS
View PUCCH Format 2 H-ARQ Indicators
View PUCCH Format 2 DM-RS Transmission Correlation Peaks
PUCCH format 2 DM-RS uplink subframe timing estimate
offset = lteULFrameOffsetPUCCH2(ue,chs,waveform,oack)
[offset,ack] = lteULFrameOffsetPUCCH2(ue,chs,waveform,oack)
[offset,ack,corr] = lteULFrameOffsetPUCCH2(ue,chs,waveform,oack)
offset = lteULFrameOffsetPUCCH2(ue,chs,waveform,oack) performs synchronization using PUCCH format 2 demodulation reference signals (DM-RS) for the time-domain waveform, waveform, given UE-specific settings, ue, PUCCH format 2 configuration chs, and the number of Hybrid ARQ indicators oack.
The returned value offset indicates the number of samples from the start of the waveform waveform to the position in that waveform where the first subframe containing the DM-RS begins.
offset provides subframe timing; frame timing can be achieved by using offset with the subframe number, ueNSubframe. This behavior is consistent with real-world operation because the base station knows when, in which subframe, to expect uplink transmissions.
[offset,ack] = lteULFrameOffsetPUCCH2(ue,chs,waveform,oack) also returns a vector ack of decoded PUCCH format 2 Hybrid ARQ indicators.
[offset,ack,corr] = lteULFrameOffsetPUCCH2(ue,chs,waveform,oack) also returns a complex matrix corr, which is used to extract the timing offset.
This example performs synchronization and uses the PUCCH format 2 DM-RS when demodulating a transmission that has been delayed by 5 samples.
Initialize ue specific parameter structure, PUCCH2 structure, UL resource grid and txAck parameter.
rgrid(ltePUCCH2DRSIndices(ue,pucch2)) = ltePUCCH2DRS(ue,pucch2,txAck);
Generate modulated waveform and add a five sample delay.
waveform = lteSCFDMAModulate(ue,rgrid);
tx = [zeros(5,1);waveform];
Use PUCCH format 2 DM-RS to estimate UL frame offset timing, then demodulate the waveform.
offset = lteULFrameOffsetPUCCH2(ue,pucch2,tx,length(txAck))
rxGrid = lteSCFDMADemodulate(ue,tx(1+offset:end));
View the Hybrid ARQ indicators for a PUCCH format 2 transmission waveform. The transmission contains PUCCH format 2 demodulation reference signal (DM-RS) symbols available for estimating the waveform timing.
Create configuration structures for ue and pucch2.
Generate Transmission Waveform
On the transmit side, populate a resource grid and generate a waveform containing PUCCH2 DM-RS.
reGrid(ltePUCCH2DRSIndices(ue,pucch2)) = ltePUCCH2DRS(ue,pucch2,txAck);
tx = lteSCFDMAModulate(ue,reGrid);
Waveform Reception
On the receive side, calculate timing offset using the PUCCH2 DM-RS symbols for the time-domain waveform and return decoded PUCCH format 2 Hybrid ARQ indicators.
[offset,ack] = lteULFrameOffsetPUCCH2(ue,pucch2,tx,length(txAck));
ack = 2x1 logical array
View the correlation peak for a transmission waveform that has been delayed. The transmission contains PUCCH format 2 demodulation reference signal (DM-RS) symbols available for estimating the waveform timing.
On the receive side, calculate timing offset using the PUCCH2 DM-RS symbols for the time-domain waveform and return the correlations for the transmit waveform and for a delayed version of the transmit waveform.
[~,ack,corr] = lteULFrameOffsetPUCCH2(ue,pucch2,tx,length(txAck));
txDelayed = [zeros(5,1); tx];
[offset,ack,corrDelayed] = lteULFrameOffsetPUCCH2(ue,pucch2,txDelayed,length(txAck));
rrxGrid = lteSCFDMADemodulate(ue,txDelayed(1+offset:end));
{N}_{\text{RB}}^{\text{UL}}
chs — PUCCH format 2 configuration
PUCCH format 2 configuration, specified as a scalar structure with the following fields.
{n}_{PUCCH}^{\left(2\right)}
{N}_{RB}^{\left(2\right)}
{N}_{cs}^{\left(1\right)}
Time-domain waveform, specified as a numeric matrix. waveform must be a NS-by-NR matrix, where NS is the number of time-domain samples and NR is the number of receive antennas. waveform should be at least one subframe long and contain the DM-RS signals.
Generate waveform by SC-FDMA modulation of a resource matrix using the lteSCFDMAModulate function, or by using one of the channel model functions, lteFadingChannel, lteHSTChannel, or lteMovingChannel.
oack — Number of uncoded Hybrid ARQ bits
Number of uncoded Hybrid ARQ bits expected, 1 (PUCCH format 2a) or 2 (PUCCH format 2b).
offset — Number of samples from the start of the waveform to the position in that waveform where the first subframe begins
Number of samples from the start of the waveform to the position in that waveform where the first subframe containing the DM-RS begins, returned as a scalar integer. offset is computed by extracting the timing of the peak of the correlation between waveform and internally generated reference waveforms containing DM-RS signals. The correlation is performed separately for each antenna and the antenna with the strongest correlation is used to compute offset. This process is repeated for either one or two Hybrid ARQ indicators combination as specified by the parameter oack. This correlation amounts to a maximum likelihood (ML) decoding of the Hybrid ARQ indicators, which are signaled on the PUCCH format 2 DM-RS.
ack — Decoded PUCCH format 2 Hybrid ARQ bits
numeric vector or matrix
Decoded PUCCH format 2 Hybrid ARQ bits, returned as a numeric vector or matrix. If multiple decoded Hybrid ARQ indicator vectors have a likelihood equal to the maximum, ack is a matrix where each column represents one of the equally likely Hybrid ARQ indicator vectors.
lteULFrameOffset | lteULFrameOffsetPUCCH1 | lteULFrameOffsetPUCCH3 | lteFadingChannel | lteMovingChannel | lteHSTChannel | lteSCFDMADemodulate
|
Create Symbolic Matrix Variables - MATLAB & Simulink - MathWorks Deutschland
Comparison Between Matrix of Symbolic Scalar Variables and Symbolic Matrix Variables
Mathematical Operations with Symbolic Matrix Variables
Create Symbolic Matrix Variable from Array of Symbolic Scalar Variables
Convert Symbolic Matrix Variable into Array of Symbolic Scalar Variables
Indexing into Symbolic Matrix Variables
Display of Operations Involving Symbolic Matrix Variables
Symbolic matrix variables represent matrices, vectors, and scalars in compact matrix notation. When mathematical formulas involve matrices and vectors, writing them using symbolic matrix variables is more concise and clear than writing them componentwise. When you do this, you can take vector-based expressions and equations from textbooks, enter them in Symbolic Math Toolbox™, perform mathematical operations on them, and derive further equations from them.
Derived equations involving symbolic matrix variables are displayed in typeset as they would be in textbooks. For example, create three symbolic matrix variables
\text{A}
\text{x}
\text{y}
by using syms. Find the differential of the expression
{\text{y}}^{T}\text{A}\text{x}
with respect to the vector
\text{x}
eq = y.'*A*x
{y}^{\mathrm{T}} A x
D = diff(eq,x)
{y}^{\mathrm{T}} A
Symbolic matrix variables are an alternative to symbolic scalar variables. The two options are of different types and displayed differently.
For example, create two 3-by-4 matrices of symbolic scalar variables by using syms. For brevity, matrices of symbolic scalar variables are sometimes called symbolic matrices. These matrices are displayed by listing their components.
\left(\begin{array}{ccc}{A}_{1,1}& {A}_{1,2}& {A}_{1,3}\\ {A}_{2,1}& {A}_{2,2}& {A}_{2,3}\end{array}\right)
\left(\begin{array}{ccc}{B}_{1,1}& {B}_{1,2}& {B}_{1,3}\\ {B}_{2,1}& {B}_{2,2}& {B}_{2,3}\end{array}\right)
A matrix of symbolic scalar variables is of type sym.
Applying symbolic math operations to these matrices can result in a complex solution expressed in terms of the matrix components. For example, multiply the matrices A and B'.
C = A*B'
\left(\begin{array}{cc}{A}_{1,1} \stackrel{‾}{{B}_{1,1}}+{A}_{1,2} \stackrel{‾}{{B}_{1,2}}+{A}_{1,3} \stackrel{‾}{{B}_{1,3}}& {A}_{1,1} \stackrel{‾}{{B}_{2,1}}+{A}_{1,2} \stackrel{‾}{{B}_{2,2}}+{A}_{1,3} \stackrel{‾}{{B}_{2,3}}\\ {A}_{2,1} \stackrel{‾}{{B}_{1,1}}+{A}_{2,2} \stackrel{‾}{{B}_{1,2}}+{A}_{2,3} \stackrel{‾}{{B}_{1,3}}& {A}_{2,1} \stackrel{‾}{{B}_{2,1}}+{A}_{2,2} \stackrel{‾}{{B}_{2,2}}+{A}_{2,3} \stackrel{‾}{{B}_{2,3}}\end{array}\right)
To create symbolic matrix variables of the same size, use the syms command followed by the variable names, their size, and the matrix keyword. Symbolic matrix variables are displayed in bold to distinguish them from symbolic scalar variables.
A
B
Symbolic matrix variables are of type symmatrix.
Applying symbolic math operations to symbolic matrix variables results in a concise display. For example, multiply A and B'.
A {\left(\stackrel{‾}{B}\right)}^{\mathrm{T}}
Symbolic matrix variables are recognized as noncommutative objects. They support common math operations, and you can use these operations to build symbolic matrix variable expressions.
A B-B A
For example, check the commutation relation for multiplication between two symbolic matrix variables.
Check the commutation relation for addition.
If an operation has any arguments of type symmatrix, the result is automatically converted to type symmatrix. For example, multiply a matrix A that is represented by symbolic matrix variable and a scalar c that is represented by symbolic scalar variable. The result is of type symmatrix.
M = c*A
c A
Multiply three matrices that are represented by symbolic matrix variables. The result X is a symmatrix object.
syms V [2 1] matrix
X = V.'*A*V
{V}^{\mathrm{T}} A V
You can pass symmatrix objects as arguments to math functions. For example, perform a mathematical operation to X by taking the differential of X with respect to V.
diff(X,V)
{V}^{\mathrm{T}} {A}^{\mathrm{T}}+{V}^{\mathrm{T}} A
You can convert an array of symbolic scalar variables to a single symbolic matrix variable using the symmatrix function. Symbolic matrix variables that are converted in this way are displayed elementwise.
B = symmatrix(A)
\begin{array}{l}{\Sigma }_{1}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_{1}=\left(\begin{array}{cccc}{A}_{1,1}& {A}_{1,2}& {A}_{1,3}& {A}_{1,4}\\ {A}_{2,1}& {A}_{2,2}& {A}_{2,3}& {A}_{2,4}\\ {A}_{3,1}& {A}_{3,2}& {A}_{3,3}& {A}_{3,4}\end{array}\right)\end{array}
You can create symbolic matrix variables, derive equations, and then convert the result to arrays of symbolic scalar variables using the symmatrix2sym function.
For example, find the matrix product of two symbolic matrix variables A and B. The result X is of type symmatrix.
A B
Convert the symbolic matrix variable X to array of symbolic scalar variables. The converted matrix Y is of type sym.
\left(\begin{array}{cc}{A}_{1,1} {B}_{1,1}+{A}_{1,2} {B}_{2,1}& {A}_{1,1} {B}_{1,2}+{A}_{1,2} {B}_{2,2}\\ {A}_{2,1} {B}_{1,1}+{A}_{2,2} {B}_{2,1}& {A}_{2,1} {B}_{1,2}+{A}_{2,2} {B}_{2,2}\end{array}\right)
Check that the product obtained by converting symbolic matrix variables is equal to the product of two arrays of symbolic scalar variables.
isequal(Y,A*B)
Indexing into a symbolic matrix variable returns corresponding matrix elements in the form of another symbolic matrix variable.
{A}_{2,3}
Alternatively, convert the symbolic matrix variable A to a matrix of symbolic scalar variables. Then, index into that matrix.
Asym = symmatrix2sym(A)
Asym =
\left(\begin{array}{ccc}{A}_{1,1}& {A}_{1,2}& {A}_{1,3}\\ {A}_{2,1}& {A}_{2,2}& {A}_{2,3}\end{array}\right)
asym = Asym(2,3)
{A}_{2,3}
class(asym)
Note that both results are equal.
isequal(a,symmatrix(asym))
Matrices like those returned by eye, zeros, and ones often have special meaning with specific notation in symbolic workflows. Declaring these matrices as symbolic matrix variables display the matrices in bold along with the matrix dimensions.
symmatrix(eye(3))
{\mathrm{I}}_{3}
symmatrix(zeros(2,3))
{\mathrm{0}}_{2,3}
symmatrix(ones(3,5))
{\mathrm{1}}_{3,5}
If the inputs to a componentwise operation in MATLAB® are symbolic matrix variables, so is the output. These operations are displayed in special notations which follow conventions from textbooks.
A\odot B
A\oslash B
B\oslash A
A.*hilb(3)
\begin{array}{l}A\odot {\Sigma }_{1}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_{1}=\left(\begin{array}{ccc}1& \frac{1}{2}& \frac{1}{3}\\ \frac{1}{2}& \frac{1}{3}& \frac{1}{4}\\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5}\end{array}\right)\end{array}
A.^(2*ones(3))
{A}^{\circ 2 {\mathrm{1}}_{3,3}}
{A}^{\circ B}
A\otimes B
\mathrm{adj}\left(A\right)
\mathrm{Tr}\left(A\right)
syms | symmatrix | symmatrix2sym
|
applyop - Maple Help
Home : Support : Online Help : Programming : Operations : Operators : applyop
apply a function to specified operand(s) of an expression
applyop( f, i, e )
applyop( f, i, e, ..., xk, ...)
specifies the operand(s) in e
optional arguments to f
The applyop command manipulates the selected parts of an expression. The first argument, f, is applied to the operands of e specified by i.
If i is an integer, applyop( f, i, e) applies f to the ith operand of e. This is equivalent to subsop( i = f(op( i, e)), e). For example, if the value of e is the sum
x+y+z
, applyop( f, 2, e) computes
x+f\left(y\right)+z
If i is a list of integers, the call applyop( f, i, e) is equivalent to subsop( i = f(op( i, e)), e). This allows you to manipulate any suboperand of an expression.
If i is a set, f is applied simultaneously to all operands of e specified in the set. Note: applyop( f, {}, e) returns e.
Any additional arguments xk are passed as additional arguments to f in the order given.
p≔{y}^{2}-2y-3
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{y}}^{\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}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}
\mathrm{applyop}\left(f,2,p\right)
{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}
\mathrm{applyop}\left(f,2,p,\mathrm{x1},\mathrm{x2}\right)
{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x2}}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}
\mathrm{applyop}\left(f,[2,2],p\right)
{\textcolor[rgb]{0,0,1}{y}}^{\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}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}
\mathrm{applyop}\left(f,{2,3},p\right)
{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-3}\right)
\mathrm{applyop}\left(\mathrm{abs},{3,[2,1]},p\right)
{\textcolor[rgb]{0,0,1}{y}}^{\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}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
e≔\left(z+1\right)\mathrm{ln}\left(z\left({z}^{2}-2\right)\right)
\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{≔}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)\right)
\mathrm{expand}\left(e\right)
\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)\right)
To expand the argument to the logarithm in e:
\mathrm{applyop}\left(\mathrm{expand},[2,1],e\right)
\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\right)
To factor the argument to the logarithm in e over R:
\mathrm{applyop}\left(\mathrm{factor},[2,1],e,\mathrm{real}\right)
\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1.414213562}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1.414213562}\right)\right)
|
Nonspherical Models - MATLAB & Simulink - MathWorks Italia
What Are Nonspherical Models?
Consider the linear time series model
{y}_{t}={X}_{t}\beta +{\epsilon }_{t}
, where yt is the response, xt is a vector of values for the r predictors, β is the vector of regression coefficients, and εt is the random innovation at time t.
Ordinary least squares (OLS) estimation and inference techniques for this framework depend on certain assumptions, e.g., homoscedastic and uncorrelated innovations. For more details on the classical linear model, see Time Series Regression I: Linear Models. If your data exhibits signs of assumption violations, then OLS estimates or inferences based on them might not be valid.
In particular, if the data is generated with an innovations process that exhibits autocorrelation or heteroscedasticity, then the model (or the residuals) are nonspherical. These characteristics are often detected through testing of model residuals (for details, see Time Series Regression VI: Residual Diagnostics).
Nonspherical residuals are often considered a sign of model misspecification, and models are revised to whiten the residuals and improve the reliability of standard estimation techniques. In some cases, however, nonspherical models must be accepted as they are, and estimated as accurately as possible using revised techniques. Cases include:
Models presented by theory
Models with predictors that are dictated by policy
Models without available data sources, for which predictor proxies must be found
A variety of alternative estimation techniques have been developed to deal with these situations.
|
RRT Planner | Qian Lin's Personal Site
C++, Geometry, ROS
Implementation of RRT Motion Planning
This is a project I completed in the summer of 2018, under Karen and Sipu’s help, in Johns Hopkins University.
This can be found on my github page.
This is an implementation of Rapid-exploring Random Tree Algorithm on ROS, in 2D plane and the robot, obstacles and world are all in a shape of ellipsoid. The main language is C++.
Motion Planning with RRT
The procedure of RRT planner is described below:
Initialize: build a RRT tree whose root is the start point (in a n-dimension space) of the motion. Every node in the tree represents a “valid” point in the C-Space, which means under that position the robot is not in collision.
Explore the C-Space and grow the tree: (loop)
Generate a random pose of the ellipsoid robot (
q_{rand}
Traverse all the node in the tree and find a
q_{nearest}
, which is nearest to the
q_{rand}
Generate a new point
q_{new}
which is between
q_{nearest}
q_{rand}
and is a metric away from
q_{nearest}
q_{new}
is valid ) THEN { add
q_{new}
to the tree}. Once the goal is achieved or the search time is more than the set upper limit, the search should be terminated.
Find the trajectory from the generated tree, if the goal is successfully achieved. Dijkstra Algorithm can be used.
Smooth the trajectory.
The Structure of My ROS Package
ellipsoid.msg: ellipsoid semi-axes length, central point ordinates value, angle
world.msg: describe the arena, obstacles and robot with ellipsoid message
position.msg: (x, y, 𝜙) value describing the ellipsoid pose
point.msg: (x, y) value describing the 2D point
trajectory.msg: an array of (x, y) describing the points on the trajectory
pointsArrary.msg: an array of (x, y) on the boundary of ellipsoid generated by the server
node.msg: the x, y values of the valid searched node, father value represents the nearest node ID
rrtGraph.msg: array of node
collision_detect.srv: return whether the state is in collision according to the world msg information and current robot position msg
ellipsoid_points.srv: return a vector of point msg according to the ellipsoid msg and step
include/rrt_implement
rrt.h: define the basic tree structure
planner.h: define the planning action
rrt.cpp
planner.cpp: plan the motion. I use Euler distance to find the nearest node —— rrtPlanner
collision_detect.cpp: provide the service of collision detection —— collision_detection_srv
environment_publisher.cpp: subscribe the world information published by node rrtPlanner, and then send that to the server ellipsoid_point_gen_srv, publish the obtained 2D points to be plotted —— environment_pub
graph_publisher.cpp: subscribe the graph information published by node rrtPlanner, and then publish—— graph_pub
trajectory_publisher.cpp: subscribe the trajectory information published by node rrtPlanner, and then send that to the server ellipsoid_point_gen_srv, publish the obtained 2D points to be plotted —— trajectory_pub
ellipsoid_point_gen.cpp: a server whose input is ellipsoid msg and position msg and output is pointsArray msg describing the boundary of the ellipsoid —— ellipsoid_point_gen_srv
RVIZ configuratoin
Fixed Frame = “/root”
after catkin_make && source devel/setup.bash
rosrun rrt_implement ellipsoid_point_gen_srv
rosrun rrt_implement collision_detection_srv
run the rrt planner
rosrun rrt_implement rrtPlanner
run the publisher
rosrun rrt_implement environment_pub
rosrun rrt_implement trajectory_pub
To achieve high speed, I did not call the collision detection libraries such as FCL. I just used a simple and rough method. Assume that $A$, $B$ is the center of two ellipsoids, respectively, and their distance is D. The “radius”s of the two ellipsoids in the direction of $\overrightarrow{AB}$ is $R$ and $r$, respectively. Thus, the requirement of the two ellipsoids is seperate is $D>R+r$; the requirement of the ellipsoid $B$ is in ellipsoid $A$ entirely is $D<R-r$.
As this is just the requirement, sometimes this method would fail.
Thanks for Prof. Chirikjian‘s instruction.
Thanks for Sipu Ruan and Karen Poblete Rodriguez‘s help.
|
REtodelta - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : LREtools : REtodelta
return the difference operator associated to the LRE
REtodelta(problem)
This routine returns the difference operator, in terms of the inert name LREtools[Delta], associated to the problem. The operator is indexed by the name of the variable from the problem.
The command with(LREtools,REtodelta) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{LREtools}\right):
\mathrm{REtodelta}\left(u\left(n+1\right)-u\left(n\right),u\left(n\right),\varnothing \right)
{\left(\textcolor[rgb]{0,0,1}{\mathrm{LREtools}}[\textcolor[rgb]{0,0,1}{\mathrm{\Delta }}]\right)}_{\textcolor[rgb]{0,0,1}{n}}
\mathrm{REtodelta}\left(\left(t+1\right)u\left(t+2\right)+\left(t+2\right)u\left(t\right),u\left(t\right),\varnothing \right)
\left(\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{\mathrm{LREtools}}[\textcolor[rgb]{0,0,1}{\mathrm{\Delta }}]\right)}_{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{\mathrm{LREtools}}[\textcolor[rgb]{0,0,1}{\mathrm{\Delta }}]\right)}_{\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
LREtools[delta]
LREtools[shift]
|
Break-even (economics) - Wikipedia
Break-even (economics)
This article is about break-evens in economics. For other uses, see Break-even (disambiguation).
