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Pitfalls of Modern Portfolio Theory | Eqtor
Pitfalls of Modern Portfolio Theory
Modern Portfolio Theory (MPT) is one of the most influential theories in investment, which explains how diversification can improve return and minimize risk. Modern portfolio theory highlights the role of correlations between assets and the benefits of splitting investments into assets that do not correlate with each other. There is merit in MPT, and it has been applied effectively in practice. However, there are some shortcomings with the theory due to the way risk is quantified, and the heavy reliance on historical performance of assets.
Here is a list of the two most serious criticisms of modern portfolio theory.
Reliance on historical performance
Asset correlations and historical returns are used in modern portfolio theory to find the optimal portfolio. This places a large reliance on historical performance data. A common adage in investing is that "Past performance is no guarantee of future results." Assets can perform in the future differently than from the past, and both the returns and relationships (correlations) between assets can change.
On the other hand, although historical performance does not predict future results with certainty, historical performance is often the best indicator of future results. When betting on horse races, you would have better luck with horses that have a track record of winning compared to horses that normally finish last.
Variance is not the right measure of risk
Modern Portfolio Theory (MPT) relies on the variance, and the correlation between assets, which are statistical measures computed from historical data. One of the major criticisms is that variance is not the right measure to use for defining risk, since it only measures the variation in the asset's return, and does not make a distinction between gains and losses.
\text{variance} = \dfrac{1}{n} \sum_{n=1}^{n} (y_i - \bar{y})^2
To compute the variance, you need to take a sample of
n
returns, find their average
\bar{y}
, then take the average of the deviations of your samples from the average. If you have data points [5%, -10%, 5%] for three months, the variance is the same as with [15%, 0%, 15%], although the second case should be viewed as less risky. In fact, in the second case, there was no loss in any month.
When two portfolios have the same level of variance and returns, they are considered equally desirable under modern portfolio theory. One portfolio may have that variance because of frequent small losses. In contrast, the other could have that variance because of rare spectacular declines. Most investors would prefer frequent small losses, which would be easier to endure.
Post-modern portfolio theory (PMPT) attempts to improve on modern portfolio theory by using the downside risk instead of the variance, which is a measure for the risk of loss.
Although the core ideas behind of modern portfolio theory are very useful, MPT is limited by measures of risk and return that do not always reflect the realities of the markets. It is important to be aware of these limitations when considering a quantitatively derived portfolio allocation. Your investment strategy should carefully consider the investment risks that are relevant to you and include a good amount of diversification and proper asset allocation.
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Dynamic Programming is a technique to solving complex problems in programming. At its core, the concept is about remembering and making use of answers to smaller “subproblems” that are already solved.
Dynamic Programming (DP) is a powerful technique that makes it possible to solve certain problems considerably more efficiently than a recursive solution would usually be.
This method essentially trades space for time. Instead of calculating all the solution’s different states (taking a lot of time but no space), we take up space for storing solutions of all the sub-problems to save time later. This is called “Memoization.”
Let’s understand this using the classic example of Fibonacci Series:
First few numbers start as:
1, 1, 2, 3, 5, 8, 13...
Figure Showing the Function Calls
As we can see in the illustration in order to calculate fib(6), fib(4) is computed 2 times, fib(3) is computed 3 times and fib(2) is computed 5 times. This causes extra function calls for things that have already been calculated, and adds extra time overhead. The problem could simply be solved by storing the results of the subproblems in an array and using the results in place of function calls.
Conveniently, the dynamic programming approach could store the prior results in just 2 variables, because each fib(n) just requires the result of the last two elements.
O(1)
Below, a simple solution is shown using recursion and contrasted to one using DP.
cout << "Fibonacci of " << n << " = " << fib(n) << "\n";
T(n) = T(n-1) + T(n-2)
O(n)
Fibonacci Series Using DP (Advanced Method)
O(n)
O(1)
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When to use linear acoustic theory instead of the nonlinear one | NiSiC
Equations of fluid dyanamics
Acoustic waves are small propagating fluctuations in pressure, particle velocity, mass density, entropy and temperature on lager and more uniform background values of these quantities. Let
p_0,\bm{v}_0,\rho{}_0, s_0, T_0
refer to the pressure, partical velocity, mass density, specific entropy density (i.e., entropy per unit mass), and absolute temperature of the background medium, respectively. Then, the total pressure
P
, particle velocity
\bm{V}
, mass density
\varGamma{}
, specific entropy density
S
T
associated with a propagating acoustic disturbance are assumed to be
\begin{array}{rcl} \left\{ P,\bm{V},\varGamma,S,T \right\} & = & \left\{ p_0,\bm{v}_0,\rho{}_0,s_0,T_0 \right\} \\ & + & \left\{ p,\bm{v},\rho,s,\theta \right\} \end{array}
Similarly, the total sound speed of the medium is given by
C=c_0+c
The local form (i.e., the Eulerian specification of the flow field) of the principle of conservation of mass is expressed by the continuity equation (without mass source):
\frac{D\varGamma}{Dt}=-\varGamma{}\nabla{}\cdot\boldsymbol{V}
Where the Material Derivative
\frac{D[\bullet]}{Dt}=\frac{\partial{}[\bullet]}{\partial{}t}+\bm{V}\cdot\left(\nabla{}[\bullet]\right)
Movement Equation
Movement Equation (or the Equation of momentum) can be derived from the Newton’s Second Law (without body force):
\varGamma\frac{D\bm{V}}{Dt}=\nabla{}\mathbb{T}
\mathbb{T}=\sigma{}_{ij}
denotes the Cauchy stress tensor - a second-order symmetric tensor consisted by 6 independent scalar variables.
The Constitutive Equation explicitly relate the Cauchy stress tensor
\sigma{}_{ij}
, the normal stress vector components
P_i
and the velocity vector component
U_i
. For isotropic Newtonian fluid, with the Volume viscosity introduced by Stokes in 1949:
\begin{cases} \sigma{}_{ij}=-P\delta{}_{ij}+T_{ij} \\ T_{ij}=\mu{}'\theta{}\delta{}_{ij}+2\mu{}d_{ij} \\ \theta{}=d_{11}+d_{22}+d_{33} \\ d_{ij}=\frac{1}{2}\left[ \frac{\partial{}U_i}{\partial{}x_j}+\frac{\partial{}U_j}{\partial{}x_i}\right] \end{cases}
\mu
is the shear viscosity, and
\mu{}'
is the volume viscosity of the medium. Here we considered the viscous effect of the medium.
\left( \frac{\partial{}^2P}{\partial{}\varGamma{}^2} \right)_{S,\rho{}_0}=2c_0^2\frac{\beta{}-1}{\rho{}_0}
\beta
is the nonlinear coefficient.
heat conduction effect
\begin{array}{rcl} P & = & p_0+c_0^2\rho{}+\frac{1}{2}\left(\frac{\partial{}^2P}{\partial{}\varGamma{}^2}\right)_{S,\rho{}_0}\rho{}^2 \\ & - & \kappa\left(\frac{1}{C_v}-\frac{1}{C_p}\right)\nabla{}\cdot{}\bm{V} \end{array}
\kappa{}
is the thermal conductivity of the medium.
\begin{array}{rcl} P & = & p_0+c_{\infty}^2\rho{}+\frac{1}{2}\left(\frac{\partial{}^2P}{\partial{}\varGamma{}^2}\right)_{S,\rho{}_0,\xi{}_0}\rho{}^2\\ & - & \frac{mc_0^2}{\tau{}_r}\int_{-\infty}^t\rho{}e^{-\left(t-t'\right)/\tau{}_r}dt' \end{array}
\tau{}_r
c_\infty
is the sound speed when
\omega{}\tau{}_r \to \infty
m
is a dimensionless parameter which describing the dispersion and dissipation characteristics of the medium.
Linearization of the control equations
It is well know that the Helmholtz Equation is derived from the nonlinear acoustic theories by serial hypothesis. The viscous effect, nonlinear effect, heatconduction effect and relaxation effect can be neglected under these assumptions.
the medium is ideal fluid.
acoustic waves is much smaller disturbance then background medium.
frequency of the acoustic wave is large enough that the propagation of the sound can be regard as an adiabatic process.
the background medium is uniformly and stationary.
How to discribe those effects
I will discuss when should we apply linear acoustic theory instead of the nonlinear one based on 1-D control equations.
Mach Number is a dimensionless quantity representing the ratio of flow velocity to the local speed of sound.
\bm{Ma}=\frac{U}{C}=\frac{u_0+u}{c_0+c}
U
is the component of total particle velocity
\bm{V}=U_i
Acoustic Reynolds Number
In fluid dynamics, the Reynolds Number is a dimensionless quantity representing the ratio of inertia force to viscous force.
\mathcal{Re}=\frac{\Gamma{}\bm{V}\cdot{}\nabla{}\bm{V}}{\mu{}\nabla{}^2\bm{V}}\approx\frac{\varGamma{}UL}{\mu}
L
is a characteristic linear dimension. Similarly, we compare the convection term with the dissipative term of the Burgers Equation and got it’s acoustic Reynolds Number. Bugers Equation (no rotation) is derived from the continuity equation, movement equation and the equation of state which considered nonlinear effect and heat conduction effect.
\begin{array}{rcl}\varGamma\frac{DU}{Dt}&=&\varGamma{}\frac{\partial{}U}{\partial{}t}+\varGamma{}U\frac{\partial{}U}{\partial{}x}\\&=&-c_0^2\frac{\partial{}\varGamma{}}{\partial{}x}-c_0^2\frac{\beta{}-1}{\rho_0}\frac{\partial{}\rho{}^2}{\partial{}x}+b\frac{\partial{}^2U}{\partial{}x^2} \end{array}
b=\mu{}'+2\mu{}+\kappa{}\left(\frac{1}{C_v}-\frac{1}{C_p}\right)
. With the monochromatic traveing wave solution, the acoustic Reynolds Number can be derived
\bm{Re}=\frac{\varGamma{}U\frac{\partial{}U}{\partial{}x}}{b\frac{\partial{}^2U}{\partial{}x^2}}\sim{}\left|\frac{\frac{U}{c_0}\omega{}\rho_0{}U}{b\omega{}^2U}\right|=\frac{U\rho_0c_0}{b\omega}
When the backgound medium is stationary, the acoustic Reynolds Number can be write as
\bm{Re}=\frac{u\rho_0c_0}{b\omega}
\bm{Ma}<0.3
, that means the total particle velocity is much smaller than sound velocity, the background medium is uncompressable (i.e., the mass density is uniform) and nearly stationary.
\mathcal{Re}\gg{}1
, that means the
\mu
is neglectable small, the medium can be considered as an ideal fluid (i.e., the viscous effect can be neglect).
\bm{Re}\gg{}1
, that means the nonlinear effect and heat conduction effect can be neglect (i.e., acoustic waves is much smaller disturbance then background medium and the propagation of the sound can be regard as an adiabatic process).
If all 3 conditions are met and we can apply linear acoustic theories?
No, the discussion of the relaxiation effect is needed.
[1] Campos, L. M. “On 36 forms of the acoustic wave equation in potential flows and inhomogeneous media.” Applied Mechanics Reviews 60.4 (2007): 149-171.
[2] Qian, zu . Fei Xian Xing Sheng Xue. Bei jing: Ke xue chu ban she, 2009. Print.
[3] Solui︠a︡n, Stepan Ivanovich. Theoretical foundations of nonlinear acoustics. Consultants Bureau, 1977.
nonlinear acoustic Acoustic Reynolds Number
1. Equations of fluid dyanamics
1.2. Movement Equation
2. Linearization of the control equations
2.1. How to discribe those effects
2.1.1. Mach Number
2.1.2. Acoustic Reynolds Number
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Binary search is an efficient method for locating an element in a sorted array that is similar to searching for a word in the dictionary. If the word to search starts with the letter
S
, one usually starts searching the dictionary around the half-way mark. Since it is commonly known that
S
is located in the second half of the alphabet, it would not make sense to start searching for
S
from the dictionary’s start.
Just like in the dictionary example, when applying binary search to an array, we repeatedly divide it into individual search intervals. Initially, the original array is the search interval. If the value of the desired item is lesser than the item in the middle of the interval, then the search interval is the array with items smaller than the middle item. Similarly, if the desired item is greater than the item at the middle point, then the search interval is the array with items larger than the middle item.
This concept is illustrated in the slides below:
The time complexity of a binary search is
O(Log(n))
. The following code snippets demonstrate the recursive implementation of binary search:
int binarySearch(int sample_arr[], int left, int right, int key) {
//Base Case: binary search executes as long right-most index is greater
// than left-most index.
// if element is present at the middle return index.
if (sample_arr[mid] == key)
if (sample_arr[mid] > key)
return binarySearch(sample_arr, left, mid - 1, key);
return binarySearch(sample_arr, mid + 1, right, key);
int sample_arr[] = {2, 5, 8, 12, 16, 23, 38, 56, 72, 91};
int arr_size = sizeof(sample_arr) / sizeof(sample_arr[0]);
int result = binarySearch(sample_arr, 0, arr_size - 1, key);
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Independent Samples Z-Test - Comparing Two Proportions (Pooled) - Q
Independent Samples Z-Test - Comparing Two Proportions (Pooled)
{\displaystyle g_{1}}
{\displaystyle g_{2}}
are the two proportions and
{\displaystyle n_{1}}
{\displaystyle n_{2}}
are the sample sizes:
{\displaystyle z={\frac {g_{1}-g_{2}}{\sqrt {d_{eff}\times g_{12}(1-g_{12})(n_{1}^{-1}+n_{2}^{-1})}}}}
{\displaystyle g_{12}={\frac {n_{1}g_{1}+n_{2}g_{2}}{n_{1}+n_{2}}}}
{\displaystyle p=2(1-\Phi (|z|))}
Retrieved from ‘https://wiki.q-researchsoftware.com/index.php?title=Independent_Samples_Z-Test_-_Comparing_Two_Proportions_(Pooled)&oldid=19934’
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The measurement methods of acoustic material properties | NiSiC
Just acoustic materials, no acoustic construction here (i.e., structural sound insulation). acoustic properties only, no internal properties of porous media.
With the in-depth study and broad application of acoustic materials, the importance of the characterising acoustic material properties has emerged and risen gradually. The absorption properties and acoustic insulation properties are most used evaluation indexes for acoustic materials. The absorption properties include absorption coefficient and acoustic impedance; the acoustic insulation properties usually evaluated by the transmission losses.
Common acoustic material properties are——Absorption Coefficient， Acoustic Impedance Ratio，Transmission Loss.
There are various methods used to measure and characterise acoustic materials.
Normal incidence (impedance or standing wave tube)
To understand and model absorptive materials better, it is necessary to measure materials in a more controlled environment to allow direct comparison between theory and experiment (numerical experiment included).
standing wave method
The steady state pressure in the tube is given by
p(z)=A\left(e^{jkz}+Re^{-jkz}\right) \tag{1}
A
is the a complex constant;
k
is the wave number; R is the the reflection coefficient;
z
is the relative position to the sample. The measurement of the minimum and maximum pressures of the standing wave is needed. The standing wave ratio
s
is defined as the ratio of
p_{max}
p_{min}
and is given by:
s=\frac{p_{max}}{p_{min}} \tag{2}
The magnitude of the reflection coefficient and the absorption coefficient can be obtained:
|R|=\frac{s-1}{s+1} \tag{3}
\alpha{}=1-|R|^2 \tag{4}
One need to measure the distance between where the first minimum pressure appears and the surface of the material
x_1
to obtain the acoustic impedance ratio
Z_s
b=\frac{2x_1}{\lambda} \tag{5}
|\xi|^2=\frac{\left(2-\alpha\right)-2\sqrt{1-\alpha}\cos{2\pi{}b}}{\left(2-\alpha\right)+2\sqrt{1-\alpha}\cos{2\pi{}b}} \tag{6}
Z_s=\xi{}Z_0=\xi{}\rho_{0}c_0 \tag{7}
\lambda
is the wave length of the single frequency wave.
Fig.1. Set-ups for impedance tube measurement (Source: Ref. 2)
The standing wave method can only test single frequency waves at a time and need longer tube to test lower frequency wave. By measuring the pressure at two points in the tube the transfer function method can set up and solve simultaneous equations for the reflection coefficient and from there get the impedance and absorption coefficient. This is the principle of the transfer function , often called the two-microphone method.
R=\frac{H_{12}e^{jkz_1}-e^{jkz_2}}{e^{-jkz_2}-H_{12}e^{-jkz_1}} \tag{8}
H_{12}=p(z_2)/p(z_1)
least mean square method
This method is essentially an adaptation of the transfer function method. It usually use three microphone to cover the frequency range an impedance tube offers.
Oblique incidence (semi-anechoic spaces)
For most practitioners, the only important measurement is that which gives the random incidence absorption coefficient that is needed for performance specifications in room designing, vegetation belt setting, and so on.
The reverberation chamber test requires large sample sizes and a specialist test room, and so is expensive to undertake. It also only gives absorption coefficients; the impedance cannot be measured. Consequently, developers of absorptive material will often use the impedance tube to build up an understanding of the material properties on small samples, before undertaking reverberation tests.
The reverberation time of a room is dependent on the total absorption in the room. Consequently, by measuring the reverberation time of a room before and after a sample of absorbent is introduced, it is possible to calculate the random incidence absorption coefficient.
The reverberation time before the sample is introduced is given by:
T_0=\frac{55.3V}{c_0\alpha_0S+4Vm_1} \tag{9}
V
is the room volume,
c_0
\alpha_0
is the average absorption coefficient of the empty room,
S
is the surface area of the room and
m_1
is the air absorption constant.
The reverberation time after the sample is introduced is given by:
T_1=\frac{55.3V}{c_0\left(\alpha_0\left[S-S_s\right]+\alpha_sS_s\right)+4Vm_1} \tag{10}
S_s
is the surface area and
\alpha_s
is the absorption coefficient of the sample. By rearranging Equations
(9)
(10)
it is possible to obtain the absorption coefficient of the sample.
[1] Chen, ke , xiang . Zeng, and you . Yang. Sheng Xue Ce Liang. Bei jing: Ji xie gong ye chu ban she, 2010. Print.
[2] Cox, Trevor J., and Peter D’Antonio. Acoustic Absorbers and Diffusers: Theory, Design and Application. CRC Press, 2009.
porous materials acoustic measurement
1. Acoustic Material Properties
2.1. Normal incidence (impedance or standing wave tube)
2.2. Oblique incidence (semi-anechoic spaces)
2.3. Reverberation room
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마승렬 *
Seungryul Ma *
{K}_{xy}{\left(x\right)}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{f}_{xy}{\left(w\right)}^{2}}{{f}_{x}\left(w\right){f}_{y}\left(w\right)}
{\varphi }_{xy}\left(w\right)={\mathrm{tan}}^{-1}\left[\frac{-{q}_{xy}\left(w\right)}{{c}_{xy}\left(w\right)}\right]
{x}_{t}=\mu +A\mathrm{cos}\text{\hspace{0.17em}}\left(wt\right)+B\mathrm{sin}\text{\hspace{0.17em}}\left(wt\right)+{\epsilon }_{t}
{x}_{t}=\mu +\sum _{i}\left[{A}_{i}\mathrm{cos}\left({w}_{i}t\right)+{B}_{i}\mathrm{sin}\left({w}_{i}t\right)\right]
{x}_{t}={\alpha }_{1}+\sum _{j=1}^{p}{\beta }_{j}{x}_{t-j}+\sum _{j=1}^{p}{\gamma }_{j}{y}_{t-j}+{u}_{1,t}
{y}_{t}={\alpha }_{2}+\sum _{j=1}^{p}{\delta }_{j}{x}_{t-j}+\sum _{j=1}^{p}{\text{λ}}_{j}{y}_{t-j}+{u}_{2,t}
{u}_{1,t}~WN\left(0,{\sigma }_{1}^{2}\right)
\begin{array}{c}{u}_{2,t}~WN\left(0,{\sigma }_{2}^{2}\right)\\ \mathrm{cov}\left({u}_{1,t},{u}_{2,t}\right)={\sigma }_{12}\end{array}
1 We reported the results estimated by using the formular which has only intercept in test equation.
2) Null hypothesis: gN has a unit root or VolN has a unit root.
1) All numbers represent probability of F-statistic in Granger causality test.
2) The dark cells show the cases of rejecting the null hypothesis under 5% level of test critical values.
3) Null hypothesis: “gN does not Granger cause VolN” or “VolN does not Granger cause gN”.
1) According to the lag(p) selection criteria, we selected VAR(2) model in Total, VAR(4) model in Smetro, VAR(3) model in 5Cities and Nmetro.
2) Numbers in the parenthesis represent t statistics.
1) Related to this, we can find the previous studies which used spectral analysis to confirm the relationship between housing price and other variables except for trading volume. Cho and Ma (2006) analyzed the relationship between housing value and interest rate and Ma (2016) analyzed the relationship among housing price, business cycle, and unsold housing.
2) In our analysis, we used apartment data as the proxy of house data considering it’s absolute weight in the trading volume and number among house types.
3) We generated standardized series
{x}_{t}^{*}
{x}_{t}^{*}=\left({x}_{t}-\mu \right)/\sigma \right)
4) Refer to Appendix 1 to see the shape of periodogram.
5) Refer to Appendix 2 to see the results of cross spectral analysis more concretely.
6) Phase indicates the timing of peaks in the time series relative to peaks in the time series at a given frequency (Warner, 1998).
7) Coherence indicates the percentage of shared variance between the two time series at a particular frequency (Warner, 1998).
8) So, the periodic observation about the change in long term trend in housing price is also needed to do comprehensive judgement about housing market business cycle in addition to the analysis of cycles in housing price and trading volume (Ma, 2016).
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Magnesiokoritnigite, Mg(AsO3OH)·H2O, from the Torrecillas mine, Iquique Province, Chile: the Mg-analogue of koritnigite | Mineralogical Magazine | GeoScienceWorld
Magnesiokoritnigite, Mg(AsO3OH)·H2O, from the Torrecillas mine, Iquique Province, Chile: the Mg-analogue of koritnigite
B. P. Nash;
B. P. Nash
M. Dini;
Pasaje San Agustin 4045, La Serena
A. A. Molina Donoso
Los Algarrobos 2986, Iquique
A. R. Kampf, B. P. Nash, M. Dini, A. A. Molina Donoso; Magnesiokoritnigite, Mg(AsO3OH)·H2O, from the Torrecillas mine, Iquique Province, Chile: the Mg-analogue of koritnigite. Mineralogical Magazine 2013;; 77 (8): 3081–3092. doi: https://doi.org/10.1180/minmag.2013.077.8.03
The new mineral magnesiokoritnigite (IMA 2013-049), ideally Mg(AsO3OH)·H2O, was found at the Torrecillas mine, Salar Grande, Iquique Province, Chile, where it occurs as a secondary alteration phase in association with anhydrite, chudobaite, halite, lavendulan, quartz and scorodite. Crystals of magnesiokoritnigite are colourless to pale-pink, thin to thick laths up to 2 mm long. Laths are elongated on [001], flattened on {010} and exhibit the forms {010}, {110}, {11İ0}, {101}, {031} and {03İ1}. The crystals also occur in dense deep-pink intergrowths. Crystals are transparent with a vitreous lustre. The mineral has a white streak, Mohs hardness of ~3, brittle tenacity, conchoidal fracture and one perfect cleavage on {101}. The measured and calculated densities are 2.95(3) and 2.935 g cm−3, respectively. Optically, magnesiokoritnigite is biaxial (+) with α = 1.579(1), β = 1.586(1) and γ = 1.620(1) (measured in white light). The measured 2V is 50(2)° and the calculated 2V is 50°. Dispersion is r < v, medium. The optical orientation is Y ≈ b; Z ^ c = 36° in obtuse β (note pseudomonoclinic symmetry). The mineral is non-pleochroic. The empirical formula, determined from electron-microprobe analyses, is (Mg0.94Cu0.03Mn0.02Ca0.01)Σ1.00As0.96O5H3.19. Magnesiokoritnigite is triclinic, P1İ, with a = 7.8702(7), b = 15.8081(6), c = 6.6389(14) Å, α = 90.814(6), β = 96.193(6), γ = 90.094(7)°, V = 821.06(19) Å3 and Z = 8. The eight strongest X-ray powder diffraction lines are [dobs Å(I)(hkl)]: 7.96(100)(020), 4.80(54)(101), 3.791(85)(2İ10,210,1İ3İ1,1İ31), 3.242(56)(01İ2,2İ2İ1,012), 3.157(92)(21İ1,230,230), 3.021(61)(14İ1,141,22İ1,221), 2.798(41)(03İ2,032) and 1.908(43)(multiple). The structure, refined to R1 = 5.74% for 2360 Fo > 4σF reflections, shows magnesiokoritnigite to be isostructural with koritnigite and cobaltkoritnigite.
Na(As,Sb)43+
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Specify LMI System at the Command Line - MATLAB & Simulink - MathWorks United Kingdom
Specify LMI System
Initializing the LMI System
Specifying the LMI Variables
Specifying Individual LMIs
This tutorial example shows how to specify LMI systems at the command line using the LMI Lab tools.
Consider a stable transfer function,
G\left(s\right)=C{\left(sI-A\right)}^{-1}B.
Suppose that G has four inputs, four outputs, and six states. Consider also a set of input/output scaling matrices D with block-diagonal structure given by:
D=\left(\begin{array}{cccc}{d}_{1}& 0& 0& 0\\ 0& {d}_{1}& 0& 0\\ 0& 0& {d}_{2}& {d}_{3}\\ 0& 0& {d}_{4}& {d}_{5}\end{array}\right).
The following problem arises in the robust stability analysis of systems with time-varying uncertainty [4]. Find, if any, a scaling D with the specified structure, such that the largest gain across frequency of
DG\left(s\right){D}^{-1}
This problem has a simple LMI formulation: There exists an adequate scaling D if the following feasibility problem has solutions. Find two symmetric matrices
X\in {R}_{6×6}
S={D}^{T}D\in {R}_{4×4}
\left(\begin{array}{cc}{A}^{T}X+XA+{C}^{T}SC& XB\\ {B}^{T}X& -S\end{array}\right)<0,
X>0,
S>1.
You can use the LMI Editor to specify the LMI problem described by these expressions, as shown in Specify LMIs with the LMI Editor GUI. Alternatively, define it at the command line using lmivar and lmiterm, as follows.
For this example, use the following values for A, B, and C.
A = [ -0.8715 0.5202 0.7474 1.0778 -0.9686 0.1005;
-0.5577 -1.0843 1.8912 0.2523 1.0641 -0.0345;
-0.2615 -1.7539 -1.5452 -0.2143 0.0923 -2.4192;
0.6087 -1.0741 0.1306 -2.5575 2.3213 0.2388;
-0.7169 0.3582 -1.4195 1.7043 -2.6530 -1.4276;
-1.2944 -0.6752 1.6983 1.6764 -0.3646 -1.7730 ];
B = [ 0 0.8998 -0.2130 0.9835;
0 -0.3001 0 -0.2977;
-1.0322 0 -1.0431 1.1437;
0 -0.3451 -0.2701 -0.5316;
-0.4189 1.0128 -0.4381 0;
0 0 -0.4087 0];
C = [ 0 2.0034 0 1.0289 0.1554 0.7135;
0.9707 0.9510 0.7059 1.4580 -1.2371 0.3174;
0 0 1.4158 0.0475 -2.1935 0.4136;
-0.4383 0.6489 -1.6045 1.7463 -0.3334 -0.5771];
Define the LMI variables X and S, and then specify the terms of each LMI.
X = lmivar(1,[6 1]);
S = lmivar(1,[2 0;2 1]);
% 1st LMI
lmiterm([1 1 1 X],1,A,'s');
lmiterm([1 1 1 S],C',C);
lmiterm([1 1 2 X],1,B);
lmiterm([1 2 2 S],-1,1);
% 2nd LMI
lmiterm([-2 1 1 X],1,1);
% 3rd LMI
lmiterm([-3 1 1 S],1,1);
lmiterm([3 1 1 0],1);
LMISYS = getlmis;
The lmivar commands define the two matrix variables, X and S. The lmiterm commands describe the terms in each LMI. getlmis returns the internal representation LMISYS of this LMI problem.
For more details on how to use these commands, see:
For more information about how lmivar updates the internal representation of the LMI problem, see How lmivar and lmiterm Manage LMI Representation.
The description of an LMI system should begin with setlmis and end with getlmis. The function setlmis initializes the LMI system description. When specifying a new system, type
To add on to an existing LMI system with internal representation LMIS0, type
setlmis(LMIS0)
The matrix variables are declared one at a time with lmivar and are characterized by their structure. To facilitate the specification of this structure, the LMI Lab offers two predefined structure types along with the means to describe more general structures:
Symmetric block diagonal structure. This corresponds to matrix variables of the form
X=\left(\begin{array}{cccc}{D}_{1}& 0& \dots & 0\\ 0& {D}_{2}& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & 0& {D}_{r}\end{array}\right)
where each diagonal block Dj is square and is either zero, a full symmetric matrix, or a scalar matrix
Dj= d × I, d ∊ R
This type encompasses ordinary symmetric matrices (single block) and scalar variables (one block of size one).
Rectangular structure. This corresponds to arbitrary rectangular matrices without any particular structure.
General structures. This third type is used to describe more sophisticated structures and/or correlations between the matrix variables. The principle is as follows: each entry of X is specified independently as either 0, xn, or –xn where xn denotes the n-th decision variable in the problem. For details on how to use Type 3, see Structured Matrix Variables as well as the lmivar entry in the reference pages.
In Specify LMI System, the matrix variables X and S are of Type 1. Indeed, both are symmetric and S inherits the block-diagonal structure of D. Specifically, S is of the form
S=\left(\begin{array}{cccc}{s}_{1}& 0& 0& 0\\ 0& {s}_{1}& 0& 0\\ 0& 0& {s}_{2}& {s}_{3}\\ 0& 0& {s}_{3}& {s}_{4}\end{array}\right).
Initialize the description and declare these two matrix variables.
lmivar(1,[6 1]); % X
lmivar(1,[2 0;2 1]); % S
In both lmivar commands, the first input specifies the structure type and the second input contains additional information about the structure of the variable:
For a matrix variable X of Type 1, this second input is a matrix with two columns and as many rows as diagonal blocks in X. The first column lists the sizes of the diagonal blocks and the second column specifies their nature with the following convention:
1: full symmetric block
0: scalar block
–1: zero block
In the second command, for instance,[2 0;2 1] means that S has two diagonal blocks, the first one being a 2-by-2 scalar block and the second one a 2-by-2 full block.
For matrix variables of Type 2, the second input of lmivar is a two-entry vector listing the row and column dimensions of the variable. For instance, a 3-by-5 rectangular matrix variable would be defined by
lmivar(2,[3 5])
For convenience, lmivar also returns a “tag” that identifies the matrix variable for subsequent reference. For instance, X and S in Specify LMI System could be defined by
The identifiers X and S are integers corresponding to the ranking of X and S in the list of matrix variables (in the order of declaration). Here their values would be X=1 and S=2. Note that these identifiers still point to X and S after deletion or instantiation of some of the matrix variables. Finally, lmivar can also return the total number of decision variables allocated so far as well as the entry-wise dependence of the matrix variable on these decision variables (see the lmivar entry in the reference pages for more details).
After declaring the matrix variables with lmivar, we are left with specifying the term content of each LMI. Recall that LMI terms fall into three categories:
The constant terms, i.e., fixed matrices like I in the left side of the LMI S > I.
The variable terms, i.e., terms involving a matrix variable. For instance, ATX and CTSC in the expression:
\left(\begin{array}{cc}{A}^{T}X+XA+{C}^{T}SC& XB\\ {B}^{T}X& -S\end{array}\right)<0
Variable terms are of the form PXQ where X is a variable and P, Q are given matrices called the left and right coefficients, respectively.
The outer factors.
When describing the term content of an LMI, specify only the terms in the blocks on or above the diagonal. The inner factors being symmetric, this is sufficient to specify the entire LMI. Specifying all blocks results in the duplication of off-diagonal terms, hence in the creation of a different LMI. Alternatively, you can describe the blocks on or below the diagonal.
LMI terms are specified one at a time with lmiterm. For instance, the LMI
\left(\begin{array}{cc}{A}^{T}X+XA+{C}^{T}SC& XB\\ {B}^{T}X& -S\end{array}\right)<0
lmiterm([1 1 1 1],1,A,'s');
lmiterm([1 1 1 2],C',C);
lmiterm([1 1 2 1],1,B);
lmiterm([1 2 2 2],-1,1);
These commands successively declare the terms ATX + XA, CTSC, XB, and –S. In each command, the first argument is a four-entry vector listing the term characteristics as follows:
The first entry indicates to which LMI the term belongs. The value m means “left side of the m-th LMI,” and −m means “right side of the m-th LMI.”
The second and third entries identify the block to which the term belongs. For instance, the vector [1 1 2 1] indicates that the term is attached to the (1, 2) block.
The last entry indicates which matrix variable is involved in the term. This entry is 0 for constant terms, k for terms involving the k-th matrix variable Xk, and −k for terms involving
{X}_{k}^{T}
(here X and S are first and second variables in the order of declaration).
Finally, the second and third arguments of lmiterm contain the numerical data (values of the constant term, outer factor, or matrix coefficients P and Q for variable terms PXQ or PXTQ). These arguments must refer to existing MATLAB® variables and be real-valued. See Complex-Valued LMIs for the specification of LMIs with complex-valued coefficients.
Some shorthand is provided to simplify term specification. First, blocks are zero by default. Second, in diagonal blocks the extra argument 's' allows you to specify the conjugated expression AXB + BTXTAT with a single lmiterm command. For instance, the first command specifies ATX + XA as the “symmetrization” of XA. Finally, scalar values are allowed as shorthand for scalar matrices, i.e., matrices of the form αI with α scalar. Thus, a constant term of the form αI can be specified as the “scalar” α. This also applies to the coefficients P and Q of variable terms. The dimensions of scalar matrices are inferred from the context and set to 1 by default. For instance, the third LMI S > I in Specify Matrix Variable Structures is described by
lmiterm([-3 1 1 2],1,1); % 1*S*1 = S
lmiterm([3 1 1 0],1); % 1*I = I
Recall that by convention S is considered as the right side of the inequality, which justifies the –3 in the first command.
Finally, to improve readability it is often convenient to attach an identifier (tag) to each LMI and matrix variable. The variable identifiers are returned by lmivar and the LMI identifiers are set by the function newlmi. These identifiers can be used in lmiterm commands to refer to a given LMI or matrix variable. For the LMI system of Specify LMI System, this would look like:
BRL = newlmi;
lmiterm([BRL 1 1 X],1,A,'s');
lmiterm([BRL 1 1 S],C',C);
lmiterm([BRL 1 2 X],1,B);
lmiterm([BRL 2 2 S],-1,1);
Xpos = newlmi;
lmiterm([-Xpos 1 1 X],1,1);
Slmi = newlmi;
lmiterm([-Slmi 1 1 S],1,1);
lmiterm([Slmi 1 1 0],1);
When the LMI system is completely specified, get the internal representation of the problem.
This returns the internal representation LMISYS of this LMI system. This MATLAB description of the problem can be forwarded to other LMI-Lab functions for subsequent processing. The command getlmis must be used only once and after declaring all matrix variables and LMI terms.
Here the identifiers X and S point to the variables X and S while the tags BRL, Xpos, and Slmi point to the first, second, and third LMI, respectively. Note that –Xpos refers to the right-hand side of the second LMI. Similarly, –X would indicate transposition of the variable X.
lmiterm | lmivar | setlmis | getlmis
How lmivar and lmiterm Manage LMI Representation
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APR to EAR Calculator
Calculate the Effective Annual Rate (EAR) using the Annual Percentage Rate (APR). You can choose the compounding period to be either monthly, quarterly, or semiannually.
Compounding period in months
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HodgeStar - Maple Help
Home : Support : Online Help : Mathematics : DifferentialGeometry : Tensor : HodgeStar
Tensor[HodgeStar] - apply the Hodge star operator to a differential form
HodgeStar(g, omega)
omega - a differential form
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{Tensor}\right):
M
g
M
\mathrm{DGsetup}\left([\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}],\mathrm{M1}\right):
g≔\mathrm{evalDG}\left(\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}+\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}+\mathrm{dx5}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx5}\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dx2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}
The standard basis
\mathrm{dx1},\mathrm{dx2}, ...,\mathrm{dx5}
g
and therefore the Hodge star is easily computed.
\mathrm{HodgeStar}\left(g,\mathrm{dx1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{dx2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}
\mathrm{HodgeStar}\left(g,\mathrm{dx2}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}
\mathrm{HodgeStar}\left(g,\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}
\mathrm{HodgeStar}\left(g,\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}
\mathrm{HodgeStar}\left(g,\left(\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx5}}
To show the dependence of the Hodge star upon the metric, we consider a general metric
g
on a 2-dimensional manifold.
\mathrm{DGsetup}\left([x,y],\mathrm{M2}\right):
g≔\mathrm{evalDG}\left(a\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+b\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)+c\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}
\mathrm{HodgeStar}\left(g,\mathrm{dx}\right)
\sqrt{\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{+}\sqrt{\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}
\mathrm{HodgeStar}\left(g,\mathrm{dy}\right)
\textcolor[rgb]{0,0,1}{-}\sqrt{\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{-}\sqrt{\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}
f≔\mathrm{HodgeStar}\left(g,\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{:=}\sqrt{\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}}
\mathrm{HodgeStar}\left(g,f\right)
\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}
The Laplacian of a function with respect to a metric
g
can be calculated using the exterior derivative operation and the Hodge star operator.
To illustrate this result, we use the Euclidean metric in polar coordinates
\left(r,\mathrm{ϑ}\right)
\mathrm{DGsetup}\left([r,\mathrm{\theta }],\mathrm{M3}\right):
g≔\mathrm{evalDG}\left(\mathrm{dr}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dr}+{r}^{2}\mathrm{dtheta}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dtheta}\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{dr}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dr}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dtheta}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dtheta}}
To simplify the definition of the Laplacian, we define the Hodge operator with
g
\mathrm{Hodge}≔f↦\left(\mathrm{HodgeStar}\left(g,f\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}0<r\right)
\textcolor[rgb]{0,0,1}{\mathrm{Hodge}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{\mathrm{DifferentialGeometry:-Tensor}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{HodgeStar}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{assuming}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{r}
To display the Laplacian of
\mathrm{φ}
in compact form we invoke the PDEtools[declare] command.
\mathrm{PDEtools}[\mathrm{declare}]\left(\mathrm{\phi }\left(r,\mathrm{\theta }\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{φ}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{θ}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{will now be displayed as}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{φ}}
Here is the formula for the Laplacian in terms of HodgeStar and ExteriorDerivative. Recall that @ is the composition of functions.
\mathrm{\Delta }≔\left(\mathrm{Hodge}@\mathrm{ExteriorDerivative}@\mathrm{Hodge}@\mathrm{ExteriorDerivative}\right)\left(\mathrm{\phi }\left(r,\mathrm{\theta }\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{Δ}}\textcolor[rgb]{0,0,1}{:=}\frac{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{φ}}}_{\textcolor[rgb]{0,0,1}{r}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{φ}}}_{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{φ}}}_{\textcolor[rgb]{0,0,1}{\mathrm{θ}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{θ}}}}{{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}}
The HodgeStar program also works in the more general context of a vector bundle
E→M
\mathrm{DGsetup}\left([x,y],[u,v,w],E\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: E}}
g≔\mathrm{evalDG}\left(\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}+\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\mathrm{dw}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw}\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{du}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{du}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dv}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dv}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dw}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dw}}
\mathrm{HodgeStar}\left(g,\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}-3\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw}+2\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw}\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{du}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dv}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dw}}
The HodgeStar operation can also be performed using an indefinite metric. The keyword argument detmetric = -1 must be used when the metric has negative determinant.
\mathrm{DGsetup}\left([\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}],\mathrm{M5}\right):
g≔\mathrm{evalDG}\left(\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}-\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dx2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}
\mathrm{HodgeStar}\left(g,\mathrm{dx1},\mathrm{detmetric}=-1\right)
\textcolor[rgb]{0,0,1}{\mathrm{dx2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx4}}
\mathrm{HodgeStar}\left(g,\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4},\mathrm{detmetric}=-1\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{dx1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx2}}
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Abstract: As the global production expanding, vertical specialization is thinner and thinner; most countries are just one “node” of product value chain, and the trade of intermediate products is the goods transferring from one node to another. We may wonder whether there is a law in the connections between the “nodes”, whether the connections are subject to political system, religious culture or geographic distance. This paper adopts a new accounting system— value added accounting system—to measure the trade flows between countries, and the ratio of value added export (VAX ratio) is used to measure the degree of processing trade. Setting 60 major economies trading with China as samples, it turns out that processing trade tend to occur between countries closed or with regional trade agreements. With time going, the influence of distance on processing trade is becoming more and more significant, naming localization.
Keywords: Processing Trade, Trade in Value Added Accounting System, Value Added to Export Ratio
{x}_{n1}
{y}_{n1}
{x}_{n1}+{x}_{n2}+\cdots +{x}_{nn}+{y}_{n1}+{y}_{n2}+\cdots +{y}_{nn}={X}_{n}
{a}_{ij}=\frac{{x}_{ij}}{{X}_{j}}
i=1,2,\cdots ,n.
\left\{\begin{array}{l}{a}_{11}{X}_{1}+{a}_{12}{X}_{2}+\cdots +{a}_{1n}{X}_{n}+{y}_{11}+{y}_{12}+\cdots +{y}_{1n}={X}_{1}\\ {a}_{21}{X}_{1}+{a}_{22}{X}_{2}+\cdots +{a}_{2n}{X}_{n}+{y}_{21}+{y}_{22}+\cdots +{y}_{2n}={X}_{2}\\ \cdots \\ {a}_{n1}{X}_{1}+{a}_{n2}{X}_{2}+\cdots +{a}_{nn}{X}_{n}+{y}_{n1}+{y}_{n2}+\cdots +{y}_{nn}={X}_{n}\end{array}
\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& & {a}_{1n}\\ {a}_{21}& {a}_{22}& \vdots & {a}_{2n}\\ & \cdots & & \cdots \\ {a}_{n1}& {a}_{n2}& \vdots & {a}_{nn}\end{array}\right]\times \left[\begin{array}{c}{X}_{1}\\ {X}_{2}\\ \cdots \\ {X}_{n}\end{array}\right]+\left[\begin{array}{c}{Y}_{1}\\ {Y}_{2}\\ \cdots \\ {Y}_{n}\end{array}\right]=\left[\begin{array}{c}{X}_{1}\\ {X}_{2}\\ \cdots \\ {X}_{n}\end{array}\right]
A=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& & {a}_{1n}\\ {a}_{21}& {a}_{22}& ⋮& {a}_{2n}\\ & \cdots & & \cdots \\ {a}_{n1}& {a}_{n2}& ⋮& {a}_{nn}\end{array}\right]
{X}_{n\times 1}={A}_{n\times n}{X}_{n\times 1}+{Y}_{n\times 1}
{X}_{n\times 1}={\left(I-{A}_{n\times n}\right)}^{-1}{Y}_{n\times 1}
{\left(I-A\right)}^{-1}
{x}_{1n}+{x}_{2n}+\cdots +{x}_{nn}+{V}_{n}={X}_{n}
\left\{\begin{array}{l}{a}_{11}{X}_{1}+{a}_{21}{X}_{1}+\cdots +{a}_{n1}{X}_{1}+{V}_{1}={X}_{1}\\ {a}_{12}{X}_{2}+{a}_{22}{X}_{2}+\cdots +{a}_{n2}{X}_{2}+{V}_{2}={X}_{2}\\ \cdots \\ {a}_{1n}{X}_{n}+{a}_{2n}{X}_{n}+\cdots +{a}_{nn}{X}_{n}+{V}_{n}={X}_{n}\end{array}
\underset{i=1}{\overset{n}{\sum }}{a}_{ij}{X}_{j}+{V}_{j}={X}_{j}
{V}_{j}=\left(1-\underset{i=1}{\overset{n}{\sum }}{a}_{ij}\right){X}_{j}
\underset{i=1}{\overset{n}{\sum }}{a}_{ij}
1-\underset{i=1}{\overset{n}{\sum }}{a}_{ij}
{a}_{cj}=\underset{i=1}{\overset{n}{\sum }}{a}_{ij}.
{\stackrel{^}{A}}_{C}=\left[\begin{array}{cccc}\underset{i=1}{\overset{n}{\sum }}{a}_{i1}& & & \\ & \underset{i=1}{\overset{n}{\sum }}{a}_{i2}& & \\ & & \ddots & \\ & & & \underset{i=1}{\overset{n}{\sum }}{a}_{in}\end{array}\right]=\left[\begin{array}{cccc}{a}_{c1}& & & \\ & {a}_{c2}& & \\ & & \ddots & \\ & & & {a}_{cn}\end{array}\right]
V=\left[\begin{array}{cccc}1-{a}_{c1}& & & \\ & 1-{a}_{c2}& & \\ & & \ddots & \\ & & & 1-{a}_{cn}\end{array}\right]\times \left[\begin{array}{c}{X}_{1}\\ {X}_{2}\\ \vdots \\ {X}_{n}\end{array}\right]=\left(I-{\stackrel{^}{A}}_{c}\right)X
VA{X}_{ij}\equiv \frac{value-addexpor{t}_{ij}}{grossexpor{t}_{ij}}\equiv \frac{netabsorptio{n}_{ij}+indirectexpor{t}_{ij}}{absorptio{n}_{ij}+reflectio{n}_{ij}+redirectio{n}_{ij}}
\Delta \text{VAX}
\begin{array}{c}\mathrm{log}{Y}_{ijt}={\alpha }_{t}^{Y}+{\beta }_{1t}^{Y}*\mathrm{log}di{s}_{ij}+{\beta }_{2t}^{Y}\text{*}\mathrm{log}gdp_o+{\beta }_{3t}^{Y}*gdp_d\\ +{\beta }_{4t}^{Y}\text{*}\mathrm{log}pop_o+{\beta }_{5t}^{Y}*\mathrm{log}pop_d+{\beta }_{6t}^{Y}*conti{g}_{ij}\\ +{\beta }_{6t}^{Y}*comlan{g}_{ij}+{\xi }_{ijt}\end{array}
{Y}_{ijt}
di{s}_{ij}
gdp_o
gdp_d
pop_o
pop_d
contig
comlang
\mathrm{log}EX
\mathrm{log}VA
{\beta }_{1t}^{EX}
{\beta }_{1t}^{VA}
Cite this paper: Xiao, S. (2018) Production Globalization or Localization?. Modern Economy, 9, 572-585. doi: 10.4236/me.2018.93037.
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Mired - Wikipedia
Unit of reciprocal color temperature
Close up of the Planckian locus in the CIE 1960 color space, with the isotherms in mireds. Note the even spacing of the isotherms when using the reciprocal temperature scale. The even spacing of the isotherms on the locus implies that the mired scale is a better measure of perceptual color difference than the temperature scale.
Close up of the Planckian locus in the CIE 1960 color space, with the isotherms in kelvins. The range of isothermal color temperatures, as in the previous diagram, is from 1000 K (1000 MK−1) to 10000 K (100 MK−1).
Contracted from the term micro reciprocal degree, the mired is a unit of measurement used to express color temperature. It is given by the formula[1]
{\displaystyle M={\frac {1\,000\,000\,{\text{K}}}{T}},}
where M is the mired numerical value desired, T is the color temperature, and "K" denotes kelvins (thereby forcing T to be expressed in terms of kelvins).
For convenience, decamireds are sometimes used, with each decamired equaling ten mireds. The SI term for this unit is the reciprocal megakelvin (MK−1), shortened to mirek, but this term has not gained traction.[2]
The use of the term mired dates back to Irwin G. Priest's observation in 1932 that the just noticeable difference between two illuminants is based on the difference of the reciprocals of their temperatures, rather than the difference in the temperatures themselves.[3]
Mired difference can be quickly approximated with such a nomogram
A blue sky, which has a color temperature T of about 25000 K, has a mired value of M = 40 mireds, while a standard electronic photography flash, having a color temperature T of 5000 K, has a mired value of M = 200 mireds.
In photography, mireds are used to indicate the color temperature shift provided by a filter or gel for a given film and light source. For instance, to use daylight film (5700 K) to take a photograph under a tungsten light source (3200 K) without introducing a color cast, one would need a corrective filter or gel providing a mired shift
{\displaystyle {\frac {10^{6}}{5700}}-{\frac {10^{6}}{3200}}\approx -137~{\text{MK}}^{-1}.}
This corresponds to a color temperature blue (CTB) filter.[4]
Color gels with negative mired values appear green or blue, while those with positive values appear amber or red.
^ more precisely:
{\textstyle T^{-1}/\mathrm {mired} =10^{6}/(T/\mathrm {K} )}
^ Ohta, Noboru; Robertson, Alan R. (2005). Colorimetry: Fundamentals and Applications. Wiley. p. 84. ISBN 0-470-09472-9.
^ Priest, Irwin G. (February 1932). "A proposed scale for use in specifying the chromaticity of incandescent illuminants and various phases of daylight" (abstract). JOSA. 23 (2): 41–45. doi:10.1364/JOSA.23.000041.
^ Brown, Blain (2002). Cinematography: Theory and Practice : Imagemaking for Cinematographers. Focal Press. p. 172. ISBN 0-240-80500-3.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mired&oldid=1070989106"
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Weighted -Statistical Convergence for Sequences of Positive Linear Operators
S. A. Mohiuddine, Abdullah Alotaibi, Bipan Hazarika, "Weighted -Statistical Convergence for Sequences of Positive Linear Operators", The Scientific World Journal, vol. 2014, Article ID 437863, 8 pages, 2014. https://doi.org/10.1155/2014/437863
S. A. Mohiuddine,1 Abdullah Alotaibi ,1 and Bipan Hazarika 2
2Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791 112, India
We introduce the notion of weighted -statistical convergence of a sequence, where represents the nonnegative regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted -statistical convergence. Further, we give a rate of weighted -statistical convergence and apply the classical Bernstein polynomial to construct an illustrative example in support of our result.
1. Background, Notations, and Preliminaries
We begin this paper by recalling the definition of natural (or asymptotic) density as follows. Suppose that and . Then is called the natural density of provided that the limit exists, where represents the number of elements in the enclosed set.
The term “statistical convergence” was first presented by Fast [1] which is a generalization of the concept of ordinary convergence. Actually, a root of the notion of statistical convergence can be detected by Zygmund [2] (also, see [3]), where he used the term “almost convergence” which turned out to be equivalent to the concept of statistical convergence. The notion of Fast was further investigated by Schoenberg [4], Šalát [5], Fridy [6], and Conner [7].
The following notion is due to Fast [1]. A sequence is said to be statistically convergent to if for every , where Equivalently, In symbol, we will write . We remark that every convergent sequence is statistically convergent but not conversely.
Let and be two sequence spaces and let be an infinite matrix. If for each in the series converges for each and the sequence belongs to , then we say that matrix maps into . By the symbol we denote the set of all matrices which map into .
A matrix (or a matrix map ) is called regular if , where the symbol denotes the spaces of all convergent sequences and for all . The well-known Silverman-Toeplitz theorem (see [8]) asserts that is regular if and only if(i) for each ;(ii) ;(iii) .
Kolk [9] extended the definition of statistical convergence with the help of nonnegative regular matrix calling it -statistical convergence. The definition of -statistical convergence is given by Kolk as follows. For any nonnegative regular matrix , we say that a sequence is -statistically convergent to provided that for every we have
In 2009, the concept of weighted statistical convergence was defined and studied by Karakaya and Chishti [10] and further modified by Mursaleen et al. [11] in 2012. In 2013, Belen and Mohiuddine [12] presented a generalization of this notion through de la Vallée-Poussin mean. Quite recently, Esi [13] defined and studied the notion statistical summability through de la Vallée-Poussin mean in probabilistic normed spaces.
Let be a sequence of nonnegative numbers such that and Let We say that the sequence is -summable to if .
The lower and upper weighted densities of are defined by respectively. We say that has weighted density, denoted by , if the limits of both of the above densities exist and are equal; that is, one writes
The sequence is said to be weighted statistically convergent (or ) to if, for every , the set has weighted density zero; that is, In this case we write .
Remark 1. If for all , then -summable is reduced to -summable (or Cesàro summable) and weighted statistical convergence is reduced to statistical convergence.
On the other hand, let us recall that is the space of all functions continuous on . We know that is a Banach space with norm Suppose that is a linear operator from into . It is clear that if implies , then the linear operator is positive on . We denote the value of at a point by . The classical Korovkin approximation theorem states the following [14].
Theorem 2. Let be a sequence of positive linear operators from into . Then, for all if and only if where and .
Many mathematicians extended the Korovkin-type approximation theorems by using various test functions in several setups, including Banach spaces, abstract Banach lattices, function spaces, and Banach algebras. Firstly, Gadjiev and Orhan [15] established classical Korovkin theorem through statistical convergence and display an interesting example in support of our result. Recently, Korovkin-type theorems have been obtained by Mohiuddine [16] for almost convergence. Korovkin-type theorems were also obtained in [17] for -statistical convergence. The authors of [18] established these types of approximation theorem in weighted spaces, where , through -summability which is stronger than ordinary convergence. For these types of approximation theorems and related concepts, one can be referred to [19–27] and references therein.
2. Korovkin-Type Theorems by Weighted -Statistical Convergence
Kolk [9] introduced the notion of -statistical convergence by considering nonnegative regular matrix instead of Cesáro matrix in the definition of statistical convergence due to Fast. Inspired from the work of Kolk, we introduce the notion of weighted -statistical convergence of a sequence and then we establish some Korovkin-type theorems by using this notion.
Definition 3. Let be a nonnegative regular matrix. A sequence of real or complex numbers is said to be weighted -statistically convergent, denoted by -convergent, to if for every where In symbol, we will write .
