text stringlengths 256 16.4k |
|---|
Polynomial basis functions for tunable gain surface - MATLAB polyBasis
shapefcn\left(x\right)=\left[x,{x}^{2},\dots ,{x}^{order}\right].
shapefcn\left(x\right)=\left[{T}_{1}\left(x\right),\dots ,{T}_{order}\left(x\right)\right].
{T}_{0}\left(x\right)=1;\text{ }{T}_{1}\left(x\right)=x;\text{ }{T}_{i+1}\left(x\right)=2x{T}_{i}\left(x\right)-{T}_{i-1}\left(x\right).
shapefcn\left(x,y,z\right)=\left[{x}^{i}{y}^{j}{z}^{k}\text{\hspace{0.17em}}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le i,j,k\le order,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i+j+k>0\right].\text{ }
K\left(x,y\right)={K}_{0}+{K}_{1}x+{K}_{2}y+{K}_{3}xy.
\left[x,y,xy\right]
\left(x,y\right) |
Home : Support : Online Help : Mathematics : Algebra : Expression Manipulation : Norm
norm of an algebraic number (or function)
Norm(a, L, K)
The Norm function is a placeholder for representing the norm of an algebraic number (or function), that is the product of its conjugates. It is used in conjunction with evala.
The call evala(Norm(a, L, K)) computes the norm of a over the algebraic number (or function) field represented by K. In case K is not specified and a is an algebraic number, the norm over the rational is computed. In case K is not specified and a is an algebraic function, the smallest possible algebraic extension of the rational numbers is chosen. The expression a is viewed as an element of the smallest field containing a and the RootOfs in L.
\mathrm{alias}\left(\mathrm{sqrt2}=\mathrm{RootOf}\left({x}^{2}-2\right)\right):
\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({y}^{2}-x+\mathrm{RootOf}\left({x}^{2}-2\right),y\right)\right):
\mathrm{evala}\left(\mathrm{Norm}\left(\mathrm{\alpha }\right)\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}
\mathrm{evala}\left(\mathrm{Norm}\left(\mathrm{\alpha },\varnothing ,\mathrm{sqrt2}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{sqrt2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}
\mathrm{evala}\left(\mathrm{Norm}\left(z-\mathrm{\alpha }\right)\right)
{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\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}{2}
The name Norm must be global.
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
\mathrm{evala}\left(\mathrm{Norm}\left(z-\mathrm{\alpha }\right)\right)
Error, (in Norm) expects its 1st argument, A, to be of type {Matrix, Vector}, but received z-alpha
\mathrm{evala}\left(:-\mathrm{Norm}\left(z-\mathrm{\alpha }\right)\right)
{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\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}{2} |
To G. H. Darwin [after 4 September 1876]1
I do not think that you will care for Zacharias’ pamphet sent by this post.—2 Hermann Müller has sent you & Frank a newspaper on the development of the instincts of bees in relation to sucking flowers which I am sure you wd not care about in the least & I have not sent it.—3
I have received a German pamphet about the idea of God & immortality & socialism under a Darwinian point of view: it is so difficult I cannot make head or tails of it;4 I at first thought of sending it you, but then thought you wd. not care for it. I could, however, send it.— Do you remember calculating for me about number of molecules in a square of
\frac{1}{1000}
of an inch, in relation to Pangenesis; well I hear from Eras. that in Encyclop. Britannica, Clark-Maxwell has been writing on this subject in relation to Pangenesis, & opposed to my views, but Eras. says he cannot understand it, & that you might on your return read it.—5 I have been grieved, my dear George, to hear how very bad you have been.6
Yours affect. | C. Darwin
The date is established by the relationship between this letter and the letter from Hermann Müller, 4 September 1876.
Otto Zacharias had written an introduction to the German translation of George’s paper on cousin marriages (G. H. Darwin 1875; G. H. Darwin 1876a, pp. iii–vi). CD’s copy of G. H. Darwin 1876a is in the Darwin Pamphlet Collection–CUL.
See letter from Hermann Müller, 4 September 1876 and n. 2. The newspaper article was the last two instalments of H. Müller 1875–6.
Wilhelm Parow sent CD a copy of his lecture Der Gottes-Begriff, Unsterblichkeit und die sittliche Idee gegenüber dem Darwinismus (The concept of God, immortality and the moral idea with respect to Darwinism; Parow 1876). CD’s annotated copy is in the Darwin Pamphlet Collection–CUL.