The break-even point (BEP) in economics, business—and specifically cost accounting—is the point at which total cost and total revenue are equal, i.e. "even". There is no net loss or gain, and one has "broken even", though opportunity costs have been paid and capital has received the risk-adjusted, expected return. In short, all costs that must be paid are paid, and there is neither profit or loss.[1][2]
The break-even point (BEP) or break-even level represents the sales amount—in either unit (quantity) or revenue (sales) terms—that is required to cover total costs, consisting of both fixed and variable costs to the company. Total profit at the break-even point is zero. It is only possible for a firm to pass the break-even point if the dollar value of sales is higher than the variable cost per unit. This means that the selling price of the goods must be higher than what the company paid for the good or its components for them to cover the initial price they paid (variable and fixed costs). Once they surpass the break-even price, the company can start making a profit.
The break-even point is one of the most commonly used concepts of financial analysis, and is not only limited to economic use, but can also be used by entrepreneurs, accountants, financial planners, managers and even marketers. Break-even points can be useful to all avenues of a business, as it allows employees to identify required outputs and work towards meeting these.
The break-even value is not a generic value as such and will vary dependent on the individual business. Some businesses may have a higher or lower break-even point. However, it is important that each business develop a break-even point calculation, as this will enable them to see the number of units they need to sell to cover their variable costs. Each sale will also make a contribution to the payment of fixed costs as well.
For example, a business that sells tables needs to make annual sales of 200 tables to break-even. At present the company is selling fewer than 200 tables and is therefore operating at a loss. As a business, they must consider increasing the number of tables they sell annually in order to make enough money to pay fixed and variable costs.
If the business does not think that they can sell the required number of units, they could consider the following options:
1. Reduce the fixed costs. This could be done through a number or negotiations, such as reductions in rent payments, or through better management of bills or other costs.
2. Reduce the variable costs, (which could be done by finding a new supplier that sells tables for less).
Either option can reduce the break-even point so the business need not sell as many tables as before, and could still pay fixed costs.
The main purpose of break-even analysis is to determine the minimum output that must be exceeded for a business to profit. It also is a rough indicator of the earnings impact of a marketing activity. A firm can analyze ideal output levels to be knowledgeable on the amount of sales and revenue that would meet and surpass the break-even point. If a business doesn't meet this level, it often becomes difficult to continue operation.
The break-even point is one of the simplest, yet least-used analytical tools. Identifying a break-even point helps provide a dynamic view of the relationships between sales, costs, and profits. For example, expressing break-even sales as a percentage of actual sales can help managers understand when to expect to break even (by linking the percent to when in the week or month this percent of sales might occur).
The break-even point is a special case of Target Income Sales, where Target Income is 0 (breaking even). This is very important for financial analysis. Any sales made past the breakeven point can be considered profit (after all initial costs have been paid)
Break-even analysis can also provide data that can be useful to the marketing department of a business as well, as it provides financial goals that the business can pass on to marketers so they can try to increase sales.
Break-even analysis can also help businesses see where they could re-structure or cut costs for optimum results. This may help the business become more effective and achieve higher returns. In many cases, if an entrepreneurial venture is seeking to get off of the ground and enter into a market it is advised that they formulate a break-even analysis to suggest to potential financial backers that the business has the potential to be viable and at what points.
In the linear Cost-Volume-Profit Analysis model (where marginal costs and marginal revenues are constant, among other assumptions), the break-even point (BEP) (in terms of Unit Sales (X)) can be directly computed in terms of Total Revenue (TR) and Total Costs (TC) as:
{\displaystyle {\begin{aligned}{\text{TR}}&={\text{TC}}\\P\times X&={\text{TFC}}+V\times X\\P\times X-V\times X&={\text{TFC}}\\\left(P-V\right)\times X&={\text{TFC}}\\X&={\frac {\text{TFC}}{P-V}}\end{aligned}}}
TFC is Total Fixed Costs,
P is Unit Sale Price, and
V is Unit Variable Cost.
The Break-Even Point can alternatively be computed as the point where Contribution equals Fixed Costs.
{\displaystyle \left(P-V\right)}
, is of interest in its own right, and is called the Unit Contribution Margin (C): it is the marginal profit per unit, or alternatively the portion of each sale that contributes to Fixed Costs. Thus the break-even point can be more simply computed as the point where Total Contribution = Total Fixed Cost:
{\displaystyle {\begin{aligned}{\text{Total Contribution}}&={\text{Total Fixed Costs}}\\{\text{Unit Contribution}}\times {\text{Number of Units}}&={\text{Total Fixed Costs}}\\{\text{Number of Units}}&={\frac {\text{Total Fixed Costs}}{\text{Unit Contribution}}}\end{aligned}}}
To calculate the break-even point in terms of revenue (a.k.a. currency units, a.k.a. sales proceeds) instead of Unit Sales (X), the above calculation can be multiplied by Price, or, equivalently, the Contribution Margin Ratio (Unit Contribution Margin over Price) can be calculated:
{\displaystyle {\text{Break-even(in Sales)}}={\frac {\text{Fixed Costs}}{C/P}}.}
R=C,
Where R is revenue generated, C is cost incurred i.e. Fixed costs + Variable Costs or
{\displaystyle {\begin{aligned}Q\times P&=\mathrm {TFC} +Q\times VC&{\text{(Price per unit)}}\\Q\times P-Q\times \mathrm {VC} &=\mathrm {TFC} \\Q\times (P-\mathrm {VC} )&=\mathrm {TFC} \\\end{aligned}}}
or, Break Even Analysis
Q = TFC/c/s ratio = Break Even
Margin of safetyEdit
Margin of safety represents the strength of the business. It enables a business to know what is the exact amount it has gained or lost and whether they are over or below the break-even point.[3] In break-even analysis, margin of safety is the extent by which actual or projected sales exceed the break-even sales.[4]
Margin of safety = (current output - breakeven output)
Margin of safety% = (current output - breakeven output)/current output × 100
When dealing with budgets you would instead replace "Current output" with "Budgeted output." If P/V ratio is given then profit/PV ratio.
Break-even analysisEdit
By inserting different prices into the formula, you will obtain a number of break-even points, one for each possible price charged. If the firm changes the selling price for its product, from $2 to $2.30, in the example above, then it would have to sell only 1000/(2.3 - 0.6)= 589 units to break even, rather than 715.
To make the results clearer, they can be graphed. To do this, draw the total cost curve (TC in the diagram), which shows the total cost associated with each possible level of output, the fixed cost curve (FC) which shows the costs that do not vary with output level, and finally the various total revenue lines (R1, R2, and R3), which show the total amount of revenue received at each output level, given the price you will be charging.
The break-even points (A,B,C) are the points of intersection between the total cost curve (TC) and a total revenue curve (R1, R2, or R3). The break-even quantity at each selling price can be read off the horizontal axis and the break-even price at each selling price can be read off the vertical axis. The total cost, total revenue, and fixed cost curves can each be constructed with simple formula. For example, the total revenue curve is simply the product of selling price times quantity for each output quantity. The data used in these formula come either from accounting records or from various estimation techniques such as regression analysis.
The Break-even analysis is only a supply
the scale of production is likely to cause fixed costs to rise.
It assumes average variable costs are constant per unit of output
(i.e., there is no change in the quantity of goods held in inventory at the beginning of the period and the quantity of goods held in inventory at the end of the period).
In multi-product companies, it assumes that the relative proportions
^ Levine, David; Michele Boldrin (2008-09-07). Against Intellectual Monopoly. Cambridge University Press. p. 312. ISBN 978-0-521-87928-6.
^ Tapang, Bienvenido, and Lorelei Mendoza. Introductory Economics. University of the Philippines, Baguio.
^ The Margin of Safety in MAAW, Chapter 11.
^ Margin of Safety Definition | Formula | Calculation | Example
Dayananda, D.; Irons, R.; Harrison, S.; Herbohn, J.; and P. Rowland, 2002, Capital Budgeting: Financial Appraisal of Investment Projects. Cambridge University Press. pp. 150.
Dean, Joel. "Cost structures of enterprises and break-even charts." The American Economic Review (1948): 153-164.
Patrick, A. W. "Some Observations on the Break-Even Chart." Accounting Review (1958): 573-580.
Tucker, Spencer A. The break-even system: A tool for profit planning. Prentice-Hall, 1963.
Tucker, Spencer A. Profit planning decisions with the break-even system. Thomond Press: distribution to the book trade in the US by Van Nostrand Reinhold, 1980.
Wikimedia Commons has media related to Break-even charts.
Example of Break Even Point using Microsoft Excel
Retrieved from "https://en.wikipedia.org/w/index.php?title=Break-even_(economics)&oldid=1073725930"
|
American Standard Code for Information Interchange (ASCII) | James's Knowledge Graph
ASCII is standard to encoding text character symbols as binary data for electronic communication. Each character is mapped to a 7-bit code, allowing for 127 characters total (
1111111_{2} = 124_{10}
). The numerical order of these codes is known as ASCIIbetical order.
The first 32 codes are reserved as control characters, such as backspace and linefeed. However, the character for Delete is the 127th code for historic reasons. The remaining codes are reserved for printable characters including punctuation, symbols, numbers, and letters.
Deeper Knowledge on American Standard Code for Information Interchange (ASCII)
Broader Topics Related to American Standard Code for Information Interchange (ASCII)
American Standard Code for Information Interchange (ASCII) Knowledge Graph
|
CARDINAL NUMBER - Encyclopedia Information
Cardinal number Information
Size of a possibly infinite set
A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.
Aleph-null, the smallest infinite cardinal
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol
{\displaystyle \aleph }
( aleph) followed by a subscript, describe the sizes of infinite sets.
{\displaystyle 0,1,2,3,\ldots ,n,\ldots ;\aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\ }
4.1 Successor cardinal
4.2 Cardinal addition
4.3 Cardinal multiplication
4.4 Cardinal exponentiation
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three. This is established by the existence of a bijection (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.
Cantor applied his concept of bijection to infinite sets [1] (for example the set of natural numbers N = {0, 1, 2, 3, ...}). Thus, he called all sets having a bijection with N denumerable (countably infinite) sets, which all share the same cardinal number. This cardinal number is called
{\displaystyle \aleph _{0}}
, aleph-null. He called the cardinal numbers of infinite sets transfinite cardinal numbers.
In his 1874 paper " On a Property of the Collection of All Real Algebraic Numbers", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of N. His proof used an argument with nested intervals, but in an 1891 paper, he proved the same result using his ingenious and much simpler diagonal argument. The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol
{\displaystyle {\mathfrak {c}}}
for it.
Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (
{\displaystyle \aleph _{0}}
, aleph-null), and that for every cardinal number there is a next-larger cardinal
{\displaystyle (\aleph _{1},\aleph _{2},\aleph _{3},\ldots ).}
His continuum hypothesis is the proposition that the cardinality
{\displaystyle {\mathfrak {c}}}
of the set of real numbers is the same as
{\displaystyle \aleph _{1}}
. This hypothesis has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions.
More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size. For example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c}, which has 3 elements.
A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y. An injective mapping identifies each element of the set X with a unique element of the set Y. This is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:
We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. By the Schroeder–Bernstein theorem, this is equivalent to there being both an injective mapping from X to Y, and an injective mapping from Y to X. We then write |X| = |Y|. The cardinal number of X itself is often defined as the least ordinal a with |a| = |X|. [2] This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
n → n + 1
With this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown. This motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a Dedekind-infinite set); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.
Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal number α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed, then a different approach is needed. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the class [X] of all sets that are equinumerous with X. This does not work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is too large to be a set. In fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X, and so by the axiom of limitation of size, [X] is a proper class. The definition does work however in type theory and in New Foundations and related systems. However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work (this is a trick due to Dana Scott: [3] it works because the collection of objects with any given rank is a set).
Formally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an injective function from X to Y. The Cantor–Bernstein–Schroeder theorem states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|. The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|. [4] [5]
Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal
{\displaystyle \aleph _{0}}
( aleph null or aleph-0, where aleph is the first letter in the Hebrew alphabet, represented
{\displaystyle \aleph }
) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality
{\displaystyle \aleph _{0}}
). The next larger cardinal is denoted by
{\displaystyle \aleph _{1}}
, and so on. For every ordinal α, there is a cardinal number
{\displaystyle \aleph _{\alpha },}
and this list exhausts all infinite cardinal numbers.
Further information: Successor cardinal
If the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that
{\displaystyle \kappa ^{+}\nleq \kappa .}
) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal.
If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}).
{\displaystyle |X|+|Y|=|X\cup Y|.}
{\displaystyle (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu )).}
{\displaystyle \kappa +\mu =\max\{\kappa ,\mu \}\,.}
The product of cardinals comes from the Cartesian product.
{\displaystyle |X|\cdot |Y|=|X\times Y|}
κ·μ = 0 → (κ = 0 or μ = 0).
{\displaystyle \kappa \cdot \mu =\max\{\kappa ,\mu \}.}
{\displaystyle |X|^{|Y|}=\left|X^{Y}\right|,}
where XY is the set of all functions from Y to X. [8]
κ0 = 1 (in particular 00 = 1), see empty function.
κμ · ν = (κμ)ν.
(κ·μ)ν = κν·μν.
Exponentiation is non-decreasing in both arguments:
(1 ≤ ν and κ ≤ μ) → (νκ ≤ νμ) and
(κ ≤ μ) → (κν ≤ μν).
Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying
{\displaystyle \nu ^{\mu }=\kappa }
{\displaystyle \kappa }
Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying
{\displaystyle \mu ^{\lambda }=\kappa }
. However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy
{\displaystyle \nu ^{\lambda }=\kappa }
The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess. [9] [10] [11]
The continuum hypothesis (CH) states that there are no cardinals strictly between
{\displaystyle \aleph _{0}}
{\displaystyle 2^{\aleph _{0}}.}
The latter cardinal number is also often denoted by
{\displaystyle {\mathfrak {c}}}
; it is the cardinality of the continuum (the set of real numbers). In this case
{\displaystyle 2^{\aleph _{0}}=\aleph _{1}.}
Similarly, the generalized continuum hypothesis (GCH) states that for every infinite cardinal
{\displaystyle \kappa }
, there are no cardinals strictly between
{\displaystyle \kappa }
{\displaystyle 2^{\kappa }}
. Both the continuum hypothesis and the generalized continuum hypothesis have been proved independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ( ZFC).
Indeed, Easton's theorem shows that, for regular cardinals
{\displaystyle \kappa }
, the only restrictions ZFC places on the cardinality of
{\displaystyle 2^{\kappa }}
{\displaystyle \kappa <\operatorname {cf} (2^{\kappa })}
, and that the exponential function is non-decreasing.
^ Dauben 1990, pg. 54
^ Weisstein, Eric W. "Cardinal Number". mathworld.wolfram.com. Retrieved 2020-09-06.
^ Deiser, Oliver (May 2010). "On the Development of the Notion of a Cardinal Number". History and Philosophy of Logic. 31 (2): 123–143. doi: 10.1080/01445340903545904. S2CID 171037224.
^ Friedrich M. Hartogs (1915), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Math. Ann., Leipzig: B. G. Teubner, Bd. 76 (4): 438–443, doi: 10.1007/bf01458215, ISSN 0025-5831, S2CID 121598654, archived from the original on 2016-04-16, retrieved 2014-02-02
^ Schindler 2014, pg. 34
^ Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315, Springer-Verlag.
^ Eduard Čech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
^ D. A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.
Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton: Princeton University Press, ISBN 0691-02447-2
Schindler, Ralf-Dieter (2014). Set theory : exploring independence and truth. Cham: Springer-Verlag. doi: 10.1007/978-3-319-06725-4. ISBN 978-3-319-06725-4.
"Cardinal number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
{\displaystyle \mathbb {N} }
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Q} }
{\displaystyle \mathbb {A} }
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {H} }
{\displaystyle \mathbb {O} }
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {S} }
Retrieved from " https://en.wikipedia.org/?title=Cardinal_number&oldid=1085271802"
Cardinal Number Videos
Cardinal Number Websites
Cardinal Number Encyclopedia Articles
|
Complexity Zoo:E - Complexity Zoo
Revision as of 07:33, 17 February 2022 by Douglas (talk | contribs)
E: Exponential Time With Linear Exponent
Equals DTIME(2O(n)).
Does not equal NP [Boo72] or PSPACE [Boo74] relative to any oracle. However, there is an oracle relative to which E is contained in NP (see ZPP), and an oracle relative to PSPACE is contained in E (by equating the former with P).
There exists a problem that is complete for E under polynomial-time Turing reductions but not polynomial-time truth-table reductions [Wat87].
Problems hard for BPP under Turing reductions have measure 1 in E [AS94].
It follows that, if the problems complete for E under Turing reductions do not have measure 1 in E, then BPP does not equal EXP.
[IT89] gave an oracle relative to which E = NE but still there is an exponential-time binary predicate whose corresponding search problem is not in E.
[BF03] gave a proof that if E = NE, then no sparse set is collapsing, where they defined a set
{\displaystyle A}
to be collapsing if
{\displaystyle A\notin {\mathsf {P}}}
and if for all
{\displaystyle B}
{\displaystyle A}
{\displaystyle B}
are Turing reducible to each other,
{\displaystyle A}
{\displaystyle B}
are many-to-one reducible to each other.
Contrast with EXP.
EE: Double-Exponential Time With Linear Exponent
Equals DTIME(22O(n)) (though some authors alternatively define it as being equal to DTIME(2O(2n))).
EE = BPE if and only if EXP = BPP [IKW01].
Contained in EEXP and NEE.
EEE: Triple-Exponential Time With Linear Exponent
Equals DTIME(222O(n)).
In contrast to the case of EE, it is not known whether EEE = BPEE implies EE = BPE [IKW01].
EESPACE: Double-Exponential Space With Linear Exponent
Equals DSPACE(22O(n)).
Is not contained in BQP/qpoly [NY03].
EEXP: Double-Exponential Time
Equals DTIME(22p(n)) for p a polynomial.
Also known as 2-EXP.
Contains EE, and is contained in NEEXP.
EH: Exponential-Time Hierarchy With Linear Exponent
Has roughly the same relationship to E as PH does to P.
More formally, EH is defined as the union of E, NE, NENP, NE with Σ2P oracle, and so on.
See [Har87] for more information.
If coNP is contained in AM[polylog], then EH collapses to S2-EXP•PNP [SS04] and indeed AMEXP [PV04].
On the other hand, coNE is contained in NE/poly, so perhaps it wouldn't be so surprising if NE collapses.
There exists an oracle relative to which EH does not contain SEH [Hem89]. EH and SEH are incomparable for all anyone knows.
ELEMENTARY: Iterated Exponential Time
Equals the union of DTIME(2n), DTIME(22n), DTIME(222n), and so on.
Contained in PR.
ELkP: Extended Low Hierarchy
An extension of LkP.
The class of problems A such that ΣkPA is contained in Σk-1PA,NP.
Defined in [BBS86].
EP: NP with 2k Accepting Paths
If the answer is 'no,' then all computation paths reject.
If the answer is 'yes,' then the number of accepting paths is a power of two.
Contained in C=P, and in ModkP for any odd k. Contains UP.
Defined in [BHR00].
EPTAS: Efficient Polynomial-Time Approximation Scheme
The class of optimization problems such that, given an instance of length n, we can find a solution within a factor 1+ε of the optimum in time f(ε)p(n), where p is a polynomial and f is arbitrary.
Contains FPTAS and is contained in PTAS.
Defined in [CT97], where the following was also shown:
If FPT = XPuniform then EPTAS = PTAS.
If EPTAS = PTAS then FPT = W[P].
If FPT is strictly contained in W[1], then there is a natural problem that is in PTAS but not in EPTAS. (See [CT97] for the statement of the problem, since it's not that natural.)
k-EQBP: Width-k Polynomial-Time Exact Quantum Branching Programs
See k-PBP for the definition of a classical branching program.
A quantum branching program is the natural quantum generalization: we have a quantum state in a Hilbert space of dimension k. Each step t consists of applying a unitary matrix U(t)(xi): that is, U(t) depends on a single bit xi of the input. (So these are the quantum analogues of so-called oblivious branching programs.) In the end we measure to decide whether to accept; there must be zero probability of error.
Defined in [AMP02], where it was also shown that NC1 is contained in 2-EQBP.
k-BQBP can be defined similarly.
EQP: Exact Quantum Polynomial-Time
The same as BQP, except that the quantum algorithm must return the correct answer with probability 1, and run in polynomial time with probability 1. Unlike bounded-error quantum computing, there is no theory of universal QTMs for exact quantum computing models. In the original definition in [BV97], each language in EQP is computed by a single QTM, equivalently to a uniform family of quantum circuits with a finite gate set K whose amplitudes can be computed in polynomial time. See EQPK. However, some results require an infinite gate set. The official definition here is that the gate set should be finite.
Without loss of generality, the amplitudes in the gate set K are algebraic numbers [ADH97].
There is an oracle that separates EQP from NP [BV97], indeed from Δ2P [GP01]. There is also an oracle relative to which EQP is not in ModpP where p is prime [GV02]. On the other hand, EQP is in LWPP [FR98].
P||NP[2k] is contained in EQP||NP[k], where "||NP[k]" denotes k nonadaptive oracle queries to NP (queries that cannot depend on the results of previous queries) [BD99].
See also ZBQP.
EQPK: Exact Quantum Polynomial-Time with Gate Set K
The set of problems that can be answered by a uniform family of polynomial-sized quantum circuits whose gates are drawn from a set K, and that return the correct answer with probability 1, and run in polynomial time with probability 1, and the allowed gates are drawn from a set K. K may be either finite or countable and enumerated. If S is a ring, the union of EQPK over all finite gate sets K whose amplitudes are in the ring R can be written EQPS.
Defined in [ADH97] in the special case of a finite set of 1-qubit gates controlled by a second qubit. It was shown there that transcendental gates may be replaced by algebraic gates without decreasing the size of EQPK.
[FR98] show that EQPQ is in LWPP. The proof can be generalized to any finite, algebraic gate set K.