Remark 4. One has the following.(i)If we take , where denotes the identity matrix, then weighted -statistical convergence of a sequence is reduced to ordinary convergence.(ii)If we take , where denotes the Cesáro matrix of order one, then weighted -statistical convergence of a sequence reduces to weighted statistical convergence.(iii)If we take and for all , then weighted -statistical convergence of a sequence reduces to statistical convergence.
Note that convergent sequence implies weighted -statistically convergent to the same value but the converse is not true in general. For example, take and for all and define a sequence by where . Then this sequence is statistically convergent to 0 but not convergent; in this case, weighted -statistical convergence of a sequence coincides with statistical convergence.
Theorem 5. Let be a nonnegative regular matrix. Consider a sequence of positive linear operators from into itself. Then, for all bounded on whole real line, if and only if
Proof. Equation (20) directly follows from (19) because each of belongs to . Consider a function . Then there is a constant such that for all . Therefore, Let be given. By hypothesis there is a such that Solving (21) and (22) and then substituting , one obtains Equation (23) can also be written as Operating to (24) since is linear and monotone, one obtains Note that is fixed, so is constant number. Thus, we obtain from (25) that The term “ ” in (26) can also be written as Now substituting the value of in (26), we get that We can rewrite the term “ ” in (28) as follows: Equation (28) with the above value of becomes Therefore, where . Taking supremum over , one obtains or where Hence, For a given , choose such that , and we will define the following sets: It is easy to see that Thus, for each , we obtain from (35) that Taking limit in (38) and also (20) gives that This yields that for all .
We also obtain the following Korovkin-type theorem for weighted statistical convergence by writing Cesáro matrix instead of nonnegative regular matrix in Theorem 5.
Theorem 6. Consider a sequence of positive linear operators from into itself. Then, for all if and only if
Proof. Following the proof of Theorem 5, one obtains and so
Equations (42)–(44) give that
Remark 7. If we replace nonnegative regular matrix by Cesáro matrix and choose for all , in Theorem 5, then we obtain Theorem 1 due to Gadjiev and Orhan [15].
Remark 8. By Theorem 2 of [10], we have that if a sequence is weighted statistically convergent to , then it is strongly -summable to provided that is bounded; that is, there exists a constant such that for all . We write for the set of all sequences which are strongly -summable to .
Theorem 9. Let be a sequence of positive linear operators which satisfies (43)-(44) of Theorem 6 and the following condition holds: Then, for any .
Proof. It follows from (49) that , for some constant and for all . Hence, for , one obtains Right hand side of (51) is constant, so is bounded. Since (49) implies (42), by Theorem 6 we get that By Remark 8, (51) and (52) together give the desired result.
3. Rate of Weighted -Statistical Convergence
First we define the rate of weighted -statistically convergent sequence as follows.
Definition 10. Let be a nonnegative regular matrix and let be a positive nonincreasing sequence. Then, a sequence is weighted -statistically convergent to with the rate of if for each where In symbol, we will write
We will prove the following auxiliary result by using the above definition.
Lemma 11. Let be a nonnegative regular matrix. Suppose that and are two positive nonincreasing sequences. Let and be two sequences such that and . Then,(i) ,(ii) ,(iii) , for any scalar ,where .
Proof. (i) Suppose that Given , define It is easy to see that This yields that holds for all . Since , (59) gives that Taking limit in (60) together with (56), we obtain Thus, Similarly, we can prove (ii) and (iii).
Recall that the modulus of continuity of in is defined by It is well known that
Theorem 12. Let be a nonnegative regular matrix. If the sequence of positive linear operators satisfies the conditions(i) ,(ii) with and ,where and are two positive nonincreasing sequences, then for all , where .
Proof. Equation (27) can be reformed into the following form: Choosing , one obtains where . For a given , we will define the following sets: It follows from (67) that holds for . Since , we obtain from (69) that Taking limit in (70) together with Lemma 11 and our hypotheses (i) and (ii), one obtains This yields
4. Example and the Concluding Remark
The operators given by where are the fundamental Bernstein polynomials defined by for any , any , and any , are called Bernstein operators and were first introduced in [28]. Let the sequence be defined by with , where is a sequence defined by That is, . Let and consider a nonnegative regular matrix . Then, Since is weighted statistically convergent to 0 but not convergent. It is not difficult to see that and the sequence satisfies conditions (20). This yields that
On the other hand, one obtains , since , and hence It follows that does not satisfy the Korovkin theorem, since and hence is not convergent. Finally, we conclude that Theorem 5 is stronger than Theorem 2.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (351/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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Bobmeyerite, a new mineral from Tiger, Arizona, USA, structurally related to cerchiaraite and ashburtonite | Mineralogical Magazine | GeoScienceWorld
A. R. Kampf *
J. J. Pluth;
J. J. Pluth
Department of Geophysical Sciences, Center for Advanced Radiation Sources, University of Chicago
5734 South Ellis Avenue, Chicago, Illinois 60637-1434
Y.-S. Chen;
Center for Advanced Radiation Sources, University of Chicago
A. C. Roberts;
R. M. Housley
A. R. Kampf, J. J. Pluth, Y.-S. Chen, A. C. Roberts, R. M. Housley; Bobmeyerite, a new mineral from Tiger, Arizona, USA, structurally related to cerchiaraite and ashburtonite. Mineralogical Magazine 2013;; 77 (1): 81–91. doi: https://doi.org/10.1180/minmag.2013.077.1.08
Bobmeyerite, Pb4(Al3Cu)(Si4O12)(S0.5Si0.5O4)(OH)7Cl(H2O)3, is a new mineral from the Mammoth–Saint Anthony mine, Tiger, Pinal County, Arizona, USA. It occurs in an oxidation zone assemblage attributed to progressive alteration and crystallization in a closed system. Other minerals in this assemblage include atacamite, caledonite, cerussite, connellite, diaboleite, fluorite, georgerobinsonite, hematite, leadhillite, matlockite, murdochite, phosgenite, pinalite, quartz, wulfenite and yedlinite. Bobmeyerite occurs as colourless to white or cream-coloured needles, up to 300 μm in length, that taper to sharp points. The streak is white and the lustre is adamantine, dull or silky. Bobmeyerite is not fluorescent. The hardness could not be determined, the tenacity is brittle and no cleavage was observed. The calculated density is 4.381 g cm−3. Bobmeyerite is biaxial (–) with α ≈ β = 1.759(2), γ = 1.756(2) (white light), it is not pleochroic; the orientation is X = c; Y or Z = a or b. Electron-microprobe analyses provided the empirical formula Pb3.80Ca0.04Al3.04
Cu0.962+
Cr0.133+
Si4.40S0.58O24.43Cl1.05F0.52H11.83. Bobmeyerite is orthorhombic (pseudotetragonal), Pnnm with unit-cell parameters a = 13.969(9), b = 14.243(10), c = 5.893(4) Å, V = 1172.5(1.4) Å3 and Z = 2. The nine strongest lines in the X-ray powder diffraction pattern, listed as [dobs(Å)(I)(hkl)], are as follows: 10.051(35)(110); 5.474(54)(011,101); 5.011(35)(220); 4.333(43)(121,211); 3.545(34)(040,400); 3.278(77)(330,231,321); 2.9656(88)(141,002,411); 2.5485(93)(051,222,501); 1.873(39)(multiple). Bobmeyerite has the same structural framework as cerchiaraite and ashburtonite. In the structure, which refined to R1 = 0.079 for 1057 reflections with F > 4σF, SiO4 tetrahedra share corners to form four-membered Si4O12 rings centred on the c axis. The rings are linked by chains of edge-sharing AlO6 octahedra running parallel to [001]. The framework thereby created contains large channels, running parallel to [001]. The Cl site is centred on the c axis alternating along [001] with the Si4O12 rings. Two non-equivalent Pb atoms are positioned around the periphery of the channels. Both are eleven-coordinate, bonding to the Cl atom on the c axis, to eight O atoms in the framework and to two O (H2O) sites in the channel. The Pb atoms are off-centre in these coordinations, as is typical of Pb2+ with stereo-active lone-electron pairs. A (S,Si,Cr)O4 group is presumed to be disordered in the channel. The name honours Robert (Bob) Owen Meyer, one of the discoverers of the new mineral.
Mammoth-Saint Anthony Mine
Tiger Arizona
bobmeyerite
GEORGEROBINSONITE, Pb4(CrO4)2(OH)2FCl, A NEW CHROMATE MINERAL FROM THE MAMMOTH – ST. ANTHONY MINE, TIGER, PINAL COUNTY, ARIZONA: DESCRIPTION AND CRYSTAL STRUCTURE
Plumbophyllite, a new species from the Blue Bell claims near Baker, San Bernardino County, California
Lead-tellurium oxysalts from Otto Mountain near Baker, California: VIII. Fuettererite, Pb3Cu62+Te6+O6(OH)7Cl5, a new mineral with double spangolite-type sheets
Rowleyite,
[Na(NH4,K)9Cl4][V25+,4+(P,As)O8]6·n[H2O,Na,NH4,K,Cl]
, a new mineral with a microporous framework structure
Lazaraskeite, Cu(C2H3O3)2, the first organic mineral containing glycolate, from the Santa Catalina Mountains, Tucson, Arizona, U.S.A.
Bluebellite and mojaveite, two new minerals from the central Mojave Desert, California, USA
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What is the effect of pressure,humidity and density on the velocity of sound in air - Physics - Propagation of Sound Waves - 12420199 | Meritnation.com
What is the effect of pressure,humidity
and density on the velocity of sound in air
c=\sqrt{\frac{\gamma P}{\rho }}\phantom{\rule{0ex}{0ex}}speed is directly proportional to the root of pressure.\phantom{\rule{0ex}{0ex}}inversely proportional to root of density.\phantom{\rule{0ex}{0ex}}as humidity increases the gamma increases hence the velocity increases
Kumaran Jash answered this
Ansaf Sabu answered this
A number of factors (altitude/pressure, temperature, and humidity) influence air density. A higher altitude, low-pressure area, higher temperature and high humidity all have one result: they lower the density of the air. And as a result of that: a reduction in aircraft and engine performance.
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Section 53.2 (0BXX): Curves and function fields—The Stacks project
Section 53.2: Curves and function fields (cite)
53.2 Curves and function fields
In this section we elaborate on the results of Varieties, Section 33.4 in the case of curves.
Lemma 53.2.1. Let $k$ be a field. Let $X$ be a curve and $Y$ a proper variety. Let $U \subset X$ be a nonempty open and let $f : U \to Y$ be a morphism. If $x \in X$ is a closed point such that $\mathcal{O}_{X, x}$ is a discrete valuation ring, then there exist an open $U \subset U' \subset X$ containing $x$ and a morphism of varieties $f' : U' \to Y$ extending $f$.
Proof. This is a special case of Morphisms, Lemma 29.42.5. $\square$
Lemma 53.2.2. Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety. The set of rational maps from $X$ to $Y$ is the same as the set of morphisms $X \to Y$.
Proof. A rational map from $X$ to $Y$ can be extended to a morphism $X \to Y$ by Lemma 53.2.1 as every local ring is a discrete valuation ring (for example by Varieties, Lemma 33.43.8). Conversely, if two morphisms $f,g: X \to Y$ are equivalent as rational maps, then $f = g$ by Morphisms, Lemma 29.7.10. $\square$
Lemma 53.2.3. Let $k$ be a field. Let $f : X \to Y$ be a nonconstant morphism of curves over $k$. If $Y$ is normal, then $f$ is flat.
Proof. Pick $x \in X$ mapping to $y \in Y$. Then $\mathcal{O}_{Y, y}$ is either a field or a discrete valuation ring (Varieties, Lemma 33.43.8). Since $f$ is nonconstant it is dominant (as it must map the generic point of $X$ to the generic point of $Y$). This implies that $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective (Morphisms, Lemma 29.8.6). Hence $\mathcal{O}_{X, x}$ is torsion free as a $\mathcal{O}_{Y, y}$-module and therefore $\mathcal{O}_{X, x}$ is flat as a $\mathcal{O}_{Y, y}$-module by More on Algebra, Lemma 15.22.10. $\square$
Lemma 53.2.4. Let $k$ be a field. Let $f : X \to Y$ be a morphism of schemes over $k$. Assume
$Y$ is separated over $k$,
$X$ is proper of dimension $\leq 1$ over $k$,
$f(Z)$ has at least two points for every irreducible component $Z \subset X$ of dimension $1$.
Then $f$ is finite.
Proof. The morphism $f$ is proper by Morphisms, Lemma 29.41.7. Thus $f(X)$ is closed and images of closed points are closed. Let $y \in Y$ be the image of a closed point in $X$. Then $f^{-1}(\{ y\} )$ is a closed subset of $X$ not containing any of the generic points of irreducible components of dimension $1$ by condition (3). It follows that $f^{-1}(\{ y\} )$ is finite. Hence $f$ is finite over an open neighbourhood of $y$ by More on Morphisms, Lemma 37.43.2 (if $Y$ is Noetherian, then you can use the easier Cohomology of Schemes, Lemma 30.21.2). Since we've seen above that there are enough of these points $y$, the proof is complete. $\square$
Lemma 53.2.5. Let $k$ be a field. Let $X \to Y$ be a morphism of varieties with $Y$ proper and $X$ a curve. There exists a factorization $X \to \overline{X} \to Y$ where $X \to \overline{X}$ is an open immersion and $\overline{X}$ is a projective curve.
Proof. This is clear from Lemma 53.2.1 and Varieties, Lemma 33.43.6. $\square$
Here is the main theorem of this section. We will say a morphism $f : X \to Y$ of varieties is constant if the image $f(X)$ consists of a single point $y$ of $Y$. If this happens then $y$ is a closed point of $Y$ (since the image of a closed point of $X$ will be a closed point of $Y$).
Theorem 53.2.6. Let $k$ be a field. The following categories are canonically equivalent
The category of finitely generated field extensions $K/k$ of transcendence degree $1$.
The category of curves and dominant rational maps.
The category of normal projective curves and nonconstant morphisms.
The category of nonsingular projective curves and nonconstant morphisms.
The category of regular projective curves and nonconstant morphisms.
The category of normal proper curves and nonconstant morphisms.
Proof. The equivalence between categories (1) and (2) is the restriction of the equivalence of Varieties, Theorem 33.4.1. Namely, a variety is a curve if and only if its function field has transcendence degree $1$, see for example Varieties, Lemma 33.20.3.
The categories in (3), (4), (5), and (6) are the same. First of all, the terms “regular” and “nonsingular” are synonyms, see Properties, Definition 28.9.1. Being normal and regular are the same thing for Noetherian $1$-dimensional schemes (Properties, Lemmas 28.9.4 and 28.12.6). See Varieties, Lemma 33.43.8 for the case of curves. Thus (3) is the same as (5). Finally, (6) is the same as (3) by Varieties, Lemma 33.43.4.
If $f : X \to Y$ is a nonconstant morphism of nonsingular projective curves, then $f$ sends the generic point $\eta $ of $X$ to the generic point $\xi $ of $Y$. Hence we obtain a morphism $k(Y) = \mathcal{O}_{Y, \xi } \to \mathcal{O}_{X, \eta } = k(X)$ in the category (1). If two morphisms $f,g: X \to Y$ gives the same morphism $k(Y) \to k(X)$, then by the equivalence between (1) and (2), $f$ and $g$ are equivalent as rational maps, so $f=g$ by Lemma 53.2.2. Conversely, suppose that we have a map $k(Y) \to k(X)$ in the category (1). Then we obtain a morphism $U \to Y$ for some nonempty open $U \subset X$. By Lemma 53.2.1 this extends to all of $X$ and we obtain a morphism in the category (5). Thus we see that there is a fully faithful functor (5)$\to $(1).
To finish the proof we have to show that every $K/k$ in (1) is the function field of a normal projective curve. We already know that $K = k(X)$ for some curve $X$. After replacing $X$ by its normalization (which is a variety birational to $X$) we may assume $X$ is normal (Varieties, Lemma 33.27.1). Then we choose $X \to \overline{X}$ with $\overline{X} \setminus X = \{ x_1, \ldots , x_ n\} $ as in Varieties, Lemma 33.43.6. Since $X$ is normal and since each of the local rings $\mathcal{O}_{\overline{X}, x_ i}$ is normal we conclude that $\overline{X}$ is a normal projective curve as desired. (Remark: We can also first compactify using Varieties, Lemma 33.43.5 and then normalize using Varieties, Lemma 33.27.1. Doing it this way we avoid using the somewhat tricky Morphisms, Lemma 29.53.16.) $\square$
Definition 53.2.7. Let $k$ be a field. Let $X$ be a curve. A nonsingular projective model of $X$ is a pair $(Y, \varphi )$ where $Y$ is a nonsingular projective curve and $\varphi : k(X) \to k(Y)$ is an isomorphism of function fields.
A nonsingular projective model is determined up to unique isomorphism by Theorem 53.2.6. Thus we often say “the nonsingular projective model”. We usually drop $\varphi $ from the notation. Warning: it needn't be the case that $Y$ is smooth over $k$ but Lemma 53.2.8 shows this can only happen in positive characteristic.
Lemma 53.2.8. Let $k$ be a field. Let $X$ be a curve and let $Y$ be the nonsingular projective model of $X$. If $k$ is perfect, then $Y$ is a smooth projective curve.
Proof. See Varieties, Lemma 33.43.8 for example. $\square$
Lemma 53.2.9. Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$. For a field extension $K/k$ denote $Y_ K$ a nonsingular projective model of $(X_ K)_{red}$.
If $X$ is proper, then $Y_ K$ is the normalization of $X_ K$.
There exists $K/k$ finite purely inseparable such that $Y_ K$ is smooth.
Whenever $Y_ K$ is smooth1 we have $H^0(Y_ K, \mathcal{O}_{Y_ K}) = K$.
\[ \xymatrix{ \Omega & K' \ar[l] \\ K \ar[u] & k \ar[l] \ar[u] } \]
of fields such that $Y_ K$ and $Y_{K'}$ are smooth, then $Y_\Omega = (Y_ K)_\Omega = (Y_{K'})_\Omega $.
Proof. Let $X'$ be a nonsingular projective model of $X$. Then $X'$ and $X$ have isomorphic nonempty open subschemes. In particular $X'$ is geometrically irreducible as $X$ is (some details omitted). Thus we may assume that $X$ is projective.
Assume $X$ is proper. Then $X_ K$ is proper and hence the normalization $(X_ K)^\nu $ is proper as a scheme finite over a proper scheme (Varieties, Lemma 33.27.1 and Morphisms, Lemmas 29.44.11 and 29.41.4). On the other hand, $X_ K$ is irreducible as $X$ is geometrically irreducible. Hence $X_ K^\nu $ is proper, normal, irreducible, and birational to $(X_ K)_{red}$. This proves (1) because a proper curve is projective (Varieties, Lemma 33.43.4).
Proof of (2). As $X$ is proper and we have (1), we can apply Varieties, Lemma 33.27.4 to find $K/k$ finite purely inseparable such that $Y_ K$ is geometrically normal. Then $Y_ K$ is geometrically regular as normal and regular are the same for curves (Properties, Lemma 28.12.6). Then $Y$ is a smooth variety by Varieties, Lemma 33.12.6.
If $Y_ K$ is geometrically reduced, then $Y_ K$ is geometrically integral (Varieties, Lemma 33.9.2) and we see that $H^0(Y_ K, \mathcal{O}_{Y_ K}) = K$ by Varieties, Lemma 33.26.2. This proves (3) because a smooth variety is geometrically reduced (even geometrically regular, see Varieties, Lemma 33.12.6).
If $Y_ K$ is smooth, then for every extension $\Omega /K$ the base change $(Y_ K)_\Omega $ is smooth over $\Omega $ (Morphisms, Lemma 29.34.5). Hence it is clear that $Y_\Omega = (Y_ K)_\Omega $. This proves (4). $\square$
[1] Or even geometrically reduced.
Comment #2057 by David Hansen on June 07, 2016 at 20:08
In Lemma 46.2.5, you should probably say that X is a curve!
Comment #4169 by Che Shen on April 14, 2019 at 18:40
In the third paragraph of theorem 0BY1, we conclude that "there is a fully faithful functor (5)→(1)". We only proved it's full, but did not prove it's faithful.
To prove faithfulness, we can add the following before "Conversely, suppose ...":
f,g: X \to Y
gives the same field homomorphism
K(X)\to K(Y)
. By 01RK,
f,g
are dominant morphisms. So by 0BXN,
f
g
are equal as ratinal maps. So they agree on an open subset
U \subset X
Y
is separated,
f = g
as morphisms.
(For the last sentence above, I think there should be a lemma in stacks project saying "If
f,g: X \to Y
agree on an open subset
U \subset X
Y
is separated, then
f=g
", but I did not find it.)
OK, thanks and thanks for sending the edits by email. The corresponding changes are here.
Comment #5048 by Raphael Rouquier on April 21, 2020 at 19:17
Maybe an irreducibility assumption is needed for the curves in Theorem 0BY1?
Apologies, by definition a curve has the required irreducibility property.
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Home : Support : Online Help : Programming : Logic : Boolean : verify : sign
verify for sign differences
verify(expr1, expr2, sign)
The verify(expr1, expr2, sign) calling sequence returns true if any of the following is true:
The arguments are equal.
Multiplying pairs of multiplicands by
-1
results in equal arguments.
{a}^{n}
{b}^{n}
n
a=b
a=-b
This verification is more memory and computationally intensive than checking whether the normal of the difference is zero, but it ensures that the two arguments have similar forms.
\mathrm{verify}\left(-\frac{x}{a-b},\frac{x}{b-a},\mathrm{sign}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
p≔\frac{3{\left(x-y\right)}^{2}\left(x-2y-xy\right)\left(x-y-4x\right)\left(x{y}^{2}+2x-2y\right)}{y-3x}
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\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}{x}\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}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\right)\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}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\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}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}
q≔\frac{3{\left(y-x\right)}^{2}\left(xy+2y-x\right)\left(4x-y-x\right)\left(2y-x{y}^{2}-2x\right)}{3x-y}
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}\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}{x}\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}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\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}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)
\mathrm{verify}\left(p,q,\mathrm{sign}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Erratum to “Compact Operators for Almost Conservative and Strongly Conservative Matrices”
S. A. Mohiuddine, M. Mursaleen, A. Alotaibi, "Erratum to “Compact Operators for Almost Conservative and Strongly Conservative Matrices”", Abstract and Applied Analysis, vol. 2014, Article ID 327852, 2 pages, 2014. https://doi.org/10.1155/2014/327852
S. A. Mohiuddine,1 M. Mursaleen,2 and A. Alotaibi1
We redefine the space and state the results of [1] in this light.
Let be a semigroup of positive regular matrices .
A bounded sequence is said to be -almost convergent to the value if and only if , as uniformly in , where and which is -transform of a sequence (see Mursaleen [2]). The number is called the generalized limit of , and we write . We write
Using the idea of -almost convergence, we define the following.
An infinite matrix is said to be -almost conservative if for all , and we denote it by . An infinite matrix is said to be -strongly conservative if for all , and we denote it by .
Now, we restate Theorem 11 and Theorem 15 of [1] as follows, respectively.
Theorem 11. Let be a -almost conservative matrix. Then, one has where .
Proof. It follows on the same lines as of Theorem 11 [1] by only replacing by .
Theorem 15. Let be a normal positive regular matrix. Let be an infinite matrix. Then, one has the following.(i)If , then (ii)If , then where for all .(iii)If , then where is the composition of the matrices and ; that is, .
Proof. It follows on the same lines as Theorem 15 of [1] by only replacing by .
Remark 1 (see [2]). If consists of the iterates of the operator defined on by , where is an injection of the set of positive integers into itself having no finite orbits, then -invariant mean is reduced to the -mean and -almost convergence is reduced to -convergence. In this case, our results are reduced to the results of [3].
S. A. Mohiuddine, M. Mursaleen, and A. Alotaibi, “Compact operators for almost conservative and strongly conservative matrices,” Abstract and Applied Analysis, vol. 2014, Article ID 567317, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
M. Mursaleen, “On
\mathcal{A}
-invariant mean and
\mathcal{A}
-almost convergence,” Analysis Mathematica, vol. 37, no. 3, pp. 173–180, 2011. View at: Publisher Site | Google Scholar | MathSciNet
M. Mursaleen and A. K. Noman, “On σ-conservative matrices and compact operators on the space Vσ,” Applied Mathematics Letters: An International Journal of Rapid Publication, vol. 24, no. 9, pp. 1554–1560, 2011. View at: Publisher Site | Google Scholar | MathSciNet
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The Application of an Aerodynamic Shroud for Axial Ventilation Fans | J. Fluids Eng. | ASME Digital Collection
D. R. Neal,
e-mail: nealdoug@msu.edu
J. F. Foss
Neal, D. R., and Foss, J. F. (January 2, 2007). "The Application of an Aerodynamic Shroud for Axial Ventilation Fans." ASME. J. Fluids Eng. June 2007; 129(6): 764–772. https://doi.org/10.1115/1.2734206
An experimental investigation of an aerodynamic shroud applied to an axial ventilation fan system is reported. The aerodynamic shroud consists of a pressurized plenum and Coanda attachment surface, which also serves as the shroud for the fan. This combination delivers a curved surface wall jet of high momentum air into the tip region of the fan and subsequently into the downstream diffuser region. Simultaneous improvements of performance and efficiency were found for a specific fan geometry with an aerodynamic shroud system when compared with a standard production fan (no shroud) system. Overall, the addition of the aerodynamic shroud was able to increase the system flowrate by 34% while simultaneously improving the efficiency by 13%. A higher efficiency condition
(+17%)
was also found that resulted in a somewhat lower improvement in flow rate
(+23%)
. These results clearly show that the best blade design for the aerodynamic shroud system is different than the best blade design for a system that does not include the aerodynamic shroud. Particle image velocimetry measurements made at the exit plane of the system’s diffuser provide insight into the mechanistic basis for the performance measurements.
aerodynamics, jets, ventilation, fans, blades, flow visualisation
Blades, Diffusers, Fans, Flow (Dynamics), Pressure, Ventilation, Design
Effect of Air Temperature on Performance of Growing-Finishing Swine
Heat Stress as it Affects Animal Production
J. Anim. Sci. (Savoy, Ill.)
Performance of Neonatal and Newly-Weaned Pigs as Affected by Temperature and Diet
Thermal Environmental Effect on Growing-Finishing Swine: Part I—Growth, Feed Intake and Heat Production
Overhults
Energy Use in Tunnel Ventilated Broiler Houses
Proceedings Livestock Environment IV
, 4th Int Symposium, Coventry, England, July 6–9, pp.
Greenhouses, Advanced Technology Protected Horticulture
Evaluating Greenhouse Mechanical Ventilation System Performance
,” Agriculture and Biological Engineering Fact Sheet I-42, The Pennsylvania State University, University Park, PA.
, 2002, M.S. thesis, Michigan State University, East Lansing, MI.
An Aerodynamic Shroud for Automotive Cooling Fans
, 2000, Directory of Agricultural Products With Certified Ratings, Air Movement and Control Asociation, Inc., Publication No.
Agriculture Ventilation Fans, Performances and Efficiencies
,” Paper No. UILU ENG 99-7001.
The Automotive Cooling Fan Research and Development Facility
Proceedings of the 3rd SAE Vehicle Thermal Management Systems Conference
, Indianapolis, IN.
A Moment-of-Momentum Flux Mass Air Flow Measurement Device
On the Accuracy and Reliability of PIV Measurements
Proceedings of the 7th International Symposium on Application of Laser Techniques to Flow Measurements
, Lisbon, Portugal, July 11–14.
An experimental Investigation of Incompressible Flow in Conical Diffusers
Modern Diffuser Design
Experiments on Conical Diffusers
,” Aeronautical Research Council Technical Report, R & M 2751.
, 1985, Laboratory Methods of Testing Fans for Rating, Air Movement and Control Association, Inc., Publication No. 210-85.
Experimental and Numerical Investigation of the Flow Field in the Tip Region of an Axial Ventilation Fan
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Use Python Script Automation to fetch images from NASA
In the previous tutorial, we learned how to download images to our local machine using NASA’s API. That was cool, right?
We are going to take this one step further and learn how to send those images as attachments in email. Then we will automate that process using a task scheduler. Please note that to download the images from NASA’s API, we will use the same code from the the previous tutorial, so read through those basics before continuing here.
How to send today’s astronomy picture by email
How to send a picture from an input date
How to send a random astronomy picture
How to send astronomy images automatically
In this tutorial, we use Google’s Gmail service and the SMTP (Simple Mail Transfer Protocol) to send emails. But Google may not allow us to do so for security reasons. So, first of all we need to change our security settings on our Google account.
You can change these settings by logging into your Google account and then navigating to the security settings.
1. Disable the 2 step verification
Our goal is to download Astronomy Picture of the Day (APOD) using the NASA API like we did in the previous tutorial. But instead of displaying it on the notebook, we are will send it to an email address. After downloading the image to our local machine, we are going to use SMTP module to send it to our email.
nasapy: We use the nasapy library to access the information provided by NASA API.
datetime: To get a date in a specific format, we’ll use a datetime library.
urllib: The image we get from NASA API will be in a form of a URL.
smtplib: We use the smtp module to create a client session object that can be used to send mail to any internet machine that listens to smtp.
MIMEMultipart: Our mail content will have simple text and attachments.
MIMEText: MIMEText is for text/html/plain text. If the message is in text form, we use this.
MIMEBase: MIMEBase is provided primarily as a convenient base class for more specific MIME-aware subclasses.
encoders: Here we import the encoders module to encode the payload or attachments while sending emails.
Step 2: Create an object of NASA class
Here we create an object called nasa. We are using the nasapy library and our registered api key. We’ll use this object to call a method of NASA class to get the information on astronomical images.
k = "523p5hPYHGzafYGLCkqa54kKMTV2vbP0XcPxkcLm"
Step 3: Get date in required format
Now to get an image for a specific date, we need to pass a date in a specific format. The required format for the date is YYYY-MM-DD. To get today’s date, we are going to use datetime.today() function.
We need to convert this tuple into a string using the function strftime(%Y-%m-%d), which convert our dates into the right format.
#Get today's date in YYYY-MM-DD format:
d = datetime.today().strftime('%Y-%m-%d')
Step 4: Get information from NASA’s API
Now that we have today’s date in the required format, we can place a request for today’s astronomical image. Here we are going to use picture_of_the_day() method to get the image data.
Note that our data is stored in apod variable.
Step 5: File name for image files
Now, we have to maintain a certain kind of name structure to save the images. So, here we will use the following structure in order: the date of the image, the title of the image, the extension of the image. We also replace the colons and spaces with an underscore.
title = d + "_" + apod["title"].replace(" ","_").replace(":","_") + ".jpg"
Now that we have enough information to download the files, we download just the image files. Also, if the hd image is not available, we will access an sd image.
if apod['media_type'] == 'image':
if 'hdurl' in apod.keys():
urllib.request.urlretrieve(url = apod["hdurl"] , filename = title)
urllib.request.urlretrieve(url = apod["url"] , filename = title)
Step 7: Email addresses
Now that we have downloaded the images, it’s time to send that image via email. There are two addresses used here: from_addr and to_addr. You can add more email addresses in to_addr separated by comma.
from_addr = "sppratik22@gmail.com"
to_addr = ["shuklapratik22@gmail.com"]
Step 8: Creating an object
In the email, we are going to send the text description and the image. Since there are multiple parts in this image, we are creating an instance of class MIMEMultipart(). This allows us to add content in multiple formats.
Step 9: Subject of the email
Now we are going to send the email with a subject. The subject here will be the file name or title. You can modify it as you would like it.
message['Subject'] = apod["title"]
Step 10: Main body of the email
In the body section of our email, we write a simple text followed by the explanation of the attached image. After writing the body, we are going to attach it to the message object using MIMEText. Notice that our text will be in the Plain text format.
body = "Hello, this email is generated using python by Pratik Shukla!\n\n\n" + apod["explanation"]
Step 11: Open the attachment
Next, we open the file that we want to attach. Here we are going to open the file in binary mode.
attachment = open(filename, "rb")
Step 12: Create object of MIMEBase
Here we’re creating an object of class MIMEBase called p. A MIME attachment with the content type application/octet-stream is a binary file.
Step 13: Add header with filename
This determines how we want to display the content. Here we are going to send the image as attachment.
p.add_header('Content-Disposition', 'attachment', filename= title)
Step 14: Attach the file to the message
Now we are going to attach the file to the main message. Here we are also encoding the attachment.
Step 15: Send the email(s)
Now that we have our content ready, we send the email using the smtp server. Here, I’m using port number 587 and Gmail. We also use the Transport Layer Security (TLS) while sending the email.
Next, we give our email address and password to log in to the gmail account. Next, we convert the whole message to a string. Lastly, we send the email using sendmail() function. After sending the email, we’re going to terminate the session using s.quit().
You should see a message: “Email sent successfully!”
s.login(from_addr, password = "Test_Password#011")
Step 16: Image not available
If the image file is not available to download, no email will send. We will simply print a message that says “Image is not available”.
print("Sorry Image not available!")
print("Email not sent!")
print(apod["url"])
You can download and run the above code file online on Google Collaboratory
Now if we want to send an image for a specific date, we just have to get the date from the user. The rest of the steps are exactly the same.
nasa = nasapy.Nasa(key="523p5hPYHGzafYGLCkqa54kKMTV2vbP0XcPxkcLm")
d = input("Enter date in YYYY-MM-DD format : ")
title = d+"_"+apod["title"].replace(" ","_").replace(":","_")+".jpg"
#Downloading file only if it's image:
body = "Hello, This is an automatic email by Pratik Shukla.\n\n\n" + apod["explanation"]
print("Image not available!")
print("Please enter some other date!!")
You can run and download the above code online on Google Collaboratory.
In the last tutorial, we use the date_range() function to send 5 images. Now, we will pick a random image from those downloaded images and send it via email.
Now here we are going to modify the code a bit. Instead of sending the explanation of the image, we are going to send a random astronomy quote from a csv file.
Note that the majority of the code remains the same as the previous tutorial. So here we’ll only discuss the code parts that vary from the previous code.
os: To get the files from a specific directory.
pandas: To read the csv file.
random: To randomly select the image and quote.
Step 2: Selecting the astronomy quote
We are going to read the csv file in a variable called data. Data has all the quotes with their index numbers. Now we are going to select a number randomly between 1 and len(data).
Now we’ll find the quote at that index value. This gives us a variable for the quote. Remember that we are going to use this quote in the body section of the email.
data = pd.read_csv("Astronomy_Quotes - Sheet1.csv")
Quote = data["Quote"][quote_index]
Step 3: Getting the image location
Our images are stored in the AAA_Images_1 folder. We get all the files available there in a list called astro_images. We can access the images like list elements.
image_dir = "AAA_Images_1"
Step 4: Select random image
Now we are going to select an image randomly. The astro_images variable has a list of all the images, so we are going to randomly select
nth
image from the list of images available. We’ll use the title/file name as subject of the email later in this tutorial.
Here we’re going to make a few variations to our previous code.
a) In the body, we attach the random quote instead of explanation of the image.
b) We look into the directory to find the image for attachment.
Take a look at the complete code below!
#Image Directory"
from_addr = 'sppratik22@gmail.com'
to_addr = ['shuklapratik22@gmail.com']
body = "Hello, This is an automatic email generated using python by Pratik Shukla.\n\n" +"Here's an astronomy quote for you:\n\n" + Quote
Output of code above
You can download and run the above code file online on Google Collaboratory.
Above learned how to send images by email using Python scripts. But what if we can automate that process? Let’s see how to automate those scripts using a task scheduler in Python.
To accomplish this, you won’t need any extra download any software. We’ll use the built-in task scheduler provided by the Windows operating system.
We’re going to use the last part of our previous tutorial to send the email. But this time, our code script will run on it’s own!
Step 1: Prepare a Python script
To run the code, we need to have the python script in .py format. If you’re using a jupyter notebook, you can easily download it from the File section.
Step 2: Opening the task scheduler
Go to the task scheduler. byfollowing these steps:
Control Panel ⇒ Administrative Tools ⇒ Task Scheduler
When you double click on the task scheduler icon, you’ll see the following screen.
Now we’ll create a task. To create a new task click on the “Create Task” option as shown below.
Now when you click on the “Create Task” option, a window will open requesting more details. First, you have to assign a name value for your task. We can write anything we want here. After that we can also add some description for our task. This is useful when we have to schedule many tasks. After that, click on the “OK” button.
Now go to the “Actions” tab. We click on the “New” button to create a new action. Action basically defines the task that we want to schedule.
After clicking on the New button, a window will open. Here in the Program/Script text box we have to add the full path where our python.exe file is located. Then we’ll add the filename that we want to run(.py). Next, we’ll add the path that leads to the python file.
Step 5: Setting time for action
Then go to the Triggers tab. Here we have to mention that the action we just created will be triggered at what time. Specify the time and date at which we want to trigger our action.
When it’s time our file will execute and if it requires any input from the user, you have to feed it with that input.
Notice that our task scheduler executed the program at 11:00 PM and we got the mail at the same time.
You can download all the executable python files here.
Congratulations of completing this fun activity with NASA, Task Schedulers, and Python. I hope this activity taught you some new things you can do with Python and inspired you to keep learning.
If you have any doubts, questions, or thoughts regarding this article, feel free to contact me at shuklapratik22@gmail.com
If you want to continue learning abut APIs and want to take these skills to the next level, check out Educative’s course Learn REST and SOAP API Test Automation in Java. You will learn how to perform REST and SOAP API test automation and learn how to write APIs and integration tests from scratch.
Continue reading about Python and APIs
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Prometheus Unbound; a lyrical drama in four acts with other poems/Prometheus Unbound - Wikisource, the free online library
Prometheus Unbound; a lyrical drama in four acts with other poems/Prometheus Unbound
For other versions of this work, see Prometheus Unbound.
A Lyrical Drama in Four Acts
Prometheus Unbound is a four-act play by Percy Bysshe Shelley first published in 1820. It is inspired by Aeschylus's Prometheus Bound and concerns Prometheus' release from captivity. However, unlike Aeschylus' version, there is no reconciliation between Prometheus and Zeus in Shelley's narrative. Shelley's play is closet drama, meaning it was not intended to be produced on the stage. In the tradition of other poets in the Romantic Poetry, Shelley wrote for the imagination, intending his play's stage to reside in the imaginations of his readers.
Warning: template has been deprecated.— Excerpted from Prometheus Unbound on Wikipedia, the free encyclopedia.
26908Prometheus Unbound — A Lyrical Drama in Four ActsPercy Bysshe Shelley
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}}
Oceanides.
Spirits. Echoes. Fawns.
Retrieved from "https://en.wikisource.org/w/index.php?title=Prometheus_Unbound;_a_lyrical_drama_in_four_acts_with_other_poems/Prometheus_Unbound&oldid=11139967"
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Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n-D Cubical Grids - LRDE
Among the different flavors of well-composednesses on cubical grids, two of them, called respectively Digital Well-Composedness (DWCness) and Well-Composedness in the sens of Alexandrov (AWCness), are known to be equivalent in 2D and in 3D. The former means that a cubical set does not contain critical configurations when the latter means that the boundary of a cubical set is made of a disjoint union of discrete surfaces. In this paper, we prove that this equivalence holds i{\displaystyle n}
-D, which is of interest because today images are not only 2D or 3D but also 4D and beyond. The main benefit of this proof is that the topological properties available for AWC sets, mainly their separation properties, are also true for DWC sets, and the properties of DWC sets are also true for AWC sets: an Euler number locally computable, equivalent connectivities from a local or global point of view... This result is also true for gray-level images thanks to cross-section topology, which means that the sets of shapes of DWC gray-level images make a tree like the ones of AWC gray-level images.
title = {Equivalence between Digital Well-Composedness and
Well-Composedness in the Sense of {A}lexandrov on {$n$-D}
Cubical Grids},
doi = {10.1007/s10851-020-00988-z},
abstract = {Among the different flavors of well-composednesses on
cubical grids, two of them, called respectively Digital
Well-Composedness (DWCness) and Well-Composedness in the
sens of Alexandrov (AWCness), are known to be equivalent in
2D and in 3D. The former means that a cubical set does not
contain critical configurations when the latter means that
the boundary of a cubical set is made of a disjoint union
of discrete surfaces. In this paper, we prove that this
equivalence holds in $n$-D, which is of interest because
today images are not only 2D or 3D but also 4D and beyond.
The main benefit of this proof is that the topological
properties available for AWC sets, mainly their separation
properties, are also true for DWC sets, and the properties
of DWC sets are also true for AWC sets: an Euler number
locally computable, equivalent connectivities from a local
or global point of view... This result is also true for
gray-level images thanks to cross-section topology, which
means that the sets of shapes of DWC gray-level images make
a tree like the ones of AWC gray-level images. }
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Engineering Acoustics/Solution Methods for Electro-Mechanical Analogies - Wikibooks, open books for an open world
Engineering Acoustics/Solution Methods for Electro-Mechanical Analogies
After drawing the electro-mechanical analogy of a mechanical system, it is always safe to check the circuit. There are two methods to accomplish this:
1 Review of Circuit Solving Methods
2 Dot Method: (Valid only for planar network)
3 Low-Frequency Limits
4 Additional Resources for solving linear circuits
Review of Circuit Solving MethodsEdit
Kirchkoff's Voltage law
"The sum of the potential drops around a loop must equal zero."
{\displaystyle v_{1}+v_{2}+v_{3}+v_{4}=0\displaystyle }
Kirchkoff's Current Law
"The Sum of the currents at a node (junction of more than two elements) must be zero"
{\displaystyle -i_{1}+i_{2}+i_{3}-i_{4}=0\displaystyle }
Hints for solving circuits:
Remember that certain elements can be combined to simplify the circuit (the combination of like elements in series and parallel)
If solving a ciruit that involves steady-state sources, use impedances. Any circuit can eventually be combined into a single impedance using the following identities:
Impedances in series:
{\displaystyle Z_{\mathrm {eq} }=Z_{1}+Z_{2}+\,\cdots \,+Z_{n}.}
Impedances in parallel:
{\displaystyle {\frac {1}{Z_{\mathrm {eq} }}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\,\cdots \,+{\frac {1}{Z_{n}}}.}
Dot Method: (Valid only for planar network)Edit
This method helps obtain the dual analog (one analog is the dual of the other). The steps for the dot product are as follows:
1) Place one dot within each loop and one outside all the loops.
2) Connect the dots. Make sure that there is only one line through each element and that no lines cross more than one element.
3) Draw in each line that crosses an element its dual element, including the source.
4) The circuit obtained should have an equivalent behavior as the dual analog of the original electro-mechanical circuit.
The parallel RLC Circuit above is equivalent to a series RLC driven by an ideal current source
Low-Frequency LimitsEdit
This method looks at the behavior of the system for very large or very small values of the parameters and compares them with the expected behavior of the mechanical system. For instance, you can compare the mobility circuit behavior of a near-infinite inductance with the mechanical system behavior of a near-infinite stiffness spring.
Capacitor Short Circuit Open Circuit
Inductor Open Circuit Closed Circuit
Resistor Open Circuit Short Circuit
Additional Resources for solving linear circuitsEdit
Thomas & Rosa, "The Analysis and Design of Linear Circuits", Wiley, 2001
Hayt, Kemmerly & Durbin, "Engineering Circuit Analysis", 6th ed., McGraw Hill, 2002
Retrieved from "https://en.wikibooks.org/w/index.php?title=Engineering_Acoustics/Solution_Methods_for_Electro-Mechanical_Analogies&oldid=3246296"
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1 Description of 2D Periodic Hill Flow
1.2.1.1 Modeling and simulation issue
1.2.1.2 Physical issue
1.2.1.3 Experimental investigation
1.2.1.4 Test case study
Description of 2D Periodic Hill Flow
This contribution presents detailed LES,DNS and experimental data for the flow over smoothly contoured constrictions in a plane channel. This configuration represents a generic case of a flow separating from a curved surface with well-defined flow conditions which makes it especially suited as benchmark case for computing separated flows and testing RANS and Hybrid LES-RANS methods.
Flow separation from curved surfaces and subsequent reattachment is a flow phenomenon often appearing in engineering applications. Its prediction is complicated by several phenomena including irregular movement of the separation and reattachment lines in space and time, strong interactions with the outer flow, transition from a boundary layer type of flow to a separated shear layer with failure of the law-of-the wall and standard model assumptions for either attached flows or free shear layers. The improvement of flow prediction by Reynolds-averaged Navier-Stokes (RANS) simulation or large-eddy simulation (LES) in such flows is dependent on reliable data of generic test cases including the main features of the respective flow phenomena.
The flow over periodically arranged hills separates from a curved surface, recirculates in the leeward side of the hill and reattaches naturally at the flat channel bottom. The challenge of this case is to predict the point of separation from that curved surface which has a strong impact on the point of reattachment. The length and height of the main recirculation bubble varies with the Reynolds number. Furthermore, a tiny recirculation zone has been detected on the top of the hill at Re=10,595 and a minor one can be found for various Re at the windward foot of the hill. Fig. 2.1 depicts streamlines of the time-averaged flow and the turbulent kinetic energy with its maximum in the free shear layer right above the mean recirculation zone.
In order to motivate why this case is especially useful for basic investigations of the performance of turbulence models - not only subgrid-scale (SGS) models but also statistical models in the RANS context -, and other issues such as wall modeling, the history of how this test case was established is briefly sketched.
Almeida et al. (1993) experimentally investigated the flow behind two-dimensional model hills. Two different configurations were considered, i.e. the flow over a single hill and the flow over periodic hills. In 1995 these experiments were chosen as the basis of a test case at the 4th ERCOFTAC/IAHR workshop held in Karlsruhe in 1995 (Rodi et al. 1995). In order to select the least demanding configuration, the periodic arrangement without side walls was considered. However, the calculations carried out for this test case highlighted a number of serious problems and open questions, see Mellen et al. (2000). This concerns the unknown influence of the side walls and hence 3D effects in the experiment not taken into account in the 2D predictions. Since the aspect ratio in the experiment was small (almost square cross-section), it was expected that the spanwise confinement provoked spanwise variations. Furthermore, the predictions at the workshop (see Rodi et al. 1995) have cast doubt on the true periodicity of the flow achieved in the experiment, leading to the conclusion that simulations and experiment were not really comparable. As a test case for LES, another critical point is the high Reynolds number. Based on the hill height h and the mean centerline velocity the Reynolds number was Re = 60,000. Since the channel height in the experiment was large (L_y = 6.071 h ), the corresponding Reynolds number based on L_y is even about six times larger resulting in high computational costs for the configuration chosen. This problem is even greater for the single hill case for which suitable experimental data are available. Theunknown effect of the side walls remains for this case. Therefore, intended as a test case for LES,a new configuration was defined by Mellen et al. (2000), which leans on the experimental setup by Almeida et al. (1993) but avoids the problems discussed above.
The re-definition of the test case allows to meet a number of desiderata for this to be a good test case for LES studies (Mellen et al. 2000, Temmerman et al. 2003). The flow has to contain the key generic phenomena of interest, whilst being amenable to a simulation at economically tolerable cost. The new geometry is sketched in Fig. 2.2. The shape of the hill is taken from the study of Almeida et al. (1993). An accurate geometric specification is available in form of a polynomial ansatz (see Section "Test Cases Studies").
Fig. 2.2 Re-defined geometry of the test case
The new configuration differs from the original setup in five aspects:
First, compared with Almeida et al.'s configuration the distance between two hill crests in streamwise direction was doubled. This increased distance allows the flow to reattach naturally between successive hills, providing a significant post-reattachment-recovery region on the flat plate between the two hills prior to the re-acceleration over the next hill. From the numerical and modeling point of view this modification means that reattachment is now strongly influenced by wall modeling, SGS modeling, and grid arrangement issues. This aspect was not obvious in the original configuration since reattachment was dictated by the presence of the windward face of the consecutive hill.
Second, the original channel height was halved. This measure reduces the computational effort and allows a higher aspect ratio L_z / L_y.
Third, the side walls existing in the original experimental setup of Almeida et al. (1993) are removed and instead periodicity in the spanwise direction is assumed. Based on additional investigations by Mellen et al. (2000) a spanwise extension of the computational domain of L_z = 4.5 h was recommended for LES or hybrid LES-RANS predictions. For a detailed discussion on this issue we refer to the sections "Test Case Studies" and "Best Practice Advice".
Fourth, the Reynolds number was reduced and set to Re = 10,595 where
{\displaystyle Re=U_{B}h/\nu }
is based on the hill height h, the bulk velocity
{\displaystyle U_{B}}
taken at the crest of the first hill and the kinematic viscosity
{\displaystyle \nu }
of the fluid.
Fifth, the flow is assumed to be periodic in the streamwise direction which represents a simple way out of the dilemma of specifying appropriate inflow boundary conditions for LES or DNS. For that purpose the increase of the distance between two consecutive hills described above is beneficial too, since it enhances the streamwise decorrelation. Thus a well-defined flow state independent of inflow conditions is achieved. For a detailed discussion on this issue we also refer to the sections "Test Case Studies" and "Best Practice Advice".
As a consequence the resulting geometrically simple test case offers a number of important features challenging from the point of view of turbulence modeling and simulation. The pressure-induced separation takes place from a continuous curved surface and reattachment is observed at the flat plate (see Fig. 2.1). Hence these flow features are sensitive to numerical and modeling aspects. Therefore, this configuration was chosen as a test case at the subsequent ERCOFTAC/IAHR/COST Workshops on Refined Turbulence Modeling held in Darmstadt in 2001 (Jakirlic et al. 2001,2002) and in Poitiers in 2002 (Manceau et al. 2002, Manceau and Bonnet 2003), respectively and in the European ATAAC project. From these workshops, predicted results using a wide variety of RANS models as well as some LES results are available which can only be partially cited in the following section. Results from the ATAAC project are available and described in D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf.
The periodic hill flow test case has been studied so far pursuing two main objectives, either the modeling and simulation issue or the physical issue. Regarding the first, it is used as a benchmark case to investigate the ability of RANS and LES to resolve separation from a curved geometry. Furthermore, the flow is also an interesting case to study the physical mechanisms of separation on curved surfaces in more detail.
Modeling and simulation issue
Besides the workshops mentioned above (Jakirlic et al. 2001,2002, Manceau et al. 2002) a few more studies on the modeling and simulation issue should be reviewed first emphasizing on LES. Temmerman and Leschziner (2001)at Imperial College, London, investigated the periodic hill flow configuration set up by Mellen et al (2000) at Re =10,595 using LES. The emphasis was on the effectiveness of different combinations of subgrid-scale models and wall functions on relatively coarse grids. The accuracy was judged by reference to a wall-resolved simulation (lower wall only) on a grid with about 4.6 million nodes. It was demonstrated that even gross-flow parameters, such as the length of the separation bubble, are very sensitive to modeling approximations (SGS and wall models) and the grid quality. A similar investigation had been carried out by Mellen et al. (2000)at Karlsruhe University assessing the impact of different SGS models and the effect of grid refinement. In the succeeding study by Temmerman et al. (2003) the previous efforts of both groups (Imperial College/Karlsruhe) were combined and a comparative investigation was carried out applying three grids, six SGS models and eight practices of approximating the near-wall region. Again the coarse-grid simulations were judged by wall-resolved simulations carried out by both groups using the fine grid mentioned above and two independent codes, these simulations to be published later by Fröhlich et al (2005). The simulations on coarse grids highlighted the outstanding importance of an adequate streamwise resolution of the flow in the vicinity of the separation line. The main reason is the high sensitivity of the reattachment position to that of the separation. Furthermore, the near-wall treatment was found to have more influence on the quality of the results obtained on coarse grids than the subgrid-scale modeling.
To evaluate the performance of wall models for LES of attached flows the turbulent plane channel flow is the standard test case. That is due to its geometrical simplicity including two homogeneous directions which allow the application of periodic boundary conditions avoiding inflow and outflow boundary conditions completely. For the development and investigation of wall models for separated flows, the channel flow with periodic constrictions has nearly reached an equivalent status and importance. Similar to the plane channel the computational setup of the hill flow is simple owing to the possibility to apply periodic boundary conditions in 2 directions. However, for the hill case the flow separates from a curved surface and a large back-flow region emerges. Further downstream the flow reattaches and is accelerated at the windward side of the hill. Therefore, the separation and reattachment process can be studied in detail and wall models developed for attached and separated flows can be evaluated based on this flow.
As mentioned above, Temmerman et al. (2003) investigated the predictive accuracy of different wall models based on this case. It was clearly shown that the predictions provided by classical wall models developed for attached flows are not satisfactory if the wall-nearest computational point is located outside the viscous sublayer. This renders the case as a sensitive platform to develop and improve wall models, see e.g. Manhart et al. (2008) and Breuer et al. (2007).
In the meantime several studies used this test case to evaluate the performance not only of LES on coarse grids but also of different kinds of hybrid LES-RANS approaches including detached-eddy simulations (DES), see, e.g. Breuer et al. (2008), Jaffrezic and Breuer (2008) and Saric et al. (2007, 2010). The latter, for example, was a collaborative effort of the DFG-CNRS group "LES of Complex Flows" involving five different flow solvers used by five different groups in order to cover a broad range of numerical methods and implementations.
Concerning the physical issue, based on the wall-resolved LES by 2 groups (Imperial College/Karlsruhe) using about 4.6 million nodes and two independent codes, a comprehensive investigation of the periodic hill flow at Re = 10,595 was carried out by Fröhlich et al (2005). For reasons given above, especially because of the existence of a distinct post-reattachment-recovery region, the chosen configuration stands out from the crowd of investigations on flows over wavy-terrain geometries (e.g. Zilker et al. 1977 or Zilker and Hanratty 1979). Fröhlich et al (2005) carried out a detailed analysis including the evaluation of the budgets for all Reynolds stress components, anisotropy measures, and spectra. Note that these budgets are available in digital form from the Imperial College work. They can be obtained from L. Temmerman (lionel.temmerman@numeca.be) or M. Leschziner (mike.leschziner@imperial.ac.uk) or downloaded from the NASA LARC database[1] (http://turbmodels.larc.nasa.gov/Other_LES_Data/2dhill_periodic.html). The emphasis was on elucidating the turbulence mechanisms associated with separation, recirculation and acceleration. The statistical data were supported by investigations on the structural features of the flow. Based on these interesting observations such as the very high level of spanwise velocity fluctuations in the post-reattachment zone on the windward hill side were explained. This phenomenon revealed to be a result of the `splatting' of large-scale eddies originating from the shear layer, convecting downstream towards the windward slope and finally impinging on the wall.