James Clerk Maxwell, who served as the physical sciences editor for the ninth edition of the Encyclopaedia Britannica, had written the article on the atom (Maxwell 1875; see also Harman ed. 1990–2002, 2: 445–84). Maxwell discussed the limits of the dimensions of organic molecules in relation to pangenesis, CD’s theory of heredity (see Maxwell 1875, p. 42). Erasmus Alvey Darwin was CD’s brother.
George periodically suffered from stomach-related illnesses and in October 1876 he was also bothered by eczema (letter from Emma Darwin to Leonard Darwin, [29 October 1876] (DAR 239.23: 1.53)).
Maxwell, James Clerk. 1875. Atom. Encyclopaedia Britannica 9th ed. 3: 36–49.
Müller, Hermann. 1875–6. Die Bedeutung der Honigbiene für unsere Blumen. Bienen-Zeitung: Organ des Vereins der deutschen Bienenwirthe 31 (1875): 81–2, 102–4, 109–11, 122–5, 138–41, 165–9; 32 (1876): 20–2, 119–23, 176–84.
Parow, Wilhelm. 1876. Der Gottes-Begriff, die Unsterblichkeit und die sittliche Idee gegenüber dem Darwinismus. Ein Vortrag. Leipzig: J. G. Findel.
Has received a baffling article on God, immortality, and socialism under a Darwinian point of view.
Clerk Maxwell has disagreed with CD on molecular calculations in relation to Pangenesis in Encyclopaedia Britannica article ["Atom", Encyclopaedia Britannica, 9th ed. (1875) 3: 36–49]. |
Create ellipsoid - MATLAB ellipsoid - MathWorks España
Create and Display Ellipsoid
Apply Translation and Rotation to Ellipsoid
Display Ellipsoids with Different Numbers of Faces
xr,yr,zr
Create ellipsoid
[X,Y,Z] = ellipsoid(xc,yc,zc,xr,yr,zr)
[X,Y,Z] = ellipsoid(xc,yc,zc,xr,yr,zr,n)
ellipsoid(___)
ellipsoid(ax,___)
[X,Y,Z] = ellipsoid(xc,yc,zc,xr,yr,zr) returns the x-, y-, and z-coordinates of an ellipsoid without drawing it. The returned ellipsoid has center coordinates at (xc,yc,zc), semiaxis lengths (xr,yr,zr), and consists of 20-by-20 faces.
The function returns the x-, y-, and z- coordinates as three 21-by-21 matrices.
To draw the ellipsoid using the returned coordinates, use the surf or mesh functions.
[X,Y,Z] = ellipsoid(xc,yc,zc,xr,yr,zr,n) returns the x-, y-, and z-coordinates of an ellipsoid with n-by-n faces. The function returns the x-, y-, and z-coordinates as three (n+1)-by-(n+1) matrices.
ellipsoid(___) plots the ellipsoid without returning the coordinates. Use this syntax with any of the previous input arguments in previous syntaxes.
ellipsoid(ax,___) plots into the axes specified by ax instead of the current axes. Specify the axes as the first input argument.
Create and plot an ellipsoid with a center at (0, –0.5, 0) and semiaxis lengths (6, 3.25, 3.25). Use axis equal to use equal data units along each coordinate direction.
ellipsoid(0,-0.5,0,6,3.25,3.25)
Generate coordinates of an ellipsoid with a center at (0, 0, 0) and semiaxis lengths (1.5, 1.5, 3).
[X,Y,Z] = ellipsoid(0,0,0,1.5,1.5,3);
Create a surface plot of the ellipsoid.
Plot a second ellipsoid with its center translated by (3, 0, 5) from the first ellipsoid. To be able to rotate the second ellipsoid in the next step, return the surface object as s.
s = surf(X+3,Y,Z+5);
Rotate the second ellipsoid by 45 degrees around its
x
-axis. The new coordinates of the translated and rotated ellipsoid are stored in s.Xdata, s.Ydata, and s.Zdata.
rotate(s,direction,45)
Display ellipsoids with center coordinates (0, 0, 0) and semiaxis lengths (2, 1, 1) with different number of faces.
Call the tiledlayout function to create a 2-by-2 tiled chart layout. Call the nexttile function to create the axes. Then, use the ellipsoid function to plot three ellipsoids with different numbers of faces. Plot the ellipsoids in different tiles of the chart by specifying the axes.
ellipsoid(ax1,0,0,0,2,1,1)
title('20-by-20 faces (Default)')
ellipsoid(ax2,0,0,0,2,1,1,50)
title('50-by-50 faces')
xc,yc,zc — Coordinates of ellipsoid center
three comma-separated scalar numbers
Coordinates of ellipsoid center, specified as three comma-separated scalar numbers.
xr,yr,zr — Principal semiaxes along x-, y-, and z-axes
Principal semiaxes along the x-, y-, and z-axes, specified as three comma-separated scalar numbers.
n — Number of faces
Number of faces, specified as a positive scalar integer.