The hidden shift problem for a vector space over Z/2 is in EQPQ by Simon's algorithm. The discrete logarithm problem over Z/p is in EQPQ-bar using infinitely many gates [MZ03].
EQTIME(f(n)): Exact Quantum f(n)-Time
Same as EQP, but with f(n)-time (for some constructible function f) rather than polynomial-time machines.
Defined in [BV97].
ESPACE: Exponential Space With Linear Exponent
Equals DSPACE(2O(n)).
If E = ESPACE then P = BPP [HY84].
Indeed if E has nonzero measure in ESPACE then P = BPP [Lut91].
ESPACE is not contained in P/poly [Kan82].
Is not contained in BQP/mpoly [NY03].
See also: EXPSPACE.
∃BPP: BPP With Existential Operator
The class of problems for which there exists a BPP machine M such that, for all inputs x,
If the answer is "yes" then there exists a y such that M(x,y) accepts.
If the answer is "no" then for all y, M(x,y) rejects.
Alternatively defined as NPBPP.
Contains NP and BPP, and is contained in MA and SBP.
∃BPP seems obviously equal to MA, yet [FFK+93] constructed an oracle relative to which they're unequal! Here is the difference: if the answer is "yes," MA requires only that there exist a y such that for at least 2/3 of random strings r, M(x,y,r) accepts (where M is a P predicate). For all other y's, the proportion of r's such that M(x,y,r) accepts can be arbitrary (say, 1/2). For ∃BPP, by contrast, the probability that M(x,y) accepts must always be either at most 1/3 or at least 2/3, for all y's.
∃NISZK: NISZK With Existential Operator
Contains NP and NISZK, and is contained in the third level of PH.
∃Reals : Problems in ETR
Contains NP and is contained in PSPACE. Equivalently, it is the problem of testing whether a given semialgebraic set is non-empty. Many problems in discrete and computational geometry are contained in this class.
EXP: Exponential Time
Equals the union of DTIME(2p(n)) over all polynomials p.
Also equals P with E oracle.
If L = P then PSPACE = EXP.
If EXP is in P/poly then EXP = MA [BFL91].
Problems complete for EXP under many-one reductions have measure 0 in EXP [May94], [JL95].
There exist oracles relative to which
EXP = NP = ZPP [Hel84a], [Hel84b], [Kur85], [Hel86],
EXP = NEXP but still P does not equal NP [Dek76],
EXP does not equal PSPACE [Dek76].
[BT04] show the following rather striking result: let A be many-one complete for EXP, and let S be any set in P of subexponential density. Then A-S is Turing-complete for EXP.
[SM03] show that if EXP has circuits of polynomial size, then P can be simulated in MAPOLYLOG such that no deterministic polynomial-time adversary can generate a list of inputs for a P problem that includes one which fails to be simulated. As a result, EXP ⊆ MA if EXP has circuits of polynomial size.
[SU05] show that EXP
{\displaystyle \not \subseteq }
NP/poly implies EXP
{\displaystyle \not \subseteq }
P||NP/poly.
In descriptive complexity EXPTIME can be defined as SO(
{\displaystyle 2^{n^{O(1)}}}
) which is also SO(LFP)
EXP/poly: Exponential Time With Polynomial-Size Advice
The class of decision problems solvable in EXP with the help of a polynomial-length advice string that depends only on the input length.
EXPSPACE: Exponential Space
Equals the union of DSPACE(2p(n)) over all polynomials p.
See also: ESPACE.
Given a first-order statement about real numbers, involving only addition and comparison (no multiplication), we can decide in EXPSPACE whether it's true or not [Ber80].
Retrieved from "https://complexityzoo.net/index.php?title=Complexity_Zoo:E&oldid=6757"
|
Mathematical Reasoning, Popular Questions: CBSE Class 11-commerce MATH, Math - Meritnation
Utkarsh Sharma asked a question
The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line
4x 5y = 20 to the circle x2 + y2 = 9 is
(A) 20(x2 + y2) 36x + 45y = 0 (B) 20(x2 + y2) + 36x 45y = 0
(C) 36(x2 + y2) 20x + 45y = 0 (D) 36(x2 + y2) + 20x 45y = 0
Taha Yaseen asked a question
Write the component statements of the following:
"All prime numbers are either even or odd".
Shri Hari Agrawal asked a question
What Is a proof for midpoint theorem ?
4) Write the component statements of the compound statement: “All prime numbers are either even or odd”
This is a logical question:
8+4+6 = .............
and explain it
2\right) The value of \frac{\mathrm{log} 49\sqrt{7}+\mathrm{log} 25\sqrt{5}-\mathrm{log} 4\sqrt{2}}{\mathrm{log} 17.5} is\phantom{\rule{0ex}{0ex}}\left(a\right) 5\phantom{\rule{0ex}{0ex}}\left(b\right) 2\phantom{\rule{0ex}{0ex}}\left(c\right) \frac{5}{2}\phantom{\rule{0ex}{0ex}}\left(d\right) \frac{3}{2}
47. The function f : [0, 3] [1, 29], defined by f(x) = 2x3 15x2 + 36x + 1, is
(A) one-one and onto (B) onto but not one-one
(C) one-one but not onto (D) neither one-one nor onto
If (a2 + b2)3 = (a3 + b3)2 and ab not = 0 then the numerical value of a/b + b/a is equal to?
Vishal Raj asked a question
anand earns rs80 in 7 hours . parmod rs 90 in 12 hours ....find ratio of earning...plzz ans. in 2 min..
integration of sec square x
http://cbse.meritnation.com/study-online/ncert-solutions/math/11/5147/mathematical-reasoning/state-the-converse-and-contrapositive-of-e
in d 2nd bit y r v changing d sentence n thn writing converse n contra positive
Varnika Dhiman asked a question
what is the negation of the sentence “Rourkela is not an industrial area in Orissa”
Akash Bhardwaj asked a question
a+b+c=4root3 and a2+b2+c2=16 then the ratio a:b:c is
Tanay G Dalvi asked a question
What is the value of log(infinity) ?
write the negation " For all a ,b belongs to I , a-b belongs to I "
Lochan Nag asked a question
If 2x=0;then what is 2=? & X=?
If, A = 1+1x 0-1
B = 1+1 (0-1)
Then, which of the following statement is true.
a) A > B b) B < A c) A= B
Aayushi Garg asked a question
Find the hcf of 300,360,240 by prime factorisation method
The square root of 5+2root6 is?
♥♪♪♥ $!y@ ♥♪♪�... asked a question
Vector equation of the plane passing through points A(a) , B(b) and parallel to the vector c is [ r b c] + [ r c a] = [ a b c] ........ prove this plz....
The expression 3(a2 + 1)2 + 2(a - 1)(a2 + 1) - 5(a - 1)2 - 4(0.75a4 + 3a -1) when simplified reduces to ?
Sarwath Fatima asked a question
Write the converse and contrapositive of if parallelogram is a square,then it is a rhombus
if root2 is an irrational number,then prove that root2+root3 is also an irrational number.
Simar Preet Kaur Bains & 2 others asked a question
? ? ? ? ?=30
Pritham A R K asked a question
If A=diag[A1,A2,A3]then for an integer n greaterthan or equal to 1 show that A^n = diag[A1,A2,A3]???
Nabhitha Balachandar asked a question
Please help me with the 39th question. It was asked in a competitive exam .
Shri Laxmi asked a question
a sports team of 11 choosing atleast 5 from class 11 and atleast 5 from class 12 is found . If there are 20 students in each class. in how many ways can the team be formed?
450?me agar train 900km cover krti h toh 1km kitne me cover karegi
Yasra Fatema asked a question
If log0.5log5(x2- 4) > log0.51 , then 'x' lies in which interval .
Reshu R asked a question
If vector a=i(cap)+j(cap)+k(cap) and vector b= j(cap)-k(cap), find a vector c such that 'a' cross 'c' is vector b and 'a' dot 'c' is 3
If N is the smallest natural number such that N+2N+3N+....+99N is a perfect square then thenumber of digits in N^2 is
the number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples
Smgsankar asked a question
26) The area of the parallelogram having a diagonal 3
\stackrel{\to }{i}+\stackrel{\to }{j}-\stackrel{\to }{k}
and a side
\stackrel{\to }{i}-3\stackrel{\to }{j}+4\stackrel{\to }{k} is
10\sqrt{3}
6\sqrt{30}
\frac{3}{2}\sqrt{30}
3\sqrt{30}
i cant understand the last example (example 3). So kindly explain in a better way..
When a right triangle of area 4 is rotated 360 degree about its longer leg,the solid that results has a volume of the solid that results when the same right triangle is rotated about its shorter leg?
can i hav correct definition and eg to undersatand easily fast of above topics
Payal Panwar asked a question
Q3. Arusshi and Devesh are making a painting. Arushi can complete the painting in 30 minutes. Both Arushi and Devesh can complete the painting together in 20 minutes. They work together for 10 minutes and they have a quarrel. At this point, Arushi goes away. In how many minutes will Devesh finish the painting ?
using a binary code what would be the sum of binary numbers 0110 and 0101
Q.20. Find the order and degree of the differential equation
{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{\frac{4}{3}}-5{\left(\frac{dy}{dx}\right)}^{5}=0
An egg vendor calls on his first customer and sells half his eggs and half an egg. To the second customer, he sells half of what he has left with and half an egg, and to the third customer , he sells half of what he was then left with and half an egg. however, he did not break any egg. If in the end, the vendor was left with three eggs the what number of eggs did he have initially?
a)26 (b)31 (c)39 (d) none of these
If we multiply a certain two digit number by the sum of its digit we get 405 if we multiply the number consisting of the same digits written in the reverse order by the sum of the digits wicked 486 find the number
Motheeswar Vickraman asked a question
Find the sum up to n terms of the sequence:
0.7, 0.77, 0.777,....
While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as
45 and 16 respectively Find the correct mean and the variance.
Ritesh asked a question
a man buys 3 cows and 8 goats in 47200. instead , if he would have bought 8 cows and 3 goats , he had to pay 53000 more. cost of one cow
42/x - 5(41/x) + 4 = 0
The number N = 6 log102 + log1031, lies between two successive integers whose sum is equal to
[ CORRECT ANS IS (B) ]
Shilpi Roy asked a question
Please give the solution. My ans. Came as 87 degrees . Where did i go wrong?
If logab=2 ; logbc=2 and log3c=3+log3a then (a+b+c) equals .... ?
Devendra Pratap Singh asked a question
What number must be sussitituted with S to make it divisible by 36?
Pls explain the answers of all the questions
Pls answer by this night as i have my exams tomorrow morning by 7 am
Rahul Chopra asked a question
Three blocks of masses m1=4kg m2=2kg and m3=4kg are connected with ideal strings over a smooth massless pulley. The acceleration of blocks will be what?? (g=10)
Sreepadh Sandilya asked a question
Volume of uniform rod, rectangle,triangle,circular ring,disc,hollow cylinder,solid cylinder
write the contrapositive of :
if a number is divisible by 9, then it is divisible by 3.
\stackrel{\to }{a} and \stackrel{\to }{b}
\stackrel{\to }{a}+\stackrel{\to }{b}
is also a unit vector, then show that the angle between
\stackrel{\to }{a} and \stackrel{\to }{b}
\frac{2\mathrm{\pi }}{3}
Fatima Abdul Kalam asked a question
What is Euler's theorem?
Please answer 9 with explanation
Shashank Kumar asked a question
consider the multiplication in decimal notation(999).(abc)=def132 determine the digits a,b,c,d,e,f
Please solve the circled question
Shivaranjani asked a question
A firm produces 50 units of a product for rs.320 and 80 units for rs.380. Considering the cost curve to be a straight-line the cost of producing 110 units to be estimated as
Find 'x':
(2x-1 x 4x+1)/8x-1 = 16
Dev Prakash Singh asked a question
A boat starts with the speed of 1 km/h.After every 1 km ,the speed of boat becomes twice.How much will be the average speed of the boat at end of journey of 2.5 km?
Find the missing number in the given series 2,7,10,22,18,37,26,?
B}52
C}46
D}42
bhagyashreeraut... asked a question
How to solve log(1+2+3)=log1+log2+log3
write the converse of the following statement ''if I slap you then you will cry''
Rohan Vaish asked a question
from where cn i study about tautology and fallacy for the upcoming jee mains
a train crosses two bridges of length 500m and 280m in 35 seconds and 24 seconds respectively. find the length of the train.
Meritnation Experts,please help me MY ACCOUNT IS HACKED!!!!!!!
Somya Sonakshi asked a question
1/3 log3M + 3 log3 N = 1 + log 0.008 5
1.) M 9 = 9 / N
2.) N9 = 9/ M
3.)M3 = 3/N
4.) N9 = 3/M
Sankar Suresh asked a question
What is the answer for this, and how did it come?
Durga Kumar asked a question
8a^2b=27ab^2=216 then ab =?
Junaid Haque asked a question
\mathbit{Q}\mathbf{.}\mathbf{10}\mathbf{.}\mathbf{ }\mathbf{ } Solve the system of equations :\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{a} x {\mathrm{log}}_{a} \left(xyz\right) = 48\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{a} y {\mathrm{log}}_{a} \left(xyz\right) = 12, a > 0, a \ne 1.\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{a} z {\mathrm{log}}_{a} \left(xyz\right) = 84
Suramya Kumar Rawath & 1 other asked a question
if a:b=c:d , prove that a square c+ a c square :b square d + b d square =(a + c)cube :( b +d)cube .
Parag asked a question
Prove that area of circle is ?r^2 and circumference is 2?rh . //
Nikita Jacqualine Ekka asked a question
A is the father of B and D is the son of C. E is the brother of A and E also has a daughter F. If C is the sister of A, then what is the relation between F, B and D?
Want ans.fast to this
Anshika Anand asked a question
a student obtained the mean n standaard deviation of 100 observation as 40 n 5.1 respectively it was later found that one observation was wrongly copied as 50 the correct figure being 40. find the correct mean and S.D.
4x^2 +2 - 9(2x^2 +2)+8=0
pls answer this question immidietly.
Raj Pawar asked a question
R : he is rich . t: he is talanted. S: he is sucessful. The symbolic form of the statement " he is neither rich or nor talanted and he is not sucessful " is
If x = 21/3 + 22/3, then the value of x3 − 6x is
Answer question 4 ,5 and 6
find the least number which when increased by 3is exactly divisible by the numbers 21 45 63 81 210
can u say which type 1 mark q will be availiable from this leson
Kashish Chauhan asked a question
if abc4d is divisible by 4 then what isare the values of d?
Jay Kamble asked a question
write the negation of the statement"the number 5 is greater than 8 "
Sai Krishna asked a question
are -10,-5, 0, 5, 20 all multiples of 5 ?
If yes, is it safe to assume that the following statement is a tautology:
"m is a multiple of n iff m/n is an integer, where m and n are integers"
Also, is tautology a noun? (have I constructed the above question sentencet correctly?)
|
Double Mersenne number - Wikipedia
(Redirected from Double Mersenne prime)
In mathematics, a double Mersenne number is a Mersenne number of the form
7, 127, 2147483647
a(n) = 2^(2^prime(n) − 1) − 1
{\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}
2 Double Mersenne primes
3 Catalan–Mersenne number conjecture
The first four terms of the sequence of double Mersenne numbers are[1] (sequence A077586 in the OEIS):
{\displaystyle M_{M_{2}}=M_{3}=7}
{\displaystyle M_{M_{3}}=M_{7}=127}
{\displaystyle M_{M_{5}}=M_{31}=2147483647}
{\displaystyle M_{M_{7}}=M_{127}=170141183460469231731687303715884105727}
A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number
{\displaystyle M_{M_{p}}}
can be prime only if Mp is itself a Mersenne prime. For the first values of p for which Mp is prime,
{\displaystyle M_{M_{p}}}
is known to be prime for p = 2, 3, 5, 7 while explicit factors of
{\displaystyle M_{M_{p}}}
have been found for p = 13, 17, 19, and 31.
{\displaystyle p}
{\displaystyle M_{p}=2^{p}-1}
{\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}
factorization of
{\displaystyle M_{M_{p}}}
2 3 prime 7
3 7 prime 127
5 31 prime 2147483647
7 127 prime 170141183460469231731687303715884105727
11 not prime not prime 47 × 131009 × 178481 × 724639 × 2529391927 × 70676429054711 × 618970019642690137449562111 × ...
13 8191 not prime 338193759479 × 210206826754181103207028761697008013415622289 × ...
17 131071 not prime 231733529 × 64296354767 × ...
19 524287 not prime 62914441 × 5746991873407 × 2106734551102073202633922471 × 824271579602877114508714150039 × 65997004087015989956123720407169 × ...
23 not prime not prime 2351 × 4513 × 13264529 × 76899609737 × ...
29 not prime not prime 1399 × 2207 × 135607 × 622577 × 16673027617 × 4126110275598714647074087 × ...
31 2147483647 not prime 295257526626031 × 87054709261955177 × 242557615644693265201 × 178021379228511215367151 × ...
37 not prime not prime
61 2305843009213693951 unknown
Thus, the smallest candidate for the next double Mersenne prime is
{\displaystyle M_{M_{61}}}
, or 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, this number is far too large for any currently known primality test. It has no prime factor below 1 × 1036.[2] There are probably no other double Mersenne primes than the four known.[1][3]
Smallest prime factor of
{\displaystyle M_{M_{p}}}
(where p is the nth prime) are
7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1 × 1036) (sequence A309130 in the OEIS)
Catalan–Mersenne number conjecture
The recursively defined sequence
{\displaystyle c_{0}=2}
{\displaystyle c_{n+1}=2^{c_{n}}-1=M_{c_{n}}}
is called the sequence of Catalan–Mersenne numbers.[4] The first terms of the sequence (sequence A007013 in the OEIS) are:
{\displaystyle c_{0}=2}
{\displaystyle c_{1}=2^{2}-1=3}
{\displaystyle c_{2}=2^{3}-1=7}
{\displaystyle c_{3}=2^{7}-1=127}
{\displaystyle c_{4}=2^{127}-1=170141183460469231731687303715884105727}
{\displaystyle c_{5}=2^{170141183460469231731687303715884105727}-1\approx 5.454\times 10^{51217599719369681875006054625051616349}\approx 10^{10^{37.7094}}}
Catalan discovered this sequence after the discovery of the primality of
{\displaystyle M_{127}=c_{4}}
by Lucas in 1876.[1][5] Catalan conjectured that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if
{\displaystyle c_{5}}
is not prime, there is a chance to discover this by computing
{\displaystyle c_{5}}
modulo some small prime
{\displaystyle p}
(using recursive modular exponentiation). If the resulting residue is zero,
{\displaystyle p}
represents a factor of
{\displaystyle c_{5}}
and thus would disprove its primality. Since
{\displaystyle c_{5}}
is a Mersenne number, such a prime factor
{\displaystyle p}
would have to be of the form
{\displaystyle 2kc_{4}+1}
. Additionally, because
{\displaystyle 2^{n}-1}
is composite whe{\displaystyle n}
is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.
In the Futurama movie The Beast with a Billion Backs, the double Mersenne number
{\displaystyle M_{M_{7}}}
is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".
^ a b c Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
^ "Double Mersenne 61 factoring status". www.doublemersennes.org. Retrieved 31 March 2022.
^ I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]
^ Weisstein, Eric W. "Catalan-Mersenne Number". MathWorld.
^ "Questions proposées". Nouvelle correspondance mathématique. 2: 94–96. 1876. (probably collected by the editor). Almost all of the questions are signed by Édouard Lucas as is number 92:
Prouver que 261 − 1 et 2127 − 1 sont des nombres premiers. (É. L.) (*).
The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows:
(*) Si l'on admet ces deux propositions, et si l'on observe que 22 − 1, 23 − 1, 27 − 1 sont aussi des nombres premiers, on a ce théorème empirique: Jusqu'à une certaine limite, si 2n − 1 est un nombre premier p, 2p − 1 est un nombre premier p', 2p' − 1 est un nombre premier p", etc. Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude: Si n est une puissance de 2, 2n + 1 est un nombre premier. (E. C.)
Dickson, L. E. (1971) [1919], History of the Theory of Numbers, New York: Chelsea Publishing .
Weisstein, Eric W. "Double Mersenne Number". MathWorld.
Tony Forbes, A search for a factor of MM61.
Status of the factorization of double Mersenne numbers
Double Mersennes Prime Search
Retrieved from "https://en.wikipedia.org/w/index.php?title=Double_Mersenne_number&oldid=1080351545#Double_Mersenne_primes"
|
A<sub>c</sub> = A<sub>i</sub> + A<sub>p</sub>, and
A<sub>r</sub> = A<sub>p</sub> (i.e., assumed that the water storage reservoir area and permeable pavement area are the same)
Then increase A<sub>r</sub> accordingly to keep R between 0 and 2, which reduces hydraulic loading and helps avoid premature clogging. This assumes that the water storage reservoir area and permeable pavement area are the same (A<sub>r</sub> = A<sub>p</sub>).
Then increase A<sub>r</sub> accordingly to keep R between 0 and 2, which reduces hydraulic loading and helps avoid premature clogging.
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
The following calculation is used to size the stone storage bed (reservoir) used as a base course. It is assumed that the footprint of the stone bed will be equal to the footprint of the pavement. The following equations are derived from the ICPI Manual <ref>Smith, D. 2006. Permeable Interlocking Concrete Pavements; Selection, Design, Construction, Maintenance. 3rd Edition. Interlocking Concrete Pavement Institute. Burlington, ON.</ref>
The following calculation is used to size the stone storage bed (reservoir) used as a base course. It is assumed that the footprint of the stone bed will be equal to the footprint of the pavement. The following equations are derived from the Interlocking Concrete Pavement Institute (ICPI) manual <ref>Smith, D. 2017. Permeable Interlocking Concrete Pavements; Selection, Design, Specifications, Construction, Maintenance. 5th Edition. Interlocking Concrete Pavement Institute. Chantilly VA</ref>
===For full infiltration design, to calculate the total depth of clear stone aggregate layers needed for the water storage reservoir===
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
Magnetic dipole - Wikipedia
Magnetic analogue of the electric dipole
The magnetic field due to natural magnetic dipoles (upper left), magnetic monopoles (upper right), an electric current in a circular loop (lower left) or in a solenoid (lower right). All generate the same field profile when the arrangement is infinitesimally small.[1]
In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a true magnetic monopole, the magnetic analogue of an electric charge, has never been observed in nature. However, magnetic monopole quasiparticles have been observed as emergent properties of certain condensed matter systems.[2] Moreover, one form of magnetic dipole moment is associated with a fundamental quantum property—the spin of elementary particles.