In addition to the numerical investigations a physical experiment has been set up in the Laboratory for Hydromechanics of the Technische Universität München to study the flow experimentally and provide reliable reference data (Rapp, 2009). In the experimental setup periodicity is achieved by an array of 10 hills in streamwise direction and a large spanwise extent of the channel. The assumption of periodicity in the experiment was checked by the pressure drop between consecutive hill tops and PIV measurements. Experimental data in a 2D cross-section were provided by PIV measurements. For further details we refer to Rapp (2009).
In conclusion, the flow over periodically arranged hills described above is a very useful benchmark test case since
it represents well-defined boundary conditions,
can be computed at affordable costs and
nevertheless inherits all important features of a flow separating from a curved surface, reattachment and recovery of the reattached flow.
The test case study comprises new well-resolved DNS and LES obtained with curvilinear and cartesian-grid codes for the described test case geometry for various Reynolds numbers and a comparison with the recently obtained detailed experimental results (Rapp 2009). The main results for Reynolds numbers up to 10,595 are published in Breuer et al (2009). Results obtained with RANS and Hybrid LES-RANS methods in the ATAAC project for the Reynolds numbers 10,595 and 37,000 are available in D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf.
↑ managed by C. Rumsey
Contributed by: (*) Christoph Rapp, (**) Michael Breuer, (*) Michael Manhart, (*) Nikolaus Peller — (*) Technische Universitat Munchen, (**) Helmut-Schmidt University Hamburg
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Range-angle-height (Blake) chart - MATLAB blakechart - MathWorks Australia
blakechart
Display Vertical Coverage Diagram
Display Vertical Coverage Diagram Specifying Maximum Range and Height
Display Vertical Coverage Diagram of Sinc Pattern Antenna
vcpangles
Range-angle-height (Blake) chart
blakechart(vcp,vcpangles)
blakechart(vcp,vcpangles,rmax,hmax)
blakechart(___,Name,Value)
blakechart(vcp,vcpangles) creates a range-angle-height plot (also called a Blake chart) for a narrowband radar antenna. This chart shows the maximum radar range as a function of target elevation. In addition, the Blake chart displays lines of constant range and lines of constant height. The input consists of the vertical coverage pattern vcp and vertical coverage pattern angles vcpangles, both produced by radarvcd.
The range in the range-height-angle chart is the propagated range and the height is relative to the origin of the ray. It is assumed that the antenna height is less than 1000 ft (about 305 meters) above ground level. Normal atmospheric refraction is taken into account using the CRPL Exponential Reference Atmosphere Model. Scattering and ducting are assumed to be negligible.
blakechart(vcp,vcpangles,rmax,hmax), in addition, specifies the maximum range and height of the Blake chart. You can specify range and height units separately in the name-value arguments RangeUnit and HeightUnit.
blakechart(___,Name,Value) allows you to specify additional input parameters using name-value arguments. You can specify multiple name-value arguments in any order with any of the previous syntaxes.
Display the vertical coverage diagram of an antenna transmitting at 100 MHz and placed 20 meters above the ground. Set the free-space range to 100 km. Use default plotting parameters.
[vcp, vcpangles] = radarvcd(freq,rng_fs,ant_height);
blakechart(vcp, vcpangles);
Display the vertical coverage diagram of an antenna transmitting at 100 MHz and placed 20 meters above the ground. Set the free-space range to 100 km. Set the maximum plotting range to 300 km and the maximum plotting height to 250 km.
hmax = 250;
Plot the range-height-angle curve of a radar having a sinc-function antenna pattern.
Specify antenna pattern
Specify the antenna pattern as a sinc function.
pat_angles = linspace(-90,90,361)';
pat_u = 1.39157/sind(90/2)*sind(pat_angles);
pat = sinc(pat_u/pi);
Specify radar and environment parameters
Set the transmitting frequency to 100 MHz, the free-space range to 100 km, the antenna tilt angle to
{0}^{\circ }
, and place the antenna 20 meters above the ground. Assume a surface roughness of one meter.
tilt_ang = 0;
surf_roughness = 1;
Create radar range-height-angle data
Obtain the vertical coverage pattern values and angles for the radar antenna.
[vcp, vcpangles] = radarvcd(freq,rng_fs,ant_height,...
'RangeUnit','km','HeightUnit','m',...
'AntennaPattern',pat,...
'PatternAngles',pat_angles,'TiltAngle',tilt_ang,...
'SurfaceHeightStandardDeviation',surf_roughness/(2*sqrt(2)));
Plot radar range-height-angle data
Set the maximum plotting range to 300 km and the maximum plotting height to 250,000 m. Choose the range units as kilometers, 'km', and the height units as meters, 'm'. Set the range and height axes scale powers to 1/2.
hmax = 250e3;
blakechart(vcp, vcpangles, rmax, hmax, 'RangeUnit','km',...
'ScalePower',1/2,'HeightUnit','m');
vcp — Vertical coverage pattern
real-valued column vector | real-valued matrix
Vertical coverage pattern, specified as a real-valued column vector or matrix. The vertical coverage pattern is the actual maximum range of the radar. Each column of vcp corresponds to an individual vertical coverage pattern. Each row of vcp corresponds to one of the angles specified in vcpangles. Values are expressed in kilometers unless you change the unit of measure using the RangeUnit name-value argument.
Example: [282.3831; 291.0502; 299.4252]
vcpangles — Vertical coverage pattern angles
Vertical coverage pattern angles, specified as a real-valued column vector. Each element of vcpangles specifies the elevation angle in degrees at which a vertical coverage pattern is measured. The set of angles ranges from –90° to 90°.
Example: [2.1480; 2.2340; 2.3199]
rmax — Maximum range of plot
Maximum range of plot, specified as a real-valued scalar. Range units are specified by the 'RangeUnit' name-value argument.
hmax — Maximum height of plot
Maximum height of plot, specified as a real-valued scalar. Height units are specified by the 'HeightUnit' name-value argument.
Example: 'RangeUnit','m'
RangeUnit — Radar range units
'km' (default) | 'nmi' | 'mi' | 'ft' | 'm' | 'kft'
Range units denoting nautical miles, miles, kilometers, feet, meters, or kilofeet. This name-value argument specifies the units for the vertical coverage pattern input argument, vcp, and the maximum range input argument rmax.
Example: 'mi'
HeightUnit — Height units
Height units, specified as one of 'nmi', 'mi', 'km', 'ft', 'm', or 'kft' denoting nautical miles, miles, kilometers, feet, meters, or kilofeet, respectively. This name-value argument specifies the units for the maximum height hmax.
Example: 'm'
ScalePower — Scale power
0.25 (default) | real-valued scalar
Scale power, specified as a scalar in the range [0, 1]. This argument specifies the range and height axis scale power.
Surface refractivity in N-units, specified as a nonnegative real-valued scalar. The surface refractivity is a parameter of the CRPL Exponential Reference Atmosphere Model used by blakechart.
Refraction exponent, specified as a nonnegative real-valued scalar. The refraction exponent is a parameter of the CRPL Exponential Reference Atmosphere Model used by blakechart.
AntennaHeight — Antenna height
Antenna height, specified as a real-valued scalar. When you provide the antenna height, the height in the Blake chart is the height above ground level. Otherwise, the height in the Blake chart is relative to the origin of the ray, and the function assumes that the antenna is less than 1000 ft (about 305 m) above ground level. Use the HeightUnit argument to specify the units of AntennaHeight.
FaceColor — Face color of vertical coverage pattern patch
color name | short name | hexadecimal color code | RGB triplet | 'none'
Face color of vertical coverage pattern patch, specified as a color name, a short name, a hexadecimal color code, an RGB triplet, or 'none'. If you specify more than one color, the number of colors must match the number of columns of vcp.
Example: 'black'
Example: [0.850 0.325 0.098]
Example: '#D95319'
EdgeColor — Edge color of vertical coverage pattern patch
Edge color of vertical coverage pattern patch, specified as a color name, a short name, a hexadecimal color code, an RGB triplet, or 'none'. If you specify more than one color, the number of colors must match the number of columns of vcp.
N=\left(n\left(h\right)-1\right)×{10}^{6}={N}_{\text{s}}{e}^{-{R}_{\text{exp}}h},
n\left(h\right)=1+\left({N}_{\text{s}}\text{\hspace{0.17em}}×{10}^{-6}\right){e}^{-{R}_{\text{exp}}h}.
el2height | height2el | height2range | height2grndrange | landroughness | radarvcd | range2height | refractionexp | searoughness
|
Topological Properties of the First Non-Local Digitally Well-Composed Interpolation on n-D Cubical Grids - LRDE
In discrete topology, we like digitally well-composed (shortly DWC) interpolations because they remove pinches in cubical images. Usual well-composed interpolations are local and sometimes self-dual (they treat in a same way dark and bright components in the image). In our case, we are particularly interested i{\displaystyle n}
-D self-dual DWC interpolations to obtain a purely self-dual tree of shapes. However, it has been proved that we cannot have a{\displaystyle n}
-D interpolation which is at the same time local, self-dualand well-composed. By removing the locality constraint, we have obtained a{\displaystyle n}
-D interpolation with many properties in practice: it is self-dual, DWC, and in-between (this last property means that it preserves the contours). Since we did not published the proofs of these results before, we propose to provide in a first time the proofs of the two last properties here (DWCness and in-betweeness) and a sketch of the proof of self-duality (the complete proof of self-duality requires more material and will come later). Some theoretical and practical results are given.
@Article{ boutry.20.jmiv.1,
title = {Topological Properties of the First Non-Local Digitally
Well-Composed Interpolation on {$n$-D} Cubical Grids},
doi = {10.1007/s10851-020-00989-y},
abstract = {In discrete topology, we like digitally well-composed
(shortly DWC) interpolations because they remove pinches in
cubical images. Usual well-composed interpolations are
local and sometimes self-dual (they treat in a same way
dark and bright components in the image). In our case, we
are particularly interested in $n$-D self-dual DWC
interpolations to obtain a purely self-dual tree of shapes.
However, it has been proved that we cannot have an $n$-D
interpolation which is at the same time local, self-dual,
and well-composed. By removing the locality constraint, we
have obtained an $n$-D interpolation with many properties
in practice: it is self-dual, DWC, and in-between (this
last property means that it preserves the contours). Since
we did not published the proofs of these results before, we
propose to provide in a first time the proofs of the two
last properties here (DWCness and in-betweeness) and a
sketch of the proof of self-duality (the complete proof of
self-duality requires more material and will come later).
Some theoretical and practical results are given. }
Retrieved from "https://www.lrde.epita.fr/index.php?title=Publications/boutry.20.jmiv.1&oldid=125851"
|
Oscillatory Criteria for Higher Order Functional Differential Equations with Damping
Peiguang Wang, Hai Cai, "Oscillatory Criteria for Higher Order Functional Differential Equations with Damping", Journal of Function Spaces, vol. 2013, Article ID 968356, 5 pages, 2013. https://doi.org/10.1155/2013/968356
Peiguang Wang1 and Hai Cai2
2College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
We investigate a class of higher order functional differential equations with damping. By using a generalized Riccati transformation and integral averaging technique, some oscillation criteria for the differential equations are established.
In this paper, we consider the following higher order functional differential equations with distributed deviating arguments of the form as follows: where is an even number, , , , for , and has the same sign as ; when they have the same sign, is nondecreasing, and the integral of (1) is a Stieltjes one.
We restrict our attention to those solutions of (1) which exist on same half liner with for any and satisfy (1). As usual, a solution of (1) is called oscillatory if the set of its zeros is unbounded from above, otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all solutions are oscillatory.
In recent years, there has been an increasing interest in studying the oscillation behavior of solutions for the differential equations with distributed deviating arguments, and a number of results have been obtained (refer to [1–3] and their references). However, to the best of our knowledge, very little is known for the case of higher order differential equations with damping. The purpose of this paper is to establish some new oscillation criteria for (1) by introducing a class of functions defined in [2] and a generalized Riccati technique.
Firstly, we define the following two class functions.
We say that a function belongs to the function class , denoted by , if , where , which satisfies , , and has the partial derivative on that is locally integrable with respect to in .
Let , . We say that a function belongs to the function class , denoted by , if for , in , has continuous partial derivative in with respect to and .
In order to prove the main theorems, we need the following lemmas.
Lemma 1 (see [4]). Let , if is of constant sign and not identically zero on any ray for , then there exists a , an integer , with even for or odd for ; and for , , , and , .
Lemma 2 (see [5]). If the function is as in Lemma 1 and for , then there exists a constant such that for sufficiently large , there exists a constant , satisfying
Lemma 3 (see [3]). Suppose that is a nonoscillatory solution of (1). If then for any large .
Theorem 4. Assume that (3) holds, and?there exists a function such that , . , , , where , , and are constants.? is nondecreasing with , , and there exist constants and such that
If there exists a function , such that for any , and , where
Proof. Suppose to the contrary that (1) has a nonoscillatory solution . Without loss of generality, we may suppose that is an eventually positive solution. From the conditions of and , there exists a , such that
By Lemma 3, there exists a such that , . Thus, we have
By Lemma 1, there exists a such that , . Further, by Lemma 2, there exist constant and a , such that
In view of , and the definition of , , we have where .
Multiplying (12) by and integrating from to , we have
Integrating by parts and using integral averaging technique, we have thus which contradicts (5). This completes the proof of Theorem 4.
If we choose , where . By Theorem 4, we have the following results.
Corollary 5. Assume that (3), , and hold. If there exist such that for each , where and are defined by , , and
If we choose , , and let , by Theorem 4, we have the following corollary.
Corollary 6. Assume that (3), , and hold. If there exists a constant such that for each , where is defined as in Corollary 5. Then (1) is oscillatory.
Theorem 7. Assume that (3) holds, and?there exist functions , such that , , , where are constants, ;?there exist constants and , such that , , where and are constants, .
If there exists a function , such that for any , , and , and (5) holds, where is defined as in Theorem 4:
Proof. Suppose to the contrary that (1) has a nonoscillatory solution . Without loss of generality, we may suppose that is an eventually positive solution. Similar to the proof of Theorem 4, there exists a , such that , , , , , and , for , . Set then
In view of , and the definition of , , we have
The following proof is similar to Theorem 4, and we omit the details. This completes the proof of Theorem 7.
Similar to Corollaries 5 and 6, we have the following corollaries.
Corollary 8. Assume that (3), , and hold. If there exist such that for each , and (16) holds, where are defined as in Corollary 5:
Corollary 9. Assume that (3), , and hold. If there exists a constant such that for each , and (18) holds. where is defined as in Corollary 8, then (1) is oscillatory.
For the case of the function with monotonicity, we have the following theorem.
Theorem 10. Assume that (3), hold, and?there exist and , such that , where is constants. , , in which and are constants, .
If there exists a function , such that for any , , and (5) holds, where is defined as in Theorem 4:
Proof. Suppose to the contrary that (1) has a nonoscillatory solution . Without loss of generality, we may suppose that is an eventually positive solution. Similar to the proof of Theorem 4, there exists a , when , and we have , , , , , and , . Set then
The following proof is similar to Theorem 4, we omit the details. This completes the proof of Theorem 10.
Corollary 11. Assume that (3), , and hold. If there exist such that for each , and (16) holds, where are defined as in Corollary 5:
Corollary 12. Assume that (3), , and hold. If there exists a constant such that for each , and (18) holds. where is defined as in Corollary 11. Then (1) is oscillatory.
Example 13. Consider the following equation where , , obviously . Choosing , , then , and
Thus, there exists a constant , such that , that is,
By Corollary 6, then (30) is oscillatory.
Example 14. Consider the following equation where , obviously , . Choosing , , , and , then . By Corollary 12, then (33) is oscillatory.
The authors would like to thank the reviewers for their valuable suggestions and comments. The research was supported by the Natural Science Foundation of China (11271106).
P. Wang and M. Wang, “Oscillation of a class of higher order neutral differential equations,” Archivum Mathematicum, vol. 40, no. 2, pp. 201–208, 2004. View at: Google Scholar | Zentralblatt MATH
Y. G. Sun, “Oscillation of second order functional differential equations with damping,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 519–526, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
P. G. Wang and X. Liu, “Interval oscillation for some kinds of even-order nonlinear differential equations,” Chinese Journal of Engineering Mathematics, vol. 22, no. 6, pp. 1031–1038, 2005. View at: Google Scholar
I. T. Kiguradze, “On the oscillatory character of solutions of the equation
{d}^{m}u/d{t}^{m}+a\left(t\right){|u|}^{n}sgnu=0\left[J\right]
,” Matematicheskil Sbornik, vol. 65, pp. 172–187, 1964 (Russian). View at: Google Scholar
C. Philos, “A new criterion for the oscillatory and asymptotic behavior of delay differential equations,” L'Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques, vol. 29, no. 7-8, pp. 367–370, 1981. View at: Google Scholar | Zentralblatt MATH
Copyright © 2013 Peiguang Wang and Hai Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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A pexider difference for a pexider functional equation | Advances in Continuous and Discrete Models | Full Text
A pexider difference for a pexider functional equation
Saeid Ostadbashi2,
Gwang Hui Kim3 &
Sooran Mahmoudfakhe2
We deal with a Pexider difference
f\left(2x+y\right)+f\left(2x-y\right)-g\left(x+y\right)-g\left(x-y\right)-2g\left(2x\right)+2g\left(x\right)
where f and g map be a given abelian group (G, +) into a sequentially complete Hausdorff topological vector space. We also investigate the Hyers-Ulam stability of the following Pexiderized functional equation
f\left(2x+y\right)+f\left(2x-y\right)=g\left(x+y\right)+g\left(x-y\right)+2g\left(2x\right)-2g\left(x\right)
in topological vector spaces.
Mathematics subject classification (2000): Primary 39B82; Secondary 34K20, 54A20.
In 1940, Ulam [1] proposed the general stability problem: Let G1 be a group, G2 be a metric group with the metric d. Given ε > 0, does there exists δ > 0 such that if a function h: G1→ G2 satisfies the inequality
d\left(h\left(xy\right)-h\left(x\right)h\left(y\right)\right)<\delta ,\phantom{\rule{1em}{0ex}}\left(x,y\in {G}_{1}\right),
then there is a homomorphism H: G 1 → G 2 with
d\left(h\left(x\right),H\left(x\right)\right)<\epsilon ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(x\in {G}_{1}\right)?
Hyers [2] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1950, Aoki [3] extended the theorem of Hyers by considering the unbounded cauchy difference inequality
∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon \left({∥x∥}^{p}+{∥y∥}^{p}\right)\phantom{\rule{1em}{0ex}}\left(\epsilon >0,p\phantom{\rule{0.3em}{0ex}}\epsilon \phantom{\rule{0.3em}{0ex}}\left[0,1\right)\right).
In 1978, Rassias [4] also generalized the Hyers' theorem for linear mappings under the assumption t ↦ f (tx) is continuous in t for each fixed x.
Recently, Adam and Czerwik [5] investigated the problem of the Hyers-Ulam stability of a generalized quadratic functional equation in linear topological spaces. Najati and Moghimi [6] investigated the Hyers-Ulam stability of the functional equation
f\left(2x+y\right)+f\left(2x-y\right)=f\left(x+y\right)+f\left(x-y\right)+2f\left(2x\right)-2f\left(x\right)
in quasi-Banach spaces. In this article, we prove that the Pexiderized functional equation
f\left(2x+y\right)+f\left(2x-y\right)=g\left(x+y\right)+g\left(x-y\right)+2g\left(2x\right)-2g\left(x\right)
is stable for functions f, g defined on an abelian group and taking values in a topological vector space.
Throughout this article, let G be an abelian group and X be a sequentially complete Hausdorff topological vector space over the field ℚ of rational numbers.
A mapping f: G → X is said to be quadratic if and only if it satisfies the following functional equation
f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)
for all x y ∈ G. A mapping f G → X is said to be additive if and only if it satisfies f (x + y) = f (x) + f (y) for all x y ε G. For a given f: G → X, we will use the following notation
Df\left(x,y\right):=f\left(2x+y\right)+f\left(2x-y\right)-f\left(x+y\right)-f\left(x-y\right)-2f\left(2x\right)+2f\left(x\right).
For given sets A B ⊆ X and a number k ∈ ℝ, we define the well known operations
A+B:=\left\{a+b:a\phantom{\rule{2.77695pt}{0ex}}\in A,b\phantom{\rule{2.77695pt}{0ex}}\in B\right\},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}kA:=\left\{ka:a\phantom{\rule{2.77695pt}{0ex}}\in A\right\}.
We denote the convex hull of a set U ⊆ X by conv(U) and by
\overline{U}
the sequential closure of U. Moreover it is well know that:
If A ⊆ X are bounded sets, then conv(A) and
\overline{A}
are bounded subsets of X.
If A, B ⊆ X and α β ∈ ℝ, then α conv(A) + β conv(B) = conv(αA + βB).
Let X1 and X2 be linear spaces over ℝ. If f: X1→ X2 is a additive (quadratic) function, then f (rx) = rf (x) (f (rx) = r2f (x)), for all x ∈ X1 and all r ∈ ℚ.
We start with the following lemma.
Lemma 2.1. Let G be a 2-divisible abelian group and B ⊆ X be a nonempty set. If the functions f, g: G → X satisfy
f\left(2x+y\right)+f\left(2x-y\right)-g\left(x+y\right)-g\left(x-y\right)-2g\left(2x\right)+2g\left(x\right)\in B
for all x, y ∈ G, then
Df\left(x,y\right)\phantom{\rule{2.77695pt}{0ex}}\in \phantom{\rule{2.77695pt}{0ex}}2\mathsf{\text{ conv}}\left(B-B\right),
Dg\left(x,y\right)\in \mathsf{\text{conv}}\left(B-B\right)
for all x, y ∈ G.
Proof. Putting y = 0 in (2.1), we get
2f\left(2x\right)-2g\left(2x\right)\in B
for all x ∈ G. If we replace x by
\frac{1}{2}x
in (2.4), then we have
f\left(x\right)-g\left(x\right)\in \frac{1}{2}B
for all x ∈ G. It follows from (2.5) and (2.1) that
\begin{array}{ll}\hfill Df\left(x,y\right)& =f\left(2x+y\right)+f\left(2x-y\right)-g\left(x+y\right)-g\left(x-y\right)-2g\left(2x\right)+2g\left(x\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\left[f\left(x+y\right)-g\left(x+y\right)\right]-\left[f\left(x-y\right)-g\left(x-y\right)\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\left[2f\left(2x\right)-2g\left(2x\right)\right]+\left[2f\left(x\right)-2g\left(x\right)\right]\phantom{\rule{2em}{0ex}}\\ \in 2\mathsf{\text{ conv}}\left(B-B\right).\phantom{\rule{2em}{0ex}}\end{array}
\begin{array}{ll}\hfill Dg\left(x,y\right)& =f\left(2x+y\right)+f\left(2x-y\right)-g\left(x+y\right)-g\left(x-y\right)-2g\left(2x\right)+2g\left(x\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\left[f\left(2x+y\right)-g\left(2x+y\right)\right]-\left[f\left(2x-y\right)-g\left(2x-y\right)\right]\phantom{\rule{2em}{0ex}}\\ \in \mathsf{\text{conv}}\left(B-B\right).\phantom{\rule{2em}{0ex}}\end{array}
Theorem 2.2. Let G be a 2-divisible abelian group and B ⊆ X be a bounded set. Suppose that the odd functions f, g: G → X satisfy (2 1) for all x, y ∈ G. Then there exists exactly one additive function
\mathcal{A}:G\to X
\mathcal{A}\left(x\right)-f\left(x\right)\in 4\overline{\mathsf{\text{conv}}\left(B-B\right)}\mathsf{\text{,}}\phantom{\rule{1em}{0ex}}\mathcal{A}\left(x\right)-g\left(x\right)\in 2\overline{\mathsf{\text{conv}}\left(B-B\right)}
for all x ∈ G. Moreover the function
\mathcal{A}
\mathcal{A}\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}f\left({2}^{n}x\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}g\left({2}^{n}x\right)
for all x ∈ G. Moreover, the convergence of the sequences are uniform on G.
Proof. By Lemma 2.1, we get (2.2). Setting y = x, y = 3x and y = 4x in (2.2), we get
f\left(3x\right)-3f\left(2x\right)+3f\left(x\right)\in 2\mathsf{\text{ conv}}\left(B-B\right),
f\left(5x\right)-f\left(4x\right)-f\left(2x\right)+f\left(x\right)\in 2\mathsf{\text{conv}}\left(B-B\right),
f\left(6x\right)-f\left(5x\right)+f\left(3x\right)-3f\left(2x\right)+2f\left(x\right)\in 2\mathsf{\text{conv}}\left(B-B\right)
for all x ∈ G. It follows from (2.7), (2.8), and (2.9) that
f\left(6x\right)-f\left(4x\right)-f\left(2x\right)\in 6\phantom{\rule{0.3em}{0ex}}\mathsf{\text{conv}}\left(B-B\right)
for all x ∈ G. So
f\left(3x\right)-f\left(2x\right)-f\left(x\right)\in 6\phantom{\rule{0.3em}{0ex}}\mathsf{\text{conv}}\left(B-B\right)
for all x ∈ G. Using (2.7) and (2.10), we obtain
\frac{\mathsf{\text{1}}}{\mathsf{\text{2}}}f\left(2x\right)-f\left(x\right)\in 2\mathsf{\text{conv}}\left(B-B\right)
for all x ∈ G. Therefore
\begin{array}{ll}\hfill \frac{1}{{2}^{n}}f\left({2}^{n}x\right)-\frac{1}{{2}^{m}}f\left({2}^{m}x\right)& =\sum _{k=m}^{n-1}\left[\frac{1}{{2}^{k+1}}f\left({2}^{k+1}x\right)-\frac{1}{{2}^{k}}f\left({2}^{k}x\right)\right]\phantom{\rule{2em}{0ex}}\\ \in \sum _{k=m}^{n-1}\frac{2}{{2}^{k}}\mathsf{\text{conv}}\left(B-B\right)\phantom{\rule{2em}{0ex}}\\ \subseteq \frac{4}{{2}^{m}}\mathsf{\text{conv}}\left(B-B\right)\phantom{\rule{2em}{0ex}}\end{array}
for all x ∈ G and all integers n > m ≥ 0. Since B is bounded, we conclude that conv(B - B) is bounded. It follows from (2.11) and boundedness of the set conv(B - B) that the sequence
\left\{\frac{1}{{2}^{n}}f\left({2}^{n}x\right)\right\}
is (uniformly) Cauchy in X for all x ∈ G. Since X is a sequential complete topological vector space, the sequence
\left\{\frac{1}{{2}^{n}}f\left({2}^{n}x\right)\right\}
is convergent for all x ∈ G, and the convergence is uniform on G. Define
{\mathcal{A}}_{1}:G\to X,\phantom{\rule{1em}{0ex}}{\mathcal{A}}_{1}\left(x\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}f\left({2}^{n}x\right).
Since conv(B - B) is bounded, it follows from (2.2) that
D{\mathcal{A}}_{1}\left(x,y\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}Df\left({2}^{n}x,{2}^{n}y\right)=0
for all x y ∈ G. So
{\mathcal{A}}_{1}
is additive (see [6]). Letting m = 0 and n →∞ in (2.11), we get
{\mathcal{A}}_{1}\left(x\right)-f\left(x\right)\in 4\overline{\mathsf{\text{conv}}\left(B-B\right)}
for all x ∈ G. Similarly as before applying (2.3) we have an additive mapping
{\mathcal{A}}_{2}:G\to X
{\mathcal{A}}_{2}\left(x\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}g\left({2}^{n}x\right)
which is satisfying
{\mathcal{A}}_{2}\left(x\right)-g\left(x\right)\in 2\overline{\mathsf{\text{conv}}\left(B-B\right)}
for all x ∈ G. Since B is bounded, it follows from (2.5) that
{\mathcal{A}}_{1}={\mathcal{A}}_{2}
\mathcal{A}:={\mathcal{A}}_{1}
, we obtain (2.6) from (2.12) and (2.13).
To prove the uniqueness of
\mathcal{A}
, suppose that there exists another additive function
{\mathcal{A}}^{\prime }
: G → X satisfying (2.6). So
{\mathcal{A}}^{\prime }\left(x\right)-\mathcal{A}\left(x\right)=\left[{\mathcal{A}}^{\prime }\left(x\right)-f\left(x\right)\right]+\left[f\left(x\right)-\mathcal{A}\left(x\right)\right]\in 8\overline{\mathsf{\text{conv}}\left(B-B\right)}
for all x ∈ G. Since
{\mathcal{A}}^{\prime }
\mathcal{A}
are additive, replacing x by 2nx implies that
{\mathcal{A}}^{\prime }\left(x\right)-\mathcal{A}\left(x\right)\in \frac{8}{{2}^{n}}\overline{\mathsf{\text{conv}}\left(B-B\right)}
for all x ∈ G and all integers n. Since
\overline{\mathsf{\text{conv}}\left(B-B\right)}
is bounded, we infer
{\mathcal{A}}^{\prime }=\mathcal{A}
. This completes the proof of theorem.
Theorem 2.3 Let G be a 2, 3-divisible abelian group and B ⊆ X be a bounded set. Suppose that the even functions f, g: G → X satisfy (2 1) for all x, y ∈ G. Then there exists exactly one quadratic function
\mathcal{Q}:G\to X
\mathcal{Q}\left(x\right)-f\left(x\right)+f\left(0\right)\in 4\overline{\mathsf{\text{conv}}\left(B-B\right)},\mathcal{Q}\left(x\right)-g\left(x\right)+g\left(0\right)\in 2\overline{\mathsf{\text{conv}}\left(B-B\right)}
for all x ∈ G. Moreover, the function
\mathcal{Q}
\mathcal{Q}\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{{4}^{n}}f\left({2}^{n}x\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{{4}^{n}}g\left({2}^{n}x\right)
Proof. By replacing y by x + y in (2.2), we get
\begin{array}{ll}\hfill f\left(3x+y\right)+f\left(x-y\right)\phantom{\rule{0.3em}{0ex}}& -f\left(2x+y\right)-f\left(y\right)\phantom{\rule{2em}{0ex}}\\ -2f\left(2x\right)+2f\left(x\right)\in 2conv\left(B-B\right)\phantom{\rule{2em}{0ex}}\end{array}
for all x, y ∈ G. Replacing y by - y in (2.14), we get
\begin{array}{ll}\hfill f\left(3x-y\right)+f\left(x+y\right)& -f\left(2x-y\right)-f\left(y\right)\phantom{\rule{2em}{0ex}}\\ -2f\left(2x\right)+2f\left(x\right)\in 2\mathsf{\text{conv}}\left(B-B\right)\phantom{\rule{2em}{0ex}}\end{array}
for all x, y ∈ G. It follows from (2.2), (2.14), and (2.15) that
\begin{array}{ll}\hfill f\left(3x+y\right)& +f\left(3x-y\right)-2f\left(y\right)\phantom{\rule{2em}{0ex}}\\ -6f\left(2x\right)+6f\left(x\right)\in 6\mathsf{\text{conv}}\left(B-B\right)\phantom{\rule{2em}{0ex}}\end{array}
for all x, y ∈ G. By letting y = 0 and y = 3x in (2.16), we get
2f\left(3x\right)-6f\left(2x\right)+6f\left(x\right)-2f\left(0\right)\in 6\mathsf{\text{conv}}\left(B-B\right),
f\left(6x\right)-2f\left(3x\right)-6f\left(2x\right)+6f\left(x\right)+f\left(0\right)\in 6\mathsf{\text{conv}}\left(B-B\right)
for all x ∈ G. Using (2.17) and (2.18), we obtain
f\left(6x\right)-4f\left(3x\right)+3f\left(0\right)\in 12\mathsf{\text{conv}}\left(B-B\right)
\frac{1}{3}x
in (2.19), then
f\left(2x\right)-4f\left(x\right)+3f\left(0\right)\in 12\mathsf{\text{conv}}\left(B-B\right)
\frac{1}{{4}^{n+1}}f\left({2}^{n+1}x\right)-\frac{1}{{4}^{n}}f\left({2}^{n}x\right)+\frac{3}{{4}^{n+1}}f\left(0\right)\in \frac{3}{{4}^{n}}\mathsf{\text{conv}}\left(B-B\right)
for all x ∈ G and all integers n. So
\begin{array}{l}\frac{1}{{4}^{n}}f\left({2}^{n}x\right)-\frac{1}{{4}^{m}}f\left({2}^{m}x\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{k=m}^{n-1}\frac{1}{{4}^{k+1}}f\left({2}^{k+1}x\right)-\frac{1}{{4}^{k}}f\left({2}^{k}x\right)\phantom{\rule{2em}{0ex}}\\ \in -\sum _{k=m}^{n-1}\frac{3}{{4}^{k+1}}f\left(0\right)+\sum _{k=m}^{n-1}\frac{3}{{4}^{k}}\mathsf{\text{conv}}\left(B-B\right)\phantom{\rule{2em}{0ex}}\\ \subseteq -\sum _{k=m}^{n-1}\frac{3}{{4}^{k+1}}f\left(0\right)+\frac{1}{{4}^{m-1}}\mathsf{\text{conv}}\left(B-B\right)\phantom{\rule{2em}{0ex}}\end{array}
for all x ∈ G and all integers n >m ≥ 0. It follows from (2.21) and boundedness of the set conv(B - B) that the sequence
\left\{\frac{1}{{4}^{n}}f\left({2}^{n}x\right)\right\}
is (uniformly) Cauchy in X for all x ∈ G. The rest of the proof is similar to proof of of Theorem 2.2.
Remark 2.4. If the functions f, g: G → X satisfy (2.1), where f is even (odd) and g is odd (even), then it is easy to show that f and g are bounded.
Ulam SM: Problem in Modern Mathematics, Science edn. Wiley, New York; 1960.
Hyers DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of linear trasformation in Banach spaces. J Math Soc Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Adam M, Czerwik S: On the stability of the quadratic functional equation in topological spaces. Banach J Math Anal 2007, 1: 245–251.
Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J Math Anal Appl 2008, 337: 399–415. 10.1016/j.jmaa.2007.03.104
Department of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, 56199-11367, Iran
Saeid Ostadbashi & Sooran Mahmoudfakhe
Department of Applied Mathematics, Kangnam University, Giheung-gu, Yongin, Gyoenggi, 446-702, Republic of Korea
Sooran Mahmoudfakhe
Najati, A., Ostadbashi, S., Kim, G.H. et al. A pexider difference for a pexider functional equation. Adv Differ Equ 2012, 26 (2012). https://doi.org/10.1186/1687-1847-2012-26
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Iron-Containing Yttria-Stabilized Zirconia System For Two-Step Thermochemical Water Splitting | J. Sol. Energy Eng. | ASME Digital Collection
Nobuyuki Gokon,
Nobuyuki Gokon
, 8050 Ikarashi 2-nocho, Nishi-ku, Niigata 950-2181, Japan
Takayuki Mizuno,
Yumiko Nakamuro,
Department of Chemistry and Chemical Engineering, Faculty of Engineering,
Nobuyuki Gokon Assistant Professor
Takayuki Mizuno Master Degree Student
Yumiko Nakamuro Master Degree Student
Tatsuya Kodama Professor
Gokon, N., Mizuno, T., Nakamuro, Y., and Kodama, T. (December 28, 2007). "Iron-Containing Yttria-Stabilized Zirconia System For Two-Step Thermochemical Water Splitting." ASME. J. Sol. Energy Eng. February 2008; 130(1): 011018. https://doi.org/10.1115/1.2807197
An iron-containing yttria-stabilized zirconia (YSZ) or Fe-YSZ was found to be a promising working redox material for the thermochemical two-step water-splitting cycle. The Fe-YSZ was formed by a high-temperature reaction between YSZ doped with more than
8mol%
Y2O3
Fe3O4
supported on the YSZ at
1400°C
in an inert atmosphere. The formed Fe-YSZ reacted with steam to generate hydrogen at
1000°C
. The oxidized Fe-YSZ was reactivated by a thermal reduction at
1400°C
in an inert atmosphere. The alternative
O2
H2
generations in the repeated two-step reactions and the X-ray diffraction and chemical analysis studies on the solid materials indicated that the two-step water splitting was associated with a redox transition between
Fe2+–Fe3+
ions in the cubic YSZ lattice.
chemical analysis, hydrogen economy, iron, reduction (chemical), solar energy concentrators, solar heating, thermochemistry, X-ray diffraction, yttrium compounds, zirconium compounds, solar chemistry, concentrated solar heat, solar hydrogen, thermochemical water splitting, redox metal oxide
Hydrogen, Iron, Water, Ions, Cycles, X-ray diffraction, High temperature
An Assessment of Solar Hydrogen Production Using the Mark 13 Hybrid Process
Economical and Technical Evaluation of UT-3 Thermochemical Hydrogen Production Process for an Industrial Scale Plant
Studies on an Iodine-Sulfur Process for Thermochemical Hydrogen Production
Proceedings of the Eighth World Hydrogen Energy Conference
Solar UT-3 Thermochemical Cycle for Hydrogen Production
Hydrogen Production from Water Utilizing Solar Heat at High Temperatures
The System Iron--Oxygen. II. Equilibrium and Thermodynamics of Liquid Oxide and Other Phases
Hydrogen Production by Solar Thermochemical Water-Splitting∕Methane-Reforming Process
Proceeding of International Solar Energy Conference (ISEC) 2003
, eds., Paper No. ISES2003-44037.
Thermochemical Hydrogen Production by a Redox System of ZrO2-Supported Co(II)-Ferrite
A Two-Step Water Splitting with Ferrite Particles and Its New Reactor Concept Using an Internally Circulating Fluidized Bed
Proceeding of ASME International Solar Energy Conference (ISEC) 2006
, Denver, Co,
, Paper No. ISEC2006-99063.
Hydrogen Production Through Two-Step Water Splitting Using YSZ(Ni, Fe) System for Solar Hydrogen Production
Preparation and Characterization of Cubic ZrO2 Stabilized by Fe(III) and Fe(II)
Iron-Zirconium Oxides, an Investigation of Structural Transformations by X-Ray Diffraction, Electron Diffraction, and Iron-57 Mossbauer Spectroscopy
Stepwise Production of CO-Rich Syngas and Hydrogen Via Solar Methane Reforming by Using a Ni(II)-Ferrite Redox System
Phase Relations in the System FeO‐Fe2O3‐ZrO2‐SiO2
Wen-Xiang
Study on the Internal Action and Existent State of ZrO2 in Fused Ion Catalysts of Different Compositions
Reactivity of Iron-Containing YSZ for a Two-Step Thermochemical Water Splitting Using Thermal Reduction Temperatures of 1400–1500°C
Reactive Fe-YSZ Coated Foam Devices for Solar Two-Step Water Splitting
Intergranular Decohesion Induced by Mobile Hydrogen in Iron with and without Segregated Carbon: First-Principles Calculations
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Sequence - Maple Help
Home : Support : Online Help : Connectivity : Mathematica Translator : Mma : Sequence
return a function containing a sequence of arguments
Sequence(arguments)
The Sequence command returns a function containing a sequence of arguments which were spliced into the function.
\mathrm{with}\left(\mathrm{MmaTranslator}[\mathrm{Mma}]\right):
f\left(a,\mathrm{Sequence}\left(b,c\right)\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
\mathrm{with}\left(\mathrm{MmaTranslator}\right):
\mathrm{FromMma}\left(\mathrm{`f\left[a, Sequence\left[b, c\right]\right]`},\mathrm{evaluate}\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
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User:Hakerh400/Bijection between reals and the powerset of naturals - Esolang
User:Hakerh400/Bijection between reals and the powerset of naturals
In this article we explain how to establish a bijection between the set of real numbers and the powerset of natural numbers.
1 The existence of a bijection
2 What is a real number
3 From naturals to reals
The existence of a bijection
If two sets have the same cardinality, then there exists a bijection between them. The set of natural numbers has cardinality
{\displaystyle \aleph _{0}}
, so its powerset has cardinality
{\displaystyle 2^{\aleph _{0}}}
. The cardinality of the set of real numbers is also
{\displaystyle 2^{\aleph _{0}}}
, so there must exist a bijection between them.
Without digging too much into the axiomatization, intuitively a real number consists of a sign, integral part and fractional part. Sign is a boolean value: positive or negative. Integral part is a natural number. Fractional part can be viewed as an infinite sequence of binary digits, or bits. However, if the sequence of bits ends with infinitely many 1s, then it also represents a different real number. For example,
{\displaystyle 0.11111{\dots }=1.00000{\dots }}
(the numbers are represented in base 2). So, not every sequence of bits uniquely represents the fractional part of a real number.
From naturals to reals
In higher-order logic, a set can be represented as a function from the type of the set elements to booleans. For each set
{\displaystyle S}
there exists a unary predicate
{\displaystyle \varphi }
{\displaystyle \forall x(x\in S\longleftrightarrow \varphi (x))}
. A subset of natural numbers can therefore be considered as an infinite sequence of boolean values, or bits. So, we want to establish a bijection between infinite sequences of bits and reals.
Given an infinite sequence of bits
{\displaystyle A}
. We want to obtain the sign
{\displaystyle S}
, integral part
{\displaystyle N}
and fractional part
{\displaystyle F}
of the corresponding real number. We split the algorithm into two cases based on whether
{\displaystyle A}
ends with infinitely many same digits (case 2) or not (case 1).
{\displaystyle A}
does not end with infinitely many same digits. Remove the first bit from
{\displaystyle A}
. If the bit is
{\displaystyle 0}
{\displaystyle S}
is positive, otherwise
{\displaystyle S}
is negative. Count the number of 1s at the beginning of
{\displaystyle A}
. It represents the integral part
{\displaystyle N}
. Now remove
{\displaystyle N+1}
bits from the beginning of
{\displaystyle A}
(the last bit after 1s must be 0, and 0 exists in the sequence because the sequence is infinite and does not end with infinitely many 1s). The remaining bits in the sequence represent fractional part
{\displaystyle F}
{\displaystyle A}
ends with infinitely many same digits. Locate where the infinite sequence of same digits begins and let it be index
{\displaystyle i}
. If the ending bits of the sequence are 0s, then
{\displaystyle S}
{\displaystyle S}
is negative. Additionally, if
{\displaystyle S}
is negative, replace all 1s starting from index
{\displaystyle i+1}
with 0s. Parse
{\displaystyle N}
{\displaystyle F}
like in the Case 1.
Hakerh400 is working on a formal proof that this algorithm indeed represents a bijection between the powerset of natural numbers and the set of all 3-tuples
{\displaystyle (S,N,F)}
{\displaystyle S}
is a boolean,
{\displaystyle N}
is a natural number and
{\displaystyle F}
is a sequence of bits that does not end with infinitely many 1s. There is also additional constraint that if
{\displaystyle F}
consists only of 0s and
{\displaystyle N=0}
{\displaystyle S}
cannot be negative.
The algorithm is proved to be incorrect. Proof
The algorithm is uncumputable, but it can be used in a formal theorem prover to construct real numbers using codatatypes. The type of real numbers can be represented as
data Real = Real (Nat -> Bool)
Retrieved from "https://esolangs.org/w/index.php?title=User:Hakerh400/Bijection_between_reals_and_the_powerset_of_naturals&oldid=90855"
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Category:Quizbank/Units - Wikiversity
Category:Quizbank/Units
< Category:Quizbank
See Quizbank for more information about these units.
See Category:Quizbank/Equations for an effort to provide equation sheets for all these exams and quizzes.
See QB for list of all quizzes.
All units strive for uniformity in format and offer classroom ready quizzes or exams ready to print in sufficiently large numbers that everything can be hidden in plane sight. In other words, the actual exams are publicly posted in sufficient quantity that no harm is done to the integrity of the test or quiz. Some of the units can be used with an overhead projector for very short "check quizzes" using strips of paper or clickers during a lecture (for more information see Quizbank/Instructions.)
Equations for Units 1 and 2: page.. pdf.. file
Electromagnetism for an introductory calculus based course (OpenStax)[edit | edit source]
Three closely related units are available:
Quizbank/Electricity_and_Magnetism_(calculus_based) is a collection of 4 question quizzes that closely follow chapter examples in OpenStax University Physics Vol. 2 Unit 2. The quizzes can be either printed out, or displayed on an overhead projector.
Quizbank/Electricity_and_Magnetism_(calculus)_Exams contains the same questions as above, except the questions have been grouped into two two cumulative tests of 10 questions each. Each test covers six chapters (5-10 and 11-16).
Quizbank/University_Physics_Semester_2 takes this sequence one step further by including material not associated with OpenStax. This material includes geometrical optics, polarization, and extra questions on
{\displaystyle {\mathcal {E}}\&{\mathcal {M}}}
, and brings us one step closer to the goal of creating a complete set of assessments for the four volumes of OpenStax physics (3 volumes of University Physics and the 1-volume College Physics textbook). Equations for this unit are under construction.
Bell' theorem[edit | edit source]
Quizbank/Bell is still under construction. I used a small number of questions from this unit to supplement Quizbank/University_Physics_Semester_2 because Bell's theorem involves polarized photons. But the unit is intended for a first year conceptual physics course (without prerequisite mathematics.)
Test Course[edit | edit source]
♦/TestCourse is used to develop the code that generates these units.
Quizbank/Equations (1 cat, 6 pgs)
Pages in category "Quizbank/Units"
Quizbank/Bell
Quizbank/calcPhyEMqAll
Quizbank/College Physics Sem 1
Quizbank/Electricity and Magnetism (calculus based)
Quizbank/Electricity and Magnetism (calculus) Exams
Quizbank/Electricity and Magnetism: Gauss' Law
Quizbank/HTW
Quizbank/University Physics Semester 2
Retrieved from "https://en.wikiversity.org/w/index.php?title=Category:Quizbank/Units&oldid=2170664"
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Compute controllability staircase form - MATLAB ctrbf - MathWorks United Kingdom
\begin{array}{ccc}\overline{A}=TA{T}^{T},& \overline{B}=TB,& \overline{C}=C{T}^{T}\end{array}
\begin{array}{ccc}\overline{A}=\left[\begin{array}{cc}{A}_{uc}& 0\\ {A}_{21}& {A}_{c}\end{array}\right],& \overline{B}=\left[\begin{array}{l}0\\ {B}_{c}\end{array}\right],& \overline{C}=\left[{C}_{nc}{C}_{c}\right]\end{array}
where (Ac, Bc) is controllable, all eigenvalues of Auc are uncontrollable, and
{C}_{c}{\left(sI-{A}_{c}\right)}^{-1}{B}_{c}=C{\left(sI-A\right)}^{-1}B
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ML Terminology - ProductizeML
You will learn: basic concepts and terms used in the AI/ML community.
Artificial Intelligence: A non-human program or model that can solve sophisticated tasks. For example, a program or model that translates text or a program or model that identifies diseases from radiologic images both exhibit artificial intelligence.
Machine Learning: A program or system that builds (trains) a predictive model from input data. The system uses the learned model to make useful predictions from new (never-before-seen) data drawn from the same distribution as the one used to train the model. Machine learning also refers to the field of study concerned with these programs or systems.
Deep Learning: Is part of a broader family of machine learning methods based on artificial neural networks that are inspired by information processing and distributed communication nodes in biological systems. The adjective "deep" in deep learning comes from the use of multiple layers in the network.
Supervised Learning: Training a model from input data and its corresponding labels. Supervised machine learning is analogous to a student learning a subject by studying a set of questions and their corresponding answers. After mastering the mapping between questions and answers, the student can then provide answers to new (never-before-seen) questions on the same topic. E.g. Linear Regression, Convolutional Neural Networks.
Semi-supervised learning: Training a model on data where some of the training examples have labels but others don’t. One technique for semi-supervised learning is to infer labels for the unlabeled examples, and then to train on the inferred labels to create a new model. Semi-supervised learning can be useful if labels are expensive to obtain but unlabeled examples are plentiful.
Unsupervised Learning: Training a model to find patterns in a dataset, typically an unlabeled dataset. E.g. K-means, Autoencoders.
Reinforcement Learning: A family of algorithms that learn an optimal policy, whose goal is to maximize return when interacting with an environment. For example, the ultimate reward of most games is victory. Reinforcement learning systems can become expert at playing complex games by evaluating sequences of previous game moves that ultimately led to wins and sequences that ultimately led to losses. E.g. DQN, A3C.
Loss function: Or also called cost function, evaluates the error between the prediction and the ground truth label in every batch. For instance:
Mean Squared Error (MSE): measures the average squared difference between the actual and predicted labels in the form of:
MSE = \frac{1}{N}\sum_{i=1}^{n}{(y_i-(mx_i+b))^2}
Data augmentation: Regularization method used to decrease the model's variance error consisting in artificially increasing the number and variance of training samples by transforming existing samples to create additional samples. For example, if images are one of the system features, data augmentation can rotate, crop, and reflect each image to produce many variants of the original, yielding more variate labeled data to decrease the model's error.
Fine-tuning: Technique used to re-train a pre-trained neural network (usually in a transfer learning setting) on a new task with training data from a new domain where the weights of some layers or the whole network may be updated.
Batch size: The batch size is attributed to the number of training samples in one forward or backward pass. It is important to highlight that the higher the batch size, the more memory will be needed.
Iterations: The number of iterations is the number forward or backward of passes: each pass using a batch size number of images.
Epoch: The number of epochs measures how many times every image has been seen during training (i.e. one epoch means that every sample has been seen once). It can be also understood as a one forward pass and one backward pass of all the training examples.
epochs = \frac{batch\_size * iterations}{training\_ images}
Decay: The weight decay is an additional weight update parameter that induces the weights to exponentially decay to zero once the update process is over.
Learning rate: The learning rate parameter defines the step size for which the weights of a model are updated regarding the stochastic gradient descent.
Accuracy: Computes the number of correct predictions divided by the total number of samples.
Sensitivity: Also known as recall, is computed as the fraction of true positives that are correctly identified.
Precision: Is computed as the fraction of retrieved instances that are relevant, the ratio between true positives and predicted positives.
Specificity: Computed as the fraction of true negatives that are correctly identified.
Confusion Matrix: Table layout that allows visualization of the performance of an algorithm.
Confusion Matrix and Metrics (Courtesy of Wikipedia)
Handling null values: Drop samples with null values or replace the null value with some predefined value (e.g. feature mean or median).
Standardization: transform our features values such that the mean of the values
is 0 and the standard deviation is 1.
Handling categorical variables: A categorical variable is a variable whose values take on the value of labels. For example, the variable may be color and may take on the values “purple,” “white,” and “black.” There are many ways to encode categorical variables, although the three most common are as follows:
One Hot Encoding: Where each label is mapped to a binary vector.
Integer Encoding: Where each unique label is mapped to an integer.
Learned Embedding: Where a distributed representation of the categories is learned.
Mean subtraction: in order to center the cloud of RGB values from input data around zero along every dimension of the image, a mean subtraction can be applied across the image features.
Image normalization: By dividing each RGB dimension of input images by its standard deviation, a normalization is obtained from its original 0 and 255-pixel values to 1 and 0 normalized values. This preprocessing technique will avoid further issues caused by poor contrast images.
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Special functions - Simple English Wikipedia, the free encyclopedia
functions which have established names and notations due to their importance in mathematics
Special functions are some mathematical functions used in mathematical analysis or physics.[1][2][3] Most of them appear in higher education. Some experts are studying numerical methods for them.[4]
In mathematics, most functions are defined as a solution of a differential equation.[1] For example, the exponential function
{\displaystyle \exp(x)}
is the solution of the ordinary differential equation
{\displaystyle y^{\prime }=y}
. Due to this relation, some mathematicians are studying the connection between ODEs and special functions.[1][5]
Gamma function, it is studied since Euler[1][6][7]
Orthogonal polynomials, these are polynomials with special properties.[8][9][10]
Matrix functions, these are studied in linear algebra and matrix analysis.[11]
For more examples, find textbooks named "special functions".
↑ 1.0 1.1 1.2 1.3 Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.
↑ Silverman, R. A. (1972). Special functions and their applications. Courier Corporation.
↑ Nikiforov, A. F., & Uvarov, V. B. (1988). Special functions of mathematical physics (Vol. 205). Basel: Birkhäuser.
↑ Gil, A., Segura, J., & Temme, N. M. (2007). Numerical methods for special functions. Society for Industrial and Applied Mathematics.
↑ Iwasaki, K., Kimura, H., Shimemura, S., & Yoshida, M. (2013). From Gauss to Painlevé: a modern theory of special functions (Vol. 16). Springer Science & Business Media.
↑ Davis, P. J. (1959). Leonhard euler's integral: A historical profile of the gamma function. The American Mathematical Monthly, 66(10), 849-869.
↑ Artin, E. (2015). The gamma function. Courier Dover Publications.
↑ Gautschi, W. (2004). Orthogonal polynomials. Oxford: Oxford University Press.
↑ Cohl, H. S., & Ismail, M. E. (Eds.). (2020). Lectures on Orthogonal Polynomials and Special Functions (Vol. 464). Cambridge University Press.
↑ Ismail, M., Ismail, M. E., & van Assche, W. (2005). Classical and quantum orthogonal polynomials in one variable (Vol. 13). Cambridge University Press.
↑ Higham, N. J. (2008). Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics.
National Institute of Standards and Technology, United States Department of Commerce. NIST Digital Library of Mathematical Functions. Archived from the original on December 13, 2018.
Eric W. Weisstein, Special Function at MathWorld.