Target axes, specified as an Axes object. If you do not specify the axes, then ellipsoid plots into the current axes.
ellipsoid generates the data using this equation:
\frac{{\left(x-xc\right)}^{2}}{x{r}^{2}}+\frac{{\left(y-yc\right)}^{2}}{y{r}^{2}}+\frac{{\left(z-zc\right)}^{2}}{z{r}^{2}}=1.
ellipsoid(0,0,0,1,1,1) is equivalent to a unit sphere.
cylinder | sphere | surf | mesh | rotate |
About Solutions of Poisson's Equation with Transition Condition in Non-Smooth Domains | EMS Press
Mathematik in den Naturwissenschaften, Leipzig, Germany
Starting from integral representations of solutions of Poisson's equation with transition condition, we study the first and second derivatives of these solutions for all dimensions
d\geq 2
. This involves derivatives of single layer potentials and Newton potentials, which we regularize smoothly. On smooth parts of the boundary of the non-smooth domains under consideration, the convergence of the first derivative of the solution is uniform; this is well known in the literature for regularizations using a sharp cut-off by balls. Close to corners etc.\ we prove convergence in
L^1
with respect to the surface measure. Furthermore we show that the second derivative of the solution is in
L^1
on the bulk.
The interface problem studied in this article is obtained from the stationary Maxwell equations in magnetostatics and was initiated by work on magnetic forces.
Anja Schlömerkemper, About Solutions of Poisson's Equation with Transition Condition in Non-Smooth Domains. Z. Anal. Anwend. 27 (2008), no. 3, pp. 253–281 |
Encode binary samples using turbo algorithm - Simulink - MathWorks France
tail1, tail2
Enable trellis termination valid ports
Encode binary samples using turbo algorithm
The LTE Turbo Encoder block implements the turbo encoder described by LTE standard TS 36.212 [1] using an interface and architecture optimized for HDL code generation and hardware deployment. The encoder is a parallel concatenated convolutional code (PCCC) with two 8-state constituent encoders and an internal interleaver. The first encoder operates on the input data stream, and the second encoder operates on an interleaved version of the input data. The block terminates each encoder output with independent tail bits. The coding rate is 1/3. The encoded output bits for each input bit are returned as a 3-by-1 vector, [S P1 P2]. In this vector, S is the systematic bit, and P1 and P2 are the parity bits from the two encoders.
The block can accept new input data after the previous frame is complete. Apply input frames with at least BlockSize + 16 idle cycles between them. The 16 cycles consists of 12 cycles for pipeline delays in the algorithm, and 4 cycles of tail bits. This latency does not vary with block size. Or, you can use the output signal ctrl.end to determine when the block is ready for new input.
This waveform shows an input frame of 40 samples, with 57 idle cycles between frames. The input and output ctrl buses are expanded to show the control signals. start and end show the frame boundaries, and valid qualifies the data samples. The optional tail1 and tail2 signals indicate the cycles when the tail bits from each encoder are valid.
data — Encoded sample stream
Encoded sample stream, returned as a 3-by-1 column vector. Each encoded sample is represented by one systematic bit and two parity bits.
tail1, tail2 — Indicate trellis termination cycles
Use the optional tail1 and tail2 output ports to indicate the location of the tail bits in the output data stream. These signals are 1 (true) for the cycles that correspond to the tail bits for each encoder.
The block returns the tail bits in the order specified by the LTE standard TS 36.212 [1]. Each encoder returns two cycles of encoded tail bits.
tail1 1 1 0 0
data [E1inK E1outK E1inK+1] [E1outK+1 E1inK+2 E1outK+2] [E2inK E2outK E2inK+1] [E2outK+1 E2inK+2 E2outK+2]
Enable these ports by selecting Enable trellis termination valid ports.
Enable trellis termination valid ports — Enable ports that indicate the tail bit output samples
When you select this parameter, the tail1 and tail2 ports appear on the block. These ports return control signals that indicate the cycles when the output samples are the tail bits for each encoder.
Use the LTE Turbo Encoder block to encode data, and how to compare the hardware-friendly design with the results from LTE Toolbox™. The workflow follows these steps:
For a hardware implementation, storing the interleave indices is not practical. Supporting the 188 LTE block sizes would require 4 Mb of memory. Therefore, the algorithm uses the interleave specification to compute the indexes from the block size. This equation defines the interleave pattern:
\prod \left(i\right)=\left({f}_{1}\cdot i+{f}_{2}\cdot {i}^{2}\right)\mathrm{mod}K
K is the block size, i = 0, 1, …, (K – 1), and f1 and f2 are defined in the LTE standard TS 36.212 [1].