Because magnetic monopoles do not exist, the magnetic field at a large distance from any static magnetic source looks like the field of a dipole with the same dipole moment. For higher-order sources (e.g. quadrupoles) with no dipole moment, their field decays towards zero with distance faster than a dipole field does.
1 External magnetic field produced by a magnetic dipole moment
2 Internal magnetic field of a dipole
3 Forces between two magnetic dipoles
4 Dipolar fields from finite sources
External magnetic field produced by a magnetic dipole moment[edit]
An electrostatic analogue for a magnetic moment: two opposing charges separated by a finite distance. Each arrow represents the direction of the field vector at that point.
The magnetic field of a current loop. The ring represents the current loop, which goes into the page at the x and comes out at the dot.
In classical physics, the magnetic field of a dipole is calculated as the limit of either a current loop or a pair of charges as the source shrinks to a point while keeping the magnetic moment m constant. For the current loop, this limit is most easily derived from the vector potential:[3]
{\displaystyle {\mathbf {A} }({\mathbf {r} })={\frac {\mu _{0}}{4\pi r^{2}}}{\frac {{\mathbf {m} }\times {\mathbf {r} }}{r}}={\frac {\mu _{0}}{4\pi }}{\frac {{\mathbf {m} }\times {\mathbf {r} }}{r^{3}}},}
where μ0 is the vacuum permeability constant and 4π r2 is the surface of a sphere of radius r. The magnetic flux density (strength of the B-field) is then[3]
{\displaystyle \mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }={\frac {\mu _{0}}{4\pi }}\left[{\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{r^{5}}}-{\frac {\mathbf {m} }{r^{3}}}\right].}
Alternatively one can obtain the scalar potential first from the magnetic pole limit,
{\displaystyle \psi ({\mathbf {r} })={\frac {{\mathbf {m} }\cdot {\mathbf {r} }}{4\pi r^{3}}},}
and hence the magnetic field strength (or strength of the H-field) is
{\displaystyle {\mathbf {H} }({\mathbf {r} })=-\nabla \psi ={\frac {1}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {m} \cdot \mathbf {\hat {r}} )-\mathbf {m} }{r^{3}}}\right]={\frac {\mathbf {B} }{\mu _{0}}}.}
The magnetic field strength is symmetric under rotations about the axis of the magnetic moment. In spherical coordinates, with
{\displaystyle \mathbf {\hat {z}} =\mathbf {\hat {r}} \cos \theta -{\boldsymbol {\hat {\theta }}}\sin \theta }
, and with the magnetic moment aligned with the z-axis, then the field strength can more simply be expressed as
{\displaystyle \mathbf {H} ({\mathbf {r} })={\frac {|\mathbf {m} |}{4\pi r^{3}}}\left(2\cos \theta \,\mathbf {\hat {r}} +\sin \theta \,{\boldsymbol {\hat {\theta }}}\right).}
Internal magnetic field of a dipole[edit]
See also: Magnetic moment § Magnetic pole definition
The two models for a dipole (current loop and magnetic poles), give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.
If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}+{\frac {8\pi }{3}}\mathbf {m} \delta (\mathbf {r} )\right],}
where δ(r) is the Dirac delta function in three dimensions. Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.
If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole-charge and distance constant, the limiting field is
{\displaystyle \mathbf {H} (\mathbf {r} )={\frac {1}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}-{\frac {4\pi }{3}}\mathbf {m} \delta (\mathbf {r} )\right].}
These fields are related by B = μ0(H + M), where
{\displaystyle \mathbf {M} (\mathbf {r} )=\mathbf {m} \delta (\mathbf {r} )}
is the magnetization.
Forces between two magnetic dipoles[edit]
See also: Force between magnets § Magnetic dipole-dipole interaction
The force F exerted by one dipole moment m1 on another m2 separated in space by a vector r can be calculated using:[4]
{\displaystyle \mathbf {F} =\nabla \left(\mathbf {m} _{2}\cdot \mathbf {B} _{1}\right),}
or [5][6]
{\displaystyle \mathbf {F} (\mathbf {r} ,\mathbf {m} _{1},\mathbf {m} _{2})={\dfrac {3\mu _{0}}{4\pi r^{5}}}\left[(\mathbf {m} _{1}\cdot \mathbf {r} )\mathbf {m} _{2}+(\mathbf {m} _{2}\cdot \mathbf {r} )\mathbf {m} _{1}+(\mathbf {m} _{1}\cdot \mathbf {m} _{2})\mathbf {r} -{\dfrac {5(\mathbf {m} _{1}\cdot \mathbf {r} )(\mathbf {m} _{2}\cdot \mathbf {r} )}{r^{2}}}\mathbf {r} \right],}
where r is the distance between dipoles. The force acting on m1 is in the opposite direction.
The torque can be obtained from the formula
{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} _{2}\times \mathbf {B} _{1}.}
Dipolar fields from finite sources[edit]
See also: Near and far field
The magnetic scalar potential ψ produced by a finite source, but external to it, can be represented by a multipole expansion. Each term in the expansion is associated with a characteristic moment and a potential having a characteristic rate of decrease with distance r from the source. Monopole moments have a 1/r rate of decrease, dipole moments have a 1/r2 rate, quadrupole moments have a 1/r3 rate, and so on. The higher the order, the faster the potential drops off. Since the lowest-order term observed in magnetic sources is the dipolar term, it dominates at large distances. Therefore, at large distances any magnetic source looks like a dipole of the same magnetic moment.
^ I.S. Grant, W.R. Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons. ISBN 978-0-471-92712-9.
^ Magnetic monopoles spotted in spin ices, September 3, 2009.
^ a b Chow 2006, pp. 146–150
^ D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education. p. 276. ISBN 978-81-7758-293-2.
^ Furlani 2001, p. 140
^ K.W. Yung; P.B. Landecker; D.D. Villani (1998). "An Analytic Solution for the Force between Two Magnetic Dipoles" (PDF). Retrieved November 24, 2012. {{cite journal}}: Cite journal requires |journal= (help)
Chow, Tai L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones & Bartlett Learning. ISBN 978-0-7637-3827-3.
Jackson, John D. (1975). Classical Electrodynamics (2nd ed.). Wiley. ISBN 0-471-43132-X.
Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications. Academic Press. ISBN 0-12-269951-3.
Schill, R. A. (2003). "General relation for the vector magnetic field of a circular current loop: A closer look". IEEE Transactions on Magnetics. 39 (2): 961–967. Bibcode:2003ITM....39..961S. doi:10.1109/TMAG.2003.808597.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Magnetic_dipole&oldid=1088848568"
|
Application of Exponential Kernel to Laplace Transform
Application of Exponential Kernel to Laplace Transform
In this paper, the exponential decreasing kernel is used in Laplace integral transform to transform a function from a certain domain to another domain. It is shown, in a rigorous way, that the Laplace transform of the delta function is exactly one half rather than one, as it is believed. In addition, when this kernel is used in integral transform of attractive and repulsive Coulomb potential, it yields a finite definite value at the point of singularity.
Kernels, Integral Transforms, Laplace Transforms, Singularity
Usually, kernels determine an implicit map that transforms a function or data from the input space to a feature space, and therefore determine its distribution in the latter space. This is usually accomplished through integral transforms. Some of the well-known kernels include the polynomial, exponential and Gaussian kernels. In particular, the exponential kernel through Laplace transform has been widely used over the years [1] - [6] . The Laplace transform is defined to transform a function from a space, say
x\in \left[0,\infty \right)
to a space, say
s\in \left(0,\infty \right)
. Finding the Laplace transform of a function and its properties is normally discussed in standard mathematical physics books [7] [8] . An interesting function (more precisely a limit of some distribution) is the Dirac delta function, which has been in use in different settings [9] - [15] . The value of the Laplace transform of Delta-function can be found in mathematical physics books [8] , where it is claimed that this value is one. We believe that the approach used to obtain this result is oversimplified and not rigorous. Therefore, one main object of this paper is to present a rigorous proof, through the use of a decreasing exponential kernel, and show that the correct value of Laplace transform of the delta function is exactly one half. The second part of this paper is to apply the decreasing exponential kernel to a discontinuous function. In particular, we consider a function with repulsive Coulomb-like form on the positive real axis, and with attractive Coulomb-like form on the negative real axis. This function is singular at the origin, and its right-hand and left-hand limits towards the origin are
+\infty
-\infty
respectively. It is shown, with this decreasing exponential kernel, that the value of this function is exactly zero which is the average between its limiting values at the origin. The last section of this paper is devoted for conclusion and discussion.
2. The Laplace Transform of Delta-Function
Consider the decreasing exponential kernel
{\text{e}}^{-s|x|}
and the delta function
\delta \left(x\right)
. Our aim is to derive the Laplace transform of
\delta \left(x\right)
by applying this kernel to the integral;
{\int }_{-\infty }^{\infty }{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x
s>0
Due to the well-known property of the delta-function, namely
{\int }_{-\infty }^{\infty }f\left(x\right)\delta \left(x-a\right)\text{d}x=f\left(a\right)
{\int }_{-\infty }^{\infty }{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x={\text{e}}^{0}=1
Splitting the integral into two parts, we get
{\int }_{-\infty }^{\infty }{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x={\int }_{-\infty }^{0}{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x+{\int }_{0}^{\infty }{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x
In the first integral on the left-hand side,
|x|=-x
, and by letting
x\to -x
{\int }_{-\infty }^{0}{\text{e}}^{-s|x|}\delta \left(x\right)={\int }_{\infty }^{0}{\text{e}}^{-sx}\delta \left(-x\right)\left(-\text{d}x\right)={\int }_{0}^{\infty }{\text{e}}^{-sx}\delta \left(x\right)\text{d}x
Note that, in the last step, we used the fact that
\delta \left(-x\right)=\delta \left(x\right)
, since it is even. So upon the substitution of Equation (5) into Equation (4), one gets
{\int }_{-\infty }^{\infty }{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x=2{\int }_{0}^{\infty }{\text{e}}^{-sx}\delta \left(x\right)\text{d}x
The Laplace transform of a function
f\left(x\right)
\mathcal{L}\left\{f\left(x\right)\right\}=f\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sx}f\left(x\right)\text{d}x
Therefore, Equation (6) yields
{\int }_{-\infty }^{\infty }{\text{e}}^{-s|x|}\delta \left(x\right)\text{d}x=2\mathcal{L}\left\{\delta \left(x\right)\right\}
Hence, the use of Equation (3) gives the Laplace transform of
\delta \left(x\right)
\mathcal{L}\left\{\delta \left(x\right)\right\}=\frac{1}{2}
The problem with the derivation of the unity value of the Laplace transform of the delta function, which is found in the literature [8] , is overlooked at the lower limit (
x=0
) in the definition of the Laplace transform. The point
x=0
separates the positive and the negative parts of the x-axis. So, when applying Equation (2), one must ensure that the point
x=a
must be totally included in the range of integration. This is not satisfied for the present case, and therefore one has to examine the whole domain of the delta function. This is the main essence of our derivation.
3. Application of the Exponential Kernel to Coulomb-Like Function
Discontinuous functions arise in some physical situations and usually one has to determine the value of this function at its point of discontinuity. Examples of these problems are the electric field at charged conducting sphere [16] , the energy loss in the two capacitor problem [17] and Fermi-Dirac distribution [18] . Here, we consider a Coulomb-like potential (attractive and repulsive on negative and positive real axis respectively). This kind of function is discontinuous at the origin. We will show that this function converges to its average value at its singular point (
r=0
In this section, we apply the decreasing exponential kernel to the Coulomb-like function which is given by
f\left(r\right)=\left\{\begin{array}{l}\frac{1}{r}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }r>0\\ -\frac{1}{r}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}r<0\end{array}
{\int }_{-\infty }^{\infty }{\text{e}}^{-s|r|}f\left(r\right)\text{d}r={\int }_{-\infty }^{0}{\text{e}}^{sr}\left(-\frac{1}{r}\right)\text{d}r+{\int }_{0}^{\infty }{\text{e}}^{-sr}\left(\frac{1}{r}\right)\text{d}r
r\to -r
in the first integral of the left-hand side of the above equation, we get
\begin{array}{c}{\int }_{-\infty }^{\infty }{\text{e}}^{-s|r|}f\left(r\right)\text{d}r=-{\int }_{\infty }^{0}{\text{e}}^{-sr}\frac{1}{r}\text{d}r+{\int }_{0}^{\infty }{\text{e}}^{-sr}\frac{1}{r}\text{d}r\\ =2{\int }_{0}^{\infty }{\text{e}}^{-sr}\frac{1}{r}\text{d}r=2\mathcal{L}\left\{\frac{1}{r}\right\}\end{array}
f\left(r\right)
is odd and the kernel is even so that the integral on the left-hand side of Equation (12) is zero. Two conclusions from the above equation are drawn: The first one is that the Laplace transform
\mathcal{L}\left(1/r\right)=0
. For the second conclusion, we first observe that the limit of the integral on the left-hand side of Equation (12) as
s\to \infty
, the kernel
{\text{e}}^{-s|r|}\to 0
except at the point
r=0
, at which it is just a constant. In this case, to ensure the vanishing of the integral on the left-hand side of Equation (12), the function
f\left(r\right)
must vanish at the origin, i.e.
f\left(0\right)=0
. It is noticed that
{\mathrm{lim}}_{r\to {r}^{+}}f\left(r\right)=\infty
{\mathrm{lim}}_{r\to {r}^{-}}f\left(r\right)=-\infty
, so that the average between these two limiting values is zero. Therefore, our second conclusion is that the value of the function at its point of discontinuity converges to its average value between its two limiting values at that point.
In this paper, a decreasing exponential kernel was used to derive the correct value of the Laplace transform of the delta function which is found to be one half. We also applied this type of kernel to a function which has a Coulomb-like form. Two conclusions of this application to such function were drawn: The first is
that the Laplace transform of
\left(\frac{1}{r}\right)
is zero and the second is that the value of
this function at its point of discontinuity is the average value between its two limiting values about that point.
AL-Jaber, S.M. (2019) Application of Exponential Kernel to Laplace Transform. Journal of Applied Mathematics and Physics, 7, 1126-1130. https://doi.org/10.4236/jamp.2019.75075
1. Tsaur, J. and Wang, P. (2014) A Universal Laplace-Transform Approach to Solving Schrodinger Equation for All Solvable Models. European Journal of Physics, 35, Article ID: 015006. https://doi.org/10.1088/0143-0807/35/1/015006
2. Pimental, D.R.M. and de Castro, A.S. (2013) A Laplace Transform Approach to the Quantum Harmonic Oscillator. European Journal of Physics, 34, 199. https://doi.org/10.1088/0143-0807/34/1/199
3. Penson, K.A. and Gorska, K. (2016) On the Properties of Laplace Transform Originating from One-Sided From Levy Stable Laws. Journal of Physics A: Mathematical and Theoretical, 49, Article ID: 065201. https://doi.org/10.1088/1751-8113/49/6/065201
4. Kryzhniy, V.V. (2006) Numerical Inversion of Laplace Transform: Analysis via Regularized Analytic Continuation. Inverse Problems, 22, 579-597. https://doi.org/10.1088/0266-5611/22/2/012
5. Viaggiu, V.V. (2017) Axial and Polar Gravitational Wave Equations in a de Sitter Expanding Universe by Laplace Transform. Classical and Quantum Gravity, 34, Article ID: 035018. https://doi.org/10.1088/1361-6382/aa5570
6. Gonzalez, F., Saiz, J.M., Moreno, F. and Valle, R.J. (1992) Application of Laplace Transform Method to Binary Mixtures of Spherical Particles in Solution For Low Scattering Intensity. Journal of Physics D: Applied Physics, 25, 357-361. https://doi.org/10.1088/0022-3727/25/3/003
7. Boas, M.L. (2005) Mathematical Methods in the Physical Sciences. Wiley & Sons, Hoboken.
8. Arfken, G.B. and Weber, H.J. (2005) Mathematical Physics for Physicists. Elsevier Academic Press, Amsterdam.
9. Chua, C.-K., Liu, T.-U. and Wong, G.-G. (2018) Time-Independent Green’s Function of a Quantum Simple Harmonic Oscillator System and Solutions with Additional Generic Delta-Function Potentials. Journal of Physics Communications, 2, Article ID: 035007. https://doi.org/10.1088/2399-6528/aa9eeb
10. Galapon, E.A. (2009) Delta-Convergent Sequences That Vanish at the Support of the Limit Delta Function. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 175201. https://doi.org/10.1088/1751-8113/42/17/175201
11. Parker, E. (2017) An Apparent Paradox Concerning the Field of a Dipole. European Journal of Physics, 38, Article ID: 025205. https://doi.org/10.1088/1361-6404/aa55a6
12. Demirlab, E. (2005) Bound States of n-Dimensional Harmonic Oscillator Decorated by Delta-Functions. Journal of Physics A: Mathematical and Theoretical, 38, 4783-4793. https://doi.org/10.1088/0305-4470/38/22/003
13. Zhong, M.A. and Yang, C.N. (2010) Bosons or Fermions in 1D Power Potential Trap with Repulsive Delta Function Interactions. Chinese Physics Letters, 27, Article ID: 090505. https://doi.org/10.1088/0256-307X/27/9/090505
14. Dmitriev, D.V. and Krivnov, V. (2018) Heisenberg-Ising Delta-Chain with Bond Alternation. Journal of Physics: Condefnsed Matter, 30, Article ID: 385803. https://doi.org/10.1088/1361-648X/aadb72
15. Tracy, C.A. and Widom, H. (2016) On the Ground State Energy on the δ-Function Bose Gas. Journal of Physics A: Mathematical and Theoretical, 49, Article ID: 294001. https://doi.org/10.1088/1751-8113/49/29/294001
16. Griffiths, D.J. (1999) Introduction to Electrodynamics. Cambridge University Press, Cambridge.
17. AL-Jaber, S.M. and Salih, S. (2000) Energy Considerations in the Two-Capacitor Problem. European Journal of Physics, 41, 341. https://doi.org/10.1088/0143-0807/21/4/307
18. Pathria, R.K. and Peale, P.D. (2011) Statistical Mechanics. Academic Press, New York.
|
Compute output, error, and weights of LMS adaptive filter - MATLAB - MathWorks Switzerland
\mathit{d}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
\mathit{y}\left(\mathit{k}\right)
\mathit{d}\left(\mathit{k}\right)
\mathit{w}\left(\mathit{k}+1\right)=\mathit{w}\left(\mathit{k}\right)+\mu \mathit{e}\left(\mathit{k}\right)\mathrm{sgn}\left(\mathit{x}\left(\mathit{k}\right)\right),
\mathrm{sgn}\left(\mathit{x}\left(\mathit{k}\right)\right)=\left\{\begin{array}{c}1,\text{\hspace{0.17em}}\mathit{x}\left(\mathit{k}\right)>0\\ 0,\text{\hspace{0.17em}}\mathit{x}\left(\mathit{k}\right)=0\\ -1,\text{\hspace{0.17em}}\mathit{x}\left(\mathit{k}\right)<0\end{array}
\mathit{w}
\mathit{x}
\mathit{e}
\mu
\mu
\mu
\mu
0<\mu <\frac{1}{\mathit{N}\left\{\mathrm{InputSignalPower}\right\}},
\mathit{N}
\mu
\left(\mu \ll 1\right)
\mathit{d}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
\mu
\mu
\mathit{w}\left(\mathit{k}+1\right)=\mathit{w}\left(\mathit{k}\right)+\mu \mathrm{sgn}\left(\mathit{e}\left(\mathit{k}\right)\right)\left(\mathit{x}\left(\mathit{k}\right)\right)
\mathrm{sgn}\left(\mathit{e}\left(\mathit{k}\right)\right)=\left\{\begin{array}{c}1,\text{\hspace{0.17em}}\mathit{e}\left(\mathit{k}\right)>0\\ 0,\text{\hspace{0.17em}}\mathit{e}\left(\mathit{k}\right)=0\\ -1,\text{\hspace{0.17em}}\mathit{e}\left(\mathit{k}\right)<0\end{array}
\mathit{w}
\mathit{x}
\mathit{e}
\mu
\mu
\mu
0<\mu <\frac{1}{\mathit{N}\left\{\mathrm{InputSignalPower}\right\}}
\mathit{N}
\mu
\left(\mu \ll 1\right)
\mathit{d}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
\mu
\mu
w\left(k+1\right)=w\left(k\right)+\mu \mathrm{sgn}\left(e\left(k\right)\right)\mathrm{sgn}\left(x\left(k\right)\right),
\mathrm{sgn}\left(\mathit{z}\left(\mathit{k}\right)\right)=\left\{\begin{array}{c}1,\text{\hspace{0.17em}}\mathit{z}\left(\mathit{k}\right)>0\\ 0,\text{\hspace{0.17em}}\mathit{z}\left(\mathit{k}\right)=0\\ -1,\text{\hspace{0.17em}}\mathit{z}\left(\mathit{k}\right)<0\end{array}
\mathit{z}\left(\mathit{k}\right)=\mathit{e}\left(\mathit{k}\right)\mathrm{sgn}\left(\mathit{x}\left(\mathit{k}\right)\right)
\mathit{w}
\mathit{x}
\mathit{e}
\mu
\mu
\mu
0<\mu <\frac{1}{\mathit{N}\left\{\mathrm{InputSignalPower}\right\}}
\mathit{N}
\mu
\left(\mu \ll 1\right)
\mathit{d}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
\mathit{x}\left(\mathit{k}\right)
Q=\frac{\mu \cdot e}{u\text{'}u}
\mu \cdot e
Q\cdot u
\mu \cdot e
Q\cdot u
\begin{array}{c}y\left(n\right)={w}^{T}\left(n-1\right)u\left(n\right)\\ e\left(n\right)=d\left(n\right)-y\left(n\right)\\ w\left(n\right)=\alpha w\left(n-1\right)+f\left(u\left(n\right),e\left(n\right),\mu \right)\end{array}
f\left(u\left(n\right),e\left(n\right),\mu \right)=\mu e\left(n\right){u}^{*}\left(n\right)
f\left(u\left(n\right),e\left(n\right),\mu \right)=\mu e\left(n\right)\frac{{u}^{\ast }\left(n\right)}{\epsilon +{u}^{H}\left(n\right)u\left(n\right)}
f\left(u\left(n\right),e\left(n\right),\mu \right)=\mu e\left(n\right)\text{sign}\left(u\left(n\right)\right)
f\left(u\left(n\right),e\left(n\right),\mu \right)=\mu \text{sign}\left(e\left(n\right)\right){u}^{*}\left(n\right)
f\left(u\left(n\right),e\left(n\right),\mu \right)=\mu \text{sign}\left(e\left(n\right)\right)\text{sign}\left(u\left(n\right)\right)
|
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
R = Ai/Ap
It is important to note that R, the ratio of impervious contributing drainage area (Ai) to permeable pavement area (Ap), should not exceed 2 to limit hydraulic loading and help avoid premature clogging. Also important to note is that the contributing drainage area should not contain pervious areas that are sources of sediment that can lead to premature clogging.