Special functions at EqWorld: The World of Mathematical Equations
Special functions and polynomials by Gerard 't Hooft and Stefan Nobbenhuis (April 8, 2013)
Numerical Methods for Special Functions, by A. Gil, J. Segura, N.M. Temme (2007).
R. Jagannathan, (P,Q)-Special Functions
Specialfunctionswiki, a wiki about special functions
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Special_functions&oldid=7101252"
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student(deprecated)/Point - Maple Help
Home : Support : Online Help : student(deprecated)/Point
check for type Point
type(P,Point)
type(P,`student/Point`)
A Point is defined as a list of coordinate values (such as
[a,b,c]
). The length of the list determines the dimension of the space.
Objects of type Point differ from objects of Maple type point (such as
{x=a,y=b,z=c}
) in that the coordinates are defined by position in the list rather than by equations. The substitution
\mathrm{subs}\left({x=a,y=b,z=c},[x,y,z]\right)
converts a point
{x=a,y=b,z=c}
[a,b,c]
The short form of this type check is only available for interactive use. It is made available through use of an alias which is defined during the initialization of the student package (that is, by performing the command with(student)). Procedures which make use of this type check must make direct use of the full type name `student/Point`.
\mathrm{with}\left(\mathrm{student}\right):
\mathrm{type}\left(a,\mathrm{Point}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left([a],\mathrm{Point}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left([a+4,b],\mathrm{Point}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left([a+4,b],\mathrm{`student/Point`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
type[point]
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Unit Root Tests - MATLAB & Simulink - MathWorks América Latina
Test Simulated Data for a Unit Root
Test Time Series Data for Unit Root
Test Stock Data for a Random Walk
This example shows how to test univariate time series models for stationarity. It shows how to simulate data from four types of models: trend stationary, difference stationary, stationary (AR(1)), and a heteroscedastic, random walk model. It also shows that the tests yield expected results.
Simulate four time series.
T = 1e3; % Sample size
t = (1:T)'; % Time multiple
rng(142857); % For reproducibility
y1 = randn(T,1) + .2*t; % Trend stationary
Mdl2 = arima(D=1,Constant=0.2,Variance=1);
y2 = simulate(Mdl2,T,Y0=0); % Difference stationary
Mdl3 = arima(AR=0.99,Constant=0.2,Variance=1);
y3 = simulate(Mdl3,T,Y0=0); % AR(1)
sigma = (sin(t/200) + 1.5)/2; % Std deviation
e = randn(T,1).*sigma; % Innovations
y4 = filter(Mdl4,e,Y0=0); % Heteroscedastic
Plot the first 100 points in each series.
y = [y1 y2 y3 y4];
plot1 = plot(y(1:100,:));
plot1(1).LineWidth = 2;
plot1(3).LineStyle = ":";
title("\bf First 100 Periods of Each Series");
legend("Trend Stationary','Difference Stationary","AR(1)", ...
"Heteroscedastic",Location="NorthWest");
All of the models appear nonstationary and behave similarly. Therefore, you might find it difficult to distinguish which series comes from which model simply by looking at their initial segments.
Plot the entire data set.
plot2 = plot(y);
title("\bf Each Entire Series");
The differences between the series are clearer here:
The trend stationary series has little deviation from its mean trend.
The difference stationary and heteroscedastic series have persistent deviations away from the trend line.
The AR(1) series exhibits long-run stationary behavior; the others grow linearly.
The difference stationary and heteroscedastic series appear similar. However, that the heteroscedastic series has much more local variability near period 300, and much less near period 900. The model variance is maximal when
\mathrm{sin}\left(t/200\right)=1
100\pi \approx 314
. The model variance is minimal when
\mathrm{sin}\left(t/200\right)=-1
300\pi \approx 942
. Therefore, the visual variability matches the model.
Use the Augmented Dicky-Fuller test on the three growing series (y1, y2, and y4) to assess whether the series have a unit root. Since the series are growing, specify that there is a trend. In this case, the null hypothesis is
{H}_{0}:{y}_{t}={y}_{t-1}+c+{b}_{1}\Delta {y}_{t-1}+{b}_{2}\Delta {y}_{t-2}+{\epsilon }_{t}
and the alternative hypothesis is
{H}_{1}:{y}_{t}=a{y}_{t-1}+c+\delta t+{b}_{1}\Delta {y}_{t-1}+{b}_{2}\Delta {y}_{t-2}+{\epsilon }_{t}
. Set the number of lags to 2 for demonstration purposes.
hY1 = adftest(y1,Model="ts",Lags=2)
hY1 = logical
hY1 = 1 indicates that there is sufficient evidence to suggest that y1 is trend stationary. This is the correct decision because y1 is trend stationary by construction.
hY2 = 0 indicates that there is not enough evidence to suggest that y2 is trend stationary. This is the correct decision since y2 is difference stationary by construction.
hY4 = 0 indicates that there is not enough evidence to suggest that y4 is trend stationary. This is the correct decision, however, the Dickey-Fuller test is not appropriate for a heteroscedastic series.
Use the Augmented Dickey-Fuller test on the AR(1) series (y3) to assess whether the series has a unit root. Since the series is not growing, specify that the series is autoregressive with a drift term. In this case, the null hypothesis is
{H}_{0}:{y}_{t}={y}_{t-1}+{b}_{1}\Delta {y}_{t-1}+{b}_{2}\Delta {y}_{t-2}+{\epsilon }_{t}
{H}_{1}:{y}_{t}=a{y}_{t-1}+{b}_{1}\Delta {y}_{t-1}+{b}_{2}\Delta {y}_{t-2}+{\epsilon }_{t}
hY3 = adftest(y3,Model="ard",Lags=2)
hY3 = 1 indicates that there is enough evidence to suggest that y3 is a stationary, autoregressive process with a drift term. This is the correct decision because y3 is an autoregressive process with a drift term by construction.
Use the KPSS test to assess whether the series are unit root nonstationary. Specify that there is a trend in the growing series (y1, y2, and y4). The KPSS test assumes the following model:
{y}_{y}={c}_{t}+\delta t+{u}_{t}
{c}_{t}={c}_{t-1}+{\epsilon }_{t},
{u}_{t}
is a stationary process and
{\epsilon }_{t}
is an independent and identically distributed process with mean 0 and variance
{\sigma }^{2}
. Whether there is a trend in the model, the null hypothesis is
{H}_{0}:{\sigma }^{2}=0
(the series is trend stationary) and the alternative hypothesis is
{H}_{1}:{\sigma }^{2}>0
(not trend stationary). Set the number of lags to 2 for demonstration purposes.
hY1 = kpsstest(y1,Lags=2,Trend=true)
hY3 = kpsstest(y3,Lags=2)
All is tests result in the correct decision.
Use the variance ratio test on al four series to assess whether the series are random walks. The null hypothesis is
{H}_{0}
Var\left(\Delta {y}_{t}\right)
is constant, and the alternative hypothesis is
{H}_{1}
Var\left(\Delta {y}_{t}\right)
is not constant. Specify that the innovations are independent and identically distributed for all but y1. Test y4 both ways.
hY1 = vratiotest(y1)
hY2 = vratiotest(y2,IID=true)
hY4NotIID = vratiotest(y4)
hY4NotIID = logical
hY4IID = vratiotest(y4,IID=true)
hY4IID = logical
All tests result in the correct decisions, except for hY4_2 = 0. This test does not reject the hypothesis that the heteroscedastic process is an IID random walk. This inconsistency might be associated with the random seed.
Alternatively, you can assess stationarity using pptest
This example shows how to test a univariate time series for a unit root. It uses wages data (1900-1970) in the manufacturing sector. The series is in the Nelson-Plosser data set.
Load the Nelson-Plosser data. Extract the nominal wages data.
wages = DataTable.WN;
Trim the NaN values from the series and the corresponding dates (this step is optional because the test ignores NaN values).
wDates = dates(isfinite(wages));
wages = wages(isfinite(wages));
plot(wDates,wages)
title('Wages')
The plot suggests exponential growth.
Transform the data using the log function to linearize the series.
logWages = log(wages);
plot(wDates,logWages)
title('Log Wages')
The plot suggests that time series has a linear trend.
Test the null hypothesis that there is no unit root (trend stationary) against the alternative hypothesis that the series is a unit root process with a trend (difference stationary). Set 'Lags',7:2:11, as suggested in Kwiatkowski et al., 1992.
[h1,pValue1] = kpsstest(logWages,'Lags',7:2:11)
h1 = 1x3 logical array
pValue1 = 1×3
kpsstest fails to reject the null hypothesis that the wage series is trend stationary.
Test the null hypothesis that the series is a unit root process (difference stationary) against the alternative hypothesis that the series is trend stationary.
[h2,pValue2] = adftest(logWages,'Model','ts')
pValue2 = 0.8327
adftest fails to reject the null hypothesis that the wage series is a unit root process.
Because the results of the two tests are inconsistent, it is unclear that the wage series has a unit root. This is a typical result of tests on many macroeconomic series.
kpsstest has a limited set of calculated critical values. When it calculates a test statistic that is outside this range, the test reports the p-value at the appropriate endpoint. So, in this case, pValue reflects the closest tabulated value. When a test statistic lies inside the span of tabulated values, kpsstest linearly interpolates the p-value.
This example shows how to assess whether a time series is a random walk. It uses market data for daily returns of stocks and cash (money market) from the period January 1, 2000 to November 7, 2005.
Extract two series to test. The first column of data is the daily return of a technology stock. The last (14th) column is the daily return for cash (the daily money market rate).
tech1 = Data(:,1);
money = Data(:,14);
The returns are the logs of the ratios of values at the end of a day over the values at the beginning of the day.
Convert the data to prices (values) instead of returns. vratiotest takes prices as inputs, as opposed to returns.
tech1 = cumsum(tech1);
money = cumsum(money);
Plot the data to see whether they appear to be stationary.
plot(Dates,tech1);
title('Log(relative stock value)')
plot(Dates,money)
title('Log(accumulated cash)')
Cash has a small variability, and appears to have long-term trends. The stock series has a good deal of variability, and no definite trend, though it appears to increase towards the end.
Test whether the stock series matches a random walk.
[h,pValue,stat,cValue,ratio] = vratiotest(tech1)
vratiotest does not reject the hypothesis that a random walk is a reasonable model for the stock series.
Test whether an i.i.d. random walk is a reasonable model for the stock series.
[h,pValue,stat,cValue,ratio] = vratiotest(tech1,'IID',true)
vratiotest rejects the hypothesis that an i.i.d. random walk is a reasonable model for the tech1 stock series at the 5% level. Thus, vratiotest indicates that the most appropriate model of the tech1 series is a heteroscedastic random walk.
Test whether the cash series matches a random walk.
[h,pValue,stat,cValue,ratio] = vratiotest(money)
pValue = 4.6093e-145
vratiotest emphatically rejects the hypothesis that a random walk is a reasonable model for the cash series (pValue = 4.6093e-145). The removal of a trend from the series does not affect the resulting statistics.
[1] Kwiatkowski, D., P. C. B. Phillips, P. Schmidt and Y. Shin. “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root.” Journal of Econometrics. Vol. 54, 1992, pp. 159–178.
adftest | kpsstest | pptest | vratiotest
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Import User List Using Csv - Oobeya Docs
Learn about how to import multiple users by using csv format file in Oobeya !
In Oobeya, you can create multiple users at once by creating a file type in CSV format containing the user information of the desired people.
Example format of CSV file as in the following:
name,surname,email
Mike,Miller,<a href="/cdn-cgi/l/email-protection" class="__cf_email__" data-cfemail="3c515557591251555050594e7c48594f485552554951125f5351">[email protected]</a>
Stephen,Curry,<a href="/cdn-cgi/l/email-protection" class="__cf_email__" data-cfemail="f4878091849c919ada978186868db4809187809d9a9d8199da979b99">[email protected]</a>
Importing the User List
Navigate to Administration Panel > Go To Admin Settings.
Click the "Import User User" button.
4. Select your CSV file and click on the 'Import' button.
Please make sure that your CSV file does not include any syntax error. Otherwise you may see an error message.
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Oracle and Prices - Integral SIZE
This page explains the working mechanism of Integral SIZE's oracle, and summarizes how the protocol make sure that the time-weighted average price we use is fair, reliable, and hard to manipulate.
What's the role of oracle in Integral SIZE?
Oracle in the Integral SIZE is a smart contract that holds the reference to the external price oracle and contains the math that swaps involve (given tokenIn calculate tokenOut etc).
Currently SIZE only supports Uniswap v2 oracle.
What is unique about Integral SIZE and its oracle?
DEXs like Uniswap relies on curves to deduce price between two assets based on their amounts in the pool. A large swap will both increase and decrease the amount of assets, thus changing the ratio/price substantially and resulting price impact.
This does not apply to Integral SIZE, whose execution prices is solely determined by the oracle. If there is enough capital in the liquidity pool, your order will be completely filled with 1 SINGLE TWAP, without experiencing any price impact.
Which oracle does Integral SIZE use?
What makes Uniswap v2 qualified for being the oracle for Integral SIZE?
Arbitrageurs help Uniswap v2's price stay close to the market price
Since arbitrageurs will trade with Uniswap if the marginal price offered by Uniswap is incorrect (by a sufficient amount to make up for the fee), the price offered by Uniswap tends to track the relative market price of the assets.
A hard-to-manipulate design
Uniswap V2 improves its oracle functionality by measuring and recording the price before the first trade of each block (or equivalently, after the last trade of the previous block). This price is more difficult to manipulate than prices during a block.
TWAP-ready
Uniswap V2 accumulates this price (known as price accumulator), by keeping track of the cumulative sum of prices at the beginning of each block in which someone interacts with the contract. Each price is weighted by the amount of time that has passed since the last block in which it was updated, according to the block timestamp. This means that the accumulator value at any given time (after being updated) should be the sum of the spot price at each second in the history of the contract.
Below is a simplified model of Uniswap V2's price accumulation in a ETH-USDC pool. Blue boxes represent the price accumulator in each block, while the white boxes represent prices of all swaps that take place in each block.
On t=0 sec, Block #1 is mined, and it contains multiple ETH-USDC swaps. The last swap of this block was executed with the price 1 ETH = 4001 USDC.
On t=12 sec, Block #2 is mined. The price accumulator of this block (denoted by a) will multiply the price of the last swap in Block #1 with the time interval between Block #1 and #2, which gives us 4001*12.
On t=24 sec, Block #3 is mined. After multiplying the price of the last swap in Block #2 with the time interval between Block #2 and #3, the price accumulator will also add its counterpart value in Block #2. Therefore, price accumulator in Block #3 is 4001*12+4004*12.
On t=37 sec, Block #4 is mined. Following the rules stated above, price accumulator of this block will be 4001*12+4004*12+4008*13.
To calculate the TWAP from t1 = 12 to t2 = 37, all we need to do is calculating
(a_{t2}-a_{t1})/(t_2-t_1)
which gives us 4006.08.
How Integral SIZE interacts with Uniswap v2 to calculate TWAP?
Once the order is submitted, it has to wait in the smart contract contract for 30 minutes before interacting with the pool. During this time, the protocol will query Uniswap v2 twice: once when the delay starts, and once when the delay ends.
After the delay time elapses, Integral SIZE will calculate the TWAP by doing the calculations above. If the actual amount of tokens you'll receive from TWAP is larger than or equal to the Minimum Received, your order will be executed at TWAP, and you will receive any positive slippage incurred. Otherwise it will be reverted to protect your funds from excessive slippage.
Why Integral SIZE Uses Uniswap v2 as Oracle instead of v3?
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heat transfer - Maple Help
Home : Support : Online Help : Science and Engineering : Units : Known Units : heat transfer
Units of Heat Transfer
Heat transfer has the dimension mass per time cubed thermodynamic temperature. The SI composite unit of heat transfer is the kilogram per second cubed kelvin.
Maple knows the units of heat transfer listed in the following table.
Ufactors
USI @
The units of heat transfer are defined as follows.
A standard U factor is defined as
1
international table Btu per hour square foot degree Fahrenheit.
An SI U factor (USI) is defined as
1
watt per square meter kelvin.
\mathrm{convert}\left('\mathrm{Ufactor}','\mathrm{dimensions}','\mathrm{base}'=\mathrm{true}\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{mass}}}{\textcolor[rgb]{0,0,1}{\mathrm{thermodynamic_temperature}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{time}}}^{\textcolor[rgb]{0,0,1}{3}}}
\mathrm{convert}\left(1,'\mathrm{units}','\mathrm{Ufactor}','\mathrm{USI}'\right)
\frac{\textcolor[rgb]{0,0,1}{52752792631}}{\textcolor[rgb]{0,0,1}{9290304000}}
\mathrm{convert}\left(1,'\mathrm{units}','\mathrm{USI}',\frac{'\mathrm{kg}'}{{'s'}^{3}'K'}\right)
\textcolor[rgb]{0,0,1}{1}
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Please solve the Q 14 Which of the following frequencies of sound you hear 2 Hz, 400 Hz, - Physics - Propagation of Sound Waves - 12300367 | Meritnation.com
Q.14. Which of the following frequencies of sound you hear? 2 Hz, 400 Hz, 34000 Hz,
{10}^{6}
Mgjjfhjguhy answered this
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Linear Matrix Inequalities - MATLAB & Simulink - MathWorks United Kingdom
LMI Features
LMIs and LMI Problems
See [9] for a good introduction to LMI concepts. Robust Control Toolbox™ software is designed as an easy and progressive gateway to the new and fast-growing field of LMIs:
For users who occasionally need to solve LMI problems, the LMI Editor and the tutorial introduction to LMI concepts and LMI solvers provide for quick and easy problem solving.
For more experienced LMI users, LMI Lab, offers a rich, flexible, and fully programmable environment to develop customized LMI-based tools.
Robust Control Toolbox LMI functionality serves two purposes:
Examples of LMI-based analysis and design tools include
Functions to analyze the robust stability and performance of uncertain systems with varying parameters (popov)
Functions to design robust control with a mix of H2, H∞, and pole placement objectives (h2hinfsyn)
Functions for synthesizing robust gain-scheduled H∞ controllers (hinfgs)
Robust Control Toolbox software implements state-of-the-art interior-point LMI solvers. While these solvers are significantly faster than classical convex optimization algorithms, you should keep in mind that the complexity of LMI computations can grow quickly with the problem order (number of states). For example, the number of operations required to solve a Riccati equation is o(n3) where n is the state dimension, while the cost of solving an equivalent “Riccati inequality” LMI is o(n6).
A linear matrix inequality (LMI) is any constraint of the form
A(x) := A0 + x1A1 + ... + xNAN < 0 (1)
x = (x1, . . . , xN) is a vector of unknown scalars (the decision or optimization variables)
A0, . . . , AN are given symmetric matrices
< 0 stands for “negative definite,” i.e., the largest eigenvalue of A(x) is negative
Note that the constraints A(x) > 0 and A(x) < B(x) are special cases of Equation 1 since they can be rewritten as –A(x) < 0 and A(x) – B(x) < 0, respectively.
The LMI of Equation 1 is a convex constraint on x since A(y) < 0 and A(z) < 0 imply that
A\left(\frac{y+z}{2}\right)<0
Its solution set, called the feasible set, is a convex subset of RN
Finding a solution x to Equation 1, if any, is a convex optimization problem.
Convexity has an important consequence: even though Equation 1 has no analytical solution in general, it can be solved numerically with guarantees of finding a solution when one exists. Note that a system of LMI constraints can be regarded as a single LMI since
\left\{\begin{array}{c}{A}_{1}\left(x\right)<0\\ ⋮\\ {A}_{K}\left(x\right)<0\end{array}
A\left(x\right):=\text{diag}\left({\text{A}}_{\text{1}}\left(x\right),\dots ,{\text{A}}_{\text{K}}\left(x\right)\right)<0
where diag (A1(x), . . . , AK(x)) denotes the block-diagonal matrix with
A1(x), . . . , AK(x) on its diagonal. Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity.
In most control applications, LMIs do not naturally arise in the canonical form of Equation 1 , but rather in the form
L(X1, . . . , Xn) < R(X1, . . . , Xn)
where L(.) and R(.) are affine functions of some structured matrix variables X1, . . . , Xn. A simple example is the Lyapunov inequality
ATX + XA < 0 (2)
where the unknown X is a symmetric matrix. Defining x1, . . . , xN as the independent scalar entries of X, this LMI could be rewritten in the form of Equation 1. Yet it is more convenient and efficient to describe it in its natural form Equation 2, which is the approach taken in the LMI Lab.
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superlist - Maple Help
Home : Support : Online Help : Programming : Logic : Boolean : verify : superlist
verify that the first list is a superlist of the second
verify(expr1, expr2, superlist)
verify(expr1, expr2, superlist(ver))
The verify(expr1, expr2, superlist) and verify(expr1, expr2, superlist(ver)) calling sequences return true if it can be determined that the first list contains the second list as a sublist, where this relationship is satisfied operandwise, either by testing with equality or using the verification ver.
If true is returned, then for one sublist of the first list, each of the pairs of operands satisfied the relation. If false is returned, then for all sublists of the first list at least one pair of operands did not satisfy the relation (a result of type verify(false)). Otherwise, FAIL is returned, indicating that for every sublist at least one pair of operands it could not be determined whether or not they satisfied the relation (a result of type verify(FAIL)).
\mathrm{verify}\left([a,b,c,d,e],[a,b,c],'\mathrm{superlist}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left([a,b,c,d,e],[a,b,c,f],'\mathrm{superlist}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left([a,b,{x}^{2}-x],[a,b,x\left(x-1\right)],'\mathrm{superlist}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left([a,b,{x}^{2}-x],[a,b,x\left(x-1\right)],'\mathrm{superlist}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left([a,b,{x}^{2}-x,c],[a,b,x\left(x-1\right)],'\mathrm{superlist}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left([0.2334,0.102221],[0.2333,0.10222],'\mathrm{superlist}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left([0.2334,0.102221],[0.2333,0.10222],'\mathrm{superlist}\left(\mathrm{float}\left({10}^{6}\right)\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Surrogate optimization for global minimization of time-consuming objective functions - MATLAB surrogateopt - MathWorks América Latina
\underset{x}{\mathrm{min}}f\left(x\right)\text{ such that }\left\{\begin{array}{l}\text{lb}\le x\le \text{ub}\\ A·x\le b\\ \text{Aeq}·x=\text{beq}\\ c\left(x\right)\le 0\\ {x}_{i}\text{ integer, }i\in \text{intcon}\text{.}\end{array}
100\left(x\left(2\right)-x\left(1{\right)}^{2}{\right)}^{2}+\left(1-x\left(1\right){\right)}^{2}
\left(x\left(1\right)-1/3{\right)}^{2}+\left(x\left(2\right)-1/3{\right)}^{2}\le \left(1/3{\right)}^{2}
c\left(x\right)\le 0
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(→Prior Art)
[[METABOLITE LABORATORIES, INC. and Competitive Technologies, Inc. v. LABORATORY CORPORATION OF AMERICA HOLDINGS (doing business as LabCorp): the CAFC case (full text)]]
[[Bendix Corp. v. Balax, Inc.]]
[[Chester v. Miller (full text)]]
[[Chester v. Miller]]
[[In re Hall (full text)]]
[[In re Hall]]
[[Philips Electric Co. v. Thermal Industries, Inc. (full text)]]
[[Philips Electric Co. v. Thermal Industries, Inc.]]
[[Universal Athletic Sales Co. v. American Gym Recreational & Athletic Equipment Corporation, Inc. (full text)]]
[[Universal Athletic Sales Co. v. American Gym Recreational & Athletic Equipment Corporation, Inc.]]
[[Hotchkiss v. Greenwood (full text)]]
[[Hotchkiss v. Greenwood]]
[[Reiner v. I. Leon Co. (full text)]]
[[Reiner v. I. Leon Co.]]
[[Pfaff v. Wells Electronics: full text]]
Justice clipart, copyright [http://etc.usf.edu/clipart/ FCIT.]
{\displaystyle {\dot {x}}=Ax+Bu}
{\displaystyle {\dot {x}}=f(x)+g(x)u}
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Comment #3822 by Johannes Anschuetz on December 03, 2018 at 10:25
The "Y" in the diagram seems to mean "S".
Comment #4927 by robot0079 on February 14, 2020 at 07:30
Here is a quick proof.
X/S
When S is a strict henselian local ring, this result follows from the fact that
X(S)\to X(k)
is surjective, where k is residue field of S. We can prove this by factoring it into composition of etale and relative affine space morphism.
Now in general case. From above we see that
X(O_{S, \tilde{s}})
is non empty, we deduce our result by limit preserving property of morphism of locally of finite presentation.
@#4927. Yes, this is an outline of a proof.
Comment #5970 by Dario Weißmann on March 11, 2021 at 10:45
typo: W should be defined as
\pi(U)\cap ...
\pi(V)\cap...
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Relaxation (iterative method) - Wikipedia
Relaxation (iterative method)
(Redirected from Relaxation method)
This article is about iterative methods for solving systems of equations. For other uses, see Relaxation (disambiguation).
In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems.[1]
Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations.[2][3] They are also used for the solution of linear equations for linear least-squares problems[4] and also for systems of linear inequalities, such as those arising in linear programming.[5][6][7] They have also been developed for solving nonlinear systems of equations.[1]
Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences.[4][3][2]
Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation, it resembles repeated application of a local smoothing filter to the solution vector. These are not to be confused with relaxation methods in mathematical optimization, which approximate a difficult problem by a simpler problem whose "relaxed" solution provides information about the solution of the original problem.[7]
1 Model problem of potential theory
2 Convergence and acceleration
Model problem of potential theory[edit]
When φ is a smooth real-valued function on the real numbers, its second derivative can be approximated by:
{\displaystyle {\frac {d^{2}\varphi (x)}{{dx}^{2}}}={\frac {\varphi (x{-}h)-2\varphi (x)+\varphi (x{+}h)}{h^{2}}}\,+\,{\mathcal {O}}(h^{2})\,.}
Using this in both dimensions for a function φ of two arguments at the point (x, y), and solving for φ(x, y), results in:
{\displaystyle \varphi (x,y)={\tfrac {1}{4}}\left(\varphi (x{+}h,y)+\varphi (x,y{+}h)+\varphi (x{-}h,y)+\varphi (x,y{-}h)\,-\,h^{2}{\nabla }^{2}\varphi (x,y)\right)\,+\,{\mathcal {O}}(h^{4})\,.}
To approximate the solution of the Poisson equation:
{\displaystyle {\nabla }^{2}\varphi =f\,}
numerically on a two-dimensional grid with grid spacing h, the relaxation method assigns the given values of function φ to the grid points near the boundary and arbitrary values to the interior grid points, and then repeatedly performs the assignment φ := φ* on the interior points, where φ* is defined by:
{\displaystyle \varphi ^{*}(x,y)={\tfrac {1}{4}}\left(\varphi (x{+}h,y)+\varphi (x,y{+}h)+\varphi (x{-}h,y)+\varphi (x,y{-}h)\,-\,h^{2}f(x,y)\right)\,,}
until convergence.[3][2]
The method, sketched here for two dimensions,[3][2] is readily generalized to other numbers of dimensions.
Convergence and acceleration[edit]
While the method converges under general conditions, it typically makes slower progress than competing methods. Nonetheless, the study of relaxation methods remains a core part of linear algebra, because the transformations of relaxation theory provide excellent preconditioners for new methods. Indeed, the choice of preconditioner is often more important than the choice of iterative method.[8]
Multigrid methods may be used to accelerate the methods. One can first compute an approximation on a coarser grid – usually the double spacing 2h – and use that solution with interpolated values for the other grid points as the initial assignment. This can then also be done recursively for the coarser computation.[8][9]
In linear systems, the two main classes of relaxation methods are stationary iterative methods, and the more general Krylov subspace methods.
The Jacobi method is a simple relaxation method.
The Gauss–Seidel method is an improvement upon the Jacobi method.
Successive over-relaxation can be applied to either of the Jacobi and Gauss–Seidel methods to speed convergence.
^ a b Ortega, J. M.; Rheinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. Classics in Applied Mathematics. Vol. 30 (Reprint of the 1970 Academic Press ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. xxvi+572. ISBN 0-89871-461-3. MR 1744713.
^ a b c d Richard S. Varga 2002 Matrix Iterative Analysis, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.
^ a b c d David M. Young, Jr. Iterative Solution of Large Linear Systems, Academic Press, 1971. (reprinted by Dover, 2003)
^ a b Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. ISBN 0-89871-321-8.
^ Murty, Katta G. (1983). "16 Iterative methods for linear inequalities and linear programs (especially 16.2 Relaxation methods, and 16.4 Sparsity-preserving iterative SOR algorithms for linear programming)". Linear programming. New York: John Wiley & Sons Inc. pp. 453–464. ISBN 0-471-09725-X. MR 0720547.
^ Goffin, J.-L. (1980). "The relaxation method for solving systems of linear inequalities". Math. Oper. Res. 5 (3): 388–414. doi:10.1287/moor.5.3.388. JSTOR 3689446. MR 0594854.
^ a b Minoux, M. (1986). Mathematical programming: Theory and algorithms. Egon Balas (foreword) (Translated by Steven Vajda from the (1983 Paris: Dunod) French ed.). Chichester: A Wiley-Interscience Publication. John Wiley & Sons, Ltd. pp. xxviii+489. ISBN 0-471-90170-9. MR 0868279. (2008 Second ed., in French: Programmation mathématique: Théorie et algorithmes. Editions Tec & Doc, Paris, 2008. xxx+711 pp. . ).
^ a b Yousef Saad, Iterative Methods for Sparse Linear Systems, 1st edition, PWS, 1996.
^ William L. Briggs, Van Emden Henson, and Steve F. McCormick (2000), A Multigrid Tutorial Archived 2006-10-06 at the Wayback Machine (2nd ed.), Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-462-1.
Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. ISBN 0-89871-321-8.
Ortega, J. M.; Rheinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. Classics in Applied Mathematics. Vol. 30 (Reprint of the 1970 Academic Press ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. xxvi+572. ISBN 0-89871-461-3. MR 1744713.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 18.3. Relaxation Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
Yousef Saad, Iterative Methods for Sparse Linear Systems, 1st edition, PWS, 1996.
Richard S. Varga 2002 Matrix Iterative Analysis, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.
David M. Young, Jr. Iterative Solution of Large Linear Systems, Academic Press, 1971. (reprinted by Dover, 2003)
Southwell, R.V. (1940) Relaxation Methods in Engineering Science. Oxford University Press, Oxford.
Southwell, R.V. (1946) Relaxation Methods in Theoretical Physics. Oxford University Press, Oxford.
John. D. Jackson (1999). Classical Electrodynamics. New Jersey: Wiley. ISBN 0-471-30932-X.
M.N.O. Sadiku (1992). Numerical Techniques in Electromagnetics. Boca Raton: CRC Pres.
P.-B. Zhou (1993). Numerical Analysis of Electromagnetic Fields. New York: Springer.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Relaxation_(iterative_method)&oldid=1014258690"
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EvoComposer: An Evolutionary Algorithm for 4-Voice Music Compositions | Evolutionary Computation | MIT Press
R. De Prisco,
R. De Prisco
Dipartimento di Informatica, University of Salerno, Fisciano (SA), 84084, Italy robdep@unisa.it
G. Zaccagnino,
G. Zaccagnino
Dipartimento di Informatica, University of Salerno, Fisciano (SA), 84084, Italy zaccagnino.gianluca@gmail.com
R. Zaccagnino
Dipartimento di Informatica, University of Salerno, Fisciano (SA), 84084, Italy rzaccagnino@unisa.it
R. De Prisco, G. Zaccagnino, R. Zaccagnino; EvoComposer: An Evolutionary Algorithm for 4-Voice Music Compositions. Evol Comput 2020; 28 (3): 489–530. doi: https://doi.org/10.1162/evco_a_00265
Evolutionary algorithms mimic evolutionary behaviors in order to solve problems. They have been successfully applied in many areas and appear to have a special relationship with creative problems; such a relationship, over the last two decades, has resulted in a long list of applications, including several in the field of music. In this article, we provide an evolutionary algorithm able to compose music. More specifically we consider the following 4-voice harmonization problem: one of the 4 voices (which are bass, tenor, alto, and soprano) is given as input and the composer has to write the other 3 voices in order to have a complete 4-voice piece of music with a 4-note chord for each input note. Solving such a problem means finding appropriate chords to use for each input note and also finding a placement of the notes within each chord so that melodic concerns are addressed. Such a problem is known as the unfigured harmonization problem. The proposed algorithm for the unfigured harmonization problem, named EvoComposer, uses a novel representation of the solutions in terms of chromosomes (that allows to handle both harmonic and nonharmonic tones), specialized operators (that exploit musical information to improve the quality of the produced individuals), and a novel hybrid multiobjective evaluation function (based on an original statistical analysis of a large corpus of Bach's music). Moreover EvoComposer is the first evolutionary algorithm for this specific problem. EvoComposer is a multiobjective evolutionary algorithm, based on the well-known NSGA-II strategy, and takes into consideration two objectives: the harmonic objective, that is finding appropriate chords, and the melodic objective, that is finding appropriate melodic lines. The composing process is totally automatic, without any human intervention. We also provide an evaluation study showing that EvoComposer outperforms other metaheuristics by producing better solutions in terms of both well-known measures of performance, such as hypervolume,
Δ
index, coverage of two sets, and standard measures of music creativity. We conjecture that a similar approach can be useful also for similar musical problems.
Evolutionary algorithms, automatic music composition, evolutionary music.
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Neutralizing Agents Odour Gas Industry
What are the Odour Neutralizers?
These products are the result of the study of natural plant substances (flowers, fruits, herbs, roots, etc.) and the development of processes of industrial biosynthesis in order to design complex mixtures of fragrances and essential oils which can neutralize any odour and gas emission coming from industrial manufacturing processes, wastewater treatment plants and zootechnics.
Unlike the traditional air fresheners, the odour neutralizers of the MasterDeo line do not mask the smells in the air but neutralize them completely.
The neutralization relies on the chemical or physical destruction of the molecules that build the odours.
We know how to eliminate the odours, without masking them.
How do we measure the odours?
What do the odour neutralizers contain?
Analytical methods: gas chromatography, mass spectrometry valid for specific substances dissolved in gaseous fluids. They do not reveal the origin of olfactory harassment .
Olfactometric method: human noses, appropriately selected assessors trained to recognize the intensity and the pleasantness of odours.
Sensory methods: electronic noses, biosensors that combine the action of olfactory receptors and physical-chemical transducers of the signal.
Olfactometric Unit (OU) expresses a value that combines the odour concentration threshold and its air dilution.
vegetable extracts of plants, herbs, flowers, fruits and essential oils, properly mixed to act specifically against any odour molecule.
common excipients (propylene glycol) and surfactants (which make them soluble in water).
How does the neutralization of odours work against some specific mixtures that contain these functional groups?
{H}_{2}S
: the process of chemical decomposition of
{H}_{2}S
occurs by chemical interaction of the amine functional group
R\text{-}N{H}_{3}
R\text{-}N{H}_{3}+{H}_{2}S
R\text{-}N{H}_{3}+SH
R\text{-}N{H}_{3}+SH+{O}_{2}+{H}_{2}O
R\text{-}N{H}_{3}+S{O}_{4}+OH
R\text{-}N{H}_{3}+S{O}_{4}+OH
R\text{-}N{H}_{3}+S{O}_{4}+{H}_{2}O
in that reaction, the hydrogen sulphide is captured by the amino functional group. With oxygen, a non-volatile compound containing sulphate is then formed.
N{H}_{3}
N{H}_{3}
is made per condensation by the functional acid group
R\text{-}COOH
R\text{-}COOH+N{H}_{3}
R\text{-}COON{H}_{4}
the ammonia is captured by the functional acid group
COOH
and a no more volatile compound containing ammonium salt is formed.
The odour neutralisation occurs with different chemical reactions for each substance which has to be treated:
Construction the foul-smelling gas reacts chemically with the neutralizing molecule and forms an odourless compound (ammonia).
Combination the foul-smelling gas is combined with a pleasant odour forming a third lightly scented or unscented product (chlorine derivatives).
Counteraction when you add to a bad smelling substance another one that opposes to it, interferes with it and finally eliminates the odour (derivatives of sulphur).
Absorption a foul-smelling substance is absorbed by the neutralizer and forms an odourless product (styrene).
Chemical reaction all the ingredients of the neutralizer interact with the bad odours and form an odourless product (mercaptans).
Collection the micelles of some ingredients capture the odour molecules in the air, they create then a heavy drop that falls to the ground and holds the odours.
Neutralization in the air: indoor application, wastewater treatment plants, sludge accumulation tanks, porous floors, toilets, urinals, scrubbers and fireplaces.
Contact neutralization: landfills, composting plants, waste disposal, sludge tanks, wastewater treatment plants, septic tanks, stables, farms, chemical toilets and well drains
The odour neutralization against the smell pollution
The problem of the odours from industrial and composting plants, landfills, wastewater treatment plants etc. is becoming more and more important. Nowadays at a national level there are no specific regulations and no limit values for the odour emissions. The current legislation about the air quality, the waste and the wastewater treatment refers however to the prevention and the reduction of odour annoyance. Some Regions have already introduced some laws and regulations regarding the reduction of the odour emissions and have set the upper limits of emissions in the atmosphere before the forthcoming adaptation to European guidelines. The “odour annoyance” is part of the regulations concerning the emissions of pollutants in the atmosphere, even though there are no benchmarks to verify the intensity of the odour emissions. The bad smell is considered as a true form of environmental pollution.
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Pendulum Motion - Maple Help
Home : Support : Online Help : Math Apps : Natural Sciences : Physics : Pendulum Motion
This demonstration shows how the length of a pendulum's arm and the acceleration due to gravity influence the speed of a pendulum's motion. You can choose any value between 5 and 30 meters for the length of the arm and any value between 1 and 20 meters per second squared for the acceleration due to gravity.
The differential equation for the angle as a function of time
t
\frac{{ⅆ}^{2}\mathrm{θ}}{ⅆ {t}^{2}}+\frac{g}{L}\mathrm{sin}\left(\mathrm{\theta }\right)=0
It is not possible to write a formula for the solution to this equation in terms of an elementary function. Instead, we use an approximation which is fairly accurate if the angle
\mathrm{\theta }
is sufficiently small (i.e. when
\mathrm{sin}\left(\mathrm{θ}\right)≈\mathrm{θ}
\frac{{ⅆ}^{2}\mathrm{θ}}{ⅆ {t}^{2}}+\frac{g}{L}\mathrm{θ}=0
Solving this differential equation allows us to find formula for the angle of the pendulum at a given time
t
\mathrm{θ}\left(t\right)={\mathrm{θ}}_{0}\cdot \mathrm{cos}\left(\sqrt{\frac{g}{L}}t\right)
and the angular speed at a given time
t
\frac{ⅆ\mathrm{θ}}{ⅆ t}={\mathrm{θ}}_{0}\cdot \sqrt{\frac{g}{L}}\mathrm{sin}\left(\sqrt{\frac{g}{L}}t\right)
{\mathrm{\theta }}_{0}
is the initial angle of the pendulum. Note that the angular frequency of the pendulum is a constant
{\mathrm{ω}}_{0}=\sqrt{\frac{g}{L}}
Length of the pendulum's arm:
\mathrm{m}
\mathrm{m} {\mathrm{s}}^{-2}
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Computational Machine Learning & Data Science | Pathbird
Computational Machine Learning & Data Science
Introduction to computational methods for identifying patterns and outliers in large data sets. Topics include the singular and eigenvalue decomposition, independent component analysis, graph analysis, clustering, linear, regularized, sparse and non-linear model fitting, deep, convolutional and recurrent neural networks. The computational textbook teaches the material to students step by step, by doing via autograded programming exercises and conceptual multiple-choice quizzes. Every codex contains an application that illustrates the ideas behind the algorithm, an exploration of why the algorithm works and when it fails (and can or cannot be fixed) as a way to understand, via mathematical principles, the strengths and weakness of the algorithms.
Raj is an Associate Professor of Electrical Engineering and Computer Science at the University of Michigan, Ann Arbor.
He received his Masters and PhD in Electrical Engineering and Computer Science at MIT as part of the MIT/WHOI Joint Program in Ocean Science and Engineering. His work is at the interface of statistical signal processing and random matrix theory with applications such as sonar, radar, wireless communications and machine learning in mind. Developing this 'living' computational book and teaching EECS 505 with it has been a career highlight.
Over the past few years of teaching the Computational Machine Learning & Data Science course at the University of Michigan, we've conceived and iterated on over two dozen distinct codices. Each codex is like a chapter of the computational book and have been organized into several units below. Select codices are available as a demo of the Pathbird platform.
Students will be introduced to the Julia programming language. By the end, students will be familiar with variables, arrays, functions, and everything else in Julia that they need to succeed in this course.
An introduction to matrix math and linear algebra.
Students will learn about vectors, matrices, arrays, and various operations on these objects.
An introduction to the concept of convolution and expressing convolution as a matrix-vector product.
An introduction to normal and non-normal matrices and the spectral theorem for normal matrices, as well as the eigenvalue decomposition and the singular value decomposition and their variational characterizations via eigshow and svdshow.
An introduction to vector spaces, subspaces, and the four fundamental subspaces of a matrix, including a discussion of basis vectors for subspaces and how the SVD of a matrix reveals these bases, as well as orthogonal projection matrices and how to efficiently compute the projection of a vector onto a matrix subspace without first computing and storing the associated projection matrix.
How to setup and solve least squares problems of the form
Ax = b
. Applications include fitting data to a higher order polynomial function, predicting search query time series results after an appropriate non-linear transformation.
How and why we need to regularize the solution of a system of equations of the form
Ax = b
. Applications include better fitting data to a higher order polynomial function, image in-painting/graffiti removal with a first difference regularizer. Discussion of how the optimal regularization coefficient is selected.
Learning to recognize sparsity in its canonical and transformed manifestations and seeing (computationally) how that helps regularize the solution of a system of equations of the form
Ax = b
in a regime where minimum norm least squares does not work well. Applications include compressed sensing, image in-painting/graffiti removal with a first difference regularizer and a discussion of how the optimal regularization coefficient is selected.
Case study: Re-engineering matrix-matrix multiplication for BIG DATA matrices
Application illustrating how we need to split matrices into blocks and write a matrix-matrix multiplication routine from scratch for settings where the matrices are too big to fit into memory.
Machine Learning for supervised and unsupervised classification
Classification via the nearest subspace algorithm
Learning to tell apart multiple classes with the nearest subspace classification algorithm.
Classification via linear discriminant analysis
Fisher's linear discriminant analysis, the generalized eigen-problem and the
Ax = b
equivalent formulation
Classification via support vector machines
First principles derivation of the support vector machine and how its loss function is different from least squares classification. Learning to tell apart multiple classes with the SVM.
False positive, false negatives and how we can change the classification output to weight one or the other. How decision theory informs the choice of decision thresholds.
How to generate inputs that can cause a classifier to give erroneous outputs; how to train an algorithm to be robust to such errors.
Unsupervised learning via the k-means and k-mediods algorithm
What is unsupervised learning and how the k-means algorithm can be used for that.
Matrix Learning Methods, Algorithms & Applications
Learning to do foreground-background subtracting using the SVD
How to use the singular value decomposition (SVD) to isolate moving objects from the background of a video. Many real-world datasets consist of dynamic signals and noise on top of some static backdrop or baseline. The SVD can isolate these components, allowing us to focus on what we are interested in and disregard the rest.
Learning to complete a matrix with missing entries using the SVD
How to use the SVD to fill in missing elements of a matrix with known rank. Random matrix theory to predict sparsity and incoherence conditions when near-perfect recovery is possible. Applications to low rank image inpainting and collaborative filtering.
Learning to optimally project data via SVD, PCA and ICA
Implement Principal Component Analysis (PCA) and Independent Component Analysis (ICA). These methods reason about covariance and kurtosis to emphasize important data components.
Learning PCA and ICA via cumulant optimization
Cumulant Theory of ICA
Learning to align mis-aligned data
Viewing PCA and ICA as cumulant maximization problems. Understanding limits of each and finding ways to encode the error of higher order ICA and why kurtosis based ICA is the one used in practice.
Learning to embed data and unsupervised learning
Learn2Embed
Learning to find communities in graphs
Spectral algorithms for finding communities in a graph via modularity maximization. Applications include finding communities in a karate club dataset and inferring baseball team division structure from a dataset of how frequently they play each other. Investigation of limits of finding community structure in the stochastic block model.
Phase transitions and robustness to outliers
Phase transitions in PCA and examination of how to reason about unlearnability of structure due outlier-induced phase transitions in the low-rank-plus-noise-plus-sparse outliers data model.
Case study: Learning to find synchronized waveforms in data
Synchronized Waveforms
Shallow Neural Networks and universal function approximation
Use Flux.jl to quickly design and train single input, single output shallow neural networks for 1-D function approximation
Use Flux.jl to quickly design and train 1-D and 2-D deep feedforward neural nets. See the power of deep nets and critically examine what it means for conv. nets to be "shift-invariant". Applications include handwritten digit recognition and 1-D signal recognition.
Use Flux.jl to quickly design and train convolutional neural nets. Applications include handwritten digit recognition.
Intro to Recurrent Nets (under development)
Use Flux.jl to quickly design and train recurrent neural nets. Learn why they are used for sequence modeling, what they can do that Conv and Deep Feedforward nets cannot. Applications include image captioning, predicting the Fibonacci sequence and time series.
Neural nets for regression and function approximation
How to use neural networks in the context of regression and function approximation
Case study: Wine classification
Using methods learned to classify red wine versus white wine
Case study: Fruit Classification
Using methods learned to classify varieties of fruit
Case study: Rock, Paper, Scissor Classification
Using methods learned to tell apart gesures for "rock", "paper" and "scissors"
How to train a network to be robust to adversarial attacks
Deep, denoising and variational autoencoders. Application includes auto-encoding of handwritten digits.
Generative adversarial networks (under development)
Theory and practice include the choice of the loss function. Application includes generating realistic handwritten digits.
Can this help me in my existing machine learning class?
There is so much to learn in machine learning and so much about it is fun and exciting! Instructors usually run out of time before they run out of topics. The codices are designed to augment your (the instructor's) voice. Machine learning is tricky because it links math with code. Students need the instructor and the instructional staff's help when they get stuck -- and they will get stuck because computational machine learning algorithms require mastery math and programming and linking math to code and vice-versa.
The codices are self-contained and in that sense they are like a textbook. The computational component makes them more than that. They can be used as homework assignments to reinforce concepts in an instructor's lecture/notes or another textbook. We use it at Michigan as a lab component to the course. The instructor has peace of mind knowing that the codices have been tested on over a thousand students -- the platform allows an instructor to scale their to hundreds. The backend support for autograding programming assignments allows the instructional staff to engage a student at the moment when they are stuck when the learner is eager and ready to learn more. Codices will amplify the instructor's voice by linking in-class theory to computational practice.
Does it scale to large courses?
YES! The reason we built Pathbird was to scale our ability to effectively teach students. We've taught 260+ students in class working on the codex live in class. (See below) The codices and platform help amplify an instructor's voice while helping solve the "assignment grading" problem for computational machine learning so the class can scale easily into the hundreds.
We are actively recruiting instructors to try the platform and the codices in their own courses as a supplement for existing course material either as homeworks or as in-class computational labs. You can express your interest using this Google form and we will get back to you as soon as possible.
The Pathbird platform is currently in beta. The price for the book includes the cost of cloud-computing, hosting and storage resources since every codex involves the learning learning by doing via computing on the platform.
The learner personalizes their book via their answers for the auto-graded programming assignments, the various free form question prompts and their ability to take notes within the platform. There is a "expert" mode for each codex which allows the learner to deepen their mastery of a code that they have already solved. Thus learners can continue learning (for a nominal fee) even when the course is completed.
The codices for this course were initially developed and used to teach EECS 505 at the University of Michigan. It has also been used to teach classes at MIT and for private training camps across the country.
Special thanks to Travis DePrato for building Pathbird from scratch. Thanks to Don Winsor for the introduction and thinking (correctly) that Travis was capable of way more than he was doing then. This way of writing and publishing a computational book would not have been possible without him.
Thanks to Jonas Kersulis for editing and proof-reading the codices. Thanks to Brian Moore for helping create an early version of the autograder -- they are what give the assignments in the codices an extra "oomph!". Thanks to Adrian Stoll r creating the API for the autograder -- this was a big step towards Pathbird.
This book would not be possible without them and the various graduate student instructors (Arvind Prasadan, David Hong, Hao Wu, Rishi Sonthalia, Dipak Narayan, Yash Sanjay Bhalgat and Raj Tejas Suryaprakash) who helped edit, test and refine the codices, right before they were about to go live to hundreds of students. Thanks to Simon Danisch for the helping start this journey in 2017 by porting my MATLAB demos to Julia and to David Sanders for the gitter post on a Jupyter forum whose response by Min Ragan-Kelley, I used to initiate the first conversation with Travis.
Thanks in particular to Gil Strang for his encouragement, feedback and support and for inspiring the idea behind the codices during the very special semester of Spring 2017 when we launched and taught 18.065 at MIT. Thanks to Jeremy Kepner for making that semester happening and for the opportunity to teach the course at the MIT Lincoln Laboratories -- seeing the excitement there for the ideas and math gave me the impetus to want to do, and reach out, more to a more diverse audience than the "typical" grad students who had taken my class. Thanks to Muralidhar Rangaswamy for the opportunity to reprise the course at AFRL Dayton and the many scientists and engineers there whose enthusiasm for the material in the codex format gave me hope that this format could succeed beyond just at Michigan.
Multiple thanks to Alan Edelman for years of encouragement and inspiration and for teaching me so much (including Julia). A learner experiencing this book by doing/coding might sometimes recognize their voice in the way I write and speak about the underlying math and code. That's no accident. This book is infused with their DNA and years of me soaking in their thoughts and ideas on so many matters, particularly on how elegant math produces elegant codes and vice versa. All they taught me about how to see math and linear algebra makes me love it, and to want to share with you in the codex way, even more.
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hemisphere - Maple Help
Home : Support : Online Help : Graphics : Packages : Plot Tools : hemisphere
generate 3-D plot object for a hemisphere
hemisphere(c, r, options)
(optional) list(realcons); center of the base circle of the hemisphere
(optional) realcons; radius of the hemisphere
(optional) equations of the form option=value where option is capped or an option listed on the plot3d/options help page
capped : true or false
Specifies whether a cap should be drawn on the open end of the hemisphere. The default value of this option is true.
The hemisphere command creates a three-dimensional plot data object, which when displayed is a hemisphere centered at c with a radius r. The default values for c and r are
[0,0,0]
1
The plot data object produced by the hemisphere command can be used in a PLOT3D data structure, or displayed using the plots[display] command.
\mathrm{with}\left(\mathrm{plottools}\right):
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{display}\left(\mathrm{hemisphere}\left([1,1,1]\right),\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{boxed},\mathrm{orientation}=[45,75],\mathrm{color}=\mathrm{green}\right)
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Home : Support : Online Help : Programming : Logic : Boolean : verify : vector
verify a relation between the elements of two vectors
verify(expr1, expr2, vector)
verify(expr1, expr2, 'vector'(ver))
anything, assumed to be of type vector
verification for the vector elements
The verify(expr1, expr2, vector) and verify(expr1, expr2, 'vector'(ver)) calling sequences return true if it can be determined that the two vectors satisfy a relation elementwise, either by testing with equality or using the verification ver.
The verification vector is symmetric and a verification 'vector'(ver) is symmetric if and only if the verification ver is symmetric.
Because vector is a Maple function, it must be enclosed in single quotes to prevent evaluation.
If either expr1 or expr2 is not of type vector, then false is returned.
Important: The command vector has been superseded by Vector.
\mathrm{u1}≔\mathrm{vector}\left(2,[1,x\left(1-x\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{u1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)\end{array}]
\mathrm{u2}≔\mathrm{vector}\left(2,[1,x-{x}^{2}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{u2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\end{array}]
\mathrm{verify}\left(\mathrm{u1},\mathrm{u2},'\mathrm{vector}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(\mathrm{u1},\mathrm{u2},'\mathrm{vector}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{v1}≔\mathrm{vector}\left(2,[0.3222,0.5001]\right)
\textcolor[rgb]{0,0,1}{\mathrm{v1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.3222}& \textcolor[rgb]{0,0,1}{0.5001}\end{array}]
\mathrm{v2}≔\mathrm{vector}\left(2,[0.3223,0.5000]\right)
\textcolor[rgb]{0,0,1}{\mathrm{v2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.3223}& \textcolor[rgb]{0,0,1}{0.5000}\end{array}]
\mathrm{verify}\left(\mathrm{v1},\mathrm{v2},'\mathrm{vector}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(\mathrm{v1},\mathrm{v2},'\mathrm{vector}\left(\mathrm{float}\left({10}^{6}\right)\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Albert algebra - Wikipedia
In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism.[1] One of them, which was first mentioned by Pascual Jordan, John von Neumann, and Eugene Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation
{\displaystyle x\circ y={\frac {1}{2}}(x\cdot y+y\cdot x),}
{\displaystyle \cdot }
denotes matrix multiplication. Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution.
Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4.[2][3] (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).[4]
The Kantor–Koecher–Tits construction applied to an Albert algebra gives a form of the E7 Lie algebra. The split Albert algebra is used in a construction of a 56-dimensional structurable algebra whose automorphism group has identity component the simply-connected algebraic group of type E6.[5]
The space of cohomological invariants of Albert algebras a field F (of characteristic not 2) with coefficients in Z/2Z is a free module over the cohomology ring of F with a basis 1, f3, f5, of degrees 0, 3, 5.[6] The cohomological invariants with 3-torsion coefficients have a basis 1, g3 of degrees 0, 3.[7] The invariants f3 and g3 are the primary components of the Rost invariant.