Calculation of the indexes is simplified based on these equations:
\begin{array}{l}\pi \left(i+1\right)=\left\{\begin{array}{l}\pi \left(i\right)+g\left(i\right)\text{ if }\pi \left(i\right)+g\left(i\right)<K\\ \pi \left(i\right)+g\left(i\right)-K\text{ otherwise}\end{array}\\ \pi \left(0\right)=0\end{array}
\begin{array}{l}g\left(i+1\right)=\left\{\begin{array}{l}g\left(i\right)+2{f}_{2}\text{ if }g\left(i\right)+2{f}_{2}<K\\ g\left(i\right)+2{f}_{2}-K\text{ otherwise}\end{array}\\ g\left(0\right)={f}_{1}+{f}_{2}\end{array}
Therefore, the block stores f1 and f2 in memory, and uses those two constants and four adders to calculate the interleave indexes.
When Block size source is set to Property, the block uses two constant coefficients to derive the read addresses for the fixed block size. When Block size source is set to Input port, the algorithm saves the 188 pairs of coefficients in a ROM (< 5 Kb). Then the block reads the matching pair at run time to derive the interleave memory read addresses.
These resource and performance data are the synthesis results from the generated HDL targeted to a Xilinx® Zynq®-7000 ZC706 board. The implementation is for a fixed block size of 6144 samples. The design achieves 312.5 MHz clock frequency.
lteTurboEncode (LTE Toolbox) | lteTurboDecode (LTE Toolbox) | lteDLSCHInfo (LTE Toolbox) |
Cox-Ingersoll-Ross (CIR) mean-reverting square root diffusion model - MATLAB - MathWorks Benelux
Create a cir Object
Cox-Ingersoll-Ross (CIR) mean-reverting square root diffusion model
Creates and displays cir objects, which derive from the sdemrd (SDE with drift rate expressed in mean-reverting form) class.
Use cir objects to simulate sample paths of NVars state variables expressed in mean-reverting drift-rate form. These state variables are driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time CIR stochastic processes with square root diffusions.
You can simulate any vector-valued CIR process of the form:
d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)d{W}_{t}
CIR = cir(Speed,Level,Sigma)
CIR = cir(___,Name,Value)
CIR = cir(Speed,Level,Sigma) creates a default CIR object.
CIR = cir(___,Name,Value) creates a CIR object with additional options specified by one or more Name,Value pair arguments.
The CIR object has the following Properties:
Although cir does not enforce restrictions on the signs of these input arguments, each argument is specified as a positive value.
If StartState is a scalar, cir applies the same initial value to all state variables on all trials.
If StartState is a column vector, cir applies a unique initial value to each state variable on all trials.
If StartState is a matrix, cir applies a unique initial value to each state variable on each trial.
F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}
G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)
simByTransition Simulate CIR sample paths with transition density
The Cox-Ingersoll-Ross (CIR) short rate class derives directly from SDE with mean-reverting drift (SDEMRD):
d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)dW
D
is a diagonal matrix whose elements are the square root of the corresponding element of the state vector.
Create a cir object to represent the model:
d{X}_{t}=0.2\left(0.1-{X}_{t}\right)dt+0.05{X}_{t}^{\frac{1}{2}}dW
When you invoke these parameters with inputs, they behave like functions, giving the impression of dynamic behavior. The parameters accept the observation time t and a state vector Xt, and return an array of appropriate dimension. Even if you originally specified an input as an array, cir treats it as a static function of time and state, by that means guaranteeing that all parameters are accessible by the same interface.
drift | diffusion | sdeddo | simulate | interpolate | simByEuler | simByTransition | nearcorr |
EUDML | Sufficient conditions for starlike functions of order . EuDML | Sufficient conditions for starlike functions of order .
Sufficient conditions for starlike functions of order
\alpha
Ravichandran, V.; Selvaraj, C.; Rajalaksmi, R.
Ravichandran, V., Selvaraj, C., and Rajalaksmi, R.. "Sufficient conditions for starlike functions of order .." JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] 3.5 (2002): Paper No. 81, 6 p., electronic only-Paper No. 81, 6 p., electronic only. <http://eudml.org/doc/123114>.
author = {Ravichandran, V., Selvaraj, C., Rajalaksmi, R.},
keywords = {starlike function of order ; univalent function; starlike function of order },
title = {Sufficient conditions for starlike functions of order .},
AU - Selvaraj, C.