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
MATHEMATICAL FINANCE - Encyclopedia Information
Mathematical finance Information
https://en.wikipedia.org/wiki/Mathematical_finance
Time deposit ( certificate of deposit)
Option ( call
Budget ( balance)
In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. [1] Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios.
French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900. But mathematical finance emerged as a discipline in the 1970s, following the work of Fischer Black, Myron Scholes and Robert Merton on option pricing theory. Mathematical investing originated from the research of mathematician Edward Thorp who used statistical methods to first invent card counting in blackjack and then applied its principles to modern systematic investing. [2]
The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory that is involved in financial mathematics. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock. See: Valuation of options; Financial modeling; Asset pricing. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results. [3]
3.3 Portfolio modelling
Environment risk-neutral probability
{\displaystyle \mathbb {Q} }
Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community. Quantitative derivatives pricing was initiated by Louis Bachelier in The Theory of Speculation ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes, the Brownian motion, and its applications to the pricing of options. [4] [5] The Brownian motion is derived using the Langevin equation and the discrete random walk. [6] Bachelier modeled the time series of changes in the logarithm of stock prices as a random walk in which the short-term changes had a finite variance. This causes longer-term changes to follow a Gaussian distribution. [7]
The theory remained dormant until Fischer Black and Myron Scholes, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, to option pricing. For this M. Scholes and R. Merton were awarded the 1997 Nobel Memorial Prize in Economic Sciences. Black was ineligible for the prize because of his death in 1995. [8]
The next important step was the fundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current price P0 of a security is arbitrage-free, and thus truly fair only if there exists a stochastic process Pt with constant expected value which describes its future evolution: [9]
{\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}
A process satisfying ( 1) is called a " martingale". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the blackboard font letter "
{\displaystyle \mathbb {Q} }
The relationship ( 1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.
Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as ( 1), a similar relationship is used to define the price of new derivatives.
Environment real-world probability
{\displaystyle \mathbb {P} }
This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "
{\displaystyle \mathbb {P} }
", as opposed to the "risk-neutral" probability "
{\displaystyle \mathbb {Q} }
" used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio. Increasingly, elements of this process are automated; see Outline of finance § Quantitative investing for a listing of relevant articles.
The portfolio-selection work of Markowitz and Sharpe introduced mathematics to investment management. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, one-period models were replaced by continuous time, Brownian-motion models, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions. [11] Furthermore, in recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of the market parameters. [12] See Financial risk management § Investment management.
Much effort has gone into the study of financial markets and how prices vary with time. Charles Dow, one of the founders of Dow Jones & Company and The Wall Street Journal, enunciated a set of ideas on the subject which are now called Dow Theory. This is the basis of the so-called technical analysis method of attempting to predict future changes. One of the tenets of "technical analysis" is that market trends give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics.[ citation needed]
Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the financial crisis of 2007–2010. Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Paul Wilmott, and by Nassim Nicholas Taleb, in his book The Black Swan. [13] Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and Emanuel Derman published the Financial Modelers' Manifesto in January 2009 [14] which addresses some of the most serious concerns. Bodies such as the Institute for New Economic Thinking are now attempting to develop new theories and methods. [15]
In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate. [16] In the 1960s it was discovered by Benoit Mandelbrot that changes in prices do not follow a Gaussian distribution, but are rather modeled better by Lévy alpha- stable distributions. [17] The scale of change, or volatility, depends on the length of the time interval to a power a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable. [13]
Survival analysis ( Proportional hazards model)
Retrieved from " https://en.wikipedia.org/?title=Mathematical_finance&oldid=1078802622"
Mathematical Finance Videos
Mathematical Finance Websites
Mathematical Finance Encyclopedia Articles
|
Physics Engine | Qian Lin's Personal Site
C++, Geometry, OpenGL
It is the final project for the course of Fundamentals of Computer Aided Design, THU 2018 Fall, programming with OpenGL.
It’s a course assignment of a Computer Aided Design couse in CMU.
The source code can be found on my github page. The other programming assignments of this course are also maintained in the repository. The detailed instruction is in addtionalMaterial.pdf and README. The source edited files are src/physics/collision*, physics*, spherebody*, spring*.
I was also selected to give a presentation about my project.
There are 6 scenes to be implemented in this project.
Collision Collision Stress Damping
Spring Rotation Newtons Cradle Rotation
Collision — Detection
Sphere & Sphere: A collision would happen when $\overrightarrow{O_1 O_2} < r_1+r_2$ and $(\vec{v}_1 - \vec{v}_2)\cdot\overrightarrow{O_1O_2} >0$.
Sphere & Plane: d<r and sphere moves towards the plane.
Sphere & Triangle: When the projection of the sphere center on the triangle’s plane falls in the triangle, i.e., $(\overrightarrow{PA_0} \times \overrightarrow{PA_1}), (\overrightarrow{PA_1}\times \overrightarrow{PA_2}), (\overrightarrow{PA_2}\times \overrightarrow{PA_0})$ have the same orientation, the same as the plane case. Otherwise, consider the projection of P on the three edges (if falls outside, the select the nearest vertex). Calculate the distance of the three selected points to the sphere center respectively; one of them is the nearest distance of the two bodies, which is also the collision point if collision happens.
Spring — Force Apply
The step() function applies forces and torques to the attached bodies.
The force is applied every frame, so we should clear the force applied in the previous frame.
SphereBody — Integration
The force varies with velocity and position. So we need to use 2-degree RK4 in Physics and call the Spring::step to update force (acceleration). The motion_damping used in SphereBody‘s step functions also need to be updated by Spring, which is a little troublesome. I just see force as a constant in every frame.
I was confused by the requirements in the given files, so I did not really realize the RK4 method. The time was limited to complete this task, so I used my friends’ method —- Although this method actually equals to
\begin{cases} v = v_0 + dt \cdot a \\ x = dt \cdot v_0 + \frac{1}{2} a\cdot dt^2 \end{cases}
, if use the latter directly, the error could not be neglected and would finally causes huge deviation. That is really a strange phenomenon and I have lost the code with the latter method. The only explanation to this I could find is the calculating precision of this data type.
Physics — Move Forward
The position and angular_position of a SphereBody::Sphere has already been updated in SphereBody, so in Physics the only thing to do is to call them. The gravity is applied in the initialization.
Fisrt, check collisions by traverse the sphere list and check them with the other bodies (including sphere).
Traverse the spring list, apply forces.
Traverse all the sphere and update them.
OpenGL and other framework
SDL1.2 (not 2.0)
libpng-1.2.57 (this version only)
GLEW, GLUT
(The makefile may need to be changed under other operation system)
make # compile with make
./physics scenes/collision_stress.scene # or other scene in scenes dir
The course also has 7 other assignments, six of which are programming work (the other one is a design work, and I even forget to hand in my design).
OpenGL can achieve a pretty beautiful effect so I put all of them here.
My work in Solar System was also selected as one of the two excellent homework by the TA.
Display a 3D Model Rotating Stars Snow Falling
Solar System Bezier Surface Lighting the 3D Model
The basic frame of those works was given by TA.
Thanks for the help of my senior Rui Wang and my classmate Ran Zhou.
|
Meyer hardness test - Wikipedia
This graph shows the differences between the Brinell hardness test and the Meyer hardness test. Notice that the Brinell test can report the same hardness value for a given specimen twice depending on the load.
The Meyer hardness test is a hardness test based upon projected area of an impression. The hardness,
{\displaystyle H}
, is defined as the maximum load,
{\displaystyle P_{\text{max}}}
divided by the projected area of the indent,
{\displaystyle A_{\text{p}}}
{\displaystyle H={\frac {P_{\text{max}}}{A_{\text{p}}}}.}
This is a more fundamental measurement of hardness than other hardness tests which are based on the surface area of an indentation. The principle behind the test is that the mean pressure required to test the material is the measurement of the hardness of the material. Units of megapascals (MPa) are frequently used for reporting Meyer hardness, but any unit of pressure can be used.[2]
The test was originally defined for spherical indenters, but can be applied to any indenter shape. It is often the definition used in nanoindentation testing.[1] An advantage of the Meyer test is that it is less sensitive to the applied load, especially compared to the Brinell hardness test. For cold worked materials the Meyer hardness is relatively constant and independent of load, whereas for the Brinell hardness test it decreases with higher loads. For annealed materials the Meyer hardness increases continuously with load due to strain hardening.[2]
Based on Meyer's law hardness values from this test can be converted into Brinell hardness values, and vice versa.[3]
The Meyer hardness test was devised by Eugene Meyer of the Materials Testing Laboratory at the Imperial School of Technology, Charlottenburg, Germany, circa 1908.[4][5]
^ a b Fischer-Cripps, Anthony C. (2011). Nanoindentation (3rd ed.). New York: Springer. ISBN 9781441998729. OCLC 756041216.
^ a b Hardness Testing, retrieved 2008-10-07 .
^ Tabor, p. 10.
^ E. Meyer, "Untersuchungen über Härteprüfung und Härte Brinell Methoden," Z. Ver. deut. Ing., 52 (1908).
^ V.E. Lysaght, Indentation Hardness Testing, New York: Reinhold Publishing Corp., 1949, p 39-47.
Tabor, David (2000). The Hardness of Metals. Oxford University Press. ISBN 0-19-850776-3. .
Retrieved from "https://en.wikipedia.org/w/index.php?title=Meyer_hardness_test&oldid=952496475"
|
Where the total area of the pavement (A<sub>c</sub>) and total depth of clear stone aggregate needed for load bearing capacity are known (i.e., storage reservoir depth is fixed) or if available space is constrained in the vertical dimension due to water table or bedrock elevation, the minimum footprint area of the water storage reservoir, A<sub>r</sub> can be calculated as follows:<br>
Where the total contributing drainage area (A<sub>c</sub>) and total depth of clear stone aggregate needed for load bearing capacity are known (i.e., storage reservoir depth is fixed) or if available space is constrained in the vertical dimension due to water table or bedrock elevation, the minimum footprint area of the water storage reservoir, A<sub>r</sub> can be calculated as follows:<br>
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
Differential Equations - Homogeneous Equations with Constant Coefficients | Brilliant Math & Science Wiki
Differential Equations - Homogeneous Equations with Constant Coefficients
Guillermo Templado, Calvin Lin, Jimin Khim, and
A real linear homogeneous differential equation of
n^\text{th}
order with constant real coefficients is an equation in the form
a_{n} \dfrac{d^ny}{dx^n} + a_{n-1} \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{1} \dfrac{dy}{dx} + a_{0} y = 0 \qquad (1)
a_{0}, a_{1} ,a_{n-1} ,\ldots , a_{n}
real constants.
Case of Distinct Real Roots
Case of Repeated Real Roots
Case of Complex Roots
\dfrac{d^ky}{dx^k} e^{rx} = r^k e^{rx}. \qquad (2)
y = e^{rx}
(1)
(2),
a_{n} r^n e^{rx} + a_{n-1} r^{n-1} e^{rx} + \cdots + a_{1} r e^{rx} + a_{0} e^{rx} = 0.
e ^ {rx} \neq 0
, we can cancel out this term to obtain the polynomial
a_ {n}r^n + a_ {n-1}r^{n-1}+ \cdots + a_ {1} r + a_ {0} = 0. \qquad (3)
y = e^{rx}
will be the solution of equation
(1)
r
is a root of this polynomial.
This equation is called the characteristic equation or auxiliary equation of the differential equation
(1).
n
r_1, r_2, \ldots, r_{n}
of the characteristic equation are real and distinct, then
y(x) = c_1 e^{r_1x} + c_2e^{r_2x} +\cdots +c_ne^{r_nx}
is a general solution of the equation
(1),
c_1, c_2, \ldots, c_n
constants.
_\square
\begin{array}{c}&y'' + 2y' - 8y = 0, &y(0) = 5, &y'(0) = - 2.\end{array}
We need to solve the characteristic equation
r^2 + 2r - 8 = 0.
r = 2, -4
are the roots of this equation. Then since
y(x) = c_1 e^{2x} + c_2 e^{-4x}
is the general solution,
y'(x) = 2 c_1 e^{2x} - 4 c_2 e^{-4x}
. Now, using the initial conditions, we have
y(0) = c_1 + c_2 = 5, y'(0) = 2c_1 - 4c_2 = -2 \Rightarrow c_1 = 3, c_2=2.
Thereofore, the desired particular solution is
y(x) = 3e^{2x} + 2e^{-4x}. \ _\square
If there are repeated real roots of the characteristic equation
(3)
, then we cannot generate
n
linearly independent solutions to equation
(1)
by the method of Theorem 1. Let us now consider the case that characteristic equation
a_ {n}r^n + a_ {n-1}r^{n-1}+ \cdots + a_ {1} r + a_ {0} = 0 \qquad (3)
has repeated real roots.
If the characteristic equation
(3)
has a repeated real root
r
k,
then part of the general solution of the differential equation corresponding to
r
(1)
(c_1 + c_ {2}x + c_ {3}x^2 + \cdots + c_ {k}x^{k-1}) \cdot e^{rx}. \ _\square
Find a general solution of
y^{(4)} + 3 \cdot y^{(3)} + 3 \cdot y'' + y' = 0.
The characteristic equation of the differential equation is
r^4 + 3 \cdot r^3 + 3 \cdot r^2 + r= r \cdot (r+1)^3=0
It has the single root
r_1 = 0,
which gives the solution
y_1 = c_1
to the general solution, and the triple root
(k=3)
r_2 = -1,
y_2 = (c_2 + c_ {3}x + c_ {4}x^2) \cdot e^{-x}.
Thus, the general solution of the differential equation is
y(x) = c_1 + (c_2 + c_ {3}x + c_ {4}x^2) \cdot e^{-x}. \ _\square
Because the coefficients of the differential equation and its characteristic equation are real, any root complex appears in complex conjugate pair
a \pm bi,
and
b
are real and i =
\sqrt{-1}.
Theorem 3: (No repeated complex roots)
(3)
has a pair of complex roots not repeated
a \pm bi
, then the relevant part to them of the general solution of equation
(1)
e^{ax} \cdot (c_1 \cos bx + c_2 \sin bx). \ _\square
y'' - 4y' + 5y = 0.
The characteristic equation is
r^2 - 4r + 5 = 0,
2 \pm i.
Thus, the general solution is
y(x) = e^{2x} \cdot (c_1 \cos x + c_2 \sin x). \ _\square
Theorem 2 holds for the case of repeated complex roots. If the conjugate pair
a \pm bi
have multiplicity
k,
then the relevant part to them of the general solution has the form
\begin{aligned} &\left(A_1 + A_ {2}x + A_ {3}x^2 + \cdots + A_ {k}x^{k-1}\right) \cdot e^{(a+bi)x} + \left(B_1 + B_ {2}x + B_ {3}x^2 + \cdots + B_ {k}x^{k-1}\right) \cdot e^{(a-bi)x}\\ &= \sum_{p=0}^{k-1}x^{p} \cdot e^{ax} \cdot (c_{p} \cos bx + d_{p} \sin bx). \qquad (4) \end{aligned}
y^{(4)} + 4 \cdot y^{(3)} + 12 \cdot y'' + 16y' + 16y = 0.
(r^2 + 2r + 4)^2 = 0,
-1 \pm i \cdot \sqrt{3}
2.
(4)
gives the general solution
y(x) =e^{-x} \cdot \left(c_1 \cos x\sqrt{3} + d_1 \sin x\sqrt{3}\right) + xe^{-x} \cdot \left(c_2 \cos x\sqrt{3} + d_2 \sin x\sqrt{3}\right). \ _\square
DIFFERENTIAL EQUATIONS - NO HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS
y_p
be a particular solution of the no homogeneous equation
a_{n}(x) \dfrac{d^ny}{dx^n} + a_{n-1}(x) \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{1}(x) \dfrac{dy}{dx} + a_{0}(x) y = f(x) \qquad (1)
in an open interval
I
f(x)
a_{i}(x)
are continuous. Let
y_1, y_2, \ldots , y_n
be linearly independent solutions of the associated differential homogeneous equation
a_{n}(x) \dfrac{d^ny}{dx^n} + a_{n-1}(x) \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{1}(x) \dfrac{dy}{dx} + a_{0}(x) y = 0. \qquad (*)
Y
is a solution of equation
(1)
I,
then there exists constants
c_1, c_2, \ldots , c_n
Y(x) = c_1 y_1(x) + c_2 y_2(x) + \cdots + c_n y_n(x) + y_p(x)
x
I.
_\square
Y
(1)
\Rightarrow
Y - y_p
solution of the associated differential equation
(*)
\Rightarrow
c_1, c_2, \ldots , c_n
Y(x) = c_1 y_1(x) + c_2 y_2(x) + \ldots + c_n y_n(x) + y_p(x)
x
I.
_\square
Let now the (???) no homogeneous differential equation of
n^\text{th}
order with constant coefficients
a_{n} \dfrac{d^ny}{dx^n} + a_{n-1} \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{1} \dfrac{dy}{dx} + a_{0} y = f(x), \qquad (1)
then the general solution of
(1)
Y = y_c + y_p,
y_c
is a general solution of the associated homogeneous equation
a_{n} \dfrac{d^ny}{dx^n} + a_{n-1} \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{1} \dfrac{dy}{dx} + a_{0} y = 0, \qquad (2)
y_p
is a simple particular solution of
(1).
The method of undetermined coefficients is applied when
f(x)
according to equation
(1)
is a finite linear combination of products of functions of three types:
A polynomial in
x.
\cos kx
\sin kx.
y'' + 3y' + 4y = 3x +2.
f(x) = 3x +2.
Let's seek
y_p(x) = Ax + B.
y'_p(x) = A
y''_p(x) = 0
y_p
will satisfy the differential equation
3A + 4(Ax + B) = 3x +2
4A = 3
3A +4B = 2,
A = \frac{3}{4}
B = \frac {-1}{16}.
y_p(x) = \frac {3}{4}x - \frac {1}{16}. \ _\square
y'' + 4y = 3x^3.
The complementary function of this equation is
y_c(x) = c_1 \cdot \cos 2x + c_2 \cdot \sin 2x
f(x) = 3x^3
. Let's seek
\begin{aligned} y_p(x) &= Ax^3 + Bx^2 + Cx + D \\ y'_p(x) &= 3Ax^2+ 2Bx + C\\ y''_p(x) &= 6Ax + 2B. \end{aligned}
\begin{aligned} y''_p + 4y_p &= (6Ax + 2B) + 4\left(Ax^3 + Bx^2 + Cx + D\right) \\ &= 4Ax^3 + 4Bx^2 + (6A +4C)x + (2B + D) \\ &= 3x^3 \\ \\ \Rightarrow 4A &= 3, ~4B = 0 , ~6A + 4C = 0, ~2B + D = 0 \\ A &= \frac {3}{4}, ~B = 0, ~C = \frac{-9}{8}, ~D = 0\\ \\ \Rightarrow y_p(x) &= \frac{3}{4}x^3 - \frac{9}{8}x. \ _\square \end{aligned}
\begin{array}{c}&y'' - 3y' + 8y = 3e^{-x} - 10 \cos 3x, &y(0) = 1, &y'(0) = 2.\end{array}
The characteristic equation
r^2 - 3r + 2 = 0
has the roots
r=1
r=2.
Then the complementary function is
y_c = c_1 e^{x} + c_2 e^{2x}.
Let's seek a particular solution
\begin{aligned} y_p &= Ae^{-x} + B\cos 3x + C\sin 3x\\ y'_p &= -Ae^{-x} - 3B\sin 3x + 3C\cos 3x\\ y''_p &= Ae^{-x} - 9B\cos 3x - 9C\sin 3x. \end{aligned}
Substituing, we have
\begin{aligned} y''_p - 3y'_p + 2y_p &=6Ae^{-x} + (-7B - 9C)\cos 3x + (9B - 7C)\sin 3x\\ &= 3e^{-x} - 10\cos 3x\\ \\ \Rightarrow 6A &= 3, ~-7B - 9C = -10, ~9B - 7C = 0 \\ A &= \frac{1}{2}, ~B = \frac{7}{13} , ~C = \frac{9}{13}\\ \\ \Rightarrow y_p(x) &= \frac{1}{2}e^{-x} + \frac{7}{13}\cos 3x + \frac{9}{13}\sin 3x. \end{aligned}
Then the general solution is
y(x) = y_c(x) + y_p(x) = c_1e^{x} + c_2e^{2x} + \frac{1}{2}e^{-x} + \frac{7}{13}\cos 3x + \frac{9}{13}\sin 3x.