Euclidean Jordan algebra for the Jordan algebras considered by Jordan, von Neumann and Wigner
Euclidean Hurwitz algebra for details of the construction of the Albert algebra for the octonions
^ Springer & Veldkamp (2000) 5.8, p.153
^ Springer & Veldkamp (2000) 7.2
^ Chevalley C, Schafer RD (February 1950). "The Exceptional Simple Lie Algebras F(4) and E(6)". Proc. Natl. Acad. Sci. U.S.A. 36 (2): 137–41. Bibcode:1950PNAS...36..137C. doi:10.1073/pnas.36.2.137. PMC 1063148. PMID 16588959.
^ Knus et al (1998) p.517
^ Skip Garibaldi (2001). "Structurable Algebras and Groups of Type E_6 and E_7". Journal of Algebra. 236 (2): 651–691. arXiv:math/9811035. doi:10.1006/jabr.2000.8514.
^ Garibaldi, Merkurjev, Serre (2003), p.50
^ Garibaldi (2009), p.20
Albert, A. Adrian (1934), "On a Certain Algebra of Quantum Mechanics", Annals of Mathematics, Second Series, 35 (1): 65–73, doi:10.2307/1968118, ISSN 0003-486X, JSTOR 1968118
Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, Providence, RI: American Mathematical Society, ISBN 978-0-8218-3287-5, MR 1999383
Garibaldi, Skip (2009). Cohomological invariants: exceptional groups and Spin groups. Memoirs of the American Mathematical Society. Vol. 200. doi:10.1090/memo/0937. ISBN 978-0-8218-4404-5.
Jordan, Pascual; Neumann, John von; Wigner, Eugene (1934), "On an Algebraic Generalization of the Quantum Mechanical Formalism", Annals of Mathematics, 35 (1): 29–64, doi:10.2307/1968117, JSTOR 1968117
Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 978-0-8218-0904-4, Zbl 0955.16001
McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/b97489, ISBN 978-0-387-95447-9, MR 2014924
Springer, Tonny A.; Veldkamp, Ferdinand D. (2000) [1963], Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66337-9, MR 1763974
Petersson, Holger P.; Racine, Michel L. (1994), "Albert algebras", in Kaup, Wilhelm (ed.), Jordan algebras. Proceedings of the conference held in Oberwolfach, Germany, August 9-15, 1992, Berlin: de Gruyter, pp. 197–207, Zbl 0810.17021
Petersson, Holger P. (2004). "Structure theorems for Jordan algebras of degree three over fields of arbitrary characteristic". Communications in Algebra. 32 (3): 1019–1049. CiteSeerX 10.1.1.496.2136. doi:10.1081/AGB-120027965. S2CID 34280968.
Albert algebra at Encyclopedia of Mathematics.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Albert_algebra&oldid=1039500973"
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m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{compactplot}\right)
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{channel}=1,\mathrm{includesignal}=[\mathrm{color}="Navy"],\mathrm{includepowerspectrum},\mathrm{colorscheme}=["Orange","SteelBlue","Navy"]\right)
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Advanced LMI Techniques - MATLAB & Simulink - MathWorks United Kingdom
Structured Matrix Variables
Specify Matrix Variable Structures
Specify Interdependent Matrix Variables
Complex-Valued LMIs
Specifying cTx Objectives for mincx
Feasibility Radius
Well-Posedness Issues
Semi-Definite B(x) in gevp Problems
Solving M + PTXQ + QTXTP < 0
This last section gives a few hints for making the most out of the LMI Lab. It is directed toward users who are comfortable with the basics, as described in Tools for Specifying and Solving LMIs.
Fairly complex matrix variable structures and interdependencies can be specified with lmivar. Recall that the symmetric block-diagonal or rectangular structures are covered by Types 1 and 2 of lmivar provided that the matrix variables are independent. To describe more complex structures or correlations between variables, you must use Type 3 and specify each entry of the matrix variables directly in terms of the free scalar variables of the problem (the so-called decision variables).
With Type 3, each entry is specified as either 0 or ±xn where xn is the n-th decision variable. The following examples illustrate how to specify nontrivial matrix variable structures with lmivar. The following examples show variable structures with uncorrelated and interdependent matrix variables.
Suppose that the variables of the problem include a 3-by-3 symmetric matrix X and a 3-by-3 symmetric Toeplitz matrix, Y, given by:
Y=\left(\begin{array}{ccc}{y}_{1}& {y}_{2}& {y}_{3}\\ {y}_{2}& {y}_{1}& {y}_{2}\\ {y}_{3}& {y}_{2}& {y}_{1}\end{array}\right).
The variable Y has three independent entries, and thus involves three decision variables. Since Y is independent of X, label these decision variables n + 1, n + 2, and n + 3, where n is the number of decision variables involved in X. To retrieve this number, define the Type 1 variable X.
[X,n] = lmivar(1,[3 1]);
The second output argument n gives the total number of decision variables used so far, which in this case is n = 6. Given this number, you can define Y.
Y = lmivar(3,n+[1 2 3;2 1 2;3 2 1]);
An equivalent expression to define Y uses the MATLAB(R) command toeplitz to generate the matrix.
Y = lmivar(3,toeplitz(n+[1 2 3]));
To confirm the variables, visualize the decision variable distributions in X and Y using decinfo.
decinfo(lmis,X)
decinfo(lmis,Y)
Consider three matrix variables, X, Y, and Z, with the following structure.
X=\left(\begin{array}{cc}x& 0\\ 0& y\end{array}\right),\phantom{\rule{1em}{0ex}}Y=\left(\begin{array}{cc}z& 0\\ 0& t\end{array}\right),\phantom{\rule{1em}{0ex}}Z=\left(\begin{array}{cc}0& -x\\ -t& 0\end{array}\right),
where x, y, z, and t are independent scalar variables. To specify such a triple, first define the two independent variables, X and Y, which are both Type 1.
[X,n,sX] = lmivar(1,[1 0;1 0]);
[Y,n,sY] = lmivar(1,[1 0;1 0]);
The third output of lmivar gives the entry-wise dependence of X and Y on the decision variables
\left({\mathit{x}}_{1},{\mathit{x}}_{2},{\mathit{x}}_{3},{\mathit{x}}_{4}\right):=\text{\hspace{0.17em}}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right).
sX = 2×2
sY = 2×2
Using lmivar, you can now specify the structure of the Type 3 variable Z in terms of the decision variables
{\mathit{x}}_{1}=\mathit{x}
{\mathit{x}}_{4}=\mathit{t}
[Z,n,sZ] = lmivar(3,[0 -sX(1,1);-sY(2,2) 0]);
Because sX(1,1) refers to
{\mathit{x}}_{1}
and sY(2,2) refers to
{\mathit{x}}_{4}
, this expression defines the variable Z as:
Z=\left(\begin{array}{cc}0& -{x}_{1}\\ -{x}_{4}& 0\end{array}\right)=\left(\begin{array}{cc}0& -x\\ -t& 0\end{array}\right).
Confirm this results by checking the entry-wise dependence of Z on its decision variables.
The LMI solvers are written for real-valued matrices and cannot directly handle LMI problems involving complex-valued matrices. However, complex-valued LMIs can be turned into real-valued LMIs by observing that a complex Hermitian matrix L(x) satisfies
L(x) < 0
\left(\begin{array}{cc}\mathrm{Re}\left(L\left(x\right)\right)& \mathrm{Im}\left(L\left(x\right)\right)\\ -\mathrm{Im}\left(L\left(x\right)\right)& \mathrm{Re}\left(L\left(x\right)\right)\end{array}\right)<0.
This suggests the following systematic procedure for turning complex LMIs into real ones:
Decompose every complex matrix variable X as
X = X1 + jX2
where X1 and X2 are real
Decompose every complex matrix coefficient A as
A = A1 + jA2
where A1 and A2 are real
Carry out all complex matrix products. This yields affine expressions in X1, X2 for the real and imaginary parts of each LMI, and an equivalent real-valued LMI is readily derived from the above observation.
For LMIs without outer factor, a streamlined version of this procedure consists of replacing any occurrence of the matrix variable X = X1 + jX2 by
\left(\begin{array}{cc}{X}_{1}& {X}_{2}\\ -{X}_{2}& {X}_{1}\end{array}\right)
and any fixed matrix A = A1 + jA2, including real ones, by
\left(\begin{array}{cc}{A}_{1}& {A}_{2}\\ -{A}_{2}& {A}_{1}\end{array}\right).
For instance, the real counterpart of the LMI system
MHXM < X, X = XH > I (1)
reads (given the decompositions M = M1 + jM2 and X = X1 + jX2 with Mj, Xj real):
\begin{array}{c}{\left(\begin{array}{cc}{M}_{1}& {M}_{2}\\ -{M}_{2}& {M}_{1}\end{array}\right)}^{T}\left(\begin{array}{cc}{X}_{1}& {X}_{2}\\ -{X}_{2}& {X}_{1}\end{array}\right)\left(\begin{array}{cc}{M}_{1}& {M}_{2}\\ -{M}_{2}& {M}_{1}\end{array}\right)<\left(\begin{array}{cc}{X}_{1}& {X}_{2}\\ -{X}_{2}& {X}_{1}\end{array}\right)\\ \left(\begin{array}{cc}{X}_{1}& {X}_{2}\\ -{X}_{2}& {X}_{1}\end{array}\right)<I.\end{array}
Note that X = XH in turn requires that
{X}_{1}={X}_{1}^{H}
{X}_{2}+{X}_{2}^{T}=0
. Consequently, X1 and X2 should be declared as symmetric and skew- symmetric matrix variables, respectively.
Assuming, for instance, that M ∊ C5×5, the LMI system (Equation 1) would be specified as follows:
M1=real(M), M2=imag(M)
bigM=[M1 M2;-M2 M1]
% declare bigX=[X1 X2;-X2 X1] with X1=X1' and X2+X2'=0:
[X1,n1,sX1] = lmivar(1,[5 1])
[X2,n2,sX2] = lmivar(3,skewdec(5,n1))
bigX = lmivar(3,[sX1 sX2;-sX2 sX1])
% describe the real counterpart of (1.12):
lmiterm([-1 1 1 bigX],1,1)
lmiterm([2 1 1 bigX],bigM',bigM)
Note the three-step declaration of the structured matrix variable bigX,
bigX=\left(\begin{array}{cc}{X}_{1}& {X}_{2}\\ -{X}_{2}& {X}_{1}\end{array}\right).
Specify X1 as a (real) symmetric matrix variable and save its structure description sX1 as well as the number n1 of decision variables used in X1.
Specify X2 as a skew-symmetric matrix variable using Type 3 of lmivar and the utility skewdec. The command skewdec(5,n1) creates a 5-by–5 skew-symmetric structure depending on the decision variables n1 + 1, n1 + 2,...
Define the structure of bigX in terms of the structures sX1 and sX2 of X1 and X2.
See Structured Matrix Variables for more details on such structure manipulations.
The LMI solver mincx minimizes linear objectives of the form cTx where x is the vector of decision variables. In most control problems, however, such objectives are expressed in terms of the matrix variables rather than of x. Examples include Trace(X) where X is a symmetric matrix variable, or uTXu where u is a given vector.
The function defcx facilitates the derivation of the c vector when the objective is an affine function of the matrix variables. For the sake of illustration, consider the linear objective
\text{Trace}\left(X\right)+{x}_{0}^{T}P{x}_{0}
where X and P are two symmetric variables and x0 is a given vector. If lmsisys is the internal representation of the LMI system and if x0, X, P have been declared by
x0 = [1;1]
P = lmivar(1,[2 1])
the c vector such that
{c}^{T}x=\text{Trace}\left(X\right)+{x}_{0}^{T}P{x}_{0}
n = decnbr(lmisys)
c = zeros(n,1)
[Xj,Pj] = defcx(lmisys,j,X,P)
c(j) = trace(Xj) + x0'*Pj*x0
The first command returns the number of decision variables in the problem and the second command dimensions c accordingly. Then the for loop performs the following operations:
Evaluate the matrix variables X and P when all entries of the decision vector x are set to zero except xj: = 1. This operation is performed by the function defcx. Apart from lmisys and j, the inputs of defcx are the identifiers X and P of the variables involved in the objective, and the outputs Xj and Pj are the corresponding values.
Evaluate the objective expression for X:= Xj and P:= Pj. This yields the j-th entry of c by definition.
In our example the result is
Other objectives are handled similarly by editing the following generic skeleton:
n = decnbr( LMI system )
[ matrix values ] = defcx( LMI system,j,
matrix identifiers)
c(j) = objective(matrix values)
When solving LMI problems with feasp, mincx, or gevp, it is possible to constrain the solution x to lie in the ball
xTx < R2
where R > 0 is called the feasibility radius. This specifies a maximum (Euclidean norm) magnitude for x and avoids getting solutions of very large norm. This may also speed up computations and improve numerical stability. Finally, the feasibility radius bound regularizes problems with redundant variable sets. In rough terms, the set of scalar variables is redundant when an equivalent problem could be formulated with a smaller number of variables.
The feasibility radius R is set by the third entry of the options vector of the LMI solvers. Its default value is R = 109. Setting R to a negative value means “no rigid bound,” in which case the feasibility radius is increased during the optimization if necessary. This “flexible bound” mode may yield solutions of large norms.
The LMI solvers used in the LMI Lab are based on interior-point optimization techniques. To compute feasible solutions, such techniques require that the system of LMI constraints be strictly feasible, that is, the feasible set has a nonempty interior. As a result, these solvers may encounter difficulty when the LMI constraints are feasible but not strictly feasible, that is, when the LMI
L(x) ≤ 0
has solutions while
For feasibility problems, this difficulty is automatically circumvented by feasp, which reformulates the problem:
L(x) ≤ 0 (2)
Minimize t subject to
Lx < t × I.
In this modified problem, the LMI constraint is always strictly feasible in x, t and the original LMI Equation 2 is feasible if and only if the global minimum tmin of Equation 2 satisfies
tmin ≤ 0
For feasible but not strictly feasible problems, however, the computational effort is typically higher as feasp strives to approach the global optimum tmin = 0 to a high accuracy.
For the LMI problems addressed by mincx and gevp, nonstrict feasibility generally causes the solvers to fail and to return an “infeasibility” diagnosis. Although there is no universal remedy for this difficulty, it is sometimes possible to eliminate underlying algebraic constraints to obtain a strictly feasible problem with fewer variables.
Another issue has to do with homogeneous feasibility problems such as
ATP + P A < 0, P > 0
While this problem is technically well-posed, the LMI optimization is likely to produce solutions close to zero (the trivial solution of the nonstrict problem). To compute a nontrivial Lyapunov matrix and easily differentiate between feasibility and infeasibility, replace the constraint P > 0-by-P > αI with α > 0. Note that this does not alter the problem due to its homogeneous nature.
Consider the generalized eigenvalue minimization problem
A(x) < λB(x), B(x) > 0, C(x) <0. (3)
Technically, the positivity of B(x) for some x ∊ Rn is required for the well-posedness of the problem and the applicability of polynomial-time interior-point methods. Hence problems where
B\left(x\right)=\left(\begin{array}{cc}{B}_{1}\left(x\right)& 0\\ 0& 0\end{array}\right)
with B1(x) > 0 strictly feasible, cannot be directly solved with gevp. A simple remedy consists of replacing the constraints
A(x) < B(x), B(x) > 0
A\left(x\right)<\left(\begin{array}{cc}Y& 0\\ 0& 0\end{array}\right),\text{ }Y<\lambda {B}_{1}\left(x\right),\text{ }{B}_{1}\left(x\right)>0
where Y is an additional symmetric variable of proper dimensions. The resulting problem is equivalent to Equation 3 and can be solved directly with gevp.
As explained in Tools for Specifying and Solving LMIs, the term-oriented description of LMIs used in the LMI Lab typically leads to higher efficiency than the canonical representation
A0 + x1A1 + ... + xNAN < 0. (4)
This is no longer true, however, when the number of variable terms is nearly equal to or greater than the number N of decision variables in the problem. If your LMI problem has few free scalar variables but many terms in each LMI, it is therefore preferable to rewrite it as Equation 4 and to specify it in this form. Each scalar variable xj is then declared independently and the LMI terms are of the form xjAj.
If M denotes the total row size of the LMI system and N the total number of scalar decision variables, the flop count per iteration for the feasp and mincx solvers is proportional to
N3 when the least-squares problem is solved via Cholesky factorization of the Hessian matrix (default) [2].
M-by-N2 when numerical instabilities warrant the use of QR factorization instead.
While the theory guarantees a worst-case iteration count proportional to M, the number of iterations actually performed grows slowly with M in most problems. Finally, while feasp and mincx are comparable in complexity, gevp typically demands more computational effort. Make sure that your LMI problem cannot be solved with mincx before using gevp.
In many output-feedback synthesis problems, the design can be performed in two steps:
Compute a closed-loop Lyapunov function via LMI optimization.
Given this Lyapunov function, derive the controller state-space matrices by solving an LMI of the form
M + PTXQ + QTXTP < 0 (5)
where M, P, Q are given matrices and X is an unstructured m-by-n matrix variable.
It turns out that a particular solution Xc of Equation 5 can be computed via simple linear algebra manipulations [1]. Typically, Xc corresponds to the center of the ellipsoid of matrices defined by Equation 5.
The function basiclmi returns the “explicit” solution Xc:
Xc = basiclmi(M,P,Q)
Since this central solution sometimes has large norm, basiclmi also offers the option of computing an approximate least-norm solution of Equation 5. This is done by
X = basiclmi(M,P,Q,'Xmin')
and involves LMI optimization to minimize ∥X ∥.
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Frame_bundle Knowpia
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bundle (where k is the rank of E).
The frame bundle
{\displaystyle {\mathcal {F}}(E)}
of the Möbius strip
{\displaystyle E}
is a non-trivial principal
{\displaystyle \mathbb {Z} /2\mathbb {Z} }
-bundle over the circle.
The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle.
Definition and constructionEdit
Let E → X be a real vector bundle of rank k over a topological space X. A frame at a point x ∈ X is an ordered basis for the vector space Ex. Equivalently, a frame can be viewed as a linear isomorphism
{\displaystyle p:\mathbf {R} ^{k}\to E_{x}.}
The set of all frames at x, denoted Fx, has a natural right action by the general linear group GL(k, R) of invertible k × k matrices: a group element g ∈ GL(k, R) acts on the frame p via composition to give a new frame
{\displaystyle p\circ g:\mathbf {R} ^{k}\to E_{x}.}
This action of GL(k, R) on Fx is both free and transitive (this follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another). As a topological space, Fx is homeomorphic to GL(k, R) although it lacks a group structure, since there is no "preferred frame". The space Fx is said to be a GL(k, R)-torsor.
The frame bundle of E, denoted by F(E) or FGL(E), is the disjoint union of all the Fx:
{\displaystyle \mathrm {F} (E)=\coprod _{x\in X}F_{x}.}
Each point in F(E) is a pair (x, p) where x is a point in X and p is a frame at x. There is a natural projection π : F(E) → X which sends (x, p) to x. The group GL(k, R) acts on F(E) on the right as above. This action is clearly free and the orbits are just the fibers of π.
The frame bundle F(E) can be given a natural topology and bundle structure determined by that of E. Let (Ui, φi) be a local trivialization of E. Then for each x ∈ Ui one has a linear isomorphism φi,x : Ex → Rk. This data determines a bijection
{\displaystyle \psi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times \mathrm {GL} (k,\mathbf {R} )}
{\displaystyle \psi _{i}(x,p)=(x,\varphi _{i,x}\circ p).}
With these bijections, each π−1(Ui) can be given the topology of Ui × GL(k, R). The topology on F(E) is the final topology coinduced by the inclusion maps π−1(Ui) → F(E).
With all of the above data the frame bundle F(E) becomes a principal fiber bundle over X with structure group GL(k, R) and local trivializations ({Ui}, {ψi}). One can check that the transition functions of F(E) are the same as those of E.
The above all works in the smooth category as well: if E is a smooth vector bundle over a smooth manifold M then the frame bundle of E can be given the structure of a smooth principal bundle over M.
Associated vector bundlesEdit
A vector bundle E and its frame bundle F(E) are associated bundles. Each one determines the other. The frame bundle F(E) can be constructed from E as above, or more abstractly using the fiber bundle construction theorem. With the latter method, F(E) is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as E but with abstract fiber GL(k, R), where the action of structure group GL(k, R) on the fiber GL(k, R) is that of left multiplication.
Given any linear representation ρ : GL(k, R) → GL(V,F) there is a vector bundle
{\displaystyle \mathrm {F} (E)\times _{\rho }V}
associated to F(E) which is given by product F(E) × V modulo the equivalence relation (pg, v) ~ (p, ρ(g)v) for all g in GL(k, R). Denote the equivalence classes by [p, v].
The vector bundle E is naturally isomorphic to the bundle F(E) ×ρ Rk where ρ is the fundamental representation of GL(k, R) on Rk. The isomorphism is given by
{\displaystyle [p,v]\mapsto p(v)}
where v is a vector in Rk and p : Rk → Ex is a frame at x. One can easily check that this map is well-defined.
Any vector bundle associated to E can be given by the above construction. For example, the dual bundle of E is given by F(E) ×ρ* (Rk)* where ρ* is the dual of the fundamental representation. Tensor bundles of E can be constructed in a similar manner.
Tangent frame bundleEdit
The tangent frame bundle (or simply the frame bundle) of a smooth manifold M is the frame bundle associated to the tangent bundle of M. The frame bundle of M is often denoted FM or GL(M) rather than F(TM). If M is n-dimensional then the tangent bundle has rank n, so the frame bundle of M is a principal GL(n, R) bundle over M.
Smooth framesEdit
Local sections of the frame bundle of M are called smooth frames on M. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in U in M which admits a smooth frame. Given a smooth frame s : U → FU, the trivialization ψ : FU → U × GL(n, R) is given by
{\displaystyle \psi (p)=(x,s(x)^{-1}\circ p)}
where p is a frame at x. It follows that a manifold is parallelizable if and only if the frame bundle of M admits a global section.
Since the tangent bundle of M is trivializable over coordinate neighborhoods of M so is the frame bundle. In fact, given any coordinate neighborhood U with coordinates (x1,…,xn) the coordinate vector fields
{\displaystyle \left({\frac {\partial }{\partial x^{1}}},\ldots ,{\frac {\partial }{\partial x^{n}}}\right)}
define a smooth frame on U. One of the advantages of working with frame bundles is that they allow one to work with frames other than coordinates frames; one can choose a frame adapted to the problem at hand. This is sometimes called the method of moving frames.
Solder formEdit
The frame bundle of a manifold M is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of M. This relationship can be expressed by means of a vector-valued 1-form on FM called the solder form (also known as the fundamental or tautological 1-form). Let x be a point of the manifold M and p a frame at x, so that
{\displaystyle p:\mathbf {R} ^{n}\to T_{x}M}
is a linear isomorphism of Rn with the tangent space of M at x. The solder form of FM is the Rn-valued 1-form θ defined by
{\displaystyle \theta _{p}(\xi )=p^{-1}\mathrm {d} \pi (\xi )}
where ξ is a tangent vector to FM at the point (x,p), and p−1 : TxM → Rn is the inverse of the frame map, and dπ is the differential of the projection map π : FM → M. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of π and right equivariant in the sense that
{\displaystyle R_{g}^{*}\theta =g^{-1}\theta }
where Rg is right translation by g ∈ GL(n, R). A form with these properties is called a basic or tensorial form on FM. Such forms are in 1-1 correspondence with TM-valued 1-forms on M which are, in turn, in 1-1 correspondence with smooth bundle maps TM → TM over M. Viewed in this light θ is just the identity map on TM.
As a naming convention, the term "tautological one-form" is usually reserved for the case where the form has a canonical definition, as it does here, while "solder form" is more appropriate for those cases where the form is not canonically defined. This convention is not being observed here.
Orthonormal frame bundleEdit
If a vector bundle E is equipped with a Riemannian bundle metric then each fiber Ex is not only a vector space but an inner product space. It is then possible to talk about the set of all of orthonormal frames for Ex. An orthonormal frame for Ex is an ordered orthonormal basis for Ex, or, equivalently, a linear isometry
{\displaystyle p:\mathbf {R} ^{k}\to E_{x}}
where Rk is equipped with the standard Euclidean metric. The orthogonal group O(k) acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right O(k)-torsor.
The orthonormal frame bundle of E, denoted FO(E), is the set of all orthonormal frames at each point x in the base space X. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank k Riemannian vector bundle E → X is a principal O(k)-bundle over X. Again, the construction works just as well in the smooth category.
If the vector bundle E is orientable then one can define the oriented orthonormal frame bundle of E, denoted FSO(E), as the principal SO(k)-bundle of all positively oriented orthonormal frames.
If M is an n-dimensional Riemannian manifold, then the orthonormal frame bundle of M, denoted FOM or O(M), is the orthonormal frame bundle associated to the tangent bundle of M (which is equipped with a Riemannian metric by definition). If M is orientable, then one also has the oriented orthonormal frame bundle FSOM.
Given a Riemannian vector bundle E, the orthonormal frame bundle is a principal O(k)-subbundle of the general linear frame bundle. In other words, the inclusion map
{\displaystyle i:{\mathrm {F} }_{\mathrm {O} }(E)\to {\mathrm {F} }_{\mathrm {GL} }(E)}
is principal bundle map. One says that FO(E) is a reduction of the structure group of FGL(E) from GL(k, R) to O(k).
G-structuresEdit
If a smooth manifold M comes with additional structure it is often natural to consider a subbundle of the full frame bundle of M which is adapted to the given structure. For example, if M is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of M. The orthonormal frame bundle is just a reduction of the structure group of FGL(M) to the orthogonal group O(n).
In general, if M is a smooth n-manifold and G is a Lie subgroup of GL(n, R) we define a G-structure on M to be a reduction of the structure group of FGL(M) to G. Explicitly, this is a principal G-bundle FG(M) over M together with a G-equivariant bundle map
{\displaystyle {\mathrm {F} }_{G}(M)\to {\mathrm {F} }_{\mathrm {GL} }(M)}
In this language, a Riemannian metric on M gives rise to an O(n)-structure on M. The following are some other examples.
Every oriented manifold has an oriented frame bundle which is just a GL+(n, R)-structure on M.
A volume form on M determines a SL(n, R)-structure on M.
A 2n-dimensional symplectic manifold has a natural Sp(2n, R)-structure.
A 2n-dimensional complex or almost complex manifold has a natural GL(n, C)-structure.
In many of these instances, a G-structure on M uniquely determines the corresponding structure on M. For example, a SL(n, R)-structure on M determines a volume form on M. However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed. A Sp(2n, R)-structure on M uniquely determines a nondegenerate 2-form on M, but for M to be symplectic, this 2-form must also be closed.
Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3
Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2008-08-02
Sternberg, S. (1983), Lectures on Differential Geometry ((2nd ed.) ed.), New York: Chelsea Publishing Co., ISBN 0-8218-1385-4
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Mushroom elbaite from the Kat Chay mine, Momeik, near Mogok, Myanmar: I. Crystal chemistry by SREF, EMPA, MAS NMR and Mössbauer spectroscopy | Mineralogical Magazine | GeoScienceWorld
A. J. Lussier;
Department of Geological Sciences, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
P. M. Aguiar;
P. M. Aguiar
Department of Chemistry, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
V. K. Michaelis;
S. Kroeker;
S. Herwig;
Y. Abdu;
F. C. Hawthorne *
A. J. Lussier, P. M. Aguiar, V. K. Michaelis, S. Kroeker, S. Herwig, Y. Abdu, F. C. Hawthorne; Mushroom elbaite from the Kat Chay mine, Momeik, near Mogok, Myanmar: I. Crystal chemistry by SREF, EMPA, MAS NMR and Mössbauer spectroscopy. Mineralogical Magazine 2008;; 72 (3): 747–761. doi: https://doi.org/10.1180/minmag.2008.072.3.747
Tourmaline from the Kat Chay mine, Momeik, near Mogok, Shan state, Myanmar, shows a variety of habits that resemble mushrooms, and it is commonly referred to as ‘mushroom tourmaline’. The structure of nine single crystals of elbaite, ranging in colour from pink to white to black and purple, extracted from two samples of mushroom tourmaline from Mogok, have been refined (SREF) to R indices of ~2.5% using graphite-monochromated Mo-Kα X-radiation. 11B and 27Al Magic Angle Spinning Nuclear Magnetic Resonance spectroscopy shows the presence of [4]B and the absence of [4]Al in samples with transition-metal content low enough to prevent paramagnetic quenching of the signal. Site populations were assigned from refined site-scattering values and unit formulae derived from electron-microprobe analyses of the crystals used for X-ray data collection. 57Fe Mössbauer spectroscopy shows that both Fe2+ and Fe3+ are present, and the site populations derived by structure refinement show that there is no Fe at the Z site; hence all Fe2+ and Fe3+ occurs at the Y site. The 57Fe Mössbauer spectra also show peaks due to intervalence charge-transfer involving Fe2+ and Fe3+ at adjacent Y sites. Calculation of the probability of the total amount of Fe occurring as Fe2+–Fe3+ pairs for a random short-range distribution is in close accord with the observed amount of Fe involved in Fe2+–Fe3+, indicating that there is no short-range order involving Fe2+ and Fe3+ in these tourmalines.
Shan State Burma
Kat Chay Mine
Crystallographic and spectroscopic characterization of Fe-bearing chromo-alumino-povondraite and its relations with oxy-chromium-dravite and oxy-dravite
TETRAHEDRALLY COORDINATED BORON IN Li-BEARING OLENITE FROM “MUSHROOM” TOURMALINE FROM MOMEIK, MYANMAR
Mn22+
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Lemma 10.136.13 (00SV)—The Stacks project
Comment #3254 by Dario Weißmann on April 11, 2018 at 16:26
\mathfrak{p}=R\cap \mathfrak{q}
. It is clear from the context, but still.
Comment #4721 by comment_bot on November 25, 2019 at 17:09
A pedantic comment: I suggest adding "such that every nonempty
R
-fiber of
S
n - c
" to the second sentence of the statement. This would make the statement less ambiguous: theoretically, in that sentence it is not clear whether we are choosing a presentation as in the definition of a "global complete intersection" or whether we are choosing an arbitrary presentation for the
R
S
(knowing that
S
happens to be a global complete intersection, as witnessed by some other presentation).
The same pedantic comment applies to other statements in this section.
Tried to improve the wording of these lemmas. See changes here.
(2) is equivalent to that
R[x_1,\ldots,x_n]/(f_1,\ldots,f_c)
R
. And since flatness is preserved under base change, we can reduce (2) to Noetherian case.
But I don't know how to reduce the case to Noetherian for (1).
The point is that
R
will be the filtered union of Noetherian rings for which the result is true. Then you use that if
R = colim_{i \in I} R_i
is a filtered colimit and if
f_1, \ldots, f_r \in R_0
0 \in I
form a regular sequence in each
R_i
i \geq 0
f_1, \ldots, f_r
form a regular sequence in
R
@#6606: Thanks and I have now added the extra arguments here.
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Paper presented at NIPS 2013 workshop: Abstraction in Decision-Makers with Limited Information Processing Capabilities – Tim Genewein
Information-optimal hierarchies for inference and decision-making
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Paper presented at NIPS 2013 workshop: Abstraction in Decision-Makers with Limited Information Processing Capabilities
Genewein T, Braun DA (2013) Abstraction in Decision-Makers with Limited Information Processing Capabilities, NIPS 2013 workshop on Planning with Information Constraints. arXiv:1312.4353
Bounded rational decision-making
Bounded rationality and abstractions
Our paper on “Abstraction in decision-makers with limited information processing capabilities” was presented at the NIPS 2013 workshop on Planning with Information Constraints - you can find the slides of my talk here.
The paper is about constructing optimal decision-makers that have limited computational capacity. As a consequence of these computational limitations we find that natural levels of abstraction emerge in the agent’s behavior.
In classical decision-making an agent is presented with an observation
w
and is then faced with the task of picking an optimal action
a^*
out of a set of actions such that the utility
U(w,a)
is maximized. This is known as the maximum-expected-utility (MEU) principle. The problem with MEU is that finding the best action out of a large set of actions can be computationally very demanding and for an agent with limited computational resources that has to react within a certain time-limit this search problem can easily become infeasible.
To overcome the intractable search-problem over actions the MEU rule can be replaced with a soft maximum rule where the agent does not seek for the single best action but rather picks the first action that fulfills a certain fidelity constraint. In other words: the agent searches through action space according to some search distribution and picks the first action that is good enough. This is the central idea behind bounded rational decision-making and it can be implemented in many different ways (heuristics, probabilistic approximation schemes, etc.). In the paper we build upon a quantitative probabilistic framework for bounded rational decision-making that is derived from first principles and (perhaps surprisingly) has strong ties to free energy minimization in thermodynamics. The resulting optimization problem for optimal acting under limited computational resources is mathematically equivalent to the problem formulation in rate distortion theory, the information-theoretic framework for lossy compression. In rate distortion theory the goal is to compress information such that a distortion measure is minimized while at the same time not exceeding a limit on the channel capacity. Analogously, the optimality principle for bounded rational acting trades off a large expected utility against low computational cost
The trade-off is a variational problem over the optimal policy
p^*(a|w)
that maps world-states
w
to actions
a
such that the expected utility is maximized while at the same time keeping the computational cost low. The computational cost arises because for each
w
the agent has to adapt its behavior from (a prior)
p(a)
to (a posterior)
p(a|w)
and the average computational effort for this adaptation is measured with the mutual information
I(w;a)
which is the average KL-divergence from prior to posterior. The factor
\beta
governs the trade-off and due to the thermodynamic roots of the principle is called the inverse temperature.
The optimization problem has a closed-form solution (actually a set of two self-consistent equations) which is well known in information theory and can be computed with the Blahut-Arimoto algorithm.
See Research for more information on the thermodynamic framework and its connection to rate distortion theory.
Abstractions are formed by reducing the information content of an entity until it only contains crucial information. For instance, consider the abstract concept of a chair: chairs can come in many different colors, materials, sizes and even different particular shapes but there is something crucial to an object to make it a chair and distinguish it from a table or a sofa. In a sense the abstract concept is a lossy compression where irrelevant details (“noise” such as color or material) have been discarded. The problem of finding optimal lossy compressions and forming abstractions are thus tightly related. The information-theoretic principle for bounded rational decision-making can be cast as a lossy compression problem, therefore the principle can also be used to form bounded-optimal abstractions. In decision-making, abstractions translate into behavior where several world-states
w
are treated as if they were the same, meaning that the agent uses the same policy
p(a|w)
for subsets of world-states.
In the paper we show that the information-theoretic principle for bounded rational acting leads to the formation of behavioral abstractions. Importantly, the inverse temperature
\beta
controls the granularity and thus the level of abstraction. We illustrate this in the paper with an example and show that the abstractions formed by the principle are shaped by the utility-function and the computational limitations of the agent. Interestingly we find that in our example a low number of levels of abstractions emerge naturally from the utility-structure where behavior within a level of abstraction does not change qualitatively (but does get more or less stochastic). In-between the levels of abstraction there are steep “phase transitions” where behavior abruptly changes qualitatively. See the paper for more details on the example.
© 2019 Tim Genewein. Powered by Jekyll, based on the Minimal Mistakes theme.
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Specify LMIs with the LMI Editor GUI - MATLAB & Simulink - MathWorks United Kingdom
The LMI Editor lmiedit is a graphical user interface (GUI) to specify LMI systems in a straightforward symbolic manner. Typing
calls up a window with several editable text areas and various buttons.
In more detail, to specify your LMI system,
Declare each matrix variable (name and structure) in the upper half of the worksheet. The structure is characterized by its type (S for symmetric block diagonal, R for unstructured, and G for other structures) and by an additional “ structure” matrix. This matrix contains specific information about the structure and corresponds to the second argument of lmivar (see Specifying the LMI Variables for details).
Please use one line per matrix variable in the text editing areas.
Specify the LMIs as MATLAB® expressions in the lower half of the worksheet. For instance, the LMI
\left(\begin{array}{cc}{A}^{T}X+XA& XB\\ {B}^{T}X& -I\end{array}\right)<0
is entered by typing
[a'*x+x*a x*b; b'*x -1] < 0
if x is the name given to the matrix variable X in the upper half of the worksheet. The left- and right-hand sides of the LMIs should be valid MATLAB expressions.
Once the LMI system is fully specified, the following tasks can be performed by clicking the corresponding button:
Visualize the sequence of lmivar/lmiterm commands needed to describe this LMI system (view commands button). Conversely, the LMI system defined by a particular sequence of lmivar/lmiterm commands can be displayed as a MATLAB expression by clicking on the describe... buttons.
Beginners can use this facility as a tutorial introduction to the lmivar and lmiterm commands.
Save the symbolic description of the LMI system (save button). This description can be reloaded later on by clicking the load button.
Read a sequence of lmivar/lmiterm commands from a file (read button). You can then click on describe the matrix variables or describe the LMIs to visualize the symbolic expression of the LMI system specified by these commands. The file should describe a single LMI system but may otherwise contain any sequence of MATLAB commands.
This feature is useful for code validation and debugging.
Write in a file the sequence of lmivar/lmiterm commands needed to describe a particular LMI system (write button).
This is helpful to develop code and prototype MATLAB functions based on the LMI Lab.
Generate the internal representation of the LMI system by clicking create. The result is written in a MATLAB variable named after the LMI system (if the “name of the LMI system” is set to mylmi, the internal representation is written in the MATLAB variable mylmi). Note that all LMI-related data should be defined in the MATLAB workspace at this stage.
The internal representation can be passed directly to the LMI solvers or any other LMI Lab function.
As with lmiterm, you can use various shortcuts when entering LMI expressions at the keyboard. For instance, zero blocks can be entered simply as 0 and need not be dimensioned. Similarly, the identity matrix can be entered as 1 without dimensioning. Finally, upper diagonal LMI blocks need not be fully specified. Rather, you can just type (*) in place of each such block.
Though fairly general, lmiedit is not as flexible as lmiterm and the following limitations should be kept in mind:
Parentheses cannot be used around matrix variables. For instance, the expression
(a*x+b)'*c + c'*(a*x+b)
is invalid when x is a variable name. By contrast,
(a+b)'*x + x'*(a+b)
Loops and if statements are ignored.
When turning lmiterm commands into a symbolic description of the LMI system, an error is issued if the first argument of lmiterm cannot be evaluated. Use the LMI and variable identifiers supplied by newlmi and lmivar to avoid such difficulties.
Users familiar with MATLAB may wonder how lmivar and lmiterm physically update the internal representation LMISYS since LMISYS is not an argument to these functions. In fact, all updating is performed through global variables for maximum speed. These global variables are initialized by setlmis, cleared by getlmis, and are not visible in the workspace. Even though this artifact is transparent from the user's viewpoint, be sure to:
Invoke getlmis only once and after completely specifying the LMI system.
Refrain from using the command clear global before the LMI system description is ended with getlmis.
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Home : Support : Online Help : Programming : Logic : Boolean : verify : boolean
return a Boolean comparison or result
verify/truefalse
return a truefalse comparison or result
verify(expr1, expr2, boolean)
verify(expr1, expr2, boolean(ver))
verify(expr1, expr2, truefalse)
verify(expr1, expr2, truefalse(ver))
The verify(expr1, expr2, boolean) and verify(expr1, expr2, truefalse) calling sequences perform an evalb comparison. That is, verify(a, b, boolean) = evalb(a = b) and verify(a, b, truefalse) = evalb(a = b).
If a verification ver is given, then boolean acts as a filter, converting anything returned by verify into either true, false, or FAIL.
The verification truefalse makes one further change, converting any FAIL results into false.
These are useful in case the result from verify is to be used as input to another procedure such as sort or select.
The verifications boolean and truefalse are symmetric and a verification boolean(ver) is symmetric if and only if the verification ver is symmetric. If a verification ver is symmetric, then this implies that the verification truefalse(ver) is symmetric.
\mathrm{evalb}\left([\mathrm{undefined}]=[\mathrm{undefined}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left([\mathrm{undefined}],[\mathrm{undefined}],'\mathrm{boolean}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left([\mathrm{undefined}],[\mathrm{undefined}],'\mathrm{list}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({\left(x-1\right)}^{2},{x}^{2}-2x+1,'\mathrm{boolean}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left([3],3,'\mathrm{list}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left([3],3,'\mathrm{boolean}\left(\mathrm{list}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(0.3232,0.3233,'\mathrm{float}\left(1\right)'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ulps}}]
\mathrm{verify}\left(0.3232,0.3233,'\mathrm{boolean}\left(\mathrm{float}\left(1\right)\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(\mathrm{sin}\left(x\right),0,\mathrm{greater_than}\right)
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
\mathrm{verify}\left(\mathrm{sin}\left(x\right),0,\mathrm{boolean}\left(\mathrm{greater_than}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
\mathrm{verify}\left(\mathrm{sin}\left(x\right),0,\mathrm{truefalse}\left(\mathrm{greater_than}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{map}\left(\mathrm{verify},[3,5,a,b,\mathrm{exp}\left(1\right),\mathrm{exp}\left(2\right)],\mathrm{\pi },\mathrm{less_than}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}]
\mathrm{map}\left(\mathrm{verify},[3,5,a,b,\mathrm{exp}\left(1\right),\mathrm{exp}\left(2\right)],\mathrm{\pi },\mathrm{truefalse}\left(\mathrm{less_than}\right)\right)
[\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}]
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Supervised Learning - ProductizeML
You will learn: most-used supervised learning algorithms such as linear regression, logistic regression, SVM, and decision trees.
Linear regression is a supervised machine learning algorithm where the predicted output value keeps a linear (continuous) relationship with the input.
One simple application of linear regression can be used to predict the economic activity of a country in relation to its population.
Figure 1: The regression line represents the continuous range of values between GDP as economic activity measurement and population associated.
We can distinguish between two types:
When just considering a single variable as input data, the linear regression line can be expressed with an equation of the form:
\sf{y = mx + b}
m
b
are the weights our algorithm will try to “learn” to provide the most accurate predictions
y
x
represents our input data and
y
represents our prediction.
Given Example 1, linear regression would let us predict the economic activity (
y
) from the number of people (
x
). For instance, for a region with 30 million people, the expected GDP value would be about $1,400 billion.
The same inverse logic could be applied if trying to predict the population (
x
) from its economic activity (
y
). In that case, we would reverse the regression equation to:
\sf{x = \frac{y -b}{m}}
Following the same idea, if several variables play a role in the linear prediction, multiple weights
w_i
will apply to each variable in the following form:
f(x, y, z) = w_0 + w_1x + w_2y + w_3z
w_i
are the coefficients the algorithm will try to “learn” to provide the most accurate predictions given
x, y
z
as input data.
R&D Investment ($M)
Sales (M units)
In Example 2, we can still predict the sales of a company given the number of employees, investment in R&D and income as follows:
sales = w_0 + w_1*employees + w_2*R\&D_{investment} + w_3 * income
In order to find the best weights
w_i
that minimizes the error between the model's predictions from its ground-truth labels, we must introduce the concept of a loss value. We could describe it as the absolute difference between the true value and the predicted one.
When this measurement is applied to a set of evidence, we call it loss function (or cost function). It can be understood as a measurement of how "good" your model is: the lower the loss, the better your model is.
For linear regression models, we typically use Mean Squared Error (MSE) as a loss function:
MSE = \frac{1}{N}\sum_{i=1}^{n}{(y_i-(mx_i+b))^2}
N
is the total number of samples.
To find the value of the parameters that minimize the cost function, there is a closed-form solution — in other words, a mathematical equation that gives the result directly. This is called the Normal Equation.
How can we compute the optimal parameters when we have a lot of data and features and they can't fit in memory or it is computationally too expensive?
The answer is via Gradient Descent, covered in the Neural Networks section.
Logistic regression is another supervised learning method to classify samples by assigning a discrete set of classes.
Similarly, as linear regression makes predictions on continuous number values, logistic regression uses a logistic Sigmoid function as a prediction value associated with multiple variables of the form:
y = \frac{1}{1 + e^{-z}}
The logistic curve of the sigmoid function for which outputs are between 0 and 1. This property allows the regression to provide a probability value.
We can distinguish between three types:
This regression type only considers situations in which the observed outcome for a dependent variable only takes two types. As an example, we might use logistic regression to classify lung cancer as benign or malignant.
In this second case, the outcome can be presented as three or more types. An example case could be predicting the type of lung cancer as e.g. "cancer A", "cancer B", or "cancer C".
Same idea as in (2) Multi-class classification, but this time the predictions are ordered with a sense of positioning. An example case could be a restaurant service rating choice among "poor", "fair", "good", and "excellent".
Support vector machines (SVMs) are a set of supervised machine learning algorithms for classification, regression, and outlier detection that are built around hyperplane separation of the data. In a two dimensional space, this separation can be understood as a simple decision boundary in the form of a line, but SVMs are also effective in high dimensional spaces.
Scikit-learn's SVM for a three class classification example.
Thanks to their great adaptation to multidimensional spaces, SVMs are often used for Natural Language Classification problems.
Decision trees are non-parametric supervised machine learning techniques that are able to make predictions by learning simple decision rules encoded in a flowchart-like structure.
These tree structures are easy to represent and interpret. Leaves on the tree represent classifications labels, non-leaf nodes are features, and branches represent conjunctions of features that lead to the classifications.
An example of a decision tree with the objective of predicting if the weather is good or not for a hike. Source: https://www.sciencedirect.com/topics/computer-science/decision-trees
When grouping multiple decision trees together, we refer to this ensemble of models as random forests.
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Reliability of Wind Turbine Technology Through Time | J. Sol. Energy Eng. | ASME Digital Collection
E. Echavarria,
E. Echavarria
, Mekelweg 2, 2628 CD Delft, The Netherlands
e-mail: e.echavarriauribe@tudelft.nl
, e.V. Königstor 59, D-34119 Kassel, Germay
e-mail: bhahn@iset.uni-kassel.de
G. J. W. van Bussel,
G. J. W. van Bussel
Echavarria, E., Hahn, B., van Bussel, G. J. W., and Tomiyama, T. (June 26, 2008). "Reliability of Wind Turbine Technology Through Time." ASME. J. Sol. Energy Eng. August 2008; 130(3): 031005. https://doi.org/10.1115/1.2936235
This study attempts to obtain more detailed knowledge of failures of wind turbines (WTs) by using the German “
250MW
Wind” test program database. Specific objectives are to show the reliability of some major components and to analyze how their design has advanced through time, what the main failures are, and which technologies have proven to work. Within the program, reports on operation and maintenance are analyzed with respect to WT type, size, and technologies used. This paper presents a comparison of component reliability through time, with respect to their technology. The results show significant differences in reliability for certain subcomponents depending on the size of the WT and especially on the type of power control. For instance, induction generators show half the annual failure rate compared to synchronous generators. The study also includes failures of other components that are affected or added due to the use of the components being analyzed. In general, the results show that failure rates of WTs decrease with time. Most of the data show a short period of “early failures” and later a long period of “random failures.” However, this is not the case for the megawatt class: As technology is introduced into the market, WTs show a longer early failure behavior, which has not yet become stable. Furthermore, large turbines, included in the database analyzed, show a significantly higher annual failure rate of components, per WT. This may be due to the immature technology of the WTs included in the database.
failure analysis, power generation faults, power generation reliability, wind turbines, reliability, maintenance, failures, wind turbines
Failure, Generators, Reliability, Wind turbines, Maintenance, Turbines, Electromagnetic induction
Solarpraxis and James & James (Science Publishers) Ltd.
, Chaps. 1–3.
James & James (Science Publishers) Ltd.
,” ISET, Kassel.
Eggersglüß: Windenergie X Praxisergebnisse, Landwirtschaftskammer Schleswig-Holstein
,” 12(4)–14(3), Denmark.
,” MS thesis, KTH School of Electrical Engineering, Stockholm.
,” TU Delft, Report No. 10048.
A Statistical Evaluation of Icing Failures in Germany’s ’250MW Wind’ Program
Influence of Wind Speed on Wind Turbine Reliability
, S. 329–332,
Improving the Reliability of Wind Turbine Generation and Its Impact on Overall Distribution Network Reliability
18th International Conference on Electrical Distribution
Machine and Converter Reliabilities in Wind Turbines
Third IET International Conference on Power Electronics, Machines and Drives
, Xiang, 2005, “
Wind Turbine Reliability, How Does It Compare With Other Embedded Generation Sources
IEE RTDN Conference
2008, NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/http://www.itl.nist.gov/div898/handbook/.
Broemm
AMSAA Reliability Growth Guide
,” Army Materiel Systems Analysis Activity Aberdeen Proving Ground, AMSAA-TR-652.
G. J.W.
Offshore Wind Energy, the Reliability Dilemma
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Speed of sound in air is 330 m s-1 and in water i 1650 m s-1 It takes 2 - Physics - Propagation of Sound Waves - 12296113 | Meritnation.com
Speed of sound in air is 330 m s-1 and in water i 1650 m s-1. It takes 2.5 second for sound to reach from one place to another. How much time will it take for sound to reach the same distance in water
Srijith Balan answered this
We know that, v= S/t or S = vt (1) (s -distance, v- velocity and t=time)
In the given problem sound has to travel the same distance in both medium i.e., vata = vwtw
tw = (vata)/vw = (330
×
2.5) / 1650 = 0.5 s
Thus solved
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CNCnutz: How to square your X and Y Axis - episode 212
How to square your X and Y Axis - episode 212
Like tramming the Z axis on your machine to be square to the tabletop, Squaring the X and Y axis is another job you need to complete for your machine to cut accurately. The biggest problem with this is it isn't as simple as grabbing a square and checking it. Machinists squares are far too small for this task and builders squares are not really accurate enough. Even if it was then on a large machine like mine even a builders square is too small and there is nowhere I can put it to reference square from anyway. This is true of most CNC machines but none the less it has to be done and done accurately.
If your X&Y are not square to one another then nothing you cut will be square.
The good news is that it takes very little in the way of technical knowledge and tools to set your axis up square and when you are finished it will be super accurate.
For this we will use Pythagoras's theorem but don't worry if you aren't good with maths.
{\displaystyle a^{2}+b^{2}=c^{2}.}
The 3,4,5 Triangle
While it is handy to know the actual theory there is an easier way to use it and that is by simply using the 3,4,5 triangle
This method is used all around the world by builders to set things up squarely when it isn't possible to use a builders square. If you wanted to set up boxing for a large pad for a building you would use the 3,4,5 triangle to lay everything out and this is also the best method for us to use on our machines. The bigger the triangle we use the better. Each side of the triangle can be multiplied by fixed amount and the result will always be correct. In the Video I multiply each side by 200mm giving me 600 x 800 x 1000mm but I could have just as easily used 8 inches which would have given me 24 x 32x 40"
If my table had been a bit larger I could have gone 3 x 4 x 5 foot. So long as the same multiplier is used for each side then it will hold true.
Now I have square what do I do with it?
This process is something you only want to do once so when you finally get it set right you need to have a simple way to resetting it each time you use your machine. There are 2 methods for doing this. You can either set your hard limits to be a reference or set up auto squaring.
Hard limit squaring
This is my preferred method to square my axis. Simply engage the Estop and manually pull the gantry toward the front of the machine until it rests on the Hard limits. Then release the Estop and it is done. Of course if you are using leadscrews then the gantry needs to be driven into the hard limits as you cannot pull against the screws like you can on a rack and pinion machine.
Squaring with homing switches
No matter how good people think having homing switches are, and I have nothing against them, you still need to go through this process to establish what is square in the first place. From there you can position the homing switches for the auto square. Don't forget to test the auto square feature after setting them up to make sure that all is working as it should.
You need to ensure that both your X and Y axis are moving exactly the correct distance or your results will be meaningless. If you find that they aren't then use the calibration tools built into Mach3 or UCCNC to correct the error. If you are using control software that doesn't have a calibration feature then manually adjust your steps per unit until the axis move the correct distance.
I recommend you use a steel ruler rather than a tape measure as a ruler is easier to manage. In the video I used a 1000mm rule so that is why I chose to make the longest side of the triangle 1000mm.
If you need to use 2 rulers as I did in the video then make sure that the units of measurement are the same on each by comparing them against one another. I have seen instances where some rulers are not the same. This isn't a problem as such so long as the divisions on the ruler are even and you use the same ruler for all measurements but it is a big problem if you change rulers part way through.
I don't recommend you use an engraving bit for this as the point on them are off centre. The pointed stick I made only took a minute to make and was ideal. I left is about 0.5mm or 0.002 of the table top while jogging around. Just high enough to get the tape under.
While I was doing this video I found that already having the cross drawn on the tape and sliding it under the point was the best way of doing it. It is unbelievably easy to do and accurate.
In the past when doing this I either jogged the point over the cross at the starting point and tried to draw the cross under the point at the end. Both methods very frustrating, time consuming and inaccurate.
Is is really handy once you have your machine square to run a cutter down the sides of your spoilboard to give yourself reference edges for setting stock on the table and other things.
I hope you find this information useful and can get your machine set up nice and square.
Don't forget to tram your Z axis as well or you will only have done half the job. For he machine to work right you need to have everything set right. If you want to know about tramming your Z axis please follow the link below.
https://www.cncnutz.com/2017/01/how-to-set-your-router-perpendicular-to.html
TEACHERLOOKUP. COM 18 April 2019 at 17:12
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Abstract Wikipedia/Wikifunctions logo concept/Vote - Meta
Abstract Wikipedia/Wikifunctions logo concept/Vote
< Abstract Wikipedia | Wikifunctions logo concept
Voting is now closed. Thank you for participating! More info
Wikifunctions needs a logo. On this page, you can cast your vote for a logo concept, before Monday March 15th 23:59 UTC.
You can vote by clicking on the proposal numbers below; you can vote for multiple candidates, but you cannot vote for the same candidate multiple times.
See the main logo page for more details about the process.
Proposal 23 [ Discussion ]
The inspiration of this logo came from combining all three characters: W, f (function) and λ (lambda).
Uses Wikimedia colors.
All three things are combined to make it look simple, and it might be easier to use a mono color variant.
Logo's idea is to incorporate some of the known coding symbology, in this case the curly brackets and semi-colon
Thought was used to make sure that logo remain legible when reduced to smaller sizes
Meant to be as simple as possible without much clutter
Proposal 3 [ Discussion ]
Using Euler's function notation f(), containing the first W of wiki
Using Wikimedia colors, the "radial gradient" is optional
Can replace the black color with white to fit on a black background
Inspired by the first proposed name of Wikifunctions, Wikilambda.
Consists of the "Wikipedia W" connected with a lambda character, representing lambda functions.
Not red, blue and green.
Symbolizes Wikimedia including functions (symbolized by lambda).
It is compatible with "Powered by" button
Possible issue: Similar with Half-Life logo (see c:File:Half-Life Lambda popr..png))
The idea was to have the logo representing a function, with the red central circle being the "code".
The blue parenthesis brings a dynamic visual for the concept.
Colors, design proportions and the visual language are meant to be in line with the family of project logos.
The lines illustrate lines of code/instructions or reference areas.
The “vertical” style holds two secrets: the shapes form an abstracted “face” and and the lines imply the three ascending strokes of a “W”.
I like the simplicity, yet it tells a story - building something for humans/humanity from diverse elements/atoms.
It works well in different scales. The logo would be inverted on a dark background. You can preview different use cases for the logo in this Google Slide deck – e.g. merchandise, favicon in the browser, presentation, how it looks animated. (I chose this format, so any could copy the slides and try out their logo for these use cases).
This logo is derived from File:W-fun.svg
The arrow (right) stands for functions, the W (below) corresponds to the first letter of Wikifunctions, and the three dots symbolize the inputs.
The colors correspond to the four common ones used in Wikimedia Logos.
The punchcard is simplified so it works at all scales (though we could perhaps simplify the favicon).
It also resembles other technology interfaces.
The dot pattern holds a special secret: the W and the number 13 reflecting that Wikifunctions is the 13th Wikimedia project. (Binary 01010111 and 1101 - the favicon could maybe just include 13 in binary dots)
Logo re-designed to move away from C like languages and be more inclusive of further languages
Logo colour scheme to follow as Wikimedia's branding
Logo overall visual structure meant to be legible at smaller sizes
Logo has some similarity to the Deutsche Bank logo
The inspiration for this logo is the commonly used function call notation of the parentheses (denoted in blue).
The red hub and spokes inside the parentheses symbolizes the function parameter which graphically represents WF (wiki functions) blended together, and also denotes the complex nature of what a function may do, as well as it looks like a dancing, joyful human figure.
The inner side of the left and right parentheses forms part of the circle. The vertical green "parentheses" completes the circle and symbolically denotes a recursive call.
Both sets of parentheses highlights the circle which denotes the globe, that unites all of humanity on this planet.
The zig zag shape references three different manifestations of functions:
electronic circuits (as the symbol for a resistor); mathematical functions (as a line graph); Wikifunctions itself (as a W).
Like Wikidata's logo, the shape is composed of red, green and blue parallel lines. But instead of being straight, the lines are interwoven like threads in a loom (maybe a Jacquard loom 😊), just as Wikifunctions will interweave raw data from Wikidata to create a rich tapestry of output.
As a small bonus, this interweaving gives the W a crossed center, like in Wikipedia's wordmark.
Resembles the Whataburger logo.
The red and the blue symbolizes the human and the algorithmic sides of the project.
The way the "ƒ" divides it in a recognizable shape symbolizes that we need a good balance between the two.
Some concerns have been noted about similarities to a former logo for FinaleMusic (current)
Inspired from Lambda alphabet (λ), insert to circle, and use Wikimedia colors.
This logo is make in such a way that the logo is easy to see in all browser, even as an favicon.
Some concerns have been noted about similarities to existing logos such as that for Racket.
Inspired from Gear and "f" icon
Used the Wikimedia colors
Graphical function (in the W shape) and equals (=) sign to capture essential concepts.
Simple shapes to scale well.
Playful. An abstract creature that can adapt to many contexts.
This represents Wikifunctions as “the union of all functions”.
⋃ is the union operator, as in
{\displaystyle \bigcup _{i=1}^{n}S_{i}}
↦ is the “maps to” function operator, as in
{\displaystyle f\colon x\mapsto x^{2}}
Inspired by mathematical function notation, with the "Wikipedia W" in place of x.
Derived from the Wikimedia Logo and kept consistent to the brand guidelines
Starting from the lambda symbol, the mathematical references include the original blue circle cut to resemble two parentheses, and the red circle is a nod to w:Dot_(diacritic)#Overdot
Could be represented directly with Unicode combination characters (λ)
It was observed in the comments a potential similarity with this logo c:File:Half-Life Lambda popr..png, although it was unknown to the author
The central part shows a pair of green parentheses (functions), surrounding two red bowls at top symbolizing the inputs, then processed in the middle in a few steps symbolized by the blue chevrons, and another red bowl at bottom symbolizing the output.
It is designed to be fully language-neutral and suitable for BiDi rendering (no need to change any letter or mirror it).
It uses the 3 main Wikimedia logo colors, but is suitable for use "as is" onlight or dark barkground, and remains identifiable if rendered at small sizes.
It should be fairly obvious that the design is based on a fusion of two letters "W" (for wiki) and "λ" (for functions).
The logo is also a carefully organized composition of four triangular arrows. Arrow is used in the design as the building block because it is commonly used to represent functions under various circumstances, such as arrow notation for mathematical functions, and arrow functions in different programming languages.
Because of its simplicity, it works well at different sizes and can be easily adapted for different applications (the "powered by" button is an example application).
Shapes in the logo are based on the design grid to create geometric harmony.
It is a geometric abstraction of the Wikipedia logo, intentionally incomplete.
The shape is in the process of forming, with the possibility of being used in animations (like “loading” patterns) to extend the visual system.
Inspiration: I once took a course for GLSL and learned that to form a circle, the computer draws many triangles, because that is the most efficient. The more triangles - the more perfect the circle. I think there is something very poetic about that. Just like how we want many contributions to our project(s) to make the world better.
Retrieved from "https://meta.wikimedia.org/w/index.php?title=Abstract_Wikipedia/Wikifunctions_logo_concept/Vote&oldid=22248824"
Wiki of functions logo proposals
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Revision as of 22:40, 21 August 2015 by MathAdmin (talk | contribs) (→The Sum of the first n Natural Numbers)
{\displaystyle n}
{\displaystyle n}
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle (n-1)^{\textrm {th}}}
case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value tha{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
|
Hyperbolic penalty value for a point with respect to a bounded region - MATLAB hyperbolicPenalty - MathWorks Australia
hyperbolicPenalty
Calculate Hyperbolic Penalty for a Point
Hyperbolic penalty value for a point with respect to a bounded region
p = hyperbolicPenalty(x,xmin,xmax)
p = hyperbolicPenalty(___,lambda,tau)
p = hyperbolicPenalty(x,xmin,xmax) calculates the nonnegative (hyperbolic) penalty vector p for the point x with respect to the region bounded by xmin and xmax. p has the same dimension as x. This syntax uses the default values of 1 and 0.1 for the lambda and tau parameters of the hyperbolic function, respectively.
p = hyperbolicPenalty(___,lambda,tau) specifies both the lambda and tau parameters of the hyperbolic function. If lambda is an empty matrix its default value is used. Likewise if tau is an empty matrix or it is omitted, its default value is used instead.
This example shows how to use the hyperbolicPenalty function to calculate the hyperbolic penalty for a given point with respect to a bounded region.
Calculate the penalty value for the point 0.1 within the interval [-2,2], using default values for the lambda and tau parameters.
hyperbolicPenalty(0.1,-2,2)
hyperbolicPenalty(4,-2,2)
Calculate the penalty value for the point 0.1 within the interval [-2,2], using a lambda parameter of 5.
hyperbolicPenalty(0.1,-2,2,5)
Calculate the penalty value for the point 4 outside the interval [-2,2], using a lambda parameter of 5.
hyperbolicPenalty(4,-2,2,5)
Calculate the penalty value for the point 4 outside the interval [-2,2], using a tau parameter of 0.5.
hyperbolicPenalty(4,-2,2,5,0.5)
Calculate the penalty value for the point [-2,0,4] with respect to the box defined by the intervals [0,1], [-1,1], and [-2,2] along the x, y, and z dimensions, respectively, using the default value for lambda and a tau parameter of 0.
hyperbolicPenalty([-2 0 4],[0 -1 -2],[1 1 2],1,0)
Calculate penalties for all the points in the vector, using default values for the lambda and tau parameters.
p = hyperbolicPenalty(x,-2,2);
Lower bounds for x, specified as a numeric scalar, vector or matrix. To use the same minimum value for all elements in x specify xmin as a scalar.
Upper bounds for x, specified as a numeric scalar, vector or matrix. To use the same maximum value for all elements in x specify xmax as a scalar.
lambda — Lambda parameter of the hyperbolic function
Lambda parameter of the hyperbolic function, specified as a scalar.
tau — Tau parameter of the hyperbolic function
Tau parameter of the hyperbolic function, specified as a scalar.
Penalty value, returned as a vector of nonnegative elements. Each element pi depends on the position of xi with respect to the interval specified by xmini and xmaxi. The hyperbolic penalty function returns the value:
p\left(x\right)=-\lambda \left(x-{x}_{\mathrm{min}}\right)+\sqrt{{\lambda }^{2}{\left(x-{x}_{\mathrm{min}}\right)}^{2}+{\tau }^{2}}-\lambda \left({x}_{\mathrm{max}}-x\right)+\sqrt{{\lambda }^{2}{\left({x}_{\mathrm{max}}-x\right)}^{2}+{\tau }^{2}}
Here, λ is the argument lambda, and τ is the argument tau. Note that for positive values of τ the returned penalty value is always positive, because on the right side of the equation the magnitude of the second term is always greater than that of the first, and the magnitude of the fourth term is always greater than that of the third. If τ is zero, then the returned penalty is zero inside the interval defined by the bounds, and it grows linearly with x outside this interval. If x is multidimensional, then the calculation is applied independently on each dimension. Penalty functions are typically used to generate negative rewards when constraints are violated, such as in generateRewardFunction.
generateRewardFunction | exteriorPenalty | barrierPenalty
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Help:Formatting - LIMSWiki
5 Other formatting and tools
6 Inserting media and tables
7 Inserting templates, citations, and category tags
10 Formatting help
[[Laboratory]]<nowiki/> equipment
Skip Level 1; it is page name level.
An article with four or more headings automatically creates a table of contents.
<xh2>Level 2</xh2>
Begin with a semicolon. One item per line; a new line can appear before the colon, but using a space before the colon improves parsing.
MediaWiki ignores single line breaks. To start a new paragraph, leave an empty line:
When used in a list, a newline ''does'' affect the layout (see above).
When used in a list, a newline does affect the layout (see above).
If necessary, you can force a line break within a paragraph with the HTML tag <br />:
without a new paragraph.<br />
Leading spaces to preserve formatting
interprets [[Help:MediaWiki basics/Introduction to MediaWiki and wikis|wiki]]
Here's a link to a page named [[Cell counter]].
You can even say [[cell counter]]s
Example: ''[[Laboratory informatics]]''.
capitalized, so [[laboratory informatics]] goes to the same place
as [[Laboratory informatics]]. Capitalization matters after the
[[Laboratory information management system#Technology]]
You can make the text appearing on an internal link different from the article title:
[[Laboratory information management system#Technology|technology of LIMS]]
If you wish to link to a category, add a colon in front:
[[:Category:LIMSwiki help documentation]]
Here's a link to a page named Cell counter. You can even say cell counters and the link will show up correctly.
You can put formatting around a link. Example: Laboratory informatics.
The first letter of articles is automatically capitalized, so laboratory informatics goes to the same place as Laboratory informatics. Capitalization matters after the first letter.
You can link to a page section by its title: Laboratory information management system#Technology
You can make the text appearing on an internal link different from the article title: technology of LIMS
If you wish to link to a category, add a colon in front: Category:LIMSwiki help documentation
http://clinfowiki.org
[http://clinfowiki.org ClinfoWiki.org]
[http://clinfowiki.org]
mailto:someone@example.com or
[mailto:someone@example.com someone]
You can make an external link just by typing a URL: http://clinfowiki.org
You can give it a title: ClinfoWiki.org
Other formatting and tools
{\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}
Comments are only visible in the edit zone.
Signing talk page comments
signature: ~~~ <br>
signature plus date/time: ~~~~ <br>
- Three tildes gives your signature: Shawn Douglas
- Four tildes give your signature plus date/time: Shawn Douglas 13:45, 23 October 2013 (EDT)
- Five tildes gives the date/time alone: 13:45, 23 October 2013 (EDT)
#REDIRECT [[Laboratory informatics]]
You use redirects most often on pages with incorrect or outdated page titles. You simply copy or remove the existing content, paste this code in, and change the internal link text to the title of the article you wish to automatically redirect users to.
#REDIRECT Laboratory informatics
Inserting media and tables
For more on these topics: Help:MediaWiki basics/Intermediate training
Inserting templates, citations, and category tags
For more on these topics: Help:MediaWiki basics/Advanced training
See the list of all HTML entities on the Wikipedia article List of XML and HTML character entity references. Additionally, MediaWiki supports two non-standard entity reference sequences: &רלמ; and &رلم; which are both considered equivalent to ‏ which is a right-to-left mark. (Used when combining right-to-left languages with left-to-right languages in the same page.)
Beyond the text formatting markup shown on this page, here are some other formatting references:
Category:LIMSwiki training
You can find more help documentation at Category:LIMSwiki help documentation.
Retrieved from "https://www.limswiki.org/index.php?title=Help:Formatting&oldid=13285"
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Polyphase FFT synthesis filter bank - MATLAB
{e}^{j{w}_{k}n},\text{ }{w}_{k}=2\pi k/M,\text{ }k=0,1,...,M-1
{H}_{0}\left(z\right)={b}_{0}+{b}_{1}{z}^{-1}+...+{b}_{N}{z}^{-N}
{H}_{0}\left(z\right)=\begin{array}{c}\left({b}_{0}+{b}_{M}{z}^{-M}+{b}_{2M}{z}^{-2M}+..+{b}_{N-M+1}{z}^{-\left(N-M+1\right)}\right)+\\ {z}^{-1}\left({b}_{1}+{b}_{M+1}{z}^{-M}+{b}_{2M+1}{z}^{-2M}+..+{b}_{N-M+2}{z}^{-\left(N-M+1\right)}\right)+\\ \begin{array}{c}⋮\\ {z}^{-\left(M-1\right)}\left({b}_{M-1}+{b}_{2M-1}{z}^{-M}+{b}_{3M-1}{z}^{-2M}+..+{b}_{N}{z}^{-\left(N-M+1\right)}\right)\end{array}\end{array}
{H}_{0}\left(z\right)={E}_{0}\left({z}^{M}\right)+{z}^{-1}{E}_{1}\left({z}^{M}\right)+...+{z}^{-\left(M-1\right)}{E}_{M-1}\left({z}^{M}\right)
{H}_{k}\left(z\right)={H}_{0}\left(z{e}^{j{w}_{k}}\right)
{H}_{k}\left(z\right)={h}_{0}+{h}_{1}{e}^{jwk}{z}^{-1}+{h}_{2}{e}^{j2wk}{z}^{-2}...+{h}_{N}{e}^{jNwk}{z}^{-N}
{H}_{k}\left(z\right)=\left[1\text{ }{e}^{j{w}_{k}}\text{ }{e}^{j2{w}_{k}}\text{ }...\text{ }{e}^{j\left(M-1\right){w}_{k}}\right]\left[\begin{array}{c}{E}_{0}\left({z}^{M}\right)\\ {z}^{-1}{E}_{1}\left({z}^{M}\right)\\ ⋮\\ {z}^{-\left(M-1\right)}{E}_{M-1}\left({z}^{M}\right)\end{array}\right]
H\left(z\right)=\left[\begin{array}{c}\begin{array}{lllll}1\hfill & 1\hfill & 1\hfill & ...\hfill & 1\hfill \end{array}\\ \begin{array}{lllll}1\hfill & {e}^{j{w}_{1}}\hfill & {e}^{j2{w}_{1}}\hfill & ...\hfill & {e}^{j\left(M-1\right){w}_{1}}\hfill \end{array}\\ ⋮\\ \begin{array}{lllll}1\hfill & {e}^{j{w}_{M-1}}\hfill & {e}^{j2{w}_{M-1}}\hfill & ...\hfill & {e}^{j\left(M-1\right){w}_{M-1}}\hfill \end{array}\end{array}\right]\left[\begin{array}{c}{E}_{0}\left({z}^{M}\right)\\ {z}^{-1}{E}_{1}\left({z}^{M}\right)\\ ⋮\\ {z}^{-\left(M-1\right)}{E}_{M-1}\left({z}^{M}\right)\end{array}\right]
H\left(z\right)=\left[\begin{array}{c}\begin{array}{lllll}1\hfill & 1\hfill & 1\hfill & ...\hfill & 1\hfill \end{array}\\ \begin{array}{lllll}1\hfill & {e}^{j{w}_{1}}\hfill & {e}^{j2{w}_{1}}\hfill & ...\hfill & {e}^{j\left(M-1\right){w}_{1}}\hfill \end{array}\\ ⋮\\ \begin{array}{lllll}1\hfill & {e}^{j{w}_{M-1}}\hfill & {e}^{j2{w}_{M-1}}\hfill & ...\hfill & {e}^{j\left(M-1\right){w}_{M-1}}\hfill \end{array}\end{array}\right]\left[\begin{array}{c}{E}_{0}\left(z\right)\\ {E}_{1}\left(z\right)\\ ⋮\\ {E}_{M-1}\left(z\right)\end{array}\right]
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Experimental and Numerical Investigation on the Influence of Thermally Induced Stress Gradients on Fatigue Life of the Nickel-Base Alloy Mar-M247 | J. Eng. Gas Turbines Power | ASME Digital Collection
Dresden 01062,
e-mail: marcus.thiele@tu-dresden.de
Stefan Eckmann,
Freiburg 79108,
e-mail: stefan.eckmann@iwm.fraunhofer.de
Material Testing Institute
e-mail: min.huang@mpa.uni-stuttgart.de
Kathrin A. Fischer,
Kathrin A. Fischer
Thiele, M., Eckmann, S., Huang, M., Gampe, U., Fischer, K. A., and Schlesinger, M. (September 29, 2020). "Experimental and Numerical Investigation on the Influence of Thermally Induced Stress Gradients on Fatigue Life of the Nickel-Base Alloy Mar-M247." ASME. J. Eng. Gas Turbines Power. October 2020; 142(10): 101009. https://doi.org/10.1115/1.4047810
Today's and future parameters of stationary gas turbines and aircraft engines require intensive and highly efficient cooling of hot gas path components. High temperature and thermally induced stress gradients with impact on fatigue life are the consequence. Thermally induced stress gradients differ from geometrically induced stress gradients with respect to stress mechanics by the independence from external loads and material mechanics by the influence of temperature on material properties and strength. Regarding the contribution and evaluation on damage, the latter characteristic feature in turbomachinery is currently not fully understood. Therefore, a test facility has been designed, setup, and reported in GTP-18-1482 for the investigation of the influence of stationary temperature, and thus thermally induced stress gradients, on the damage evolution of cooled high-temperature components. To achieve high temperature and thermally induced stress gradients, large heat fluxes are required. A unique radiation heating has been developed allowing very high heat fluxes of
q˙
≥ 1.5 MW/m2 for testing of hollow cylindrical specimens. The conventional cast nickel-base alloy Mar-M247 has been chosen to study the influence of thermally induced stress gradients on fatigue life. The low-cycle fatigue testing of the hollow cylindrical specimens has been conducted both with and without superimposed stationary temperature gradients. In addition, complex low-cycle fatigue (CLCF) tests with symmetric and nonsymmetric loading conditions have been performed to provide the necessary database for the adaptation of a viscoplastic deformation model. To calculate the local stress–strain field and service life of the test specimens, linear elastic and viscoplastic finite element studies have been performed and were assessed by means of a fracture mechanics-based lifetime model. The test results show the considerable influence of the temperature gradient on the low-cycle fatigue life for the investigated material. Both the radial temperature variation over the specimen wall with a hot outer surface and a cooled inner surface as well as the thermally induced stresses are stated to be the main drivers for the change in low-cycle fatigue life. The test results enhance the understanding of fatigue-damage mechanisms under local unsteady conditions and can be used as a basis for improved service life predictions.
Damage, Stress, Temperature, Temperature gradient, Low cycle fatigue, Deformation, Nickel, Fracture (Materials), Alloys, Fatigue life
Multiaxial Thermo‐Mechanical Fatigue on Material Systems for Gas Turbines
Singheiser
.10.1051/epjconf/20123302001
Stresses Due to Temperature Gradients in Ceramic‐Matrix‐Composite Aerospace Components
,” Chinese Society for Composite Materials,
, accessed Dec. 10, 2018, https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/25534/1/ICCM21%286%29fullpaperFormatted.pdf
Rissverhalten unter anisothermen Beanspruchungsbedingungen - Berechnungsverfahren für Nickelbasislegierungen: Abschlussbericht zum AiF-Forschungsvorhaben Nr. 15 525 N
FVV, Frankfurt am Main
TMF-Rissverhalten II, Zeitabhängiges Rissverhalten Unter Anisothermen Beanspruchungen -Entwicklung Und Validierung Von Berechnungsmethoden: Abschlussbericht Zum AiF-Forschungsvorhaben Nr. 17809 N
Modelling the Fatigue Life of Automobile Components
Fatigue 2002—Proceedings of the Eighth International Fatigue Congress
Tables of Hutchinson-Rice-Rosengren Singular Field Quantities
, Report No. MRL E-147.
Advances in Fracture Research 84—Proc. of ICF6
Engineering Approach for Elastic-Plastic Fracture Analysis
Report No. NP-1931
The Penny-Shaped Crack and the Plane Strain Crack in an Infinite Body of Power Law Material
von Hartrott
Verbesserte Methoden Zur Lebensdauerberechnung Von Abgasturbolader-Radialverdichterrädern Aus Hochwarmfesten Aluminiumlegierungen: Abschlussbericht Zum FW-Forschungsvorhaben Nr. 897
Creep Processes in MAR-M247 Nickel-Base Superalloy
Materials Structure & Micromechanics of Fracture VIII: Selected, Peer Reviewed Papers From the Eighth International Conference on Materials Structure & Micromechanics of Fracture (MSMF-8)
.https://www.scientific.net/SSP.258.603#:~:text=Abstract%3A,the%20creep%20deformation%20and%20fracture.
Accelerated Material Data Generation for Viscoplastic Material Models Based on Complex LCF and Incremental Creep Tests
Investigation of Isothermal and Thermo-Mechanical Fatigue Behavior of the Nickel-Base Superalloy IN738 LC Using Standardized and Advanced Test Methods
, Standard No. 2368–04.
Validated Code-of-Practice for Strain-Controlled Thermo-Mechanical Fatigue Testing
Pettenll, The Netherlands
.https://www.researchgate.net/publication/265152601_Validated_Code-of-Practice_for_Strain-Controlled_Thermo-Mechanical_Fatigue_Testing
Hierarchical Crystal Plasticity FE Model for Nickel-Based Superalloys: Sub-Grain Microstructures to Polycrystalline Aggregates
Ginzbursky
Gas Temperature Measurement in Engine Conditions Using Uniform Crystal Temperature Sensors (UCTS)
Intermediate Temperature, Low-Cycle Fatigue Behavior of Coated and Uncoated Nickel Base Superalloys in Air and Corrosive Sulfate Environments
Low-Cycle Fatigue of Nickel-Based Superalloy Hastelloy X at Elevated Temperatures
|
Heat Transfer in Rotating, Trailing Edge, Converging Channels With Full- and Partial-Height Strip-Fins | J. Turbomach. | ASME Digital Collection
Izzet Sahin 1
1Corresponding author. Email: izzetsahin@tamu.edu
Sahin, I., Chen, I., Wright, L. M., Han, J., Xu, H., and Fox, M. (March 4, 2022). "Heat Transfer in Rotating, Trailing Edge, Converging Channels With Full- and Partial-Height Strip-Fins." ASME. J. Turbomach. September 2022; 144(9): 091009. https://doi.org/10.1115/1.4053492
A wide variety of pin-fins have been used to enhance heat transfer in internal cooling channels. However, due to their large blockage in the flow direction, they result in an undesirable high pressure drop. This experimental study aims to reduce pressure drop while increasing the heat transfer surface area by utilizing strip-fins in converging internal cooling channels. The channel is designed with a trapezoidal cross section, converges in both transverse and longitudinal directions, and is also skewed
β=120deg
with respect to the direction of rotation to model a trailing edge cooling channel. Only the leading and trailing surfaces of the channel are instrumented, and each surface is divided into 18 isolated copper plates to measure the regionally averaged heat transfer coefficient. Utilizing pressure taps at the inlet and outlet of the channel, the pressure drop is obtained. Three staggered arrays of strip-fins are investigated: one full-height configuration and two partial fin height arrangements (Sz = 2 mm and 1 mm). In all cases, the strip-fins are 2 mm wide (W) and 10 mm long (Lf) in the flow direction. The fins are spaced such that Sy/Lf = 1 in the streamwise direction. However, due to the convergence, the spanwise spacing, Sx/W, was varied from 8 to 6.2 along the channel. The rotation number of the channel varied up to 0.21 by ranging the inlet Reynolds number from 10,000 to 40,000 and rotation speed from 0 to 300 rpm. It is found that the full-height strip-fin channel results in a more nonuniform spanwise heat transfer distribution than the partial-height strip-fin channel. Both trailing and leading surface heat transfer coefficients are enhanced under rotation conditions. The 2 mm height partial strip-fin channel provided the best thermal performance, and it is comparable to the performance of the converging channels with partial-length circular pins. The strip-fin channel can be a design option when the pressure drop penalty is a major concern.
boundary layer development, turbomachinery blading design
Fins, Flow (Dynamics), Heat transfer, Pressure, Rotation, Strips, Reynolds number, Pressure drop
ASME-JSME Therm. Eng. Joint Conf.
,” Heat Transfer 1986;
. https://ui.adsabs.harvard.edu/abs/1986hetr.conf.2915S/abstract
, “Blade Trailing Edge Heat Transfer,” ASME Paper No. 80-GT-45.
, “Row Resolved Heat Transfer Variations in Pin-Fin Arrays Including Effects of Non-Uniform Arrays and Flow Convergence,” ASME Paper No. 86-GT-132.
Copyright © 2022 Solar Turbines Incorporated
Prediction Relations for Pressure Drop Across Finned Sections
|
QuickSort is an in-place sorting algorithm with worst-case time complexity of
n^{2}
QuickSort can be implemented iteratively and recursively. We’ll mainly focus on the recursive implementation, as it is far more convenient, intuitive and simplistic. Iterative implementation is generally unrecommended. Arrays are used as an example here, but it can be implemented on other data structures (like linked lists) as well.
The algorithm can be broken down into 3 parts.
Partitioning the array about the pivot
Passing the smaller arrays to the recursive calls
Joining the sorted arrays that are returned from the recursive call, and the pivot
In the above illustration, we used the first element of the array as a pivot (though any of the elements can be taken).
public static int[] QuickSort(int[] arr, int elements) {
if(elements < 2){ //Base Case
int current_position=0; //position of pivot element
int temp; //a temporary variable to assist in swapping
for(int i=1; i<elements; i++) //Partitioning loop
if(arr[i] <= arr[0])
arr[i] = arr[current_position];
arr[current_position] = temp;
arr[0] = arr[current_position];
arr[current_position] = temp; //Brings pivot to it's appropriate position
int[] left = QuickSort(arr,current_position); //sorts the elements to the left of pivot
int[] arr2 = Arrays.copyOfRange(arr, current_position+1, elements);//separates elements right of pivot
int[] right = QuickSort(arr2, elements-current_position-1); //sorts the elements to the right of pivot
int[] final_array = new int[elements]; //final array, to merge everything together
for(int i=0; i<current_position; i++)
final_array[i] = left[i];
final_array[current_position] = arr[current_position];
for(int i=current_position+1; i<elements; i++)
final_array[i] = right[i-current_position-1];
return final_array;
int[] array = {4,2,7,3,1,6}; //array to be sorted
System.out.print("Original Array: [");
array = QuickSort(array, array.length);//sorting
System.out.print("Sorted Array: ["); //printing the sorted array
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SAMDuino - FabLab KAMAKURA
FabLab KAMAKURA SAMDuino
SAMDuino SAMDuino 目次
images for milling with SRM-20
MattairTech MT-D21E(rev B)
Arduino Zero version
MattaireTech MT-D21E version
examples - Arduino Zero
hello echo (serial communication)
serialUSB for Native USB port
Serial1 for 0(RX)/1(TX)
stepper motor (WIP)
examples - MattairTech MT-D21E (WIP)
SAMDuino#
Author : Jun Kawahara (Fablab Kamakura)
SAMDuino is an ATSAMD21E breakout board, compatible with Arduino Zero. The board is powered by Microchip’s SAMD21E microcontroller, a 32-bit ARM Cortex® M0+ core. It runs at 3.3 V.
images for milling with SRM-20#
image resolution: 1600 dpi(Enter 3200 dpi in mods settings because of mac-mods problem, e)
trace (crystal FC-135 version)
1 6MM_SWITCH S1 OMRON SWITCH
1 LM1117 SOT223 U1 Voltage Regulator
2 0 RES-US1206FAB R1206FAB R4, R6
1 0.1 uF CAP-UNPOLARIZEDFAB C1206FAB C5
1 10K RES-US1206FAB R1206FAB R3
2 10pF CAP-UNPOLARIZEDFAB C1206FAB C1, C2
2 10uF CAP-UNPOLARIZEDFAB C1206FAB C3, C4
1 1uF CAP-UNPOLARIZEDFAB C1206FAB C6
2 510 RES-US1206FAB R1206FAB R1, R2
1 FC-135 FC-135 X1 Epson’s FC-135 Thin SMD High-Frequency Crystal unit
2 LEDFAB1206 LEDFAB1206 LED1206FAB U2, U
1 SAMD21E18A SAMD21EXXX-A TQFP32-08 U2
1 ZX62R-B-5P ZX62R-B-5P HRS_ZX62R-B-5P J1 Conn Micro USB 2.0 Type B
pin mapping#
Pin mapping depends on which bootlaoder you’ll burn onto SAMDuino.
Arduino Zero#
MattairTech MT-D21E(rev B)#
burning a bootloader#
SAMDuino (a target board)
programmer: CMSIS-DAP compliant adaptor(a programmer; hello.CMSIS-DAP.10.D11C)
50-mil(1.27 mm) 10-pin mini squid cable with 10 x 100-mil(2.54 mm) sockets
Arduino Zero version#
Connect your host PC and programmer. Make sure that your host PC can recognize your programmer as a USB device.
In Arduino IDE, go to Tools > Board and select “Arduino Zero (Programming Port),” also Tools > Programmer and select “Atmel-ICE.”
Now Burn Bootloader!
After verbose messages, Arduino IDE will be able to detect SAMDuino as an “Arduino Zero(Native USB Port)“
When you upload a sketch through USB, you have to select a board “Arduino Zero (Native USB Port)”
MattaireTech MT-D21E version#
Install board package for ATSAMD21, if neccessary. Open Arduino > preferences… Add following URL to Additional Boards Manager URLs section.
http://www.mattairtech.com/software/arduino/package_MattairTech_index.json
Go Tools > Board > Board Mangers…, search mattair and install MattairTech SAM D|L|C core for Arduino.
In Arduino IDE, go to Tools > Board and select “MattairTech MT-D21E (rev B)“
Microcontrller and Clock Srouce depend on your board design. Here I used SAMD21E18A and “32KHZ_CRYSTAL.”
Then, Tools > Programmer and select “Atmel-ICE.”
4. After a long verbose messages, “MattairTech Xeno Mini” will show up on your port.
It’s not “MT-D21E,” but your host PC is able to recognize your board as a USB device, named “MT-D21E.”
examples - Arduino Zero#
In Arduino IDE, Open File > Examples > 01.Basics > Blink, and upload. A built-in LED at pin 13 will blink.
hello echo (serial communication)#
Arduino Zero has at least two serial ports, SerialUSB for Native USB port (USB CDC) and Serial1 for 0(RX)/1(TX). Usual Serial which we use with Arduino Uno doesn’t work here unless if you define so.
serialUSB for Native USB port#
As you can see in Neil’s example code below, hello.D21E.echo.ino, in embdded programming week, he uses SerialUSB, not a familiar Serial.
// hello.D21E.echo.ino
// ATSAMD21E USB echo hello-world
SerialUSB.begin(0);
chr = SerialUSB.read();
SerialUSB.print("hello.D21E.echo: you typed \"");
SerialUSB.println("\"");
Connect your host PC and a SAMDuino with a USB cable, and upload the sketch. Open a terminal and type a command like an image below, then type a byte character. If SAMDuino echoed back the character, the serial communication through Native USB port worked.
Serial1 for 0(RX)/1(TX)#
Open hello.D21E.echo.ino with Arduino IDE.
In the sketch, replace SerialUSB with Serial1. Set a baudrate because “0” doesn’t work.
SerialUSB.begin(0) -> Serial1.begin(9600)
Serial1.begin(9600); //set baudrate
if (Serial1.available()) { // SerialUSB -> Serial1
chr = Serial1.read();
Serial1.print("hello.SAMDuino.echo: you typed \"");
Serial1.print(buffer);
3. Connect a UART-USB converter and SAMDuino with jumper wires.
- UART-USB converter TX <-> SAMDuino RX(0) - UART-USB converter RX <-> SAMDuino TX(1)
Open a terminal and type a command like below. Serial port device depends on your UART-USB converter.
python -m serial.tools.miniterm /dev/cu.usbserial-A105196J
Type a byte character. If SAMDuino echoed back the character, the serial communication through Native USB port worked.
In Arduino IDE, Open File > Examples > Servo > Sweep.
Wire between SAMDuino and a servo as follows.
servo 5 V -> SAMDuino VBUS
servo GND -> SAMDuino GND
serov signal -> SAMDuino pin 9
Upload the sketch. You’ll see the servo rotate like in the video below.
NeoPixel#
Adafruit NeoPixel Library supports ATSAMD21 (Arduino Zero/M0 and other SAMD21 boards) @ 48 MHz. I also confirmed that FastLED worked with SAMD21.
Here’s an example, NeoPixel-Knightrider which uses Adafruit Neopixel Library.
// 888b 888 8888888b. d8b 888
// 8888b 888 888 Y88b Y8P 888
// 88888b 888 888 888 888
// 888Y88b 888 .d88b. .d88b. 888 d88P 888 888 888 .d88b. 888
// 888 Y88b888 d8P Y8b d88""88b 8888888P" 888 `Y8bd8P' d8P Y8b 888
// 888 Y88888 88888888 888 888 888 888 X88K 88888888 888
// 888 Y8888 Y8b. Y88..88P 888 888 .d8""8b. Y8b. 888
// 888 Y888 "Y8888 "Y88P" 888 888 888 888 "Y8888 888
// NeoPixel-KnightRider
// A highly configurable Knight Rider (larson display) routine for your NeoPixels
// (WS2812 RGB LED)
// Copyright (c) 2013 Technobly - technobly@gmail.com - August 13th 2013
// ASCII GEN http://patorjk.com/software/taag/#p=display&f=Colossal&t=NeoPixel
// SETUP YOUR OUTPUT PIN AND NUMBER OF PIXELS
clearStrip(); // Initialize all pixels to 'off'
knightRider(3, 35, 4, 0xFF0000); // Cycles, Speed, Width, RGB Color (red)
// Cycles - one cycle is scanning through all pixels left then right (or right then left)
// Speed - how fast one cycle is (32 with 16 pixels is default KnightRider speed)
// Width - how wide the trail effect is on the fading out LEDs. The original display used
// light bulbs, so they have a persistance when turning off. This creates a trail.
// Effective range is 2 - 8, 4 is default for 16 pixels. Play with this.
// Color - 32-bit packed RGB color value. All pixels will be this color.
// knightRider(cycles, speed, width, color);
void knightRider(uint16_t cycles, uint16_t speed, uint8_t width, uint32_t color) {
uint32_t old_val[NUM_PIXELS]; // up to 256 lights!
// Larson time baby!
for (int count = 1; count<NUM_PIXELS; count++) {
strip.setPixelColor(count, color);
old_val[count] = color;
for(int x = count; x>0; x--) {
old_val[x-1] = dimColor(old_val[x-1], width);
strip.setPixelColor(x-1, old_val[x-1]);
for (int count = NUM_PIXELS-1; count>=0; count--) {
for(int x = count; x<=NUM_PIXELS ;x++) {
strip.setPixelColor(x+1, old_val[x+1]);
for( int i = 0; i<NUM_PIXELS; i++){
uint32_t dimColor(uint32_t color, uint8_t width) {
return (((color&0xFF0000)/width)&0xFF0000) + (((color&0x00FF00)/width)&0x00FF00) + (((color&0x0000FF)/width)&0x0000FF);
2. Upload the sketch and you’ll see the NeoPixel ring grow like in the video below.
LCD (I2C)#
need a level shifter
stepper motor (WIP)#
examples - MattairTech MT-D21E (WIP)#
Crystal/FC-135 version: sch, brd, and lbr
trace image (1600 dpi. Because of mac-mods problem, enter 3200 dpi in mods settings)
Minimal ATSAMD21 Computer - Technoblogy
Arduino Zero - Arduino.cc
Burning Zero bootloader with Beaglebone as SWD programmer
Using ATSAMD21 SERCOM for more SPI, I2C and Serial ports - Adafruit
Serial Communications - Arduino Cookbook, 3rd Edition by Michael Margolis, Brian Jepson, Nicholas Robert Weldin(O’REILLY)
Samd- Arduino Library List
前 USB-D11C-serial
次 Barduino 2.0
|
Gridded data interpolation - MATLAB - MathWorks América Latina
3-D Interpolation Using Full Grid vs. Grid Vectors
Interpolation with Default Grid
2-D Interpolation Over Finer Grid
Interpolate Multiple Sets of Values on Same Grid
Interpolate multiple data sets simultaneously
Gridded data interpolation
Use griddedInterpolant to perform interpolation on a 1-D, 2-D, 3-D, or N-D gridded data set. griddedInterpolant returns the interpolant F for the given data set. You can evaluate F at a set of query points, such as (xq,yq) in 2-D, to produce interpolated values vq = F(xq,yq).
Use scatteredInterpolant to perform interpolation with scattered data.
F = griddedInterpolant creates an empty gridded data interpolant object.
F = griddedInterpolant(x,v) creates a 1-D interpolant from a vector of sample points x and corresponding values v.
F = griddedInterpolant(X1,X2,...,Xn,V) creates a 2-D, 3-D, or N-D interpolant using a full grid of sample points passed as a set of n-dimensional arrays X1,X2,...,Xn. The V array contains the sample values associated with the point locations in X1,X2,...,Xn. Each of the arrays X1,X2,...,Xn must be the same size as V.
F = griddedInterpolant(V) uses the default grid to create the interpolant. When you use this syntax, griddedInterpolant defines the grid as a set of points whose spacing is 1 and range is [1, size(V,i)] in the ith dimension. Use this syntax when you want to conserve memory and are not concerned about the absolute distances between points.
F = griddedInterpolant(gridVecs,V) specifies a cell array gridVecs that contains n grid vectors to describe an n-dimensional grid of sample points. Use this syntax when you want to use a specific grid and also conserve memory.
F = griddedInterpolant(___,Method) specifies the interpolation method: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'makima', or 'spline'. You can specify Method as the last input argument in any of the previous syntaxes.
F = griddedInterpolant(___,Method,ExtrapolationMethod) specifies both the interpolation and extrapolation methods. griddedInterpolant uses ExtrapolationMethod to estimate the value when your query points fall outside the domain of your sample points.
Sample points, specified as a vector. x and v must be the same size. The sample points in x must be unique.
Sample values, specified as a vector, matrix, or multidimensional array. The elements of v are the values that correspond to the sample points in x.
To interpolate using a single set of values, x and v must be vectors of the same length.
To interpolate using multiple sets of values, v can be an array with extra dimensions compared to x. The size of the first dimension of v must match the number of sample points in x, and each column in v defines a separate set of 1-D values. For example, if x is a column vector with 10 elements, you can specify v as a 10-by-4 matrix to interpolate using four different sets of values.
X1, X2, Xn — Sample points in full grid form
Sample points in full grid form, specified as separate n-dimensional arrays. The sample points must be unique and sorted. You can create the arrays X1,X2,...,Xn using the ndgrid function. These arrays are all the same size, and each one is the same size as V.
gridVecs — Sample points in grid vector form
cell array of grid vectors
Sample points in grid vector form, specified as a cell array of grid vectors {xg1,xg2,...,xgn}. The sample points must be unique and sorted. The vectors must specify a grid that is the same size as V. In other words, size(V) = [length(xg1) length(xg2),...,length(xgn)]. Use this form as an alternative to the full grid to save memory when your grid is very large.
Sample values, specified as an array. The elements of V are the values that correspond to the sample points. The first N dimensions of V must have the same sizes as the corresponding dimensions in the full grid of sample points, where N is the number of dimensions of the grid.
To interpolate using a single set of values, specify V as an array with the same size as the full grid of sample points. For example, if the sample points form a grid with size 100-by-100, you can specify the values with a matrix of the same size.
To interpolate using multiple sets of values, specify V as an array with extra dimensions compared to the grid of sample points. The extra dimensions define multiple values at each sample point. For example, if the sample points form a grid with size 100-by-100, you can specify the values as an array with size 100-by-100-by-4 to interpolate using four different sets of 100-by-100 values.
'linear' (default) | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'spline' | 'makima'
'linear' (default) Linear interpolation. The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension. C0
'nearest' Nearest neighbor interpolation. The interpolated value at a query point is the value at the nearest sample grid point. Discontinuous
'next' Next neighbor interpolation (for 1-D only). The interpolated value at a query point is the value at the next sample grid point. Discontinuous
'previous' Previous neighbor interpolation (for 1-D only). The interpolated value at a query point is the value at the previous sample grid point. Discontinuous
'pchip' Shape-preserving piecewise cubic interpolation (for 1-D only). The interpolated value at a query point is based on a shape-preserving piecewise cubic interpolation of the values at neighboring grid points. C1
'cubic' Cubic interpolation. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic convolution. C1
Grid must have uniform spacing, although the spacing in each dimension does not have to be the same
'spline' Cubic spline interpolation. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic spline using not-a-knot end conditions. C2
Requires 4 points in each dimension
'linear' (default) | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'spline' | 'makima' | 'none'
Extrapolation method, specified as 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'spline', or 'makima'. In addition, you can specify 'none' if you want queries outside the domain of your grid to return NaN values.
If you omit ExtrapolationMethod, the default is the value you specify for Method. If you omit both the Method and ExtrapolationMethod arguments, both values default to 'linear'.
GridVectors — Grid vectors
Grid vectors, specified as a cell array {xg1,xg2,...,xgn}. These vectors specify the grid points (locations) for the values in Values. The grid points must be unique.
Function values at sample points, specified as an array of values associated with the grid points in GridVectors.
Interpolation method, specified as a character vector. Method can be: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'spline', or 'makima'. See Method for descriptions of these methods.
Extrapolation method, specified as a character vector. ExtrapolationMethod can be: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'spline', 'makima', or 'none'. A value of 'none' indicates that extrapolation is disabled. The default value is the value of Method.
Use griddedInterpolant to create the interpolant, F. Then you can evaluate F at specific points using any of the following syntaxes:
Vq = F(Xq) specifies the query points in the matrix Xq. Each row of Xq contains the coordinates of a query point.
Vq = F(xq1,xq2,...,xqn) specifies the query points xq1,xq2,...,xqn as column vectors of length m representing m points scattered in n-dimensional space.
Vq = F(Xq1,Xq2,...,Xqn) specifies the query points using the n-dimensional arrays Xq1,Xq2,...,Xqn, which define a full grid of points.
Vq = F({xgq1,xgq2,...,xgqn}) specifies the query points as grid vectors. Use this syntax to conserve memory when you want to query a large grid of points.
Use griddedInterpolant to interpolate a 1-D data set.
Create a vector of scattered sample points v. The points are sampled at random 1-D locations between 0 and 20.
Create a gridded interpolant object for the data. By default, griddedInterpolant uses the 'linear' interpolation method.
Query the interpolant F at 500 uniformly spaced points between 0 and 20. Plot the interpolated results (xq,vq) on top of the original data (x,v).
Interpolate 3-D data using two methods to specify the query points.
Create and plot a 3-D data set representing the function
\mathit{z}\left(\mathit{x},\mathit{y}\right)=\frac{\mathrm{sin}\left({\mathit{x}}^{2}+{\mathit{y}}^{2}\right)}{{\mathit{x}}^{2}+{\mathit{y}}^{2}}
evaluated at a set of gridded sample points in the range [-5,5].
Create a gridded interpolant object for the data.
Use a finer mesh to query the interpolant and improve the resolution.
In cases where there are a lot of sample points or query points, and where memory usage becomes a concern, you can use grid vectors to improve memory usage.
When you specify grid vectors instead of using ndgrid to create the full grid, griddedInterpolant avoids forming the full query grid to carry out the calculations.
When you pass grid vectors, they are normally grouped together as cells in a cell array, {xg1, xg2, ..., xgn}. The grid vectors are a compact way to represent the points of the full grid.
Alternatively, execute the previous commands using grid vectors.
Use the default grid to perform a quick interpolation on a set of sample points. The default grid uses unit-spaced points, so this interpolation is useful when the exact xy spacing between the sample points is not important.
Create a matrix of sample function values and plot them against the default grid.
Interpolate the data using the default grid.
Query the interpolant and plot the results.
Interpolate coarsely sampled data using a full grid with spacing of 0.5.
Define the sample points as a full grid with range [1, 10] in both dimensions.
f\left(x,y\right)={x}^{2}+{y}^{2}
at the grid points.
Create the interpolant, specifying cubic interpolation.
Define a full grid of query points with 0.5 spacing and evaluate the interpolant at those points. Then plot the result.
Compare results of querying the interpolant outside the domain of F using the 'pchip' and 'nearest' extrapolation methods.
Create the interpolant, specifying 'pchip' as the interpolation method and 'nearest' as the extrapolation method.
Query the interpolant, and include points outside the domain of F.
Query the interpolant at the same points again, this time using the pchip extrapolation method.
Use griddedInterpolant to interpolate three different sets of values at the same query points.
Create a grid of sample points with
-5\le \mathit{X}\le 5
-3\le \mathit{Y}\le 3
Evaluate three different functions at the query points, and then concatenate the values into a 3-D array. The size of V is the same as the X and Y grids in the first two dimensions, but the size of the extra dimension reflects the number of values associated with each sample point (in this case, three).
Create an interpolant using the sample points and associated values.
Create a grid of query points with a finer mesh size compared to the sample points.
Interpolate all three sets of values at the query points.
Compare the original data with the interpolated results.
It is quicker to evaluate a griddedInterpolant object F at many different sets of query points than it is to compute the interpolations separately using interp1, interp2, interp3, or interpn. For example:
R2021a: Interpolate multiple data sets simultaneously
Support added to interpolate multiple data sets on the same grid at the same query points. For example, if you specify a 2-D grid, a 3-D array of values at the grid points, and a 2-D collection of query points, then griddedInterpolant returns the interpolated values at the query points for each 2-D page in the 3-D array of values.
Previously, this functionality was available in interp1 for 1-D interpolation, but this improvement to griddedInterpolant adds support for N-D multivalued interpolation.
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Remove LMI from system of LMIs - MATLAB dellmi - MathWorks United Kingdom
dellmi
Remove LMI from system of LMIs
newsys = dellmi(lmisys,n)
dellmi deletes the n-th LMI from the system of LMIs described in lmisys. The updated system is returned in newsys.
The ranking n is relative to the order in which the LMIs were declared and corresponds to the identifier returned by newlmi. Since this ranking is not modified by deletions, it is safer to refer to the remaining LMIs by their identifiers. Finally, matrix variables that only appeared in the deleted LMI are removed from the problem.
Suppose that the three LMIs
\begin{array}{l}{A}_{1}^{T}{X}_{1}+{X}_{1}{A}_{1}+{Q}_{1}<0\\ {A}_{2}^{T}{X}_{2}+{X}_{2}{A}_{2}+{Q}_{2}<0\\ {A}_{3}^{T}{X}_{3}+{X}_{3}{A}_{3}+{Q}_{3}<0\end{array}
have been declared in this order, labeled LMI1, LMI2, LMI3 with newlmi, and stored in lmisys. To delete the second LMI, type
lmis = dellmi(lmisys,LMI2)
lmis now describes the system of LMIs
\begin{array}{l}{A}_{1}^{T}{X}_{1}+{X}_{1}{A}_{1}+{Q}_{1}<0\\ {A}_{3}^{T}{X}_{3}+{X}_{3}{A}_{3}+{Q}_{3}<0\end{array}
and the second variable X2 has been removed from the problem since it no longer appears in the system.
To further delete LMI3 from the system, type
lmis = dellmi(lmis,LMI3)
lmis = dellmi(lmis,3)
Note that the system has retained its original ranking after the first deletion.
newlmi | lmiedit | lmiinfo
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Folium_of_Descartes Knowpia
In geometry, the folium of Descartes is an algebraic curve defined by the equation
The folium of Descartes (green) with asymptote (blue) when
{\displaystyle a=1}
{\displaystyle x^{3}+y^{3}-3axy=0\,}
The name comes from the Latin word folium which means "leaf".
The curve was first proposed and studied by René Descartes in 1638.[1] Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do.[2] Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.[3]
Graphing the curveEdit
Folium of Descartes in polar coordinates
The folium of Descartes can be expressed in polar coordinates as
{\displaystyle r={\frac {3a\sin \theta \cos \theta }{\sin ^{3}\theta +\cos ^{3}\theta }},}
which is plotted on the left. This is equivalent to[4]
{\displaystyle r={\frac {3a\sec \theta \tan \theta }{1+\tan ^{3}\theta }}.}
Another technique is to write
{\displaystyle y=px}
{\displaystyle x}
{\displaystyle y}
{\displaystyle p}
. This yields the rational parametric equations:[5]
{\displaystyle x={{3ap} \over {1+p^{3}}},\,y={{3ap^{2}} \over {1+p^{3}}}}
We can see that the parameter is related to the position on the curve as follows:
{\displaystyle p<-1}
{\displaystyle x>0}
{\displaystyle y<0}
: the right, lower, "wing".
{\displaystyle -1<p<0}
{\displaystyle x<0}
{\displaystyle y>0}
: the left, upper "wing".
{\displaystyle p>0}
{\displaystyle x>0}
{\displaystyle y>0}
: the loop of the curve.
Another way of plotting the function can be derived from symmetry over
{\displaystyle y=x}
. The symmetry can be seen directly from its equation (x and y can be interchanged). By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis.
This operation is equivalent to a substitution:
{\displaystyle x={{u+v} \over {\sqrt {2}}},\,y={{u-v} \over {\sqrt {2}}}}
{\displaystyle v=\pm u{\sqrt {\frac {3a{\sqrt {2}}-2u}{6u+3a{\sqrt {2}}}}}}
Plotting in the Cartesian system of
{\displaystyle (u,v)}
gives the folium rotated by 45° and therefore symmetric by
{\displaystyle u}
It forms a loop in the first quadrant with a double point at the origin and asymptote
{\displaystyle x+y+a=0\,}
It is symmetrical about the line
{\displaystyle y=x}
. As such, the two intersect at the origin and at the point
{\displaystyle (3a/2,3a/2)}
Implicit differentiation gives the formula for the slope of the tangent line to this curve to be[3]
{\displaystyle {\frac {dy}{dx}}={\frac {ay-x^{2}}{y^{2}-ax}}}
Using either one of the polar representations above, the area of the interior of the loop is found to be
{\displaystyle 3a^{2}/2}
. Moreover, the area between the "wings" of the curve and its slanted asymptote is also
{\displaystyle 3a^{2}/2}
Relationship to the trisectrix of MaclaurinEdit
The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation
{\displaystyle x^{3}+y^{3}=3axy\,}
and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting
{\displaystyle x={{X+Y} \over {\sqrt {2}}},y={{X-Y} \over {\sqrt {2}}}.}
{\displaystyle X,Y}
plane the equation is
{\displaystyle 2X(X^{2}+3Y^{2})=3{\sqrt {2}}a(X^{2}-Y^{2})}
If we stretch the curve in the
{\displaystyle Y}
direction by a factor of
{\displaystyle {\sqrt {3}}}
{\displaystyle 2X(X^{2}+Y^{2})=a{\sqrt {2}}(3X^{2}-Y^{2}),}
which is the equation of the trisectrix of Maclaurin.
^ a b "Folium of Descartes". Encyclopedia of Mathematics. June 5, 2020. Retrieved January 30, 2021. {{cite web}}: CS1 maint: url-status (link)
^ Simmons, p. 101
^ a b Stewart, James (2012). "Section 3.5: Implicit Differentiation". Calculus: Early Transcendentals. United States of America: Cengage Learning. pp. 209–11. ISBN 978-0-538-49790-9.
^ Stewart, James (2012). "Chapter 10: Parametric Equations and Polar Coordinates". Calculus: Early Transcendentals (7th ed.). Cengage Learning. p. 687. ISBN 978-0-538-49790-9.
^ "DiffGeom3: Parametrized curves and algebraic curves". N J Wildberger, University of New South Wales. Archived from the original on 2021-12-21. Retrieved 5 September 2013.
J. Dennis Lawrence: A catalog of special plane curves, 1972, Dover Publications. ISBN 0-486-60288-5, pp. 106–108
George F. Simmons: Calculus Gems: Brief Lives and Memorable Mathematics, New York 1992, McGraw-Hill, xiv,355. ISBN 0-07-057566-5; new edition 2007, The Mathematical Association of America (MAA)
Wikimedia Commons has media related to Folium of Descartes.