The initial conditions lead to
\begin{aligned} y(0) &= c_1 + c_2 + \frac{1}{2} + \frac{7}{12} = 1\\ y'(0) &= c_1 + 2c_2 - \frac{1}{2} + \frac{27}{13} = 2 \\ \Rightarrow c_1 &= - \frac{1}{2}, ~c_2 = \frac{6}{13}. \end{aligned}
Therefore, the solution sought is
y(x) = - \frac{1}{2}e^{x} + \frac{6}{13}e^{2x} + \frac{1}{2}e^{-x} + \frac{7}{13}\cos 3x + \frac{9}{13}\sin 3x. \ _\square
Editorial: PHH Ecuaciones diferenciales elementales y problemas con condiciones en la frontera authors: C.H Edwards, David Penney
Cite as: Differential Equations - Homogeneous Equations with Constant Coefficients. Brilliant.org. Retrieved from https://brilliant.org/wiki/differential-equations-homogeneous-equations-with/
|
Problems Based On Java | Brilliant Math & Science Wiki
Problems Based On Java
What is Java? Java is a popular, third-generation programming language, which can be used to do any of the thousands of things that a computer software can do. With the features it offers, Java has become the language of choice for Internet and Internet applications. Java plays an important role in the proper functioning of many software-based devices attached to a network.
Java is a very vast and huge topic. But in this wiki, we are going to discuss only the various types of problems which are the very basics for a programming language. Although this is a small topic without this knowledge, it is almost impossible to understand Java programming language. In each section, before going to the problems, there is theory part which will help you solve the problems.
The operations being carried out are represented by operators. The operations (specific tasks) are represented by operators and the objects of the operations are referred to as operands.
There are many types of operators and the main ones are discussed here.
1. \color{#20A900}\text{ Arithmetic Operators}
Operators that act upon two operands are referred to as binary operators.
To do arithmetic, Java uses operators. It provides operators for
5
basic arithmetic calculations : addition
(+)
, subtraction
(-)
, multiplication
(\times)
, division
(\div)
, and modulus (%).
Modulus operator is used to find the remainder.
If the keyword
int
is used before any operation, you should write the answer only in integer and not in decimal. However, when either of the keywords
float
double
is used, you should write the answer only in decimals.
float
is used for or 2 decimal places and
double
is used for more than 2 decimal places.
Find all the values of all 5 arithmetic operations for the numbers
5
3
. Assume that the keyword used is
int
for all.
5 + 3 = 8
5 - 3 = 2
5 \times 3 = 15
5 \div 3 = 1
Modulus: 5 % 3 = 2
a = 10
b = 15,
int (a + b)
double (b \div a)
float (a \times b)
double (b % a)
int (a + b) = 10 + 15 = 25
double (b \div a) = 15 \div 10 = 1.500
float (a \times b) = 10 \times 15 = 150.0
double
(b % a) =
15
10 = 5.000
Note: If you got the answer as an integer but the keyword
float
double
is used, you must enter the answer only as decimals.
Operators that act on one operand are referred to as unary operators.
How to use unary operators is explained in the following example:
a = 5 \implies +a = 5
b = 0 \implies +b = 0
c = -4 \implies +c = -4.
\boxed{a = n,~\text{then}~+a = +n}.
d = 5 \implies -d = -5
e = 0 \implies -e = 0
f = -4 \implies -f = 4.
\boxed{a = n,~\text{then}~-a = -n}.
a = 9
b = -6,
(+a) + (-b).
\begin{aligned} + a &= +9\\\\ -b &= -(-6) \\&= +6\\\\ (+a) + (-b) &= 9 + 6 \\&= 15. \end{aligned}
Java includes two useful operators not generally found in other computer languages (except C and C++). These are increment and decrement operators,
++
--.
++
adds 1 to its operand, and the operator
--
subtracts 1. In other words,
\begin{aligned} a = a + 1 &\text{ is the same as } ++a\ \text{ or }\ a++\\ a = a - 1 &\text{ is the same as } --a\ \text{ or }\ a--. \end{aligned}
However, both the increment and decrement operators come in two varieties: prefix and postfix.
Prefix increment and decrement operators
(
++a \text{ or } --a)
The prefix increment or decrement operator follows the "change-then-use rule"; that is, they first change (add or subtract) the value of their operand, and then use the new value in calculating the problem. If you encounter with a prefix operator, the best thing to do is to first increase the value by 1 if it is an increment or decrease the value by 1 if it is a decrement operator.
Postfix increment and decrement operators
(
a++ \text{ or } a--)
The postfix increment or decrement operator follows the "use-then-change rule"; that is, they first use the value of their operand in evaluating the expression, and then change (add or subtract) the operand value. If you encounter with a postfix operator for the first time, do not change the value of operand, but if you encounter it for the second time, then change (add or subtract) the value.
You will better understand this concept by the following examples:
x = (++y) + 2y
if initially
y = 6.
In this problem, only prefix operator is used, so we can solve this easily. First, we will change the value of
y:
++y = y + 1 = 6 + 1 = 7.
x = 7 + 2(7) = 21.\ _\square
x = (--y) + 3y
y = 10.
In this problem, too, only prefix operator is used, so we can solve this easily:
\begin{aligned} --y &= y - 1 \\&= 10 - 1 \\&= 9\\\\ x &= 9 + 3(9) \\&= 36.\ _\square \end{aligned}
x = (y++) + 2y
x = 6.
Here, we have to be careful because postfix operator is used. So, for the first operand of
y
the value remains the same, but for the next operand of
y
the value increments (1 is added):
y++ = 6\ \text{ but }\ 2y = 2(6 + 1) = 2(7) = 14.
x = (y++) + 2y = 6 + 14 = 20.\ _\square
x = (y--) + 2y
x = 10.
Here, we also have to be careful because postfix operator is used. So, for the first operand of
y
y
the value decrements (1 is added):
y-- = 10\ \text{ but }\ 2y = 2(10 + 1) = 2(9) = 18.
x = (y++) + 2y = 10 + 18 = 28.\ _\square
Not only this we can also use multiplication and division operators and also we can use both prefix and postfix operators in one problem.
x = (y++) \times (5 + 2y)
y = 5
Initially,
y = 5
(y++) = 5
2(y) = 2(5 + 1) = 2(6) = 12.
x = 5 \times (5 + 12)= 5 \times 17 = 85.\ _\square
x = (++y) \times (y++) \times \big(1 + (--y)\big)
y = 3.
y = 3
(++y) = 4, (y++) = 4.
(y--)
3
but not directly. In the before operand the operator is postfix so now the value should be
5
. But there is also a prefix operator for this
y,
\big((1 + (--y)\big) = 1 + 5 - 1 = 5.
x = 4 \times 4 \times 5 = 80.\ _\square
Cite as: Problems Based On Java. Brilliant.org. Retrieved from https://brilliant.org/wiki/problems-based-on-java/
|
Where the total contributing drainage area (A<sub>c</sub>) and total depth of clear stone aggregate needed for load bearing capacity of the pavement are known (i.e., storage reservoir depth is fixed) or if available space is constrained in the vertical dimension due to water table or bedrock elevation, the minimum footprint area of the water storage reservoir, A<sub>r</sub> can be calculated as follows:<br>
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
|
Fourier transform - SEG Wiki
The transfer function of the Nth-order causal FIR filter now can be written down by induction as
{\displaystyle {\begin{aligned}B\left(f\right)={\frac {\rm {Output}}{\rm {Input}}}={\frac {b_{0}e^{j2\pi jn\Delta t}+\dots \ +b_{\rm {l}}e^{i2\pi f\left(n-{\rm {l}}\right)\Delta t}}{e^{i2\pi fn}}},\end{aligned}}}
{\displaystyle {\begin{aligned}B\left(f\right)=b_{0}+b_{\rm {l}}e^{-i2\pi f\Delta {\rm {t}}}+\dots +b_{N}e^{-i2\pi j\Delta tN}.\end{aligned}}}
Our results can be tabulated in the form of Table 1. Definitions are important. Unfortunately, the Z-transform and the Fourier transform are defined in different ways depending on the convention used. Now we must take a moment to reflect on conventions. We will discuss three conventions: (1) the mathematics convention, (2) the electrical engineering convention, and (3) the hybrid convention.
Table 1. Filters and transfer functions.
Causal FIR filter
Corresponding transfer function
{\displaystyle B\left(Z\right)=b_{0}}
{\displaystyle B(f)=b_{0}}
{\displaystyle B\left(Z\right)=Z}
{\displaystyle B\left(f\right)=e^{-i2\pi f\Delta t}}
{\displaystyle B\left(Z\right)=b_{\rm {1}}Z}
{\displaystyle B\left(f\right)=b_{1}e^{-i2\pi f{\Delta }_{t}}}
{\displaystyle B\left(Z\right)=b_{0}+b_{\rm {1}}Z}
{\displaystyle B\left(f\cdot \right)=b_{0}+b_{1}e^{-i2\pi f\Delta t}}
{\displaystyle B\left(Z\right)=b_{0}+b_{\rm {1}}Z++b_{N}Z^{N}}
{\displaystyle B\left(j\right)=b_{0}+b_{1}e^{-i2\pi f{\Delta }t}+\dots +b_{\rm {N}}e^{-i2\pi f{\Delta t}N}}
Their counterparts are the gasoline automobile, the electric automobile, and the hybrid auto-mobile.
1) Under the mathematics convention, the generating function (i.e., the Z-transform with a capital Z) is defined in the way originally given by Euler, namely
{\displaystyle {\begin{aligned}B\left(Z\right)=b_{0}+b_{\rm {l}}Z+\dots +b_{N}Z^{N},\end{aligned}}}
and the Fourier transform is defined in the way originally given by Fourier, namely
{\displaystyle {\begin{aligned}B_{M{\rm {A}}TH}\left(f\right)=b_{0}+b_{1}e^{i2\pi f\Delta t}+\dots +b_{N}e^{i2\pi f\Delta fN}.\end{aligned}}}
In the mathematics convention, the exponents in both transforms (equations 14 and 15 ) are positive.
2) Under the electrical engineering convention, the z-transform with a lowercase z is defined as
{\displaystyle {\begin{aligned}B_{EE}\left(z\right)=b_{0}+b_{1}z^{-{l}}+\dots +b_{N}z^{-N},\end{aligned}}}
and the Fourier transform is defined as
{\displaystyle {\begin{aligned}B\left(f\right)=b_{0}+b_{1}e^{-i2\pi f\Delta t}+\dots +b_{N}e^{-i2\pi f\Delta tN}.\end{aligned}}}
In the electrical engineering convention, the exponents in both transforms (equations 16 and 17 ) are negative.
3) Under the hybrid convention, the Z-transform with a capital Z is defined as in mathematics, namely
{\displaystyle {\begin{aligned}B\left(Z\right)=b_{0}+b_{l}Z+\dots +b_{N}Z^{N},\end{aligned}}}
and the Fourier transform is defined as in electrical engineering, namely
{\displaystyle {\begin{aligned}B\left(f\right)=b_{0}+b_{l}e^{-i2\pi f\Delta t}+\dots +b_{N}e^{-i2\pi j\Delta tN}.\end{aligned}}}
In the hybrid convention, the exponents in the Z-transform (equation 18 ) are positive, and the exponents in the Fourier transform (equation 19 ) are negative.
In this book, we use the hybrid convention. Working with negative powers of Z is cumbersome. The hybrid convention avoids this contingency but keeps the form of Fourier transform that electrical engineers use. Under the hybrid convention, the transfer function of a filter is obtained formally by the substitution of
{\displaystyle Z=e^{-i2\pi f\Delta t}}
in the filter’s Z-transform. In other words, the transfer function is the (electrical engineering) Fourier transform of the impulse-response function. We notice that except for the case of the constant filter
{\displaystyle b_{0}}
, the transfer function always depends on the frequency f.
Much can be learned from considering the unit-delay filter, which can be represented as
{\displaystyle Z=e^{i\psi (f)}}
{\displaystyle {\psi (f)}=-2\pi f\Delta t}
is the phase lead. The phase lag
{\displaystyle \varphi \left(f\right)}
is defined as the negative of the phase lead - that is,
{\displaystyle \phi \left(f\right)=-\psi (f)}
. Thus, the unit-delay filter can be written as
{\displaystyle Z=e^{-i2\pi f\Delta t}=e^{-i\phi (f)}}
. Physical systems involve delay, so phase lag rather than phase lead becomes the natural choice. Electrical engineers sometimes but not always refer to phase lag simply as phase. However, wherever we use the word phase, we explicitly mean phase lag.
As we have seen, the transfer function is the (electrical engineering) Fourier transform of the impulse-response function. In polar form, the transfer function can be written as
{\displaystyle {\begin{aligned}B\left(f\right)=|B\left(f\right){|}e^{-j\psi \left(f\right)},\end{aligned}}}
{\displaystyle {|}B\left(f\right){|}}
{\displaystyle \varphi (f)}
represent, respectively, the magnitude spectrum and phase-lag spectrum (or simply the phase spectrum) of the filter. For example, the transfer function of the filter
{\displaystyle b_{0}+b_{1}Z}
{\displaystyle {\begin{aligned}B\left(f\right)=b_{0}+b_{1}e^{-j2\pi f\Delta {\rm {t}}}=b_{0}+b_{l}{\rm {\ cos\ 2}}\pi f\Delta t-ib_{l}{\rm {\ sin\ 2}}\pi f\Delta t.\end{aligned}}}
Now B(f) (for a fixed value of f) is the vector that is the sum of the vectors
{\displaystyle b_{0}}
{\displaystyle b_{1}e^{-i2\psi \Delta t}}
Figure 6. Depiction of the transfer function.
{\displaystyle {|}B\left(f\right){|}}
of the vector B(f) is
{\displaystyle {\begin{aligned}{|}B\left(f\right){|=}{\sqrt {{\left(b_{0}+b_{1}{\rm {\ cos\ 2}}\pi f\Delta t\right)}^{2}+{\left(b_{1}{\rm {\ sin\ 2}}\pi f\Delta t\right)}^{2}}}={\sqrt {b_{0}^{2}+2b_{0}b_{1}{\rm {\ cos\ 2}}\pi f\Delta t+b_{1}^{2}}}.\end{aligned}}}
This quantity is the magnitude spectrum of the filter
{\displaystyle b_{0}+b_{1}Z}
{\displaystyle \varphi (f)}
, which is a function of f, is the phase lag (simply called the phase). Thus, the function
{\displaystyle {\begin{aligned}\varphi \left(f\right)=-{\rm {tan}}^{-{1}}\left[{\frac {-b_{1}{\rm {\ sin\ 2}}\pi f\Delta t}{b_{0}+b_{1}{\rm {\ cos\ 2}}\pi f\Delta t}}\right]={\rm {tan}}^{-{1}}\left[{\frac {b_{1}{\rm {\ sin\ 2}}\pi f\Delta t}{b_{0}+b_{1}{\rm {\ cos\ 2}}\pi f\Delta t}}\right]\end{aligned}}}
yields the phase-spectrum of the filter
{\displaystyle b_{0}+b_{1}Z}
. We see that both the magnitude and the phase spectra are functions of the frequency f.
At this point, we must introduce a word of warning. The arctangent function is not a one-to-one function but instead is a many-valued function. Thus, a computer usually picks the value of the arctangent function that lies in the range from 0 to
{\displaystyle 2\pi }
. When we use such a program, we do not necessarily obtain the phase
{\displaystyle \varphi (f)}
; instead, we might obtain
{\displaystyle \varphi (f)}
reduced or augmented by
{\displaystyle 2\pi k}
, where k is an integer so determined that the computed value lies in the range from 0 to
{\displaystyle 2\pi }
. This result is called the wrapped phase spectrum; the true phase spectrum can be obtained by a computer process known as phase unwrapping.
Magnitude spectrum and phase spectrum Minimum-phase spectrum
Retrieved from "https://wiki.seg.org/index.php?title=Fourier_transform/en&oldid=167265"
|
Fixed-Point Operator Code Replacement - MATLAB & Simulink - MathWorks Benelux
Common Ways to Match Fixed-Point Operator Entries
Fixed-Point Numbers and Arithmetic
Data Type Conversion (Cast)
If you have a Fixed-Point Designer™ license, you can define fixed-point operator code replacement entries to match:
A binary-point-only scaling combination on the operator inputs and output.
A slope bias scaling combination on the operator inputs and output.
Relative scaling or net slope between multiplication or division operator inputs and output. Use one of these methods to map a range of slope and bias values to a replacement function for multiplication or division.
Equal slope and zero net bias across addition or subtraction operator inputs and output. Use this method to disregard specific slope and bias values and map relative slope and bias values to a replacement function for addition or subtraction.
The following table maps common ways to match fixed-point operator code replacement entries with the associated fixed-point parameters that you specify in a code replacement table definition file.
Minimally specify parameters
A specific binary-point-only scaling combination on the operator inputs and output.
createAndAddConceptualArg function:
CheckSlope: Specify the value true.
CheckBias: Specify the value true.
DataTypeMode (or DataType/Scaling equivalent): Specify fixed-point binary-point-only scaling.
FractionLength: Specify a fraction length (for example, 3).
A specific slope bias scaling combination on the operator inputs and output.
DataTypeMode (or DataType/Scaling equivalent): Specify fixed-point [slope bias] scaling.
Slope (or SlopeAdjustmentFactor/ FixedExponent equivalent): Specify a slope value (for example, 15).
Bias: Specify a bias value (for example, 2).
Net slope between operator inputs and output (multiplication and division).
setTflCOperationEntryParameters function:
NetSlopeAdjustmentFactor: Specify the slope adjustment factor (F) part of the net slope, F2E (for example, 1.0).
NetFixedExponent: Specify the fixed exponent (E) part of the net slope, F2E (for example, -3.0).
CheckSlope: Specify the value false.
CheckBias: Specify the value false.
DataType: Specify the value 'Fixed'.
Relative scaling between operator inputs and output (multiplication and division).
RelativeScalingFactorF: Specify the slope adjustment factor (F) part of the relative scaling factor, F2E (for example, 1.0).
RelativeScalingFactorE: Specify the fixed exponent (E) part of the relative scaling factor, F2E (for example, -3.0).
Equal slope and zero net bias across operator inputs and output (addition and subtraction).
SlopesMustBeTheSame: Specify the value true.
MustHaveZeroNetBias: Specify the value true.
Fixed-point numbers use integers and integer arithmetic to represent real numbers and arithmetic with the following encoding scheme:
V=\stackrel{˜}{V}=SQ+B
V
\stackrel{˜}{V}
is the approximate real-world value that results from fixed-point representation.
Q
is an integer that encodes
\stackrel{˜}{V}
, referred to as the quantized integer.
S
is a coefficient of
Q
, referred to as the slope.
B
is an additive correction, referred to as the bias.
The general equation for an operation between fixed-point operands is:
\left({S}_{O}{Q}_{O}+{B}_{O}\right)=\left({S}_{1}{Q}_{1}+{B}_{1}\right)<op>\left({S}_{2}{Q}_{2}+{B}_{2}\right)
The objective of fixed-point operator replacement is to replace an operator that accepts and returns fixed-point or integer inputs and output with a function that accepts and returns built-in C numeric data types. The following sections provide additional programming information for each supported operator.
The operation V0 = V1 + V2 implies that
{Q}_{0}=\left(\frac{{S}_{1}}{{S}_{0}}\right){Q}_{1}+\left(\frac{{S}_{2}}{{S}_{0}}\right){Q}_{2}+\left(\frac{{B}_{1}+{B}_{2}-{B}_{0}}{{S}_{0}}\right)
If an addition replacement function is defined such that the scaling on the operands and sum are equal and the net bias
\left(\frac{{B}_{1}+{B}_{2}-{B}_{0}}{{S}_{0}}\right)
is zero (for example, a function s8_add_s8_s8 that adds two signed 8-bit values and produces a signed 8-bit result), then the operator entry must set the operator entry parameters SlopesMustBeTheSame and MustHaveZeroNetBias to true. To match for replacement, the slopes must be the same for all addition conceptual arguments. (For parameter descriptions, see the reference page for the function setTflCOperationEntryParameters.)
The operation V0 = V1 − V2 implies that
{Q}_{0}=\left(\frac{{S}_{1}}{{S}_{0}}\right){Q}_{1}-\left(\frac{{S}_{2}}{{S}_{0}}\right){Q}_{2}+\left(\frac{{B}_{1}-{B}_{2}-{B}_{0}}{{S}_{0}}\right)
If a subtraction replacement function is defined such that the scaling on the operands and difference are equal and the net bias
\left(\frac{{B}_{1}-{B}_{2}-{B}_{0}}{{S}_{0}}\right)
is zero (for example, a function s8_sub_s8_s8 that subtracts two signed 8-bit values and produces a signed 8-bit result), then the operator entry must set the operator entry parameters SlopesMustBeTheSame and MustHaveZeroNetBias to true. To match for replacement, the slopes must be the same for all subtraction conceptual arguments. (For parameter descriptions, see the reference page for the function setTflCOperationEntryParameters.)
There are different ways to specify multiplication replacements. The most direct way is to specify an exact match of the input and output types. This is feasible if a model contains only a few known slope and bias combinations. Use the TflCOperationEntry class and specify the exact values of slope and bias on each argument. For scenarios where there are numerous slope/bias combinations, it is not feasible to specify each value with a different entry. Use a net slope entry or create a custom entry.
The operation V0 = V1 * V2 implies, for binary-point-only scaling, that
\begin{array}{l}{S}_{0}{Q}_{0}=\left({S}_{1}{Q}_{1}\right)\left({S}_{2}{Q}_{2}\right)\\ {Q}_{0}=\left(\frac{{S}_{1}{S}_{2}}{{S}_{0}}\right){Q}_{1}{Q}_{2}\\ {Q}_{0}={S}_{n}{Q}_{1}{Q}_{2}\end{array}
where Sn is the net slope.
It is common to replace all multiplication operations that have a net slope of 1.0 with a function that performs C-style multiplication. For example, to replace all signed 8-bit multiplications that have a net scaling of 1.0 with the s8_mul_s8_u8_ replacement function, the operator entry must define a net slope factor, F2E. You specify the values for F and E using operator entry parameters NetSlopeAdjustmentFactor and NetFixedExponent. For the s8_mul_s8_u8 function, set NetSlopeAdjustmentFactor to 1 and NetFixedExponent to 0.0. Also, set the operator entry parameter SlopesMustBeTheSame to false and the parameter MustHaveZeroNetBias to true. To match for replacement, the biases must be zero for all multiplication conceptual arguments. (For parameter descriptions, see the reference page for the function setTflCOperationEntryParameters.)