Weisstein, Eric W. "Folium of Descartes". MathWorld.
"Folium of Descartes" at MacTutor's Famous Curves Index
"Cartesian Folium" at MathCurve
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How to Find the Determinant of a 3X3 Matrix: 12 Steps
1 Finding the Determinant
2 Making the Problem Easier
The determinant of a matrix is frequently used in calculus, linear algebra, and advanced geometry. Finding the determinant of a matrix can be confusing at first, but it gets easier once you do it a few times.
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Write your 3 x 3 matrix. We'll start with a 3 x 3 matrix A, and try to find its determinant |A|. Here's the general matrix notation we'll be using, and our example matrix:[1] X Research source
{\displaystyle M={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&5&3\\2&4&7\\4&6&2\end{pmatrix}}}
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Choose a single row or column. This will be your reference row or column. You'll get the same answer no matter which one you choose. For now, just pick the first row. Later, we'll give some advice on how to choose the easiest option to calculate.[2] X Research source
Let's choose the first row of our example matrix A. Circle the 1 5 3. In general terms, circle a11 a12 a13.
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Cross out the row and column of your first element. Look at the row or column you circled and select the first element. Draw a line through its row and column. You should be left with four numbers. We'll treat these as a 2 x 2 matrix.[3] X Research source
In our example, our reference row is 1 5 3. The first element is in row 1 and column 1. Cross out all of row 1 and column 1. Write the remaining elements as a 2 x 2 matrix:
Find the determinant of the 2 x 2 matrix. Remember, the matrix
{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
has a determinant of ad - bc. You may have learned this by drawing an X across the 2 x 2 matrix. Multiply the two numbers connected by the \ of the X. Then subtract the product of the two numbers connected by the /. Use this formula to calculate the determinate of the matrix you just found.[4] X Research source
In our example, the determinant of the matrix
{\displaystyle {\begin{pmatrix}4&7\\6&2\end{pmatrix}}}
This determinant is called the minor of the element we chose in our original matrix.[5] X Research source In this case, we just found the minor of a11.
Multiply the answer by your chosen element. Remember, you selected an element from your reference row (or column) when you decided which row and column to cross out. Multiply this element by the determinant you just calculated for the 2x2 matrix.[6] X Research source
In our example, we selected a11, which had a value of 1. Multiply this by -34 (the determinant of the 2x2) to get 1*-34 = -34.
Determine the sign of your answer. Next, you'll multiply your answer either by 1 or by -1 to get the cofactor of your chosen element. Which you use depends on where the element was placed in the 3x3 matrix. Memorize this simple sign chart to track which element causes which:
Since we chose a11, marked with a +, we multiply the number by +1. (In other words, leave it alone.) The answer is still -34.
Alternatively, you can find the sign with the formula (-1)i+j, where i and j are the element's row and column.[7] X Research source
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Repeat this process for the second element in your reference row or column. Return to the original 3x3 matrix, with the row or column you circled earlier. Repeat the same process with this element:[8] X Research source
Cross out the row and column of that element. In our case, select element a12 (with a value of 5). Cross out row one (1 5 3) and column two
{\displaystyle {\begin{pmatrix}5\\4\\6\end{pmatrix}}}
Treat the remaining elements as a 2x2 matrix. In our example, the matrix is
{\displaystyle {\begin{pmatrix}2&7\\4&2\end{pmatrix}}}
Find the determinant of this 2x2 matrix. Use the ad - bc formula. (2*2 - 7*4 = -24)
Multiply by the chosen element of the 3x3 matrix. -24 * 5 = -120
Determine whether to multiply by -1. Use the sign chart or the (-1)ij formula. We chose element a12, which is - on the sign chart. We must change the sign of our answer: (-1)*(-120) = 120.
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Repeat with the third element. You have one more cofactor to find. Calculate i for the third term in your reference row or column. Here's a quick rundown of how you'd calculate the cofactor of a13 in our example:
Cross out row 1 and column 3 to get
{\displaystyle {\begin{pmatrix}2&4\\4&6\end{pmatrix}}}
Its determinant is 2*6 - 4*4 = -4.
Multiply by element a13: -4 * 3 = -12.
Element a13 is + on the sign chart, so the answer is -12.
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Add your three results together. This is the final step. You've calculated three cofactors, one for each element in a single row or column. Add these together and you've found the determinant of the 3x3 matrix.
In our example the determinant is -34 + 120 + -12 = 74.
Making the Problem Easier
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Pick the reference with the most zeroes. Remember, you can pick any row or column as your reference. You'll get the same answer no matter which you pick. If you pick a row or column with zeros, you only need to calculate the cofactor for the nonzero elements. Here's why:[9] X Research source
Let's say you pick row 2, with elements a21, a22, and a23. To solve this problem, we'll be looking at three different 2x2 matrices. Let's call them A21, A22, and A23.
The determinant of the 3x3 matrix is a21|A21| - a22|A22| + a23|A23|.
If terms a22 and a23 are both 0, our formula becomes a21|A21| - 0*|A22| + 0*|A23| = a21|A21| - 0 + 0 = a21|A21|. Now we only have to calculate the cofactor of a single element.
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Use row addition to make the matrix easier. If you take the values of one row and add them to a different row, the determinant of the matrix does not change. The same is true of columns. You can do this repeatedly — or multiply the values by a constant before adding — to get as many zeroes in the matrix as possible. This can save you a lot of time.
For example, say you have a 3 x 3 matrix:
{\displaystyle {\begin{pmatrix}9&-1&2\\3&1&0\\7&5&-2\end{pmatrix}}}
In order to cancel out the 9 in position a11, we can multiply the second row by -3 and add the result to the first. The new first row is [9 -1 2] + [-9 -3 0] = [0 -4 2].
The new matrix is
{\displaystyle {\begin{pmatrix}0&-4&2\\3&1&0\\7&5&-2\end{pmatrix}}}
Try to use the same trick with columns to turn a12 into a 0 as well.
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Learn the shortcut for triangular matrices. In these special cases, the determinant is simply the product of the elements along the main diagonal, from a11 in the top left to a33 in the lower right. We're still talking about 3x3 matrices, but "triangular" ones have special patterns of nonzero values:[10] X Research source
Upper triangular matrix: All the non-zero elements are on or above the main diagonal. Everything below is a zero.
Lower triangular matrix: All the non-zero elements are on or below the main diagonal.
Diagonal matrix: All the non-zero elements are on the main diagonal. (A subset of the above.)
Why is the formula for the determinant (b^2-4ac)^(1/2) instead of ad-bc?
I think the OP was confused. They were referring to the discriminant, something you use in the quadratic formula. The formula for the determinant is different for every matrix, but for a 3x3 one is very hard to type out. It might be easier to Google it.
How do I adjoin a matrix?
The adjoint of a square matrix is the transpose of the matrix Cij (cofactor of the original matrix).
What is the formula for the determinant?
The formula to find the determinant for a quadratic formula is (b^2-4ac), which is all in a square root.
If all elements of a row or column are 0, the determinant of that matrix is 0.
This method extends to square matrices of any size. For example, if using this for a 4x4 matrix, your "crossing out" leaves you with a 3x3 matrix, for which you calculate the determinate as described above. Be warned, this gets very tedious by hand!
Solve Matrices
Divide Matrices
Reduce a Matrix to Row Echelon Form
Understand the Basics of Matrices
Solve a 2x3 Matrix
Use Cramer's Rule
↑ https://www.mathsisfun.com/algebra/matrix-determinant.html
↑ https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/deter/deter.html
↑ https://www.hec.ca/en/cams/help/topics/Matrix_determinants.pdf
↑ https://www.purplemath.com/modules/minors.htm
↑ http://mathonline.wikidot.com/triangular-matrices
1. Write your 3 x 3 matrix.
2. Choose a single row or column.
3. Cross out the row and column of your first element.
4. Find the determinant of the 2 x 2 matrix.
5. Multiply the answer by your chosen element.
6. Find the sign of your answer (+ or -) using the formula (-1)*(i+j), where i and j are the element's row and column. The formula will tell you whether your answer is positive or negative.
7. Repeat this process for the second element in your reference row or column.
8. Repeat with the third element.
9. Add your three results together.
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Bilinear form - Wikipedia
In mathematics, a bilinear form on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars) is a bilinear map V × V → K. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:
B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)
B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)
The dot product on
{\displaystyle \mathbb {R} ^{n}}
is an example of a bilinear form.[1]
The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.
When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
2 Maps to the dual space
3 Symmetric, skew-symmetric and alternating forms
4 Derived quadratic form
5 Reflexivity and orthogonality
7 Relation to tensor products
8 On normed vector spaces
9 Generalization to modules
Coordinate representationEdit
Let V ≅ Kn be an n-dimensional vector space with basis {e1, …, en}.
The n × n matrix A, defined by Aij = B(ei, ej) is called the matrix of the bilinear form on the basis {e1, …, en}.
If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents another vector w, then:
{\displaystyle B(\mathbf {v} ,\mathbf {w} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.}
A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all congruent. More precisely, if {f1, …, fn} is another basis of V, then
{\displaystyle \mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},}
{\displaystyle S_{i,j}}
form an invertible matrix S. Then, the matrix of the bilinear form on the new basis is STAS.
Maps to the dual spaceEdit
Every bilinear form B on V defines a pair of linear maps from V to its dual space V∗. Define B1, B2: V → V∗ by
B1(v)(w) = B(v, w)
B2(v)(w) = B(w, v)
This is often denoted as
B1(v) = B(v, ⋅)
B2(v) = B(⋅, v)
where the dot ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying).
For a finite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
{\displaystyle B(x,y)=0}
{\displaystyle y\in V}
implies that x = 0 and
{\displaystyle B(x,y)=0}
{\displaystyle x\in V}
implies that y = 0.
The corresponding notion for a module over a commutative ring is that a bilinear form is unimodular if V → V∗ is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing B(x, y) = 2xy is nondegenerate but not unimodular, as the induced map from V = Z to V∗ = Z is multiplication by 2.
If V is finite-dimensional then one can identify V with its double dual V∗∗. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V∗∗). Given B one can define the transpose of B to be the bilinear form given by
tB(v, w) = B(w, v).
The left radical and right radical of the form B are the kernels of B1 and B2 respectively;[2] they are the vectors orthogonal to the whole space on the left and on the right.[3]
If V is finite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to dim(V) then B1 and B2 are linear isomorphisms from V to V∗. In this case B is nondegenerate. By the rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the definition of nondegeneracy:
Definition: B is nondegenerate if B(v, w) = 0 for all w implies v = 0.
Given any linear map A : V → V∗ one can obtain a bilinear form B on V via
B(v, w) = A(v)(w).
This form will be nondegenerate if and only if A is an isomorphism.
If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example B(x, y) = 2xy over the integers.
Symmetric, skew-symmetric and alternating formsEdit
We define a bilinear form to be
symmetric if B(v, w) = B(w, v) for all v, w in V;
alternating if B(v, v) = 0 for all v in V;
skew-symmetric or antisymmetric if B(v, w) = −B(w, v) for all v, w in V;
Every alternating form is skew-symmetric.
This can be seen by expanding B(v + w, v + w).
If the characteristic of K is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, char(K) = 2 then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.
A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(K) ≠ 2).
A bilinear form is symmetric if and only if the maps B1, B2: V → V∗ are equal, and skew-symmetric if and only if they are negatives of one another. If char(K) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
{\displaystyle B^{+}={\tfrac {1}{2}}(B+{}^{\text{t}}B)\qquad B^{-}={\tfrac {1}{2}}(B-{}^{\text{t}}B),}
where tB is the transpose of B (defined above).
Derived quadratic formEdit
For any bilinear form B : V × V → K, there exists an associated quadratic form Q : V → K defined by Q : V → K : v ↦ B(v, v).
When char(K) ≠ 2, the quadratic form Q is determined by the symmetric part of the bilinear form B and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.
When char(K) = 2 and dim V > 1, this correspondence between quadratic forms and symmetric bilinear forms breaks down.
Reflexivity and orthogonalityEdit
Definition: A bilinear form B : V × V → K is called reflexive if B(v, w) = 0 implies B(w, v) = 0 for all v, w in V.
Definition: Let B : V × V → K be a reflexive bilinear form. v, w in V are orthogonal with respect to B if B(v, w) = 0.
A bilinear form B is reflexive if and only if it is either symmetric or alternating.[4] In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the kernel or the radical of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if Ax = 0 ⇔ xTA = 0. The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Suppose W is a subspace. Define the orthogonal complement[5]
{\displaystyle W^{\perp }=\left\{\mathbf {v} \mid B(\mathbf {v} ,\mathbf {w} )=0{\text{ for all }}\mathbf {w} \in W\right\}.}
For a non-degenerate form on a finite-dimensional space, the map V/W → W⊥ is bijective, and the dimension of W⊥ is dim(V) − dim(W).
Different spacesEdit
Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field
B : V × W → K.
Here we still have induced linear mappings from V to W∗, and from W to V∗. It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, B is said to be a perfect pairing.
In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance Z × Z → Z via (x, y) ↦ 2xy is nondegenerate, but induces multiplication by 2 on the map Z → Z∗.
Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".[6] To define them he uses diagonal matrices Aij having only +1 or −1 for non-zero elements. Some of the "inner products" are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a general field K, the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form
{\displaystyle \sum _{k=1}^{p}x_{k}y_{k}-\sum _{k=p+1}^{n}x_{k}y_{k}}
is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:[7]
Some of the real symmetric cases are very important. The positive definite case R(n, 0) is called Euclidean space, while the case of a single minus, R(n−1, 1) is called Lorentzian space. If n = 4, then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case R(p, p) will be referred to as the split-case.
Relation to tensor productsEdit
By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on V and linear maps V ⊗ V → K. If B is a bilinear form on V the corresponding linear map is given by
In the other direction, if F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w.
The set of all linear maps V ⊗ V → K is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of (V ⊗ V)∗ which (when V is finite-dimensional) is canonically isomorphic to V∗ ⊗ V∗.
Likewise, symmetric bilinear forms may be thought of as elements of Sym2(V∗) (the second symmetric power of V∗), and alternating bilinear forms as elements of Λ2V∗ (the second exterior power of V∗).
On normed vector spacesEdit
Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is bounded, if there is a constant C such that for all u, v ∈ V,
{\displaystyle B(\mathbf {u} ,\mathbf {v} )\leq C\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is elliptic, or coercive, if there is a constant c > 0 such that for all u ∈ V,
{\displaystyle B(\mathbf {u} ,\mathbf {u} )\geq c\left\|\mathbf {u} \right\|^{2}.}
Generalization to modulesEdit
Given a ring R and a right R-module M and its dual module M∗, a mapping B : M∗ × M → R is called a bilinear form if
B(u + v, x) = B(u, x) + B(v, x)
B(u, x + y) = B(u, x) + B(u, y)
B(αu, xβ) = αB(u, x)β
for all u, v ∈ M∗, all x, y ∈ M and all α, β ∈ R.
The mapping ⟨⋅,⋅⟩ : M∗ × M → R : (u, x) ↦ u(x) is known as the natural pairing, also called the canonical bilinear form on M∗ × M.[8]
A linear map S : M∗ → M∗ : u ↦ S(u) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨S(u), x⟩, and a linear map T : M → M : x ↦ T(x) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨u, T(x)⟩.
Conversely, a bilinear form B : M∗ × M → R induces the R-linear maps S : M∗ → M∗ : u ↦ (x ↦ B(u, x)) and T′ : M → M∗∗ : x ↦ (u ↦ B(u, x)). Here, M∗∗ denotes the double dual of M.
^ "Chapter 3. Bilinear forms — Lecture notes for MA1212" (PDF). 2021-01-16.
^ Zhelobenko 2006, p. 11.
^ Adkins & Weintraub 1992, p. 359.
Adkins, William A.; Weintraub, Steven H. (1992), Algebra: An Approach via Module Theory, Graduate Texts in Mathematics, vol. 136, Springer-Verlag, ISBN 3-540-97839-9, Zbl 0768.00003
Bourbaki, N. (1970), Algebra, Springer
Cooperstein, Bruce (2010), "Ch 8: Bilinear Forms and Maps", Advanced Linear Algebra, CRC Press, pp. 249–88, ISBN 978-1-4398-2966-0
Grove, Larry C. (1997), Groups and characters, Wiley-Interscience, ISBN 978-0-471-16340-4
Halmos, Paul R. (1974), Finite-dimensional vector spaces, Undergraduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3, Zbl 0288.15002
Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner Product Spaces", Spinors and calibrations, Academic Press, pp. 19–40, ISBN 0-12-329650-1
Popov, V. L. (1987), "Bilinear form", in Hazewinkel, M. (ed.), Encyclopedia of Mathematics, vol. 1, Kluwer Academic Publishers, pp. 390–392 . Also: Bilinear form, p. 390, at Google Books
Jacobson, Nathan (2009), Basic Algebra, vol. I (2nd ed.), ISBN 978-0-486-47189-1
Milnor, J.; Husemoller, D. (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016
Porteous, Ian R. (1995), Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, vol. 50, Cambridge University Press, ISBN 978-0-521-55177-9
Shafarevich, I. R.; A. O. Remizov (2012), Linear Algebra and Geometry, Springer, ISBN 978-3-642-30993-9
Shilov, Georgi E. (1977), Silverman, Richard A. (ed.), Linear Algebra, Dover, ISBN 0-486-63518-X
Zhelobenko, Dmitriĭ Petrovich (2006), Principal Structures and Methods of Representation Theory, Translations of Mathematical Monographs, American Mathematical Society, ISBN 0-8218-3731-1
Wikimedia Commons has media related to Bilinear forms.
"Bilinear form", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
"Bilinear form". PlanetMath.
This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bilinear_form&oldid=1087043004"
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OrderClassNumber - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : OrderClassNumber
compute the order profile of the elements of a finite group
compute the order class number of a finite group
compute the order rank of a finite group
OrderClassProfile( G, opts )
OrderClassPolynomial( G, x )
OrderClassNumber( G )
OrderRank( G )
option of the form output = "list", output = "collected" (the default), or output = "multiset"
The order class profile of a finite group G is the sequence of orders of elements of G, including their multiplicities.
The OrderClassProfile( G ) command computes the order class profile of a finite group G. By default, this is returned as a list of pairs of the form [ order, multiplicity ]. The sorted list of element orders can be returned by using the 'output' = "list" option. To produce, instead, a MultiSet, use the 'output' = "multiset" option.
The OrderClassPolynomial( G, x ) command returns a polynomial encoding of the order class data of the finite group G. It is a univariate polynomial in the indeterminate x for which the coefficient of x^k is equal to the number of elements of order k in G.
The order class number of a finite group
G
is the number of order classes of elements of
G
The OrderClassNumber( G ) command returns the order class number of the finite group G.
The order rank of a finite group
G
is the number of distinct order class lengths of
G
1
The OrderRank( G ) command returns the order rank of the finite group G.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔\mathrm{Alt}\left(4\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{A}}}_{\textcolor[rgb]{0,0,1}{4}}
\mathrm{OrderClassProfile}\left(G\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]]
\mathrm{OrderClassProfile}\left(G,'\mathrm{output}'="list"\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
\mathrm{OrderClassProfile}\left(G,'\mathrm{output}'="multiset"\right)
{[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]}
\mathrm{OrderClassNumber}\left(G\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{OrderRank}\left(G\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{OrderClassPolynomial}\left(\mathrm{Symm}\left(6\right),'x'\right)
\textcolor[rgb]{0,0,1}{240}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{144}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{180}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{80}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{75}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
The GroupTheory[OrderClassProfile] command was introduced in Maple 2019.
The GroupTheory[OrderClassPolynomial] and GroupTheory[OrderClassNumber] commands were introduced in Maple 2020.
The GroupTheory[OrderRank] command was introduced in Maple 2021.
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HilbertPolynomial - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Polynomials : Groebner : HilbertPolynomial
compute Hilbert series
compute Hilbert polynomial
HilbertSeries(J, X, s, characteristic=p)
HilbertPolynomial(J, X, s, characteristic=p)
(optional) a variable to use for the series or polynomial
The HilbertSeries command computes the Hilbert series of the ideal generated by J, which is defined as
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{n=0}^{\mathrm{\infty }}\left({c}_{n}-{c}_{n-1}\right){s}^{n}
{c}_{n}
is the dimension of the vector space of normal forms (with respect to J) of polynomials of degree less than or equal to
n
. The output is a rational function of the form
H\left(s\right)=\frac{P\left(s\right)}{{\left(1-s\right)}^{d}}
d
is the HilbertDimension of J. The Hilbert polynomial is defined as the polynomial asymptotically equal to
{c}_{n}-{c}_{n-1}
In the case of skew polynomials, the invariants that are returned are those of the left ideal generated by J.
The variables of the system can be specified using an optional second argument X. If X is a ShortMonomialOrder then a Groebner basis of J with respect to X is computed. Be aware that if X is not a graded monomial order (that is, tdeg or grlex) then the result may be incorrect. By default, X is the set of all indeterminates not appearing inside a RootOf or radical when J is a list or set, or PolynomialIdeals[IdealInfo][Variables](J) if J is an ideal.
The variable for the Hilbert series or polynomial can be specified with an optional third argument s. If this is omitted, the global name 'Z' is used.
The algorithms for HilbertSeries and HilbertPolynomial use the leading monomials of a total degree Groebner basis for J. To access this functionality directly (as part of a program), make J the list or set of leading monomials. The commands will detect this case and execute their algorithms with minimal overhead.
Note that the hilbertseries and hilbertpoly commands are deprecated. They may not be supported in a future Maple release.
\mathrm{with}\left(\mathrm{Groebner}\right):
F≔[{x}^{31}-{x}^{6}-x-y,{x}^{8}-z,{x}^{10}-t]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{31}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{-}\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}{x}}^{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{10}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{t}]
h≔\mathrm{HilbertSeries}\left(F,{t,x,y,z},s\right)
\textcolor[rgb]{0,0,1}{h}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}}
\mathrm{HilbertPolynomial}\left(F,{t,x,y,z},n\right)
\textcolor[rgb]{0,0,1}{31}
\mathrm{series}\left(h,s=0,10\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{19}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{30}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{32}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{31}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{31}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{31}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{31}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{10}}\right)
The system below is not holonomic, in the sense that the Hilbert dimension is greater than the number of polynomial variables (x and y).
\mathrm{with}\left(\mathrm{Ore_algebra}\right):
A≔\mathrm{diff_algebra}\left([\mathrm{Dx},x],[\mathrm{Dy},y],\mathrm{polynom}={x,y}\right):
T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Dx},\mathrm{Dy},x,y\right)\right):
p≔{x}^{2}-x+y:
F≔[p\mathrm{Dx}+\mathrm{diff}\left(p,x\right),p\mathrm{Dy}+\mathrm{diff}\left(p,y\right)]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
G≔\mathrm{Basis}\left(F,T\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}]
\mathrm{HilbertDimension}\left(F,T\right)
\textcolor[rgb]{0,0,1}{3}
p≔\mathrm{HilbertPolynomial}\left(F,T,n\right)
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}
h≔\mathrm{HilbertSeries}\left(F,T,s\right)
\textcolor[rgb]{0,0,1}{h}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}{{\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\right)}^{\textcolor[rgb]{0,0,1}{3}}}
\mathrm{series}\left(h,s\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{18}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{28}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{40}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{6}}\right)
\mathrm{series}\left(\mathrm{add}\left(\mathrm{eval}\left(p,n=i\right),i=0..6\right),s\right)
\textcolor[rgb]{0,0,1}{154}
Neither x nor y can be eliminated.
\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(F,\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left([x],[\mathrm{Dx},\mathrm{Dy},y]\right)\right)\right),x\right)
[]
\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(F,\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left([y],[\mathrm{Dx},\mathrm{Dy},x]\right)\right)\right),y\right)
[]
f≔\frac{1}{p}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}}
\left({x}^{2}-x+y\right)\mathrm{diff}\left(f,x,y\right)+2\left(2x-1\right)\mathrm{diff}\left(f,y\right)
\textcolor[rgb]{0,0,1}{0}
The system becomes holonomic when we add the polynomial above. The Hilbert dimension is now 2 and both x and y can be eliminated.
\mathrm{F2}≔[\mathrm{op}\left(F\right),\left({x}^{2}-x+y\right)\mathrm{Dx}\mathrm{Dy}+2\left(2x-1\right)\mathrm{Dy}]
\textcolor[rgb]{0,0,1}{\mathrm{F2}}\textcolor[rgb]{0,0,1}{≔}[\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{+}\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}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}]
\mathrm{HilbertDimension}\left(\mathrm{F2},T\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{HilbertPolynomial}\left(\mathrm{F2},T,n\right)
\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}
h≔\mathrm{HilbertSeries}\left(\mathrm{F2},T,s\right)
\textcolor[rgb]{0,0,1}{h}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{s}}^{\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}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}{{\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{series}\left(h,s=0\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{6}}\right)
\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(\mathrm{F2},\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left([x],[\mathrm{Dx},\mathrm{Dy},y]\right)\right)\right),x\right)
[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Dy}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{Dx}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{Dy}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}]
\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(\mathrm{F2},\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left([y],[\mathrm{Dx},\mathrm{Dy},x]\right)\right)\right),y\right)
[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}]
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Lemma 10.137.10 (00TA)—The Stacks project
Lemma 10.137.10. Let $R \to S$ be a smooth ring map. There exists an open covering of $\mathop{\mathrm{Spec}}(S)$ by standard opens $D(g)$ such that each $S_ g$ is standard smooth over $R$. In particular $R \to S$ is syntomic.
Proof. Choose a presentation $\alpha : R[x_1, \ldots , x_ n] \to S$ with kernel $I = (f_1, \ldots , f_ m)$. For every subset $E \subset \{ 1, \ldots , m\} $ consider the open subset $U_ E$ where the classes $f_ e, e\in E$ freely generate the finite projective $S$-module $I/I^2$, see Lemma 10.79.4. We may cover $\mathop{\mathrm{Spec}}(S)$ by standard opens $D(g)$ each completely contained in one of the opens $U_ E$. For such a $g$ we look at the presentation
\[ \beta : R[x_1, \ldots , x_ n, x_{n + 1}] \longrightarrow S_ g \]
mapping $x_{n + 1}$ to $1/g$. Setting $J = \mathop{\mathrm{Ker}}(\beta )$ we use Lemma 10.134.12 to see that $J/J^2 \cong (I/I^2)_ g \oplus S_ g$ is free. We may and do replace $S$ by $S_ g$. Then using Lemma 10.136.6 we may assume we have a presentation $\alpha : R[x_1, \ldots , x_ n] \to S$ with kernel $I = (f_1, \ldots , f_ c)$ such that $I/I^2$ is free on the classes of $f_1, \ldots , f_ c$.
Using the presentation $\alpha $ obtained at the end of the previous paragraph, we more or less repeat this argument with the basis elements $\text{d}x_1, \ldots , \text{d}x_ n$ of $\Omega _{R[x_1, \ldots , x_ n]/R}$. Namely, for any subset $E \subset \{ 1, \ldots , n\} $ of cardinality $c$ we may consider the open subset $U_ E$ of $\mathop{\mathrm{Spec}}(S)$ where the differential of $\mathop{N\! L}\nolimits (\alpha )$ composed with the projection
\[ S^{\oplus c} \cong I/I^2 \longrightarrow \Omega _{R[x_1, \ldots , x_ n]/R} \otimes _{R[x_1, \ldots , x_ n]} S \longrightarrow \bigoplus \nolimits _{i \in E} S\text{d}x_ i \]
is an isomorphism. Again we may find a covering of $\mathop{\mathrm{Spec}}(S)$ by (finitely many) standard opens $D(g)$ such that each $D(g)$ is completely contained in one of the opens $U_ E$. By renumbering, we may assume $E = \{ 1, \ldots , c\} $. For a $g$ with $D(g) \subset U_ E$ we look at the presentation
\[ \beta : R[x_1, \ldots , x_ n, x_{n + 1}] \to S_ g \]
mapping $x_{n + 1}$ to $1/g$. Setting $J = \mathop{\mathrm{Ker}}(\beta )$ we conclude from Lemma 10.134.12 that $J = (f_1, \ldots , f_ c, fx_{n + 1} - 1)$ where $\alpha (f) = g$ and that the composition
\[ J/J^2 \longrightarrow \Omega _{R[x_1, \ldots , x_{n + 1}]/R} \otimes _{R[x_1, \ldots , x_{n + 1}]} S_ g \longrightarrow \bigoplus \nolimits _{i = 1}^ c S_ g\text{d}x_ i \oplus S_ g \text{d}x_{n + 1} \]
is an isomorphism. Reordering the coordinates as $x_1, \ldots , x_ c, x_{n + 1}, x_{c + 1}, \ldots , x_ n$ we conclude that $S_ g$ is standard smooth over $R$ as desired.
This finishes the proof as standard smooth algebras are syntomic (Lemmas 10.137.7 and 10.136.14) and being syntomic over $R$ is local on $S$ (Lemma 10.136.4). $\square$
It seems like, at the end of the first paragraph,
S
is implicitly being replaced by
S_g
, because the result 07CF applies to
S_g
, not (obviously) to
S
itself. Maybe this should be made explicit?
I don't understand a part in the proof. Why do the
U_E
\text{Spec}(S)
? I don't think this follows from Lemma 10.78.3. We only get that the
U_E
are open. However we only need
I/I^2
to be free on
D(g)
and this is exactly Lemma 10.77.2 (the finite locally free part).
The second time we (implicitly) apply 10.78.3 it works because
I/I^2\to \bigoplus Sdx_i
stays injective upon tensoring with
\kappa(\mathfrak{p})
R\to S
is smooth. Maybe we could mention this?
These opens cover because to see this we can check over the residue fields of the primes and then we reduce to the statement: if
v_1, \ldots, v_m
generate a vector space
V
, then a subset of
\{v_1, \ldots, v_m\}
V
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Home : Support : Online Help : Mathematics : Algebra : Expression Manipulation : Combining : exponentials
combine exponentials
combine(f, exp)
Expressions involving exponentials are combined by applying the following transformations:
{ⅇ}^{x}{ⅇ}^{y}\to {ⅇ}^{x+y}
{{ⅇ}^{x}}^{y}\to {ⅇ}^{xy}
\mathrm{exp}\left(x+n\mathrm{ln}\left(y\right)\right)\to {y}^{n}{ⅇ}^{x}\mathrm{where}n\mathrm{is}\mathrm{an}\mathrm{integer}
\mathrm{combine}\left(\mathrm{exp}\left(x\right)\mathrm{exp}\left(-x\right),\mathrm{exp}\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{combine}\left({\left(\mathrm{exp}\left(5\right)\right)}^{2}\mathrm{exp}\left(3\right),\mathrm{exp}\right)
{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{13}}
\mathrm{combine}\left(\mathrm{exp}\left(x+3\mathrm{ln}\left(y\right)\right),\mathrm{exp}\right)
{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}
\mathrm{combine}\left(\mathrm{exp}\left(x+\frac{3}{2}\mathrm{ln}\left(y\right)\right),\mathrm{exp}\right)
{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)}{\textcolor[rgb]{0,0,1}{2}}}
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2013 Cycle classes and the syntomic regulator
\mathsc{V}=Spec\left(R\right)
R
\left(0,p\right)
R
\mathsc{X}
{reg}_{syn}:{CH}^{i}\left(\mathsc{X}∕\mathsc{V},2i-n\right)\to {H}_{syn}^{n}\left(\mathsc{X},i\right)
\mathsc{X}
R
Bruno Chiarellotto. Alice Ciccioni. Nicola Mazzari. "Cycle classes and the syntomic regulator." Algebra Number Theory 7 (3) 533 - 566, 2013. https://doi.org/10.2140/ant.2013.7.533
Received: 12 October 2010; Revised: 22 December 2011; Accepted: 3 May 2012; Published: 2013
Keywords: cycles , de Rham cohomology , regulator map , rigid cohomology , syntomic cohomology
Bruno Chiarellotto, Alice Ciccioni, Nicola Mazzari "Cycle classes and the syntomic regulator," Algebra & Number Theory, Algebra Number Theory 7(3), 533-566, (2013)
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Bernoulli's inequality - Wikipedia
Inequality about exponentiations of 1+x
An illustration of Bernoulli's inequality, with the graphs of
{\displaystyle y=(1+x)^{r}}
{\displaystyle y=1+rx}
{\displaystyle r=3.}
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x. It is often employed in real analysis. It has several useful variants:[1]
{\displaystyle (1+x)^{r}\geq 1+rx}
for every integer r ≥ 0 and real number x ≥ −1. The inequality is strict if x ≠ 0 and r ≥ 2.
{\displaystyle (1+x)^{r}\geq 1+rx}
for every even integer r ≥ 0 and every real number x.
{\displaystyle (1+x)^{r}\geq 1+rx}
for every real number r ≥ 1 and x ≥ −1. The inequalities are strict if x ≠ 0 and r ≠ 0, 1.
{\displaystyle (1+x)^{r}\leq 1+rx}
for every real number 0 ≤ r ≤ 1 and x ≥ −1.
2 Proof for integer exponent
3.1 Generalization of exponent
3.2 Generalization of base
6 Alternative proof
Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often.[2]
According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963), p. 177, the inequality is actually due to Sluse in his Mesolabum (1668 edition), Chapter IV "De maximis & minimis".[2]
Proof for integer exponent[edit]
Bernoulli's inequality can be proved for the case in which r is an integer, using mathematical induction in the following form:
we prove the inequality for
{\displaystyle r\in \{0,1\}}
from validity for some r we deduce validity for r + 2.
{\displaystyle (1+x)^{0}\geq 1+0x}
is equivalent to 1 ≥ 1 which is true.
Similarly, for r = 1 we have
{\displaystyle (1+x)^{r}=1+x\geq 1+x=1+rx.}
Now suppose the statement is true for r = k:
{\displaystyle (1+x)^{k}\geq 1+kx.}
{\displaystyle {\begin{aligned}(1+x)^{k+2}&=(1+x)^{k}(1+x)^{2}\\&\geq (1+kx)\left(1+2x+x^{2}\right)\qquad \qquad \qquad {\text{ by hypothesis and }}(1+x)^{2}\geq 0\\&=1+2x+x^{2}+kx+2kx^{2}+kx^{3}\\&=1+(k+2)x+kx^{2}(x+2)+x^{2}\\&\geq 1+(k+2)x\end{aligned}}}
{\displaystyle x^{2}\geq 0}
{\displaystyle x+2\geq 0}
. By the modified induction we conclude the statement is true for every non-negative integer r.
Generalization of exponent[edit]
The exponent r can be generalized to an arbitrary real number as follows: if x > −1, then
{\displaystyle (1+x)^{r}\geq 1+rx}
for r ≤ 0 or r ≥ 1, and
{\displaystyle (1+x)^{r}\leq 1+rx}
for 0 ≤ r ≤ 1.
This generalization can be proved by comparing derivatives. The strict versions of these inequalities require x ≠ 0 and r ≠ 0, 1.
Generalization of base[edit]
{\displaystyle (1+x)^{n}}
the inequality holds also in the form
{\displaystyle (1+x_{1})(1+x_{2})\dots (1+x_{r})\geq 1+x_{1}+x_{2}+\dots +x_{r}}
{\displaystyle x_{1},x_{2},\dots ,x_{r}}
are real numbers, all greater than -1, all with the same sign. Bernoulli's inequality is a special case when
{\displaystyle x_{1}=x_{2}=\dots =x_{r}=x}
. This generalized inequality can be proved by mathematical induction.
In the first step we take
{\displaystyle n=1}
. In this case the inequality
{\displaystyle 1+x_{1}\geq 1+x_{1}}
is obviously true.
In the second step we assume validity of the inequality fo{\displaystyle r}
numbers and deduce validity for
{\displaystyle r+1}
{\displaystyle (1+x_{1})(1+x_{2})\dots (1+x_{r})\geq 1+x_{1}+x_{2}+\dots +x_{r}}
is valid. After multiplying both sides with a positive number
{\displaystyle (x_{r+1}+1)}
{\displaystyle {\begin{alignedat}{2}(1+x_{1})(1+x_{2})\dots (1+x_{r})(1+x_{r+1})\geq &(1+x_{1}+x_{2}+\dots +x_{r})(1+x_{r+1})\\\geq &(1+x_{1}+x_{2}+\dots +x_{r})\cdot 1+(1+x_{1}+x_{2}+\dots +x_{r})\cdot x_{r+1}\\\geq &(1+x_{1}+x_{2}+\dots +x_{r})+x_{r+1}+x_{1}x_{r+1}+x_{2}x_{r+1}+\dots +x_{r}x_{r+1}\\\end{alignedat}}}
{\displaystyle x_{1},x_{2},\dots x_{r},x_{r+1}}
all have the same sign, the products
{\displaystyle x_{1}x_{r+1},x_{2}x_{r+1},\dots x_{r}x_{r+1}}
are all positive numbers. So the quantity on the right-hand side can be bounded as follows:
{\displaystyle (1+x_{1}+x_{2}+\dots +x_{r})+x_{r+1}+x_{1}x_{r+1}+x_{2}x_{r+1}+\dots +x_{r}x_{r+1}\geq 1+x_{1}+x_{2}+\dots +x_{r}+x_{r+1},}
The following inequality estimates the r-th power of 1 + x from the other side. For any real numbers x, r with r > 0, one has
{\displaystyle (1+x)^{r}\leq e^{rx},}
where e = 2.718.... This may be proved using the inequality (1 + 1/k)k < e.
An alternative form of Bernoulli's inequality for
{\displaystyle t\geq 1}
{\displaystyle 0\leq x\leq 1}
{\displaystyle (1-x)^{t}\geq 1-xt.}
This can be proved (for any integer t) by using the formula for geometric series: (using y = 1 − x)
{\displaystyle t=1+1+\dots +1\geq 1+y+y^{2}+\ldots +y^{t-1}={\frac {1-y^{t}}{1-y}},}
{\displaystyle xt\geq 1-(1-x)^{t}.}
Alternative proof[edit]
Using AM-GM
An elementary proof for
{\displaystyle 0\leq r\leq 1}
and x ≥ -1 can be given using weighted AM-GM.
{\displaystyle \lambda _{1},\lambda _{2}}
be two non-negative real constants. By weighted AM-GM on
{\displaystyle 1,1+x}
{\displaystyle \lambda _{1},\lambda _{2}}
{\displaystyle {\dfrac {\lambda _{1}\cdot 1+\lambda _{2}\cdot (1+x)}{\lambda _{1}+\lambda _{2}}}\geq {\sqrt[{\lambda _{1}+\lambda _{2}}]{(1+x)^{\lambda _{2}}}}.}
{\displaystyle {\dfrac {\lambda _{1}\cdot 1+\lambda _{2}\cdot (1+x)}{\lambda _{1}+\lambda _{2}}}={\dfrac {\lambda _{1}+\lambda _{2}+\lambda _{2}x}{\lambda _{1}+\lambda _{2}}}=1+{\dfrac {\lambda _{2}}{\lambda _{1}+\lambda _{2}}}x}
{\displaystyle {\sqrt[{\lambda _{1}+\lambda _{2}}]{(1+x)^{\lambda _{2}}}}=(1+x)^{\frac {\lambda _{2}}{\lambda _{1}+\lambda _{2}}},}
so our inequality is equivalent to
{\displaystyle 1+{\dfrac {\lambda _{2}}{\lambda _{1}+\lambda _{2}}}x\geq (1+x)^{\frac {\lambda _{2}}{\lambda _{1}+\lambda _{2}}}.}
{\displaystyle r={\dfrac {\lambda _{2}}{\lambda _{1}+\lambda _{2}}}}
(bearing in mind that this implies
{\displaystyle 0\leq r\leq 1}
) our inequality turns into
{\displaystyle 1+rx\geq (1+x)^{r}}
which is Bernoulli's inequality.
{\displaystyle (1+x)^{r}\geq 1+rx}
{\displaystyle (1+x)^{r}-1-rx\geq 0,}
and by the formula for geometric series (using y = 1 + x) we get
{\displaystyle (1+x)^{r}-1=y^{r}-1=\left(\sum _{k=0}^{r-1}y^{k}\right)\cdot (y-1)=\left(\sum _{k=0}^{r-1}(1+x)^{k}\right)\cdot x}
{\displaystyle (1+x)^{r}-1-rx=\left(\left(\sum _{k=0}^{r-1}(1+x)^{k}\right)-r\right)\cdot x=\left(\sum _{k=0}^{r-1}\left((1+x)^{k}-1\right)\right)\cdot x\geq 0.}
{\displaystyle x\geq 0}
then by monotony of the powers each summand
{\displaystyle (1+x)^{k}-1=(1+x)^{k}-1^{k}\geq 0}
, and therefore their sum is greater
{\displaystyle 0}
and hence the product on the LHS of (4).
{\displaystyle 0\geq x\geq -2}
then by the same arguments
{\displaystyle 1\geq (1+x)^{k}}
and thus all addends
{\displaystyle (1+x)^{k}-1}
are non-positive and hence so is their sum. Since the product of two non-positive numbers is non-negative, we get again (4).
One can prove Bernoulli's inequality for x ≥ 0 using the binomial theorem. It is true trivially for r = 0, so suppose r is a positive integer. Then
{\displaystyle (1+x)^{r}=1+rx+{\tbinom {r}{2}}x^{2}+...+{\tbinom {r}{r}}x^{r}.}
{\displaystyle {\tbinom {r}{2}}x^{2}+...+{\tbinom {r}{r}}x^{r}\geq 0,}
{\displaystyle (1+x)^{r}\geq 1+rx}
^ Brannan, D. A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 20. ISBN 9781139458955.
^ a b mathematics – First use of Bernoulli's inequality and its name – History of Science and Mathematics Stack Exchange
Carothers, N.L. (2000). Real analysis. Cambridge: Cambridge University Press. p. 9. ISBN 978-0-521-49756-5.
Bullen, P. S. (2003). Handbook of means and their inequalities. Dordercht [u.a.]: Kluwer Academic Publ. p. 4. ISBN 978-1-4020-1522-9.
Zaidman, S. (1997). Advanced calculus : an introduction to mathematical analysis. River Edge, NJ: World Scientific. p. 32. ISBN 978-981-02-2704-3.
Weisstein, Eric W. "Bernoulli Inequality". MathWorld.
Bernoulli Inequality by Chris Boucher, Wolfram Demonstrations Project.
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bernoulli%27s_inequality&oldid=1089761401"
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Deborah number - Wikipedia
The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a fluid-like material can act solid when it is deformed rapidly enough. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay.
3 Time-temperature superposition
The Deborah number is the ratio of fundamentally different characteristic times. The Deborah number is defined as the ratio of the time it takes for a material to adjust to applied stresses or deformations, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material:
{\displaystyle \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}},}
where tc stands for the relaxation time and tp for the "time of observation", typically taken to be the time scale of the process.[1]
The numerator, relaxation time, is the time needed for a reference amount of deformation to occur under a suddenly applied reference load (a more fluid-like material will therefore require less time to flow, giving a lower Deborah number relative to a solid subjected to the same loading rate).
The denominator, material time,[2] is the amount of time required to reach a given reference strain (a faster loading rate will therefore reach the reference strain sooner, giving a higher Deborah number).
Equivalently, the relaxation time is the time required for the stress induced, by a suddenly applied reference strain, to reduce by a certain reference amount. The relaxation time is actually based on the rate of relaxation that exists at the moment of the suddenly applied load.
This incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior enters the non-Newtonian regime, increasingly dominated by elasticity and demonstrating solidlike behavior.[3][4]
For example, for a Hookean elastic solid, the relaxation time tc will be infinite and it will vanish for a Newtonian viscous fluid. For liquid water, tc is typically 10−12 s, for lubricating oils passing through gear teeth at high pressure it is of the order of 10−6 s and for polymers undergoing plastics processing, the relaxation time will be of the order of a few seconds. Therefore, depending on the situation, these liquids may exhibit elastic properties, departing from purely viscous behavior.[5]
While De is similar to the Weissenberg number and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.[1]
The Deborah number was originally proposed by Markus Reiner, a professor at Technion in Israel, who chose the name inspired by a verse in the Bible, stating "The mountains flowed before the Lord" in a song by the prophet Deborah in the Book of Judges;[6] הָרִ֥ים נָזְל֖וּ מִפְּנֵ֣י יְהוָ֑ה hā-rîm nāzəlū mippənê Yahweh).[3][7]
Time-temperature superposition[edit]
The Deborah number is particularly useful in conceptualizing the time–temperature superposition principle. Time-temperature superposition has to do with altering experimental time scales using reference temperatures to extrapolate temperature-dependent mechanical properties of polymers. A material at low temperature with a long experimental or relaxation time behaves like the same material at high temperature and short experimental or relaxation time if the Deborah number remains the same. This can be particularly useful when working with materials which relax on a long time scale under a certain temperature. The practical application of this idea arises in the Williams–Landel–Ferry equation. Time-temperature superposition avoids the inefficiency of measuring a polymer's behavior over long periods of time at a specified temperature by utilizing the Deborah number.[8]
^ a b Poole, R J (2012). "The Deborah and Weissenberg numbers" (PDF). Rheology Bulletin. 53 (2): 32–39.
^ Franck, A. "Viscoelasticity and dynamic mechanical testing" (PDF). TA Instruments. TA Instruments Germany. Retrieved 26 March 2019.
^ a b Reiner, M. (1964), "The Deborah Number", Physics Today, 17 (1): 62, Bibcode:1964PhT....17a..62R, doi:10.1063/1.3051374
^ The Deborah Number Archived 2011-04-13 at the Wayback Machine
^ Barnes, H.A.; Hutton, J.F.; Walters, K. (1989). An introduction to rheology (5. impr. ed.). Amsterdam: Elsevier. pp. 5–6. ISBN 978-0-444-87140-4.
^ Millgram, Hillel I. (2018). Judges and Saviors, Deborah and Samson: Reflections of a World in Chaos. Hamilton Books. pp. 123–. ISBN 978-0-7618-6990-0.
^ Rudin, Alfred, and Phillip Choi. The Elements of Polymer Science and Engineering. 3rd. Oxford: Academic Press, 2013. Print. Page 221.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Deborah_number&oldid=1042505725"
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Section 15.66 (0651): Tor dimension—The Stacks project
Section 15.66: Tor dimension (cite)
15.66 Tor dimension
Instead of resolving by projective modules we can look at resolutions by flat modules. This leads to the following concept.
Definition 15.66.1. Let $R$ be a ring. Denote $D(R)$ its derived category. Let $a, b \in \mathbf{Z}$.
An object $K^\bullet $ of $D(R)$ has tor-amplitude in $[a, b]$ if $H^ i(K^\bullet \otimes _ R^\mathbf {L} M) = 0$ for all $R$-modules $M$ and all $i \not\in [a, b]$.
An object $K^\bullet $ of $D(R)$ has finite tor dimension if it has tor-amplitude in $[a, b]$ for some $a, b$.
An $R$-module $M$ has tor dimension $\leq d$ if $M[0]$ as an object of $D(R)$ has tor-amplitude in $[-d, 0]$.
An $R$-module $M$ has finite tor dimension if $M[0]$ as an object of $D(R)$ has finite tor dimension.
We observe that if $K^\bullet $ has finite tor dimension, then $K^\bullet \in D^ b(R)$.
Lemma 15.66.2. Let $R$ be a ring. Let $K^\bullet $ be a bounded above complex of flat $R$-modules with tor-amplitude in $[a, b]$. Then $\mathop{\mathrm{Coker}}(d_ K^{a - 1})$ is a flat $R$-module.
Proof. As $K^\bullet $ is a bounded above complex of flat modules we see that $K^\bullet \otimes _ R M = K^\bullet \otimes _ R^{\mathbf{L}} M$. Hence for every $R$-module $M$ the sequence
\[ K^{a - 2} \otimes _ R M \to K^{a - 1} \otimes _ R M \to K^ a \otimes _ R M \]
is exact in the middle. Since $K^{a - 2} \to K^{a - 1} \to K^ a \to \mathop{\mathrm{Coker}}(d_ K^{a - 1}) \to 0$ is a flat resolution this implies that $\text{Tor}_1^ R(\mathop{\mathrm{Coker}}(d_ K^{a - 1}), M) = 0$ for all $R$-modules $M$. This means that $\mathop{\mathrm{Coker}}(d_ K^{a - 1})$ is flat, see Algebra, Lemma 10.75.8. $\square$
Lemma 15.66.3. Let $R$ be a ring. Let $K^\bullet $ be an object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent
$K^\bullet $ has tor-amplitude in $[a, b]$.
$K^\bullet $ is quasi-isomorphic to a complex $E^\bullet $ of flat $R$-modules with $E^ i = 0$ for $i \not\in [a, b]$.
Proof. If (2) holds, then we may compute $K^\bullet \otimes _ R^\mathbf {L} M = E^\bullet \otimes _ R M$ and it is clear that (1) holds. Assume that (1) holds. We may replace $K^\bullet $ by a projective resolution with $K^ i = 0$ for $i > b$. See Derived Categories, Lemma 13.19.3. Set $E^\bullet = \tau _{\geq a}K^\bullet $. Everything is clear except that $E^ a$ is flat which follows immediately from Lemma 15.66.2 and the definitions. $\square$
Lemma 15.66.4. Let $R$ be a ring. Let $a \in \mathbf{Z}$ and let $K$ be an object of $D(R)$. The following are equivalent
$K$ has tor-amplitude in $[a, \infty ]$, and
$K$ is quasi-isomorphic to a K-flat complex $E^\bullet $ whose terms are flat $R$-modules with $E^ i = 0$ for $i \not\in [a, \infty ]$.
Proof. The implication (2) $\Rightarrow $ (1) is immediate. Assume (1) holds. First we choose a K-flat complex $K^\bullet $ with flat terms representing $K$, see Lemma 15.59.10. For any $R$-module $M$ the cohomology of
\[ K^{n - 1} \otimes _ R M \to K^ n \otimes _ R M \to K^{n + 1} \otimes _ R M \]
computes $H^ n(K \otimes _ R^\mathbf {L} M)$. This is always zero for $n < a$. Hence if we apply Lemma 15.66.2 to the complex $\ldots \to K^{a - 1} \to K^ a \to K^{a + 1}$ we conclude that $N = \mathop{\mathrm{Coker}}(K^{a - 1} \to K^ a)$ is a flat $R$-module. We set
\[ E^\bullet = \tau _{\geq a}K^\bullet = (\ldots \to 0 \to N \to K^{a + 1} \to \ldots ) \]
The kernel $L^\bullet $ of $K^\bullet \to E^\bullet $ is the complex
\[ L^\bullet = (\ldots \to K^{a - 1} \to I \to 0 \to \ldots ) \]
where $I \subset K^ a$ is the image of $K^{a - 1} \to K^ a$. Since we have the short exact sequence $0 \to I \to K^ a \to N \to 0$ we see that $I$ is a flat $R$-module. Thus $L^\bullet $ is a bounded above complex of flat modules, hence K-flat by Lemma 15.59.7. It follows that $E^\bullet $ is K-flat by Lemma 15.59.6. $\square$
Lemma 15.66.5. Let $R$ be a ring. Let $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ be a distinguished triangle in $D(R)$. Let $a, b \in \mathbf{Z}$.
If $K^\bullet $ has tor-amplitude in $[a + 1, b + 1]$ and $L^\bullet $ has tor-amplitude in $[a, b]$ then $M^\bullet $ has tor-amplitude in $[a, b]$.
If $K^\bullet , M^\bullet $ have tor-amplitude in $[a, b]$, then $L^\bullet $ has tor-amplitude in $[a, b]$.
If $L^\bullet $ has tor-amplitude in $[a + 1, b + 1]$ and $M^\bullet $ has tor-amplitude in $[a, b]$, then $K^\bullet $ has tor-amplitude in $[a + 1, b + 1]$.
Proof. Omitted. Hint: This just follows from the long exact cohomology sequence associated to a distinguished triangle and the fact that $- \otimes _ R^{\mathbf{L}} M$ preserves distinguished triangles. The easiest one to prove is (2) and the others follow from it by translation. $\square$
Lemma 15.66.6. Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \geq 0$. The following are equivalent
$M$ has tor dimension $\leq d$, and
there exists a resolution
\[ 0 \to F_ d \to \ldots \to F_1 \to F_0 \to M \to 0 \]
with $F_ i$ a flat $R$-module.
In particular an $R$-module has tor dimension $0$ if and only if it is a flat $R$-module.
Proof. Assume (2). Then the complex $E^\bullet $ with $E^{-i} = F_ i$ is quasi-isomorphic to $M$. Hence the Tor dimension of $M$ is at most $d$ by Lemma 15.66.3. Conversely, assume (1). Let $P^\bullet \to M$ be a projective resolution of $M$. By Lemma 15.66.2 we see that $\tau _{\geq -d}P^\bullet $ is a flat resolution of $M$ of length $d$, i.e., (2) holds. $\square$
Lemma 15.66.7. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. If $K^\bullet \oplus L^\bullet $ has tor amplitude in $[a, b]$ so do $K^\bullet $ and $L^\bullet $.
Lemma 15.66.8. Let $R$ be a ring. Let $K^\bullet $ be a bounded complex of $R$-modules such that $K^ i$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet $ has tor amplitude in $[a, b]$. In particular if $K^\bullet $ is a finite complex of $R$-modules of finite tor dimension, then $K^\bullet $ has finite tor dimension.
Proof. Follows by induction on the length of the finite complex: use Lemma 15.66.5 and the stupid truncations. $\square$
Lemma 15.66.9. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet \in D^ b(R)$ such that $H^ i(K^\bullet )$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet $ has tor amplitude in $[a, b]$. In particular if $K^\bullet \in D^ b(R)$ and all its cohomology groups have finite tor dimension then $K^\bullet $ has finite tor dimension.