When an operator entry specifies NetSlopeAdjustmentFactor and NetFixedExponent, matching entries must have arguments with zero bias.
There are different ways to specify division replacements. The most direct way is to specify an exact match of the input and output types. This is feasible if a model contains only a few known slope and bias combinations. Use the TflCOperationEntry class and specify the exact values of slope and bias on each argument. For scenarios where there are numerous slope/bias combinations, it is not feasible to specify each value with a different entry. Use a net slope entry or create a custom entry (see Customize Match and Replacement Process).
The operation V0 = (V1 / V2) implies, for binary-point-only scaling, that
\begin{array}{l}{S}_{0}{Q}_{0}=\left(\frac{{S}_{1}{Q}_{1}}{{S}_{2}{Q}_{2}}\right)\\ {Q}_{0}={S}_{n}\left(\frac{{Q}_{1}}{{Q}_{2}}\right)\end{array}
It is common to replace all division operations that have a net slope of 1.0 with a function that performs C-style division. For example, to replace all signed 8-bit divisions that have a net scaling of 1.0 with the s8_mul_s8_u8_ replacement function, the operator entry must define a net slope factor, F2E. You specify the values for F and E using operator entry parameters NetSlopeAdjustmentFactor and NetFixedExponent. For the s16_netslope0p5_div_s16_s16 function, you would set NetSlopeAdjustmentFactor to 1 and NetFixedExponent to 0.0. Also, set the operator entry parameter SlopesMustBeTheSame to false and the parameter MustHaveZeroNetBias to true. To match for replacement, the biases must be zero for all division conceptual arguments. (For parameter descriptions, see the reference page for the function setTflCOperationEntryParameters.)
The data type conversion operation V0 = V1 implies, for binary-point-only scaling, that
\begin{array}{l}{Q}_{0}=\left(\frac{{S}_{1}}{{S}_{0}}\right){Q}_{1}\\ {Q}_{0}={S}_{n}{Q}_{1}\end{array}
where Sn is the net slope. Set the operator entry parameter SlopesMustBeTheSame to false and the parameter MustHaveZeroNetBias to true. To match for replacement, the biases must be zero for all cast conceptual arguments. (For parameter descriptions, see the reference page for the function setTflCOperationEntryParameters.)
The shift left or shift right operation V0 = (V1 / 2n) implies, for binary-point-only scaling, that
\begin{array}{l}{S}_{0}{Q}_{0}=\left(\frac{{S}_{1}{Q}_{1}}{{2}^{n}}\right)\\ {Q}_{0}=\left(\frac{{S}_{1}}{{S}_{0}}\right)\left(\frac{{Q}_{1}}{{2}^{n}}\right)\\ {Q}_{0}={S}_{n}\left(\frac{{Q}_{1}}{{2}^{n}}\right)\end{array}
where Sn is the net slope. Set the operator entry parameter SlopesMustBeTheSame to false and the parameter MustHaveZeroNetBias to true. To match for replacement, the biases must be zero for all shift conceptual arguments. (For parameter descriptions, see the reference page for the function setTflCOperationEntryParameters.)
|
AFFINE GEOMETRY - Encyclopedia Information
Affine geometry Information
https://en.wikipedia.org/wiki/Affine_geometry
Euclidean geometry without distance and angles
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting" [1] [2]) the metric notions of distance and angle.
As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (Given a line L and a point P not on L, there is exactly one line parallel to L that passes through P.) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines.
Affine geometry can be developed in two ways that are essentially equivalent. [3]
In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
In more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). By choosing any point as "origin", the points are in one-to-one correspondence with the vectors, but there is no preferred choice for the origin; thus an affine space may be viewed as obtained from its associated vector space by "forgetting" the origin (zero vector).
The idea of forgetting the metric can be applied in the theory of manifolds. That is developed in the article on the affine connection.
2 Systems of axioms
2.1 Pappus' law
2.2 Ordered structure
2.3 Ternary rings
3 Affine transformations
5 Projective view
In 1748, Leonhard Euler introduced the term affine [4] [5] (Latin affinis, "related") in his book Introductio in analysin infinitorum (volume 2, chapter XVIII). In 1827, August Möbius wrote on affine geometry in his Der barycentrische Calcul (chapter 3).
After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry. [6]
In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtraction [7] at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote: [8]
Weyl's geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of parallel transport [...using] worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a null-vector; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a neighboring point.
Systems of axioms
Several axiomatic approaches to affine geometry have been put forward:
Pappus' law
Pappus's law: if the red lines are parallel and the blue lines are parallel, then the dotted black lines must be parallel.
As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise: [9] [10]
{\displaystyle A,B,C}
are on one line and
{\displaystyle A',B',C'}
on another. If the lines
{\displaystyle AB'}
{\displaystyle A'B}
are parallel and the lines
{\displaystyle BC'}
{\displaystyle B'C}
are parallel, then the lines
{\displaystyle CA'}
{\displaystyle C'A}
The full axiom system proposed has point, line, and line containing point as primitive notions:
Two points are contained in just one line.
For any line l and any point P, not on l, there is just one line containing P and not containing any point of l. This line is said to be parallel to l.
There are at least three points not belonging to one line.
According to H. S. M. Coxeter:
The interest of these five axioms is enhanced by the fact that they can be developed into a vast body of propositions, holding not only in Euclidean geometry but also in Minkowski’s geometry of time and space (in the simple case of 1 + 1 dimensions, whereas the special theory of relativity needs 1 + 3). The extension to either Euclidean or Minkowskian geometry is achieved by adding various further axioms of orthogonality, etc. [11]
The various types of affine geometry correspond to what interpretation is taken for rotation. Euclidean geometry corresponds to the ordinary idea of rotation, while Minkowski's geometry corresponds to hyperbolic rotation. With respect to perpendicular lines, they remain perpendicular when the plane is subjected to ordinary rotation. In the Minkowski geometry, lines that are hyperbolic-orthogonal remain in that relation when the plane is subjected to hyperbolic rotation.
An axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms: [12]
( Affine axiom of parallelism) Given a point A and a line r not through A, there is at most one line through A which does not meet r.
( Desargues) Given seven distinct points
{\displaystyle A,A',B,B',C,C',O}
{\displaystyle AA'}
{\displaystyle BB'}
{\displaystyle CC'}
are distinct lines through
{\displaystyle O}
{\displaystyle AB}
{\displaystyle A'B'}
{\displaystyle BC}
{\displaystyle B'C'}
{\displaystyle AC}
{\displaystyle A'C'}
The affine concept of parallelism forms an equivalence relation on lines. Since the axioms of ordered geometry as presented here include properties that imply the structure of the real numbers, those properties carry over here so that this is an axiomatization of affine geometry over the field of real numbers.
Ternary rings
Main article: Planar ternary ring
The first non-Desarguesian plane was noted by David Hilbert in his Foundations of Geometry. [13] The Moulton plane is a standard illustration. In order to provide a context for such geometry as well as those where Desargues theorem is valid, the concept of a ternary ring was developed by Marshall Hall.
In this approach affine planes are constructed from ordered pairs taken from a ternary ring. A plane is said to have the "minor affine Desargues property" when two triangles in parallel perspective, having two parallel sides, must also have the third sides parallel. If this property holds in the affine plane defined by a ternary ring, then there is an equivalence relation between "vectors" defined by pairs of points from the plane. [14] Furthermore, the vectors form an abelian group under addition; the ternary ring is linear and satisfies right distributivity:
(a + b) c = ac + bc.
Main article: Affine transformation
Geometrically, affine transformations (affinities) preserve collinearity: so they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines.
We identify as affine theorems any geometric result that is invariant under the affine group (in Felix Klein's Erlangen programme this is its underlying group of symmetry transformations for affine geometry). Consider in a vector space V, the general linear group GL(V). It is not the whole affine group because we must allow also translations by vectors v in V. (Such a translation maps any w in V to w + v.) The affine group is generated by the general linear group and the translations and is in fact their semidirect product
{\displaystyle V\rtimes \mathrm {GL} (V)}
. (Here we think of V as a group under its operation of addition, and use the defining representation of GL(V) on V to define the semidirect product.)
For example, the theorem from the plane geometry of triangles about the concurrence of the lines joining each vertex to the midpoint of the opposite side (at the centroid or barycenter) depends on the notions of mid-point and centroid as affine invariants. Other examples include the theorems of Ceva and Menelaus.
Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form an envelope inside the triangle. The ratio of the area of the envelope to the area of the triangle is affine invariant, and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give
{\displaystyle {\tfrac {3}{4}}\log _{e}(2)-{\tfrac {1}{2}},}
i.e. 0.019860... or less than 2%, for all triangles.
Familiar formulas such as half the base times the height for the area of a triangle, or a third the base times the height for the volume of a pyramid, are likewise affine invariants. While the latter is less obvious than the former for the general case, it is easily seen for the one-sixth of the unit cube formed by a face (area 1) and the midpoint of the cube (height 1/2). Hence it holds for all pyramids, even slanting ones whose apex is not directly above the center of the base, and those with base a parallelogram instead of a square. The formula further generalizes to pyramids whose base can be dissected into parallelograms, including cones by allowing infinitely many parallelograms (with due attention to convergence). The same approach shows that a four-dimensional pyramid has 4D hypervolume one quarter the 3D volume of its parallelepiped base times the height, and so on for higher dimensions.
Two types of affine transformation are used in kinematics, both classical and modern. Velocity v is described using length and direction, where length is presumed unbounded. This variety of kinematics, styled as Galilean or Newtonian, uses coordinates of absolute space and time. The shear mapping of a plane with an axis for each represents coordinate change for an observer moving with velocity v in a resting frame of reference. [15]
Finite light speed, first noted by the delay in appearance of the moons of Jupiter, requires a modern kinematics. The method involves rapidity instead of velocity, and substitutes squeeze mapping for the shear mapping used earlier. This affine geometry was developed synthetically in 1912. [16] [17] to express the special theory of relativity. In 1984, "the affine plane associated to the Lorentzian vector space L2" was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry". [18]
Affine geometry can be viewed as the geometry of an affine space of a given dimension n, coordinatized over a field K. There is also (in two dimensions) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry. In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, (x + y + z)/3, ix + (1 − i)y, etc.
Synthetically, affine planes are 2-dimensional affine geometries defined in terms of the relations between points and lines (or sometimes, in higher dimensions, hyperplanes). Defining affine (and projective) geometries as configurations of points and lines (or hyperplanes) instead of using coordinates, one gets examples with no coordinate fields. A major property is that all such examples have dimension 2. Finite examples in dimension 2 ( finite affine planes) have been valuable in the study of configurations in infinite affine spaces, in group theory, and in combinatorics.
Despite being less general than the configurational approach, the other approaches discussed have been very successful in illuminating the parts of geometry that are related to symmetry.
Projective view
In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity. [19] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality. [20] In this viewpoint, an affine transformation is a projective transformation that does not permute finite points with points at infinity, and affine transformation geometry is the study of geometrical properties through the action of the group of affine transformations.
^ Artin, Emil (1988), Geometric Algebra, Wiley Classics Library, New York: John Wiley & Sons Inc., pp. x+214, doi: 10.1002/9781118164518, ISBN 0-471-60839-4, MR 1009557 (Reprint of the 1957 original; A Wiley-Interscience Publication)
^ Miller, Jeff. "Earliest Known Uses of Some of the Words of Mathematics (A)".
^ Blaschke, Wilhelm (1954). Analytische Geometrie. Basel: Birkhauser. p. 31.
^ Coxeter, H. S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons. pp. 191. ISBN 0-471-50458-0.
^ Hermann Weyl (1918) Raum, Zeit, Materie. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0-486-60267-2 . See Chapter 1 §2 Foundations of Affine Geometry, pp 16–27
^ E. T. Whittaker (1958). From Euclid to Eddington: a study of conceptions of the external world, Dover Publications, p. 130.
^ Veblen 1918: p. 103 (figure), and p. 118 (exercise 3).
^ Coxeter 1955, The Affine Plane, § 2: Affine geometry as an independent system
^ Coxeter 1955, Affine plane, p. 8
^ Coxeter, Introduction to Geometry, p. 192
^ David Hilbert, 1980 (1899). The Foundations of Geometry, 2nd ed., Chicago: Open Court, weblink from Project Gutenberg, p. 74.
^ Rafael Artzy (1965). Linear Geometry, Addison-Wesley, p. 213.
^ Abstract Algebra/Shear and Slope at Wikibooks
^ Edwin B. Wilson & Gilbert N. Lewis (1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences 48:387–507
^ Graciela S. Birman & Katsumi Nomizu (1984). "Trigonometry in Lorentzian geometry", American Mathematical Monthly 91(9):543–9, Lorentzian affine plane: p. 544
^ H. S. M. Coxeter (1942). Non-Euclidean Geometry, University of Toronto Press, pp. 18, 19.
Emil Artin (1957) Geometric Algebra, chapter 2: "Affine and projective geometry", via Internet Archive
Wikimedia Commons has media related to Affine geometry.
Retrieved from " https://en.wikipedia.org/?title=Affine_geometry&oldid=1046610587"
Affine Geometry Videos
Affine Geometry Websites
Affine Geometry Encyclopedia Articles
|
Polylogarithmic function - Wikipedia
Not to be confused with Polylogarithm.
A polylogarithmic function in n is a polynomial in the logarithm of n,
{\displaystyle a_{k}(\log n)^{k}+\cdots +a_{1}(\log n)+a_{0}.}
{\displaystyle \log ^{k}n}
is often used as a shorthand for
{\displaystyle (\log n)^{k}}
, analogous to
{\displaystyle \sin ^{2}\theta }
{\displaystyle (\sin \theta )^{2}}
In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms (e.g., "it has polylogarithmic order").
All polylogarithmic functions of
{\displaystyle n}
{\displaystyle o(n^{\varepsilon })}
for every exponent ε > 0 (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the soft O notation Õ(n).
Black, Paul E. (2004-12-17). "polylogarithmic". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved 2010-01-10.
This theoretical computer science–related article is a stub. You can help Wikipedia by expanding it.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Polylogarithmic_function&oldid=930217441"
|
Robot Dynamics - MATLAB & Simulink - MathWorks Italia
This topic details the different elements, properties, and equations of rigid body robot dynamics. Robot dynamics are the relationship between the forces acting on a robot and the resulting motion of the robot. In Robotics System Toolbox™, manipulator dynamics information is contained within a rigidBodyTree object, which specifies the rigid bodies, attachment points, and inertial parameters for both kinematics and dynamics calculations.
To use dynamics object functions, you must set the DataFormat property of the rigidBodyTree object to "row" or "column". These setting accept inputs and return outputs as row or column vectors, respectively, for relevant robotics calculations, such as robot configurations or joint torques.
\frac{d}{dt}\left[\begin{array}{c}q\\ \stackrel{˙}{q}\end{array}\right]=\left[\begin{array}{c}\stackrel{˙}{q}\\ M{\left(q\right)}^{-1}\left(-C\left(q,\stackrel{˙}{q}\right)\stackrel{˙}{q}-G\left(q\right)-J{\left(q\right)}^{T}{F}_{Ext}+\tau \right)\end{array}\right]
M\left(q\right)\stackrel{¨}{q}=-C\left(q,\stackrel{˙}{q}\right)\stackrel{˙}{q}-G\left(q\right)-J{\left(q\right)}^{T}{F}_{Ext}+\tau
M\left(q\right)
C\left(q,\stackrel{˙}{q}\right)
\stackrel{˙}{q}
G\left(q\right)
J\left(q\right)
{F}_{Ext}
\tau
q,\stackrel{˙}{q},\stackrel{¨}{q}
forwardDynamics | inverseDynamics | externalForce | geometricJacobian | gravityTorque | centerOfMass | massMatrix | velocityProduct
rigidBodyTree | jointSpaceMotionModel | taskSpaceMotionModel | inverseKinematics
|
Flow_stress Knowpia
In materials science the flow stress, typically denoted as Yf (or
{\displaystyle \sigma _{\text{f}}}
), is defined as the instantaneous value of stress required to continue plastically deforming a material - to keep it flowing. It is most commonly, though not exclusively, used in reference to metals. On a stress-strain curve, the flow stress can be found anywhere within the plastic regime; more explicitly, a flow stress can be found for any value of strain between and including yield point (
{\displaystyle \sigma _{\text{y}}}
) and excluding fracture (
{\displaystyle \sigma _{\text{F}}}
{\displaystyle \sigma _{\text{y}}\leq Y_{\text{f}}<\sigma _{\text{F}}}
The flow stress changes as deformation proceeds and usually increases as strain accumulates due to work hardening, although the flow stress could decrease due to any recovery process. In continuum mechanics, the flow stress for a given material will vary with changes in temperature,
{\displaystyle T}
, strain,
{\displaystyle \varepsilon }
, and strain-rate,
{\displaystyle {\dot {\varepsilon }}}
; therefore it can be written as some function of those properties:[1]
{\displaystyle Y_{\text{f}}=f(\varepsilon ,{\dot {\varepsilon }},T)}
The exact equation to represent flow stress depends on the particular material and plasticity model being used. Hollomon's equation is commonly used to represent the behavior seen in a stress-strain plot during work hardening:[2]
{\displaystyle Y_{\text{f}}=K\varepsilon _{\text{p}}^{\text{n}}}
{\displaystyle Y_{\text{f}}}
is flow stress,
{\displaystyle K}
is a strength coefficient,
{\displaystyle \varepsilon _{\text{p}}}
is the plastic strain, and
{\displaystyle n}
is the strain hardening exponent. Note that this is an empirical relation and does not model the relation at other temperatures or strain-rates (though the behavior may be similar).
Generally, raising the temperature of an alloy above 0.5 Tm results in the plastic deformation mechanisms being controlled by strain-rate sensitivity, whereas at room temperature metals are generally strain-dependent. Other models may also include the effects of strain gradients.[3] Independent of test conditions, the flow stress is also affected by: chemical composition, purity, crystal structure, phase constitution, microstructure, grain size, and prior strain.[4]
The flow stress is an important parameter in the fatigue failure of ductile materials. Fatigue failure is caused by crack propagation in materials under a varying load, typically a cyclically varying load. The rate of crack propagation is inversely proportional to the flow stress of the material.
^ Saha, P. (Pradip) (2000). Aluminum extrusion technology. Materials Park, OH: ASM International. p. 25. ISBN 9781615032457. OCLC 760887055.
^ Mikell P. Groover, 2007, "Fundamentals of Modern Manufacturing; Materials, Processes, and Systems," Third Edition, John Wiley & Sons Inc.
^ Soboyejo, W. O. (2003). Mechanical properties of engineered materials. Marcel Dekker. pp. 222–228. ISBN 9780824789008. OCLC 649666171.
^ "Metal technical and business papers and mill process modeling". 2014-08-26. Archived from the original on 2014-08-26. Retrieved 2019-11-20.
|
Identifying Pattern Relationships Practice Problems Online | Brilliant
Which of the options best describes the relationship in this sequence:
2, 5, 11, 23, 47, \ldots
Square the preceding term and add 1 Add 3 to the preceding term Multiply the preceding term by 3 and subtract 1 Multiply the preceding term by 2 and add 1
1, 3, 7, 13, 21, \ldots
Add increasing even numbers to the preceding term Multiply the preceding term by 2 and add 1 Square the preceding term and add 1 Multiply the preceding term by 2 and subtract 1
3, 6, 4, 8, 6, 12, 10, 20, \ldots
Square the preceding term and add 1 Alternately, either multiply by 2 or subtract 2 None of the rest Multiply the preceding term by 2
3, 7, 11, 15, 19, \ldots
Add 2 to the preceding term Multiply the preceding term by 2 and add 1 Multiply the preceding term by 2 and subtract 1 Add 4 to the preceding term
2, 3, 8, 63, 3968, \ldots
Square the preceding term and add 1 Multiply the preceding term by 5 and subtract 7 Multiply the preceding term by 7 and subtract 5 Square the preceding term and subtract 1
|
From J. B. Innes 19 February [1862]1
Milton Brodie | Forres. | NB
You must not suppose we only think of you and yours when some fact of natural history turns up, for indeed we often think and speak of our kind friends in the South, and some times Stephens gives us a bulletin2 We were sorry the last reported some of your party indisposed3 I hope you have forgotten all about this long ago.
My gardener has got a bird the offspring of a male mule between a canary and green finch, and a hen canary. He says he is quite sure that papa was a mule, though he is not quite sure whether it was half greenfinch or chaffinch. It was reared by a labourer who was then in this garden, and he persisted in putting it with the canary in spite of all assurances that they would not breed, and this bird is the result. Probably you know plenty such cases, but it is new to me— If you want any thing looked after up here in Earth air or water tell me and we will do our little utmost.
We have had very mild weather no frost to Johnny’s sorrow as he wants to skate and has only had them on once for a short morning when rain came—4 today it has been quite warm.
We saw the announcement of Mrs Langton’s death.5 I know you were prepared for and expecting it and believe she had been in much suffering. We have been all as well as usual. Johnny has not tired of his home pursuits yet, and looks forward to some swimming in the sea when hot weather comes He likes his tutor and works pretty willingly. Eliza is much as usual and has been once out to dinner, a mighty feat for her, but I fear she will not repeat it very often.6
You will be all gay with the Exhibition. We hear so much of it, that I suppose some of us at least must struggle up to see it before it closes.7
With all our best regards to your circle | Believe me Dear Darwin | Yours faithfully | J. B. Innes
End of letter: ‘11 10
\frac{1}{4}
| 2’ ink
The year is established by the reference to the death of Charlotte Langton, Emma Darwin’s sister (see n. 5, below).
Thomas Sellwood Stephens was the curate at Down (Post Office directory of the six home counties 1862). Innes, who was the incumbent of the parish, moved to his family’s ancestral home, Milton Brodie, near Forres, Scotland, in January 1862 (see letter from J. B. Innes, 2 January [1862], and letter to J. B. Innes, [3] January [1862]).