Proof. Follows by induction on the length of the finite complex: use Lemma 15.66.5 and the canonical truncations. $\square$
Lemma 15.66.10. Let $A \to B$ be a ring map. Let $K^\bullet $ and $L^\bullet $ be complexes of $B$-modules. Let $a, b, c, d \in \mathbf{Z}$. If
$K^\bullet $ as a complex of $B$-modules has tor amplitude in $[a, b]$,
$L^\bullet $ as a complex of $A$-modules has tor amplitude in $[c, d]$,
then $K^\bullet \otimes ^\mathbf {L}_ B L^\bullet $ as a complex of $A$-modules has tor amplitude in $[a + c, b + d]$.
Proof. We may assume that $K^\bullet $ is a complex of flat $B$-modules with $K^ i = 0$ for $i \not\in [a, b]$, see Lemma 15.66.3. Let $M$ be an $A$-module. Choose a free resolution $F^\bullet \to M$. Then
\[ (K^\bullet \otimes _ B^\mathbf {L} L^\bullet ) \otimes _ A^{\mathbf{L}} M = \text{Tot}(\text{Tot}(K^\bullet \otimes _ B L^\bullet ) \otimes _ A F^\bullet ) = \text{Tot}(K^\bullet \otimes _ B \text{Tot}(L^\bullet \otimes _ A F^\bullet )) \]
see Homology, Remark 12.18.4 for the second equality. By assumption (2) the complex $\text{Tot}(L^\bullet \otimes _ A F^\bullet )$ has nonzero cohomology only in degrees $[c, d]$. Hence the spectral sequence of Homology, Lemma 12.25.1 for the double complex $K^\bullet \otimes _ B \text{Tot}(L^\bullet \otimes _ A F^\bullet )$ proves that $(K^\bullet \otimes _ B^\mathbf {L} L^\bullet ) \otimes _ A^{\mathbf{L}} M$ has nonzero cohomology only in degrees $[a + c, b + d]$. $\square$
Lemma 15.66.11. Let $A \to B$ be a ring map. Assume that $B$ is flat as an $A$-module. Let $K^\bullet $ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. If $K^\bullet $ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\bullet $ as a complex of $A$-modules has tor amplitude in $[a, b]$.
Proof. This is a special case of Lemma 15.66.10, but can also be seen directly as follows. We have $K^\bullet \otimes _ A^{\mathbf{L}} M = K^\bullet \otimes _ B^{\mathbf{L}} (M \otimes _ A B)$ since any projective resolution of $K^\bullet $ as a complex of $B$-modules is a flat resolution of $K^\bullet $ as a complex of $A$-modules and can be used to compute $K^\bullet \otimes _ A^{\mathbf{L}} M$. $\square$
Lemma 15.66.12. Let $A \to B$ be a ring map. Assume that $B$ has tor dimension $\leq d$ as an $A$-module. Let $K^\bullet $ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. If $K^\bullet $ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\bullet $ as a complex of $A$-modules has tor amplitude in $[a - d, b]$.
Proof. This is a special case of Lemma 15.66.10, but can also be seen directly as follows. Let $M$ be an $A$-module. Choose a free resolution $F^\bullet \to M$. Then
\[ K^\bullet \otimes _ A^{\mathbf{L}} M = \text{Tot}(K^\bullet \otimes _ A F^\bullet ) = \text{Tot}(K^\bullet \otimes _ B (F^\bullet \otimes _ A B)) = K^\bullet \otimes _ B^{\mathbf{L}} (M \otimes _ A^{\mathbf{L}} B). \]
By our assumption on $B$ as an $A$-module we see that $M \otimes _ A^{\mathbf{L}} B$ has cohomology only in degrees $-d, -d + 1, \ldots , 0$. Because $K^\bullet $ has tor amplitude in $[a, b]$ we see from the spectral sequence in Example 15.62.4 that $K^\bullet \otimes _ B^{\mathbf{L}} (M \otimes _ A^{\mathbf{L}} B)$ has cohomology only in degrees $[-d + a, b]$ as desired. $\square$
Lemma 15.66.13. Let $A \to B$ be a ring map. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $A$-modules with tor amplitude in $[a, b]$. Then $K^\bullet \otimes _ A^{\mathbf{L}} B$ as a complex of $B$-modules has tor amplitude in $[a, b]$.
Proof. By Lemma 15.66.3 we can find a quasi-isomorphism $E^\bullet \to K^\bullet $ where $E^\bullet $ is a complex of flat $A$-modules with $E^ i = 0$ for $i \not\in [a, b]$. Then $E^\bullet \otimes _ A B$ computes $K^\bullet \otimes _ A ^{\mathbf{L}} B$ by construction and each $E^ i \otimes _ A B$ is a flat $B$-module by Algebra, Lemma 10.39.7. Hence we conclude by Lemma 15.66.3. $\square$
Lemma 15.66.14. Let $A \to B$ be a flat ring map. Let $d \geq 0$. Let $M$ be an $A$-module of tor dimension $\leq d$. Then $M \otimes _ A B$ is a $B$-module of tor dimension $\leq d$.
Proof. Immediate consequence of Lemma 15.66.13 and the fact that $M \otimes _ A^{\mathbf{L}} B = M \otimes _ A B$ because $B$ is flat over $A$. $\square$
Lemma 15.66.15. Let $A \to B$ be a ring map. Let $K^\bullet $ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. The following are equivalent
$K^\bullet $ has tor amplitude in $[a, b]$ as a complex of $A$-modules,
$K^\bullet _\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every prime $\mathfrak q \subset B$ with $\mathfrak p = A \cap \mathfrak q$,
$K^\bullet _\mathfrak m$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every maximal ideal $\mathfrak m \subset B$ with $\mathfrak p = A \cap \mathfrak m$.
Proof. Assume (3) and let $M$ be an $A$-module. Then $H^ i = H^ i(K^\bullet \otimes _ A^\mathbf {L} M)$ is a $B$-module and $(H^ i)_\mathfrak m = H^ i(K^\bullet _\mathfrak m \otimes _{A_\mathfrak p}^\mathbf {L} M_\mathfrak p)$. Hence $H^ i = 0$ for $i \not\in [a, b]$ by Algebra, Lemma 10.23.1. Thus (3) $\Rightarrow $ (1). We omit the proofs of (1) $\Rightarrow $ (2) and (2) $\Rightarrow $ (3). $\square$
Lemma 15.66.16. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be elements which generate the unit ideal. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $R$-modules. If for each $i$ the complex $K^\bullet \otimes _ R R_{f_ i}$ has tor amplitude in $[a, b]$, then $K^\bullet $ has tor amplitude in $[a, b]$.
Proof. This follows immediately from Lemma 15.66.15 but can also be seen directly as follows. Note that $- \otimes _ R R_{f_ i}$ is an exact functor and that therefore
\[ H^ i(K^\bullet )_{f_ i} = H^ i(K^\bullet ) \otimes _ R R_{f_ i} = H^ i(K^\bullet \otimes _ R R_{f_ i}). \]
and similarly for every $R$-module $M$ we have
\[ H^ i(K^\bullet \otimes _ R^{\mathbf{L}} M)_{f_ i} = H^ i(K^\bullet \otimes _ R^{\mathbf{L}} M) \otimes _ R R_{f_ i} = H^ i(K^\bullet \otimes _ R R_{f_ i} \otimes _{R_{f_ i}}^{\mathbf{L}} M_{f_ i}). \]
Hence the result follows from the fact that an $R$-module $N$ is zero if and only if $N_{f_ i}$ is zero for each $i$, see Algebra, Lemma 10.23.2. $\square$
Lemma 15.66.17. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $R$-modules. Let $R \to R'$ be a faithfully flat ring map. If the complex $K^\bullet \otimes _ R R'$ has tor amplitude in $[a, b]$, then $K^\bullet $ has tor amplitude in $[a, b]$.
Proof. Let $M$ be an $R$-module. Since $R \to R'$ is flat we see that
\[ (M \otimes _ R^{\mathbf{L}} K^\bullet ) \otimes _ R R' = ((M \otimes _ R R') \otimes _{R'}^{\mathbf{L}} (K^\bullet \otimes _ R R') \]
and taking cohomology commutes with tensoring with $R'$. Hence $\text{Tor}_ i^ R(M, K^\bullet ) \otimes _ R R' = \text{Tor}_ i^{R'}(M \otimes _ R R', K^\bullet \otimes _ R R')$. Since $R \to R'$ is faithfully flat, the vanishing of $\text{Tor}_ i^{R'}(M \otimes _ R R', K^\bullet \otimes _ R R')$ for $i \not\in [a, b]$ implies the same thing for $\text{Tor}_ i^ R(M, K^\bullet )$. $\square$
Lemma 15.66.18. Given ring maps $R \to A \to B$ with $A \to B$ faithfully flat and $K \in D(A)$ the tor amplitude of $K$ over $R$ is the same as the tor amplitude of $K \otimes _ A^\mathbf {L} B$ over $R$.
Proof. This is true because for an $R$-module $M$ we have $H^ i(K \otimes _ R^\mathbf {L} M) \otimes _ A B = H^ i((K \otimes _ A^\mathbf {L} B) \otimes _ R^\mathbf {L} M)$ for all $i$. Namely, represent $K$ by a complex $K^\bullet $ of $A$-modules and choose a free resolution $F^\bullet \to M$. Then we have the equality
\[ \text{Tot}(K^\bullet \otimes _ A B \otimes _ R F^\bullet ) = \text{Tot}(K^\bullet \otimes _ R F^\bullet ) \otimes _ A B \]
The cohomology groups of the left hand side are $H^ i((K \otimes _ A^\mathbf {L} B) \otimes _ R^\mathbf {L} M)$ and on the right hand side we obtain $H^ i(K \otimes _ R^\mathbf {L} M) \otimes _ A B$. $\square$
Lemma 15.66.19. Let $R$ be a ring of finite global dimension $d$. Then
every module has tor dimension $\leq d$,
a complex of $R$-modules $K^\bullet $ with $H^ i(K^\bullet ) \not= 0$ only if $i \in [a, b]$ has tor amplitude in $[a - d, b]$, and
a complex of $R$-modules $K^\bullet $ has finite tor dimension if and only if $K^\bullet \in D^ b(R)$.
Proof. The assumption on $R$ means that every module has a finite projective resolution of length at most $d$, in particular every module has tor dimension $\leq d$. The second statement follows from Lemma 15.66.9 and the definitions. The third statement is a rephrasing of the second. $\square$
Comment #2148 by Juanyong Wang on July 24, 2016 at 19:39
A tiny suggesion: in Lemma 15.56.8 (resp. Lemma 15.56.9), i.e. Tag 066H (resp. Tag 066I), I think it would be better to add
[0]
K^i
H^i(K^{\bullet})
) to make things more precise, since they are à priori just
R
-modules, not (though there is a natural way to put them) in the derived category. This may be stupid, but I really have the mistaken impression that the
K^i
represents for
K^{i}[-i]
when I fisrt look at this proposition.
Well, I am going to leave it as is for now, although of course you are right. My usual convention is that I am going to think of an abelian category as a full subcategory of its derived category by putting the objects in degree
0
. So really the mistake is to use
M[0]
in Definition 15.66.1 but I am too lazy to change it.
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Implement overlap-save method of frequency-domain filtering - Simulink - MathWorks Deutschland
Overlap-Save FFT Filter (Obsolete)
Implement overlap-save method of frequency-domain filtering
The Overlap-Save FFT Filter block has been replaced with the Frequency-Domain FIR Filter block. Existing instances of the Overlap-Save FFT Filter block continue to run.
The Overlap-Save FFT Filter block uses an FFT to implement the overlap-save method, a technique that combines successive frequency-domain filtered sections of an input sequence.
The block accepts vector or matrix inputs, and treats each column of the input as an individual channel. The block unbuffers the input data into row vectors such that the length of the output vector is equal to the number of channels in the input. The data output rate of the block is M times faster than its data input rate, where M is the length of the columns in the input (frame-size).
Overlapping sections of input u are circularly convolved with the FIR filter coefficients
H\left(z\right)=B\left(z\right)={b}_{1}+{b}_{2}{z}^{-1}+\dots +{b}_{n+1}{z}^{-n}
The numerator coefficients for H(z) are specified as a vector by the FIR coefficients parameter. The coefficient vector, b = [b(1) b(2) ... b(n+1)], can be generated by one of the filter design functions in the Signal Processing Toolbox™ product, such as fir1. All filter states are internally initialized to zero.
When either the filter coefficients or the inputs to the block are complex, the Output parameter should be set to Complex. Otherwise, the default Output setting, Real, instructs the block to take only the real part of the solution.
The circular convolution of each section is computed by multiplying the FFTs of the input section and filter coefficients, and computing the inverse FFT of the product.
y = ifft(fft(u(i:i+(L-1)),nfft) .* fft(b,nfft))
where you specify nfft in the FFT size parameter as a power of two value greater (typically much greater) than n+1. Values for FFT size that are not powers of two are rounded upwards to the nearest power-of-two value to obtain nfft.
The first n points of the circular convolution are invalid and are discarded. The Overlap-Save FFT Filter block outputs the remaining nfft-n points, which are equivalent to the linear convolution.
In single-tasking operation, the Overlap-Save FFT Filter block has a latency of nfft-n+1 samples. The first nfft-n+1 consecutive outputs from the block are zero; the first filtered input value appears at the output as sample nfft-n+2.
In multitasking operation, the Overlap-Save FFT Filter block has a latency of 2*(nfft-n+1) samples. The first 2*(nfft-n+1) consecutive outputs from the block are zero; the first filtered input value appears at the output as sample 2*(nfft-n)+3.
For more information on latency and the Simulink® environment tasking modes, see Excess Algorithmic Delay (Tasking Latency) and Time-Based Scheduling and Code Generation (Simulink Coder).
The size of the FFT, which should be a power of two value greater than the length of the specified FIR filter.
FIR coefficients
The filter numerator coefficients.
The complexity of the output; Real or Complex. When the input signal or the filter coefficients are complex, this should be set to Complex.
Oppenheim, A. V. and R. W. Schafer. Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1989.
Proakis, J. and D. Manolakis. Digital Signal Processing. 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
Overlap-Add FFT Filter DSP System Toolbox
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Undecidable problem — Wikipedia Republished // WIKI 2
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run.
Undecidable and Decidable problem
Lecture 38/65: The Undecidability of the Halting Problem
Undecidability Tangent (History of Undecidability Part 1) - Computerphile
2 Example: the halting problem in computability theory
3 Relationship with Gödel's incompleteness theorem
4 Examples of undecidable problems
5 Examples of undecidable statements
A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns yes. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is phrased in terms of subsets of the natural numbers.
Formally, a decision problem is a subset of the natural numbers. The corresponding informal problem is that of deciding whether a given number is in the set. A decision problem A is called decidable or effectively solvable if A is a recursive set and undecidable otherwise. A problem is called partially decidable, semi-decidable, solvable, or provable if A is a recursively enumerable set.[1]
In computability theory, the halting problem is a decision problem which can be stated as follows:
Alan Turing proved in 1936 that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist. Hence, the halting problem is undecidable for Turing machines.
Undecidable problems can be related to different topics, such as logic, abstract machines or topology. Since there are uncountably many undecidable problems,[2] any list, even one of infinite length, is necessarily incomplete.
See also: List of statements independent of ZFC and Independence (mathematical logic)
There are two distinct senses of the word "undecidable" in contemporary use. The first of these is the sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set. The connection between these two is that if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective formal system which proves for every question A in the problem either "the answer to A is yes" or "the answer to A is no".
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools.
One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by Max Dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent. This was shown to be the case in 1952.
The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proven from ZFC.
In 1970, Russian mathematician Yuri Matiyasevich showed that Hilbert's Tenth Problem, posed in 1900 as a challenge to the next century of mathematicians, cannot be solved. Hilbert's challenge sought an algorithm which finds all solutions of a Diophantine equation. A Diophantine equation is a more general case of Fermat's Last Theorem; we seek the integer roots of a polynomial in any number of variables with integer coefficients. Since we have only one equation but n variables, infinitely many solutions exist (and are easy to find) in the complex plane; however, the problem becomes impossible if solutions are constrained to integer values only. Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking Gödel's Incompleteness Theorem.[3]
In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable, in the second sense of the term. This result was later generalized by Rice's theorem.
In 1973, Saharon Shelah showed the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory.[4]
In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the Ramsey theorem, is undecidable in the axiomatization of arithmetic given by the Peano axioms but can be proven to be true in the larger system of second-order arithmetic.
Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
Goodstein's theorem is a statement about the Ramsey theory of the natural numbers that Kirby and Paris showed is undecidable in Peano arithmetic.
Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's theorem states that for any theory that can represent enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.
In 2007, researchers Kurtz and Simon, building on earlier work by J.H. Conway in the 1970s, proved that a natural generalization of the Collatz problem is undecidable.[5]
Decidability (logic)
Proof of impossibility
^ There are uncountably many subsets of
{\displaystyle \{0,1\}^{*}}
, only countably many of which can be decided by algorithms. However, also only countably many decision problems can be stated in any language.
^ Matiyasevich, Yuri (1970). Диофантовость перечислимых множеств [Enumerable sets are Diophantine]. Doklady Akademii Nauk SSSR (in Russian). 191: 279–282.
^ Shelah, Saharon (1974). "Infinite Abelian groups, Whitehead problem and some constructions". Israel Journal of Mathematics. 18 (3): 243–256. doi:10.1007/BF02757281. MR 0357114.
^ Kurtz, Stuart A.; Simon, Janos, "The Undecidability of the Generalized Collatz Problem", in Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. ISBN 3-540-72503-2. doi:10.1007/978-3-540-72504-6_49
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Normal Edge-Transitive Cayley Graphs of the Group
A. Assari, F. Sheikhmiri, "Normal Edge-Transitive Cayley Graphs of the Group ", International Journal of Combinatorics, vol. 2014, Article ID 628214, 4 pages, 2014. https://doi.org/10.1155/2014/628214
A. Assari1 and F. Sheikhmiri1
1Department of Basic Science, Jundi-Shapur University of Technology, Dezful 64616-18674, Iran
A Cayley graph of a group is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group .
Let be a group and let be a subset of . The Cayley graph of with respect to is a graph with the group as the vertex set and is an arc of the graph, if and only if for some we have . We denote the Cayley graph by . Since is a group, the Cayley graph cannot have any parallel edges. In addition we assume that the subset does not contain the identity element of the group to avoid having loops in the Cayley graph and also to be an inverse closed set; that is, , to have an undirected Cayley graph. Therefore, we focus on simple Cayley graphs.
Let be a simple graph with vertex set and edge set . In this paper, we denote the edge, joining the vertices and by . An automorphism of the graph is a permutation on the vertex set of which preserves the edges. The set of all automorphisms of forms a group with the composition of maps as the binary operation and is denoted by . Since is a permutation group of , it acts on as well as in the usual way. is called vertex-transitive, edge-transitive, or arc-transitive, iff acts transitively on the set of vertices, edges, or arcs of , respectively. is called half-arc-transitive iff it is vertex- and edge-transitive, but not arc-transitive.
Let be a finite group and let be an inverse closed subset of that does not contain the identity. Set . is connected iff is a generating set of the group . For , define the mapping by , for all . for every , and hence is a regular subgroup of isomorphic to , forcing to be a vertex-transitive graph.
Let . Normalizer of in is equal to where denotes the semidirect product of two groups [1].
In [2], the graph is called normal if is a normal subgroup of . And after that the normality of Cayley graphs has been extensively studied from different points of view. Among them, finding and classifying the normal Cayley graphs were an essential problem, since in normal Cayley graphs we know the exact automorphism group of the graph.
It was conjectured in [2] that most Cayley graphs are normal. For example, in [3] the authors determined all possible nonnormal (and, as a consequence, normal) Cayley graphs of groups of order , and lots of other authors have done similar works for groups of orders , , , , and [4–8].
Another concept which was similar to the above one is introduced by Praeger in [9] in which a Cayley graph of a group with respect to is called normal edge-transitive or arc-transitive if acts on the edges or arcs of , respectively, and it is called normal half-arc-transitive Cayley graph if it is normal edge-transitive Cayley graph which is not normal arc-transitive. Obviously, any normal edge-transitive Cayley graph is edge-transitive. Thus, this concept talks about the symmetric properties of a Cayley graph.
The latter concepts also were considered very much in the literature. For example, Alaeiyan in [10] found a class of normal edge-transitive Cayley graphs of abelian groups or, in [11], authors found all normal edge-transitive Cayley graphs of order and as a consequence found a class of normal half-arc-transitive Cayley graphs which rarely happens. This motivated us to consider the Cayley graphs of groups and classify all normal edge-transitive Cayley graphs of groups .
One of the principle theorems that helps us to connect the group properties of a group and normal edge-transitivity of is the following theorem which is proved in [9].
Theorem 1. Let be a connected Cayley graph on . Then, is normal edge-transitive if and only if is either transitive on or has two orbits in in the form of and , where is a nonempty subset of such that .
The following corollary comes from Theorem 1 which is also mentioned in [11].
Corollary 2. Let and let be the subset of all involutions of the group . If does not generate the group and is connected normal edge-transitive, then the valency of is even.
Theorem 1 with a result in [12] characterizes arc-transitive and half-arc-transitive Cayley graphs as described in the following.
Theorem 3. Let be a connected normal edge-transitive Cayley graph. is normal arc-transitive if acts transitively on and is normal half-arc-transitive if , where is an orbit of the action of on .
Group has the presentation which we can write its elements as follows: One can see that the group has order .
In Theorem 1, there are some relations between the normal edge-transitive Cayley graphs and the automorphism group of the relying group. Therefore, first we will find the automorphism group of the group .
Theorem 4. Every automorphism of the group is in the form of which sends to and to , or sends to and to , where for and , .
Proof. First of all we find some relations between elements of the group .
Since , equivalently Hence, we have One can check by induction that, for and , we will have If we denote the order of element of by , then, for , we have
For finding the order of elements of the form for , we have to observe that, for odd and arbitrary , we have and and, for even and arbitrary , we have , , and which can be obtained from (6).
Therefore, if the order of is , for odd , then for some integer . Since order of is , we should have But should be the least integer satisfying the condition and implies ; that is, where is the greatest common divisor.
Similar argument can apply for in the case is odd to obtain
Now for even , assume for some integer and for some integer . Thus, we have Compare it with the order of ; we get Since , one can conclude that , where is the least common multiple, and finally we obtain Similar argument can be discussed to prove Suppose is an automorphism of the group ; thus, preserves the order of elements; hence, and .
By the order of elements of , if we define the following sets by the order of elements of the group , we can say that and . Let and . But all of the following cases which may happen for and yield a contradiction. (1) and contradicts with the fact that is a generating set of .(2) and . The equation implies which also implies or . But none of them can occur, since .(3) and . Thus, for some odd and for some . Hence, is even and is odd and we have But because is an automorphism and hence yields , which is a contradiction.(4) and . We can write , for odd , and , for and . Thus, and, hence, are odd and we obtain But we have . Thus, we should have ; that is, , which is a contradiction.Therefore, the only cases that may happen are the cases , and , and the theorem is proved.
And from Theorem 4, we can obtain that the automorphism group of the group is as follows:
Now, we are ready to find the orbits of the group under the action of .
Lemma 5. In the action of group on , each element falls into one of the following orbits, depending on : (I); (II); (III); (IV).
Proof. By Relation (6), one can show that, for odd , if is even, then , and if is odd, then for . Therefore, images of elements of under and in are as follows (recall that , ).
If is even, then
If is even or odd, then and . Therefore, the orbit of for an even is and for odd is . The orbit of for an even is and for odd is . And finally the orbit of is .
Now we bring a sufficient condition under which a Cayley graph of group can be normal edge-transitive.
Theorem 6. If is connected normal edge-transitive Cayley graph, then is even, greater than 2, and is contained in .
Proof. From Corollary 2, is even.
Since is normal edge-transitive, by Theorem 1, acts transitively on or where is an orbit of the action of . Since is a subgroup of , thus or, in the latter case, should be contained in one of orbits of on . But, from Lemma 5, we have 4 kinds of orbits in this action. But from (6), if and are even, . Thus, is even and is not generated by such elements. Therefore, if is contained in an orbit of kind I, II, or IV, will not generate . Therefore, will not be connected. Thus, is contained in III.
Theorem 6 has a critical corollary.
Corollary 7. is never connected normal half arc-transitive Cayley graph.
Proof. For odd and , we have . From Theorem 6, contains such elements. Since is closed under inversion, is union of for some odd and some . is an automorphism of which preserves , for all and , and hence preserves ; that is, . Accordingly, if we set , then each element of is sent to an element of by . is not an orbit of the action of on .
By Theorems 1 and 6 and Corollary 7 we can obtain the following theorem.
Theorem 8. is connected normal edge-transitive Cayley graph if is even, greater than 2, is contained in , and acts transitively on .
The authors would like to thank anonymous referees for their useful comments and suggestions.
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Copyright © 2014 A. Assari and F. Sheikhmiri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations | J. Fluids Eng. | ASME Digital Collection
LBM Fluid Dynamics Laboratory
, 3-2-1 Mitahora-higashi, Gifu 502-0003, Japan
e-mail: watari-minoru@kvd.biglobe.ne.jp
Watari, M. (October 6, 2010). "Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations." ASME. J. Fluids Eng. October 2010; 132(10): 101401. https://doi.org/10.1115/1.4002359
Relationship between accuracy and number of velocity particles in velocity slip phenomena was investigated by numerical simulations and theoretical considerations. Two types of 2D models were used: the octagon family and the D2Q9 model. Models have to possess the following four prerequisites to accurately simulate the velocity slip phenomena: (a) equivalency to the Navier–Stokes equations in the N-S flow area, (b) conservation of momentum flow
Pxy
in the whole area, (c) appropriate relaxation process in the Knudsen layer, and (d) capability to properly express the mass and momentum flows on the wall. Both the octagon family and the D2Q9 model satisfy conditions (a) and (b). However, models with fewer velocity particles do not sufficiently satisfy conditions (c) and (d). The D2Q9 model fails to represent a relaxation process in the Knudsen layer and shows a considerable fluctuation in the velocity slip due to the model’s angle to the wall. To perform an accurate velocity slip simulation, models with sufficient velocity particles, such as the triple octagon model with moving particles of 24 directions, are desirable.
finite difference methods, flow simulation, Knudsen flow, lattice Boltzmann methods, Navier-Stokes equations, slip flow
Flow (Dynamics), Momentum, Particulate matter, Relaxation (Physics), Simulation, Simulation results, Computer simulation, Reflection, Lattice Boltzmann methods, Navier-Stokes equations, Equilibrium (Physics)
Lattice-Boltzmann Simulations of Fluid Flows in MEMS
Ansumali
Sbragaglia
Lattice Boltzmann Method for Gaseous Microflows Using Kinetic Theory Boundary Conditions
Lattice Boltzmann Simulation of Rarefied Gas Flows in Microchannels
Diffuse-Reflection Boundary Conditions for a Thermal Lattice Boltzmann Model in Two Dimensions: Evidence of Temperature Jump And Slip Velocity in Microchannels
Boundary Conditions for the Upwind Finite Difference Lattice Boltzmann Model: Evidence of Slip Velocity in Micro-Channel Flow
Kinetic Lattice Boltzmann Method for Microscale Gas Flows: Issues on Boundary Condition, Relaxation Time, and Regularization
Efficient Kinetic Method for Fluid Simulation Beyond the Navier-Stokes Equation
Lattice Boltzmann Equation With Multiple Effective Relaxation Times for Gaseous Microscale Flow
Possibility of Constructing a Multispeed Bhatnagar-Gross-Krook Thermal Model of the Lattice Boltzmann Method
Velocity Slip and Temperature Jump Simulations by the Three-Dimensional Thermal Finite-Difference Lattice Boltzmann Method
Supersonic Flow Simulations by a Three-Dimensional Multispeed Thermal Model of the Finite Difference Lattice Boltzmann Method
Two-Dimensional Thermal Model of the Finite-Difference Lattice Boltzmann Method With High Spatial Isotropy
On Stresses in Rarefied Gases Arising From Inequalities of Temperature
Kinetic Theory Analysis of Linearized Rayleigh Problem
Some Studies on Rarefied Gas Flows
Some Remarks on Knudsen Layer
Effects of Accommodation Coefficient on Shear Flow of Rarefied Gas
Dynamics of Rarefied Gas Flows: Asymptotic and Numerical Analyses of the Boltzmann Equation
, Reno, NV, Jan. 8–11, AIAA Paper No. 2001-0874.
Numerical Analysis of the Shear and Thermal Creep Flows on a Rarefied Gas Over a Plane Wall on the Basis of the Linearized Boltzmann Equation for Hard-Sphere Molecules
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Trirectangular tetrahedron - WikiMili, The Best Wikipedia Reader
A trirectangular tetrahedron can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin, like:
and x/a+y/b+z/c<1
In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the right angle of the trirectangular tetrahedron and the face opposite it is called the base. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron.
Integer edges
Integer faces
Only the bifurcating graph of the
{\displaystyle B_{3}}
Affine Coxeter group has a Trirectangular tetrahedron fundamental domain.
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume
{\displaystyle V={\frac {abc}{6}}.}
The altitude h satisfies [1]
{\displaystyle {\frac {1}{h^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}.}
{\displaystyle T_{0}}
of the base is given by [2]
{\displaystyle T_{0}={\frac {abc}{2h}}.}
Main article: De Gua's theorem
If the area of the base is
{\displaystyle T_{0}}
and the areas of the three other (right-angled) faces are
{\displaystyle T_{1}}
{\displaystyle T_{2}}
{\displaystyle T_{3}}
{\displaystyle T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.}
This is a generalization of the Pythagorean theorem to a tetrahedron.
Trirectangular bipyramid with edges (240, 117, 44, 125, 244, 267, 44, 117, 240)
The area of the base (a,b,c) is always (Gua) an irrational number. Thus a trirectangular tetrahedron with integer edges is never a perfect body. The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational edges, faces and volume, but the inner space-diagonal between the two trirectangular vertices is still irrational. The later one is the double of the altitude of the trirectangular tetrahedron and a rational part of the (proved) [3] irrational space-diagonal of the related Euler-brick (bc, ca, ab).
Trirectangular tetrahedrons with integer legs
{\displaystyle a,b,c}
and sides
{\displaystyle d={\sqrt {b^{2}+c^{2}}},e={\sqrt {a^{2}+c^{2}}},f={\sqrt {a^{2}+b^{2}}}}
of the base triangle exist, e.g.
{\displaystyle a=240,b=117,c=44,d=125,e=244,f=267}
(discovered 1719 by Halcke). Here are a few more examples with integer legs and sides.
240 117 44 125 244 267 275 252 240 348 365 373 480 234 88 250 488 534 550 504 480 696 730 746 693 480 140 500 707 843 720 351 132 375 732 801 720 132 85 157 725 732 792 231 160 281 808 825 825 756 720 1044 1095 1119 960 468 176 500 976 1068 1100 1008 960 1392 1460 1492 1155 1100 1008 1492 1533 1595 1200 585 220 625 1220 1335 1375 1260 1200 1740 1825 1865 1386 960 280 1000 1414 1686 1440 702 264 750 1464 1602 1440 264 170 314 1450 1464
Notice that some of these are multiples of smaller ones. Note also A031173.
Trirectangular tetrahedrons with integer faces
{\displaystyle T_{c},T_{a},T_{b},T_{0}}
and altitude h exist, e.g.
{\displaystyle a=42,b=28,c=14,T_{c}=588,T_{a}=196,T_{b}=294,T_{0}=686,h=12}
without or
{\displaystyle a=156,b=80,c=65,T_{c}=6240,T_{a}=2600,T_{b}=5070,T_{0}=8450,h=48}
{\displaystyle a,b,c}
Irregular tetrahedra
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
A right triangle or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle, is a triangle in which one angle is a right angle or two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
In geometry, Heron's formula, named after Hero of Alexandria, gives the area of a triangle when the lengths of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first.
In English outside North America, a convex quadrilateral in Euclidean geometry, with at least one pair of parallel sides, is referred to as a trapezium ; in American and Canadian English this is usually referred to as a trapezoid. The parallel sides are called the bases of the trapezoid. The other two sides are called the legs if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below.
In mathematics, a quadratic irrational number is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as
The square root of 2 is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or , and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.
In geometry, a rhombohedron is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.
High-leg delta is a type of electrical service connection for three-phase electric power installations. It is used when both single and three-phase power is desired to be supplied from a three phase transformer. The three-phase power is connected in the delta configuration, and the center point of one phase is grounded. This creates both a split-phase single phase supply and three-phase. It is called "orange leg" because the wire is color-coded orange. By convention, the high leg is usually set in the center lug in the involved panel, regardless of the L1-L2-L3 designation at the transformer.
In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron, and tetramonohedron.
In geometry, the spiral of Theodorus is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots.
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the legs a, b and the hypotenuse c, often called the Pythagorean equation:
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term Heronian triangle; Somos uses it to refer to triangles whose ratios of sides are rational; Conway and Guy define a rational triangle as one with rational sides and rational angles measured in degrees—in which case the only rational triangle is the rational-sided equilateral triangle.
An acute triangle is a triangle with three acute angles. An obtuse triangle is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.
↑ Eves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41.
↑ Gutierrez, Antonio, "Right Triangle Formulas",
↑ Walter Wyss, "No Perfect Cuboid", arXiv : 1506.02215
Weisstein, Eric W. "Trirectangular tetrahedron". MathWorld .
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Privacy Preserving Predictive Modeling GANs | Machine Learning Medium
In the current era of increasing data sharing and ubiquity of machine learning there is a very little focus on privacy of the subjects whose data is being shared at such large scale. While most data released promises anonymization of the PIIs, there is much work in literature to point of that simple anonymization techniques fail to mask users if there is an access to auxilliary dataset in which such features are not masked.
In order to fix this issue there has been study in the field of privacy preservation for the datasets in public domain to ensure public trust in sharing their information which eventually is the root to building amazing machine learning models and drawing insights from big data.
This paper discusses a new architecture of GAN which tries to achieve this very objective of anonymization of the private data but in a deep learning setting where the work in overseen by an objective function for the encoder which embeds in itself the notion of anonymizing sensitive columns while trying to maximize the predivity of non-sensitive data columns.
The GAN framework presented consists of three components (explained in detail in later sections):
Ally that predicts the desired variables,
Adversary that predicts the sensitive data.
The objective of the GAN framework is two-fold:
learn low-dimensional representation of data points (users in this case) that excels at classifying a desired task (whether a user will answer quiz question correctly here)
prevent an adversary from recovering sensitive data (each users identity in this case)
Netflix dataset one of the famous datasets which is used for various starter tutorials on recommendation system. The history of privacy breach in this particular case is related to this seemly harmless data. While the company had ensured data anonymization to prevent breach of privacy. It was later discovered that it is easy to locate the users with good accuracy using auxilliary data from other related datasets such as IMDB which does not anonymize its dataset. Similarly there are cases in the domain of insurance companies where it has been proven that reverse engineering is a viable effort despite the anonymization.
It has similarly been shown that one can identify anonymous users in online social networks with upto 90% accuracy. These metrics point to the fact that it is possible for an attacker to uncover sensitive attributes from user data.
One of the popular work that has caught traction in this field is called Differential Privacy (DP), which proposes to add random noise to raw data, where the noise (generally from Laplacian Distribution) level controls the trade off between predictive quality and user privacy. But it has been found that this mechanism also reduces the utility of the data for predictive modeling and increases the sample complexity
The GAN model presented in this particular work is an effort to achieve the privacy (to prevent de-anonymization) while preserving the predictive aspects of the dataset (to overcome the drawbacks of techniques like DP).
The authors apply this GAN architecture in a online MOOC setting. The objectives of the work include:
Use the student data to predict whether or not they will answer a quiz correctly.
Ensure that the encoded data does not achieve a good convergence on sensitive data such as user identity.
The work is different from DP in two key aspects:
It is data-dependent, i.e., it learns representations from user data
Directly uses raw user data without relying on feature engineering
The objective of the GAN is to generate representations that minimize the loss function of the ally while maximizing the loss function of adversary.
One key advantage mentioned about the architecture is that it is model agnostic, i.e. each module can instantiate a specific differential function (e.g. neural networks) based on the needs of the particular application
Datasets and Objectives
This particular paper presents the empirical results on dataset from the course Networks: Friends, Money and Bytes on Coursera MOOC platform. This has a total of 92 in-video quiz questions among 20 lectures. Each lecture has 4-5 videos. A total of 314,632 clickstreams were recorded for 3976 unique students.
Two types of data are collected about students:
Video-watching clickstream: behavior is recorded as a sequence of clickstreams based on actions available in the scrub bar.
Question submissions: answers submitted by a student to an in-video question
The final objective is defined as a mapping from student’s interaction (clickstream) on a video to their performance on questions (data acquired regarding question submissions)
The data collected can have both time-varying as well as static attributes. Time varying attributes include the series of clickstream before a question is answered, while the static attribute will included metrics like fraction of course completed, amount of time spent etc.
Accuracy on binary prediction of questions answered
AUC-ROC curve to assess the tradeoff between true and false positive rates
K Ranks and Mean average precision at K ([email protected]) to measure performance of privacy preservation
Only one baseline benchmark is included in the work which is Laplace Mechanism in DP (Differential Privacy) which simply adds Laplace noise to the data.
The new architecture outperforms DP in terms of prediction task on question answers. It actually performs slightly better than the original features themselves.
\alpha \to 1
in the GAN architecture, encoder is biased towards prediction than sensitive data obfuscation which is theoretically correct.
\epsilon
in DP means adding smaller noise component to the actual data, and it can be seen that models are better at predictive performance under such a setting.
Larger sizes of encoding dimension ensures more preserved information towards both prediction and sensitive data with identical
\alpha
values. This confirms the fact that the size of representations controls the amount of information contained in data representation.
Raw clickstream data with LSTM performs better than the hand-crafted features in terms of the tradeoff between prediction quality vs user privacy.
Follow-up Citations
J. Bennett, S. Lanning et al., “The netflix prize,” in Proceedings of KDD cup and workshop, vol. 2007. New York, NY, USA, 2007, p. 35.
A. Narayanan and V. Shmatikov, “Robust de-anonymization of large sparse datasets,” in Security and Privacy, 2008. SP 2008. IEEE Symposium on. IEEE, 2008, pp. 111–125.
“De-anonymizing social networks,” in Security and Privacy, 2009 30th IEEE Symposium on. IEEE, 2009, pp. 173–187.
Dwork, F. McSherry, K. Nissim, and A. Smith, “Calibrating noise to sensitivity in private data analysis,” in Proc. Theory of Cryptography Conference, Mar. 2006, pp. 265–284.
“Calibrating noise to sensitivity in private data analysis,” in Theory of Cryptography Conference. Springer, 2006, pp. 265–284.
C. Huang, P. Kairouzyz, X. Chen, S. L., and R. Rajagopal, “Context-aware generative adversarial privacy,” arXiv preprint arXiv:1710.09549, Dec. 2017.
Learning Informative and Private Representations via Generative Adversarial Networks
Social Learning Networks Generative Adversarial Networks
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Twelve horses raced in the CPM Derby.
How many ways could the horses finish in the top three places?
1320
If you have not already done so, write your answer to part (a) as a fraction with factorials.
\frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}
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Transmutation - Trava Knight NFT
Transmutation allows Knights owners to easily transmute 2 Trava Amoury NFTs, which are the same rarity and type, to get another type of item in the same rarity.
After the Transmutation, the EXP of the new item will be calculated as the average of the two Trava armoury items transmuted.
A certain amount of time is required to do transmutation.
Those transmuting their items need to pay a small fee in $TRAVA, which depends on the rarity of each item.
Unlike the trade-up mechanism, in Transmutation, there is no failure rate which means users can always transmute their items successfully.
Check the detailed information about transmutation time and fee in the following table:
Forgery time: The period it takes to transmute the armoury item.
Fee: An transmutation fee of Trava Knights need to pay
For example, you transmute one diamond Shield with 100,000 EXP and one diamond Shield with 50,000 EXP to get a diamond Weapon. In this case, the Trava Knights need to pay 5000 $TRAVA and wait for 16 hours to obtain the new item. The EXP for the newly transmuted is calculated as follows:
EXP = \frac{100000+50000} {2} = 75000 (points)
Step 1: Go to: https://nft-app.trava.finance/inventory/transmutation
Choose 2 armoury items of the same type and rarity.
The transmutation fee and EXP will be shown
Click “Transmute”
Choose the armoury item you would like to transmute to
Click “Confirm” to approve the transaction
Wait for a certain amount of time to complete transmutation
Click “Claim” to add the newly transmuted item to your wallet
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Store output power and phase information for amplifiers or mixers - MATLAB - MathWorks France
Store Output Power and Phase Specifications of RF Circuit Object.
Store output power and phase information for amplifiers or mixers
Use the power class to store output power and phase specifications for a circuit object.
h = rfdata.power
h = rfdata.power(`property1',value1,'property2',value2,...)
h = rfdata.powerreturns a data object for the Pin/Pout power data, h, whose properties all have their default values.
h = rfdata.power(`property1',value1,'property2',value2,...) sets properties using one or more name-value pairs. You can specify multiple name-value pairs. Enclose each property name in a quote
Frequency data , specified as a M-element vector in hertz. The values must be positive and correspond to the power data in 'Phase', 'Pin', and 'Pout' properties. The order of frequencies is equal to the order of the phase and power values. By default, this property is empty.
Phase — Phase shift data
M-element cell
Phase shift data, specified as a M-element cell in degrees. . The values correspond to the frequencies stored in the 'Freq' property. The values within each element correspond to the input power values stored in the 'Pin' property. The default value is 1.
Pin — Input power data
M-element cell in watts
Input power data , specified as a M-element vector cell in watts. The values correspond to the frequencies stored in the 'Freq' property. For example,
{P}_{in}=\left\{\left[A\right];\left[B\right];\left[C\right]\right\};
where A, B, and C are column vectors that contain the first three frequencies stored in the 'Freq' property.
Pout — Output power data
Output power data, specified as a M-element vector in watts. The values correspond to the frequencies stored in the 'Freq' property. The values within each element correspond to the input power values stored in the 'Pin' property. The default value is 1.
'Power data' | 1-by-N character array | string
Create an object to store output power and phase specifications using rfdata.power.
powerdata = rfdata.power
powerdata =
rfdata.power with properties:
Freq: []
Pin: {[1 10]}
Pout: {[1 10]}
Phase: {}
Name: 'Power data'
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Musical_isomorphism Knowpia
{\displaystyle \mathrm {T} M}
{\displaystyle \mathrm {T} ^{*}M}
{\displaystyle \flat }
{\displaystyle \sharp }
{\displaystyle \mathrm {T} ^{*}M}
{\displaystyle X^{\flat }:=g_{ij}X^{i}\,\mathbf {e} ^{j}=X_{j}\,\mathbf {e} ^{j}.}
{\displaystyle X^{\flat }(Y)=\langle X,Y\rangle }
{\displaystyle \omega ^{\sharp }:=g^{ij}\omega _{i}\mathbf {e} _{j}=\omega ^{j}\mathbf {e} _{j},}
{\displaystyle {\bigl \langle }\omega ^{\sharp },Y{\bigr \rangle }=\omega (Y),}
{\displaystyle \flat :{\rm {T}}M\to {\rm {T}}^{*}M,\qquad \sharp :{\rm {T}}^{*}M\to {\rm {T}}M.}
Extension to tensor productsEdit
{\displaystyle \bigotimes ^{k}{\rm {T}}M,\qquad \bigotimes ^{k}{\rm {T}}^{*}M.}
{\displaystyle X^{\sharp }=g^{jk}X_{ij}\,{\rm {e}}^{i}\otimes {\rm {e}}_{k}.}
Extension to k-vectors and k-formsEdit
{\displaystyle \flat :{\bigwedge }V\to {\bigwedge }^{*}V,\qquad \sharp :{\bigwedge }^{*}V\to {\bigwedge }V,}
{\displaystyle (X\wedge \ldots \wedge Z)^{\flat }=X^{\flat }\wedge \ldots \wedge Z^{\flat },\qquad (\alpha \wedge \ldots \wedge \gamma )^{\sharp }=\alpha ^{\sharp }\wedge \ldots \wedge \gamma ^{\sharp }.}
{\displaystyle Y^{\sharp }=(Y_{i\dots k}\mathbf {e} ^{i}\otimes \dots \otimes \mathbf {e} ^{k})^{\sharp }=g^{ir}\dots g^{kt}\,Y_{i\dots k}\,\mathbf {e} _{r}\otimes \dots \otimes \mathbf {e} _{t}.}
Trace of a tensor through a metric tensorEdit
{\displaystyle \operatorname {tr} _{g}(X):=\operatorname {tr} (X^{\sharp })=\operatorname {tr} (g^{jk}X_{ij}\,{\bf {e}}^{i}\otimes {\bf {e}}_{k})=g^{ji}X_{ij}=g^{ij}X_{ij}.}
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JAN MALTE LICHTENBERG · JANUARY 7, 2022
SARSA (vs. Q-learning)
This is the fourth post of the blog series Reinforcement learning: line by line. Here we take a look at the tabular Sarsa algorithm and compare it to the Q-learning algorithm (discussed in the previous post). In case you are completely new to reinforcement learning (RL), see here for an informal introduction.
The interactive sketch below shows an implementation of the tabular Sarsa algorithm applied to a version of a simple game, called the Pancakes Gridworld. See here for more information about the Pancakes Gridworld.
SARSA: State, Action, Reward, State, Action
The tabular Sarsa algorithm is conceptually very similar to the Q-learning algorithm in that, in every time step, the agent uses only information from the current transition to improve its action-value estimates. The main conceptual difference between the two algorithms is the update rule (line 6 of the pseudo code), as discussed further below in more detail.
One technical consequence of using this other update rule is that the Sarsa algorithm has a slightly different algorithmic structure. Specifically, the Sarsa algorithm requires in each iteration the current state
S
, the current action
A
, the reward
R
, the next state
S'
, and the next action
A'
. The Q-learning update, on the other hand, only requires the variables
S, A, R, S'
. This leads to a different structure of learning updates:
\text{Q-learning: } \ \rlap{$\overbrace{\phantom{S_0, A_0, R_1, S_1}}^{1^\text{st} \text{update}}$} S_0, A_0, R_1, \rlap{$\underbrace{\phantom{S_1, A_1, R_1, S_2}}_{2^\text{nd} \text{update}}$} S_1, A_1, R_2, \rlap{$\overbrace{\phantom{S_2, A_2, R_3, S_3}}^{3^\text{rd} \text{update}}$} S_2, A_2, R_3, \rlap{$\underbrace{\phantom{S_3, A_3, R_4, S_4}}_{4^\text{th} \text{update}}$} S_3, A_3, R_4, S_4, A_4, \dots
\text{Sarsa: } \ \rlap{$\overbrace{\phantom{S_0, A_0, R_1, S_1, A_1}}^{1^\text{st} \text{update}}$} S_0, A_0, R_1, \rlap{$\underbrace{\phantom{S_1, A_1, R_1, S_2, A_2}}_{2^\text{nd} \text{update}}$} S_1, A_1, R_2, \rlap{$\overbrace{\phantom{S_2, A_2, R_3, S_3, A_3}}^{3^\text{rd} \text{update}}$} S_2, A_2, R_3, \rlap{$\underbrace{\phantom{S_3, A_3, R_4, S_4, A_4}}_{4^\text{th} \text{update}}$} S_3, A_3, R_4, S_4, A_4, \dots
For the Q-learning algorithm only state is shared between two successive updates. For the Sarsa algorithm, both state and action are shared between successive updates. This difference is the reason for the different “rhythm” in the algorithm structure, as further explained in the algorithm description. In particular, for the Sarsa algorithm we always have to carry around two actions (current action
A
and next action
A'
), whereas the Q-learning algorithm requires only one action variable (
A
The remainder of this section goes through the Sarsa pseudo code, line by line. It focuses on the differences to the Q-learning pseudo code, which was explained here (pseudo code lines that are identical in both algorithms are grayed out).
0: \text{Loop for each episode:}
(“outer loop”, same as in Q-learning)
1: S \leftarrow s_0
(“set agent to starting state” same as in Q-learning)
2: \text{Select action } A \text{ from state } S \text{ using } \epsilon\text{-greedy policy}
This is similar to Q-learning's code line 3, yet here we select an action already before entering the inner loop (code line 4). This is the different “rhythm” I talked about earlier: the Sarsa algorithm requires the current action and the next action for its update.
3: \text{Loop until a terminal state is reached:}
(“inner loop”, same as in Q-learning)
4: \text{Take action } A, \text{ observe reward } R \text{ and new state } S'
(same as in Q-learning)
5: \text{Select action } A' \text{ from state } S' \text{ using } \epsilon\text{-greedy policy}
Here the next action
A'
is selected according to an
\epsilon
-greedy policy, based on the current Q-value estimates.
6: Q(S, A) \leftarrow (1 - \alpha) \ Q(S, A) + \alpha \ [R + \gamma Q(S', A')]
The learning update. The same general intuition from the Q-learning update applies here as well (the new estimate is a weighted average of the old estimate and a Bellman estimate, as described here). The difference is that for Sarsa the Bellman estimate is not based on the Bellman equation of the optimal value function (as it is for Q-learning), but on the (deterministic) Bellman equation of the agent's current policy
\pi
q_{\pi}(s, a) = r(s, a) + q_{\pi}(s', a') , \quad \text{ for all } s \in \mathcal{S} \text{ and } a \in \mathcal{A},
s'(s, a)
is the next state as determined by the transition function, and
a' = \pi(s')
is the action chosen by the current policy in the next state. The Bellman equation looks slightly more complicated if the policy and/or environment is/are stochastic, yet the intuition remains the same. They are provided in Sutton & Barto's book. The Sarsa update is said to be on-policy because the Bellman estimate uses the Q-value of the next action that is actually chosen by the current policy. The Q-learning update, on the other hand, is said to be off-policy, because it always considers the maximum Q-value in the next state, regardless of which action the agent chooses next.
7: S \leftarrow S'; A \leftarrow A'
The “next state” becomes the “current state” of the next iteration, just as in Q-learning. In addition, the “next action” becomes the “current action”.
Q-learning vs. Sarsa
What difference does it make to use the Sarsa update rule in place of the Q-learning update rule? Let's first compare the policies learned by the two algorithms in the Pancakes Gridworld and then try to generalize our findings.
You might have noticed that the Pancakes Gridworld in the sketch above is different from the one that was used in the Q-learning post. The version used in the present post imitates the “Cliff Walking” example from Sutton & Barto's book. Example 6.6. (page 132) in Sutton & Barto (2018). It's a great example to show the difference between on-policy and off-policy learning for an explorative, stochastic policy (for example, the
\epsilon
-greedy policy ), as you will see further below.
The following sketch shows a Q-learning agent in the same Pancakes environment that was used in the Sarsa-sketch at the top of this post.
If you run both algorithms for long enough, the typical (non-exploring) trajectories taken by the respective agents will eventually look as follows. Assuming that both algorithms used a constant, small exploration parameter
\epsilon
during the learning process and then turned off exploration (i.e.,
\epsilon=0
) to create these trajectories.
Why are the policies learned by Q-learning and Sarsa different?
The different policies are a direct consequence of the different update rules used by the two algorithms. In what follows, we'll consider an example transition and compare the corresponding learning updates.
Assume that the agent has already interacted with the environment and has learned the following value estimates (hover your mouse above a grid cell to see numerical values).
Furthermore assume that the agent transitioned from
s_5
s_6
and then makes an exploratory Recall that sometimes the agent needs to randomly explore the environment during the learning process, as described here. move and selects action “south”, or more formally:
S = s_5, \ A = east, \ R = -1, \ S' = s_6, \ A' = south.
The Sarsa update for this transition is given by
Q(s_5, east) \leftarrow (1 - \alpha) \ Q(s_5, east) + \alpha \ [-1 + \gamma Q(s_6, \textcolor{blue}{south})],
whereas the Q-learning update is given by
Q(s_5, east) \leftarrow (1 - \alpha) \ Q(s_5, east) + \alpha \ [-1 + \gamma Q(s_6, \textcolor{red}{east})].
To put it in words, the Sarsa update uses the value corresponding to the “on-policy action” for its update. Q-learning, on the other hand, uses the value-maximizing action (in
s_6
, the action “
east
” has the highest value according to the current estimates). If we plug in the actual values (and setting
\alpha = 0.2
\gamma = 1
), we see a big difference in the new value estimates for
Q(s_5, east)
, depending on which update rule is used: For Sarsa, the new value is given by
Q(s_5, east) = 0.8 \times (-1.82) + 0.2 \times [-1 + (\textcolor{blue}{-36.00})] = -8.86.
For Q-learning, the new value estimate for
Q(s_5, east)
Q(s_5, east) = 0.8 \times (-1.82) + 0.2 \times [-1 + \textcolor{red}{1.22}] = -1.41.
If compared with the value estimate of
Q(s_5, east)
before the update (
=-1.82
), the Q-learning update increases the estimate but the Sarsa update decreases the estimate. Importantly, the new value-maximizing action in state
s_5
north
for the Sarsa agent, whereas it stays
east
for the Q-learning agent. This is consistent with the typical trajectories of both algorithms shown further above.
So... which policy is optimal?
It depends! The Sarsa algorithm learns the optimal policy for an agent that never ceases to explore. For an exploring agent it is dangerous to walk too close to the mushrooms and it thus pays off to make the two extra steps required to take the middle lane (
s_{10}
s_{14}
) further away from the mushrooms.
The Q-learning algorithm, on the other hand, learns the optimal policy for an agent that at some point stops exploration and thus can safely walk close to the mushrooms without ever accidentally eating them.
In the particular setting of the Pancakes world studied here, nothing speaks against
stopping exploration at some point after the environment has been thoroughly explored. Therefore, the policy learned by the Q-learning algorithm seems like the better option. However, in general, one could think of environments that change over time (for example, the positions of gold and/or mushrooms could change over time) and thus may require the agent to continue exploring forever. In such a case the Sarsa algorithm might be the better option.
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