CD reported that three of his sons were ill in bed in the letter to J. B. Innes, [3] January [1862]. Subsequently, many more members of the household were afflicted with influenza (see letter to John Lubbock, 23 January [1862], and letter to J. B. Innes, 24 February [1862]).
Innes refers to his son, John William Brodie Innes.
According to Emma Darwin’s diary (DAR 242), Charlotte Langton died on 2 January 1862. Her death was announced in The Times, 6 January 1862, p. 1.
Innes refers to his wife, Eliza Mary Brodie Innes.
The International Exhibition opened at South Kensington on 1 May 1862 (The Times, 2 May 1862, pp. 11–12). Regular descriptions of the plans for the exhibition appeared in the London papers throughout the first half of 1862.
Reports on a bird, offspring of a male mule between a canary and greenfinch, and a hen canary.
|
Hess’s Law - Course Hero
General Chemistry/Enthalpy and Bond Strength/Hess’s Law
A chemical reaction can occur in multiple steps and through different pathways. The exact path taken for a reaction or the order of the steps does not affect the reaction enthalpy. Enthalpy is a state function, which means it is independent of the path taken.
Hess's law, named for Swiss-Russian chemist Germain Hess (1802–50), states that in a multistep reaction, the reaction enthalpy is equal to the sum of the reaction enthalpies for each individual step. Hess's law means that enthalpies of reaction can be added to each other.
Consider the reaction for the combustion of methane:
{\rm{CH}}_{4}(g)+2{\rm{O}}_{2}(g)\rightarrow{\rm{CO}}_{2}(g)+2{\rm{H}}_{2}{\rm{O}}(l)
At room temperature water (H2O) is a liquid, not a gas, so the enthalpy of water changing from a liquid to a gas needs to be factored in. It is possible to calculate the enthalpy of reaction for this reaction using two separate reactions used before:
\begin{aligned}{\rm{CH}}_{4}(g)+2{\rm{O}}_{2}(g)&\rightarrow{\rm{CO}}_{2}(g)+2{\rm H}_{2}{\rm{O}}(g)&&\Delta H=-802.3\;{\rm{kJ}}\\{\rm{H}}_{2}{\rm{O}}(l)&\rightarrow{\rm{H}}_{2}{\rm{O}}(g)&&\Delta H=44\;{\rm{kJ}}\end{aligned}
The second reaction provides the energy needed to convert 1 mol of liquid water into 1 mol of water vapor. The reverse reaction has the same amount of energy with the opposite sign.
{\rm{H}}_{2}{\rm{O}}(g)\rightarrow{\rm{H}}_{2}{\rm{O}}(l)\;\;\;\;\;\Delta H=-44\;{\rm{kJ}}
Furthermore, 2 mol water vapor must be converted to liquid water.
2{\rm{H}}_{2}{\rm{O}}(g)\rightarrow2{\rm{H}}_{2}{\rm{O}}(l)\;\;\;\;\;\Delta H=-88\;{\rm{kJ}}
Now, add the reaction enthalpies of these two reactions.
\begin{aligned}&{\rm{CH}}_{4}(g)+2{\rm O}_{2}(g)\rightarrow{\rm{CO}}_{2}(g)+2{\rm H}_{2}{\rm O}(g)\;\;\;\;\;&&\Delta H=-802.3\;{\rm{kJ}}\\&2{\rm H}_{2}{\rm O}(g)\rightarrow2{\rm H}_{2}{\rm O}(l)\;\;\;\;\;&&\Delta H=-88\;{\rm{kJ}}\\&{\rm{CH}}_{4}(g)+2{\rm O}_{2}(g)+2{\rm H}_{2}{\rm O}(g)\rightarrow{\rm{CO}}_{2}(g)+2{\rm H}_{2}{\rm O}(g)+2{\rm H}_{2}{\rm O}(l)\;\;\;\;\;&&\Delta H=-802.3\;{\rm{kJ}}+-88\;{\rm{kJ}}=-890.3\;{\rm{kJ}}\end{aligned}
Because 2H2O(g) is located on both sides of the yields arrow, it can be removed from the equation.
{\rm{CH}}_{4}(g)+2{\rm O}_{2}(g)\rightarrow{\rm{CO}}_{2}(g)+2{\rm H}_{2}{\rm O}(l)\;\;\;\;\;\Delta H=-890.3\;{\rm{kJ}}
Hess's law allows scientists to use known reaction enthalpies to calculate the reaction enthalpy for an unknown reaction. This means scientists do not need to measure the reaction enthalpy of each reaction separately. As long as there is a reaction path with known reaction enthalpies, the enthalpy for the reaction can be calculated.
Hess's law also allows scientists to calculate enthalpies for reactions that do not normally occur.
Calculation of Enthalpy Change for the Formation of Carbon Monoxide
The reaction
{\rm{C}}(s)+{\textstyle\frac12}{\rm{O}}_{2}(g)\rightarrow{\rm{CO}}(g)
does not occur under normal conditions.
Carbon and carbon monoxide both undergo combustion with oxygen to form carbon dioxide. These reactions occur, and their reaction enthalpies are measured:
\begin{gathered}{\rm{C}}(s)+{\rm{O}}_{2}(g)\rightarrow{\rm{CO}}_{2}(g)\;\;\;\;\;\Delta H=-393.5\;{\rm{kJ}}\;\;\;\;\;\text{(Reaction 1)}\\\\{\rm{CO}}(g)+{\textstyle\frac12}{\rm{O}}_{2}(g)\rightarrow{\rm{CO}}_{2}(g)\;\;\;\;\;\Delta H=-283.0\;{\rm{kJ}}\;\;\;\;\;\text{(Reaction 2)}\end{gathered}
Use these reactions to calculate
\Delta H
{\rm{C}}(s)+{\frac12}{\rm{O}}_{2}(g)\rightarrow{\rm{CO}}(g)\;\;\;\;\;\text{(Reaction 3)}
The goal is to construct reaction 3 using reactions 1 and 2. Check the products. Reaction 3 has CO(g) as a product. Reaction 2 has CO(g) as a reactant. The first step is to reverse reaction 2 so that carbon monoxide becomes a product, as in reaction 3. Note that this reverses the sign of
\Delta H
{\rm{CO}}_{2}(g)\rightarrow{\rm{CO}}(g)+{\textstyle\frac12}{\rm {O}}_{2}(g)\;\;\;\;\;\Delta H=283.0\;{\rm{kJ}}\;\;\;\;\;\text{(Reaction 4)}
Now add reaction 1 and reaction 4.
\begin{gathered}{\rm{C}}(s)+{\rm{O}}_{2}(g)+{\rm{CO}}_{2}(g)\rightarrow{\rm{CO}}_{2}(g)+{\rm{CO}}(g)+{\textstyle\frac12}{\rm{O}}_{2}(g)\\\\{\Delta H}=-393.5\;{\rm{kJ}}+283.0\;{\rm{kJ}}=-110.5\;{\rm{kJ}}\end{gathered}
The term CO2(g) appears on both sides of the reaction arrow, so it can be eliminated. There is 1 mol of O2(g) on the reactant side and half a mole of O2(g) on the product side. We can eliminate half a mole of O2(g) from both sides as well. The remainder is the net reaction.
{\rm{C}}(s)+{\textstyle\frac12}{\rm{O}}_{2}(g)\rightarrow{\rm{CO}}(g)\;\;\;\;\;\Delta H=-110.5\;{\rm{kJ}}
<Enthalpies of Reaction>Enthalpies of Formation
|
Find equation of circle tangent to 3x+y+3=0 at (-3, 6) tangent to x+3y-7=0 - Maths - Conic Sections - 10580819 | Meritnation.com
Find equation of circle tangent to 3x+y+3=0 at (-3, 6)& tangent to x+3y-7=0
The circle is \mathrm{tan}gent to 3x+y+3=0\phantom{\rule{0ex}{0ex}}eqn of normal through centre is\phantom{\rule{0ex}{0ex}}x-3y=k\phantom{\rule{0ex}{0ex}}it passes through \left(-3,6\right)\phantom{\rule{0ex}{0ex}}so x-3y=-21\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}let the centre be \left(3k-21,k\right)\phantom{\rule{0ex}{0ex}}centre is equidis\mathrm{tan}t from \mathrm{tan}gents 3x+y+3=0 and x+3y-7=0\phantom{\rule{0ex}{0ex}}so\phantom{\rule{0ex}{0ex}}\frac{\left|3\left(3k-21\right)+k+3\right|}{\sqrt{{3}^{2}+{1}^{2}}}=\frac{\left|\left(3k-21\right)+3k-7\right|}{\sqrt{{1}^{2}+{3}^{2}}}\phantom{\rule{0ex}{0ex}}\frac{\left|10k-60\right|}{\sqrt{10}}=\frac{\left|6k-28\right|}{\sqrt{10}}\phantom{\rule{0ex}{0ex}}\left|10k-60\right|=\left|6k-28\right|\phantom{\rule{0ex}{0ex}}10k-60=±\left(6k-28\right)\phantom{\rule{0ex}{0ex}}solving we get k=8 and 11/2\phantom{\rule{0ex}{0ex}}centre is \left(3,8\right) and \left(-9/2,11/2\right) radius is 2\sqrt{10} and \sqrt{10}/2\phantom{\rule{0ex}{0ex}}circles are\phantom{\rule{0ex}{0ex}}\left(x-3{\right)}^{2}+\left(y-8{\right)}^{2}=40\phantom{\rule{0ex}{0ex}}\left(x+9/2{\right)}^{2}+\left(y-11/2{\right)}^{2}=10/4\phantom{\rule{0ex}{0ex}}
ugytuytb
|
Jira Align | James's Knowledge Graph
Forecasting in JiraAlign
Jira Align provides two potential inputs to use in forecasting: Member Weeks, and Team Weeks. Member weeks (MW) are the number of team members that will work on a feature multiplied by the number of weeks they believe it would take if they focused on it completely. Team Weeks (TW) are the number of weeks an entire agile team believes it would take to complete a feature if they focused on it completely. Both data points are used in forecasting and high-level planning in Jira Align, however member weeks is believed to be more accurate because it takes into account that different members of the same team can work on separate features in parallel.
Member Weeks Example: An agile team estimates that 3 of its developers can complete the shopping cart feature in about two weeks, if they fully focus on that feature. Three developers times two weeks equals 6 member weeks, or
3 developers * 2 weeks = 6 member weeks
Broader Topics Related to Jira Align
A product and project management framework for large enterprises
Jira Align Knowledge Graph
|
PDF - Curvenote Docs
After creating an article in Curvenote, you can export and download your document as a PDF using a variety of professional templates!
#📺 Video Demo
#Exit Draft Mode
Only a saved version of an article can be exported for download. If you are editing a draft you will need to exit draft mode (indicated in the footer👇) . Learn more Drafts & Versions
To export and download your article with the most recent changes made in draft mode:
Click SAVE VERSION in the header👆
This will create a new version of your article
To export and download the most recently saved version of your article - any changes made in draft mode will not be included:
Click STOP EDITING in the header👆
This will return you to a view of the last saved version of your article
#Export and Download PDF
💡 Tip - Only owners or collaborators can download articles.
You can now export and download your article! To do this:
Click the download ⬇ icon.
Hover over the thumbnail on the left for an expanded preview of the template layout.
Complete the template specific instructions for Template Options.
Each template has a variety of required and optional options to include such as author name, affiliation, email, etc. Learn more Template Options.
Other requirements such as abstracts and acknowledgements use tagged content. Learn more Tagging Blocks.
Your article export is now processing.
You can exit the window during processing. You will notified when your export is complete.
In the original Exports pop-up:
Click the ☁️⬇️ icon.
If you have closed the pop-up:
Click the download ⬇ icon**:**.
\textcircled{\checkmark}
Download PDF.
You can also download the log file for the PDF export.
Your exported article will be available for download by you or any of your collaborators until you save a new version.
You will need to repeat this process for that version.
Don’t see the template you need? Request a new template via email or add a template on Github!
|
Angles of a Knoop hardness test indenter
The Knoop hardness test /kəˈnuːp/ is a microhardness test – a test for mechanical hardness used particularly for very brittle materials or thin sheets, where only a small indentation may be made for testing purposes. A pyramidal diamond point is pressed into the polished surface of the test material with a known (often 100g) load, for a specified dwell time, and the resulting indentation is measured using a microscope. The geometry of this indenter is an extended pyramid with the length to width ratio being 7:1 and respective face angles are 172 degrees for the long edge and 130 degrees for the short edge. The depth of the indentation can be approximated as 1/30 of the long dimension.[1] The Knoop hardness HK or KHN is then given by the formula:
Comparison between the Mohs and the Knoop scales
Tooth enamel 343
{\displaystyle HK={{{\textrm {load}}({\mbox{kgf}})} \over {{\textrm {impression\ area}}({\mbox{mm}}^{2})}}={P \over {C_{p}L^{2}}}}
Cp = correction factor related to the shape of the indenter, ideally 0.070279
HK values are typically in the range from 100 to 1000, when specified in the conventional units of kgf·mm−2. The SI unit, pascals, are sometimes used instead: 1 kgf·mm−2 = 9.80665 MPa.
The test was developed by Frederick Knoop[2] and colleagues at the National Bureau of Standards (now NIST) of the United States in 1939, and is defined by the ASTM E384 standard.
The advantages of the test are that only a very small sample of material is required, and that it is valid for a wide range of test forces. The main disadvantages are the difficulty of using a microscope to measure the indentation (with an accuracy of 0.5 micrometre), and the time needed to prepare the sample and apply the indenter.
Variables such as load, temperature, and environment, may affect this procedure, which have been examined in detail.[3]
Knoop hardness of ceramics
^ "Microhardness Test", Surface Engineering Forum
^ F. Knoop, C.G. Peters and W.B. Emerson (1939). "A Sensitive Pyramidal-Diamond Tool for Indentation Measurements". Journal of Research of the National Bureau of Standards. 23 (1): 39–61 (Research Paper RP1220). doi:10.6028/jres.023.022.
^ Czemuska, J. T. (1984). Proc. Br. Ceram. Soc. 34: 145–156. {{cite journal}}: Missing or empty |title= (help)
Dental hardness tables
Retrieved from "https://en.wikipedia.org/w/index.php?title=Knoop_hardness_test&oldid=995602881"
|
Transfer Function Models - MATLAB & Simulink
Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials. The model order is equal to the order of the denominator polynomial. The roots of the denominator polynomial are referred to as the model poles. The roots of the numerator polynomial are referred to as the model zeros.
The parameters of a transfer function model are its poles, zeros, and transport delays.
In continuous time, a transfer function model has the following form:
Y\left(s\right)=\frac{num\left(s\right)}{den\left(s\right)}U\left(s\right)+E\left(s\right)
Here, Y(s), U(s), and E(s) represent the Laplace transforms of the output, input, and noise, respectively. num(s) and den(s) represent the numerator and denominator polynomials that define the relationship between the input and the output.
For more information, see What are Transfer Function Models?
Set Transfer Function Model Options
tfestOptions Option set for tfest
Transfer Function Model Basics
Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials.
Characteristics of estimation data for transfer function identification.
Estimate Transfer Function Models
Estimate Transfer Function Models with Unknown Transport Delays
This example shows how to estimate a transfer function model with unknown transport delays and apply an upper bound on the unknown transport delays.
Frequency Domain Troubleshooting
Improve frequency-domain model estimation by preprocessing data and applying frequency-dependent weighting filters.
Specify the values and constraints for the numerator, denominator and transport delays.
Specifying Initial Conditions for Iterative Estimation of Transfer Functions
Specify how initial conditions are handled during model estimation in the app and at the command line.
|
Model antenna or antenna array accounting for incident power wave (RX) and radiated power wave (TX) - Simulink - MathWorks Benelux
\begin{array}{l}TX\left(:,:,1\right)=T{X}_{\theta }=\frac{{E}_{\theta }}{\sqrt{{\eta }_{0}}}·\sqrt{4\pi }·R·{e}^{j\gamma R}\\ TX\left(:,:,2\right)=T{X}_{\phi }=\frac{{E}_{\varphi }}{\sqrt{{\eta }_{0}}}·\sqrt{4\pi }·R·{e}^{j\gamma R}\end{array}
\gamma =\frac{j\omega }{c}
{‖TX‖}^{2}={|T{X}_{\theta }|}^{2}+{|T{X}_{\varphi }|}^{2}=EIRP={P}_{t}{G}_{t}
TX\left(:,:\right)=\sqrt{{G}_{t}{R}_{e}\left\{{Z}_{in}\right\}}.{I}_{in}
pl=\frac{\lambda }{4\pi R}·{e}^{-j\gamma R}
\begin{array}{l}RX\left(:,:,1\right)=R{X}_{\theta }=T{X}_{\theta }·pl=\frac{{E}_{\theta }}{\sqrt{{\eta }_{0}}}·\frac{\lambda }{\sqrt{4\pi }}\\ RX\left(:,:,2\right)=R{X}_{\varphi }=T{X}_{\varphi }·pl=\frac{{E}_{\varphi }}{\sqrt{{\eta }_{0}}}·\frac{\lambda }{\sqrt{4\pi }}\end{array}
{‖RX‖}^{2}={|R{X}_{\theta }|}^{2}+{|R{X}_{\varphi }|}^{2}
{P}_{r}={‖RX‖}^{2}{G}_{r}
|
Elementary geometric representation of the formulas of the special theory of relativity - Wikisource, the free online library
Elementary geometric representation of the formulas of the special theory of relativity (1921)
by Paul Gruner, translated from French by Wikisource
In French: Représentation géométrique élémentaire des formules de la théorie de la relativité, Archives des sciences physiques et naturelles (5) 3: 295–296, Scans
1576016Elementary geometric representation of the formulas of the special theory of relativityPaul Gruner1921
Gruner, P. and Sauter J. (Berne). – Elementary geometric representation of the formulas of the special theory of relativity.
The theory of special relativity, applied to two one-dimensional systems, moving relatively to each other with velocity
{\displaystyle v}
, gives the following formulas:
{\displaystyle x'=\beta (x-\alpha ct)\quad ct'=\beta (ct-\alpha x),}
{\displaystyle v=\alpha \cdot c,\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}}
The geometric representation given in a general manner by Minkowski, becomes particularly simple and elegant by choosing the axes of
{\displaystyle x}
{\displaystyle t}
for two mutually orthogonal systems.
From the attached figure, the
{\displaystyle OT}
axis is perpendicular to axis
{\displaystyle OX'}
, and axis
{\displaystyle OT'}
is rotated by an angle
{\displaystyle \varphi }
{\displaystyle \sin \varphi =\alpha ;\quad \beta ={\frac {1}{\cos \varphi }};\quad \alpha \beta =\tan \varphi .}
{\displaystyle c=1}
, we immediately find that the coordinates
{\displaystyle x,t,x',t'}
{\displaystyle {\mathsf {P}}}
satisfy the requirements of the theory of relativity:
{\displaystyle x'={\frac {x}{\cos \varphi }}-t\cdot \tan \varphi ;\quad t'={\frac {t}{\cos \varphi }}-x\cdot \tan \varphi .}
With this mode of representation which contains no imaginary quantity, it is easy and simple to graphically demonstrate the different results of the theory of relativity (length contraction, dilatation of clocks, change in mass, energy, volume, etc. ).
Furthermore, the figure immediately gives the covariant
{\displaystyle (\xi ,\tau ,\xi ',\tau ')}
and contravariant
{\displaystyle (x,t,x',t')}
components of a vector
{\displaystyle {\mathsf {R}}}
; it is easy to find geometrically the law of the invariance of the square of the vector:
{\displaystyle {\mathsf {R}}^{2}=x\xi +t\tau =x'\xi '+t'\tau '}
Retrieved from "https://en.wikisource.org/w/index.php?title=Translation:Elementary_geometric_representation_of_the_formulas_of_the_special_theory_of_relativity&oldid=11637342"
|
Transform IIR lowpass filter to complex bandstop filter - MATLAB iirlp2bsc
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{-1}+\cdots +{b}_{n}{z}^{-n}}{{a}_{0}+{a}_{1}{z}^{-1}+\cdots +{a}_{n}{z}^{-n}},
b=\left[\begin{array}{ccccc}{b}_{01}& {b}_{11}& {b}_{21}& ...& {b}_{Q1}\\ {b}_{02}& {b}_{12}& {b}_{22}& ...& {b}_{Q2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {b}_{0P}& {b}_{1P}& {b}_{2P}& \cdots & {b}_{QP}\end{array}\right]
H\left(z\right)=\prod _{k=1}^{P}{H}_{k}\left(z\right)=\prod _{k=1}^{P}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}+\cdots +{b}_{Qk}{z}^{-Q}}{{a}_{0k}+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}+\cdots +{a}_{Qk}{z}^{-Q}},
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{-1}+\cdots +{b}_{n}{z}^{-n}}{{a}_{0}+{a}_{1}{z}^{-1}+\cdots +{a}_{n}{z}^{-n}},
a=\left[\begin{array}{ccccc}{a}_{01}& {a}_{11}& {a}_{21}& \cdots & {a}_{Q1}\\ {a}_{02}& {a}_{12}& {a}_{22}& \cdots & {a}_{Q2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {a}_{0P}& {a}_{1P}& {a}_{2P}& \cdots & {a}_{QP}\end{array}\right]
H\left(z\right)=\prod _{k=1}^{P}{H}_{k}\left(z\right)=\prod _{k=1}^{P}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}+\cdots +{b}_{Qk}{z}^{-Q}}{{a}_{0k}+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}+\cdots +{a}_{Qk}{z}^{-Q}},
|
Return measurement residual and residual covariance when using extended or unscented Kalman filter - MATLAB residual - MathWorks España
\mathrm{mu}
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k-1|k-1\right]
x\left[k\right]=\sqrt{x\left[k-1\right]+u\left[k-1\right]}+w\left[k-1\right]
y\left[k\right]=x\left[k\right]+2*u\left[k\right]+v\left[k{\right]}^{2}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.