AU - Rajalaksmi, R.
TI - Sufficient conditions for starlike functions of order .
KW - starlike function of order ; univalent function; starlike function of order
starlike function of order
\alpha
, univalent function, starlike function of order
\alpha
Articles by Selvaraj
Articles by Rajalaksmi |
To Francis Darwin 30 [May 1876]1
Hopedene
You cd. make aggregation less intense by using much weaker sol. of C. of Ammonia; or by putting minute splinters of glass on glands.2
If your case of Teazle holds good it is a wonderful discovery. Try whether pure water or weak infusion of raw meat will bring out the protoplasmic masses.3 The closest analogy seems to me that of an independent Amœba or Foraminiferous animal &c which feeds by involving at any point of its gelatinous body particles of organic matter & then rejecting them— A mass of rotting insects would give such particles.— Perhaps this is your view. But I do not understand what you mean by a resinous secretion becoming slimy, or about living insects being caught. I would work at this subject, if I were you, to the point of death.4
If an Amœba-like mass comes out of cells & catches dead particles & digest them, it wd. beat all to fits true digesting plants.
I never saw anything come out of quadrifids of Utricularia & I cd. hardly have failed to see them, as I was on look out for secretion.5 It wd be a grand discovery. Could you chop up or pound excessively fine raw meat, or better
\frac{1}{2}
decayed meat & colour the particles first, & then you cd see them in the protoplasmic masses; for surely you could hardly expect (unless there is a distinct hole) that they shd. be withdrawn within cells of glands.— The case is grand—
Are any orifices or orifice visible in cut-off summit of gland? For heaven sake report progress of your work.—
I see in last G. Chronicle another man denies that Dionæa profits by absorption & digestion, which he does not deny.6 It seems to me a monstrous conclusion— But this subject ought to be investigated Especially effects on Seed-bearing— Teazles good. for this—
Yours affecly— | C. Darwin
No doubt marginal glands of Drosera answer to glands on serratures of other leaves. Probably glands wd. be found on apices of spikes of Dionæa in bud-state.7
The month and year are established by the relationship between this letter and the letters from Francis Darwin, [28 May 1876] and [29 May 1876]. CD evidently wrote ‘Monday’ in error; 30 May 1876 was a Tuesday.
CD’s advice was in response to Francis Darwin’s complaint about aggregated protoplasm forming motionless masses (see letter from Francis Darwin, [29 May 1876]). CD refers to a solution of carbonate of ammonia.
Francis had discovered that protoplasmic filaments protruded from the glandular hairs lining the cups of the common or fuller’s teasel (Dipsacus sylvestris, a synonym of D. fullonum). See letter from Francis Darwin, [28 May 1876] and n. 3.
See letter from Francis Darwin, [28 May 1876] and n. 3. An amoeba was considered to be a small mass of undifferentiated protoplasm. Foraminifera are unicellular marine protozoans; they are typically found near the bottom of the sea (Lipps et al. 2011). In his printed paper, Francis described the filaments as having a ‘gelatinous consistence’ (F. Darwin 1877b, p. 248).
The quadrifids of Utricularia (bladderwort) are four divergent arms in the bladders (see Insectivorous plants, pp. 402–4).
There was a brief note on Dionaea muscipula (Venus fly trap) in Gardeners’ Chronicle, 27 May 1876, p. 689.
See letter from Francis Darwin, [28 May 1876] and n. 4. Drosera is the genus of sundews. |
Continuum mechanics/Functions - Wikiversity
3 One-to-one mapping (injection)
4 Onto mapping (surjection)
5 One-to-one and onto mapping (bijection)
6 Image, pre-image, and inverse functions
7 Identity map
9 Isomorphism and Homeomorphism
10 Continuously differentiable functions
10.1 C-1 Functions
10.2 C0 Functions
10.3 C∞ Functions
11 Sobolev spaces of functions
-- Back to Nonlinear finite elements --
There are certain terms involving relationships between functions that you will often encounter in papers dealing with finite elements and continuum mechanics. We list some of the basic terms that you will see. More details can be found in books on advanced calculus and functional analysis.
{\displaystyle A}
{\displaystyle B}
be two sets. A function is a rule that assigns to each
{\displaystyle a\in A}
{\displaystyle B}
. A function is usually denoted by
{\displaystyle f:A\rightarrow B~~~~(f~{\text{takes}}~A~{\text{to}}~B)~.}
Sometimes, one also writes
{\displaystyle a\mapsto f(a)~~~~(a~{\text{is mapped to the element}}~f(a))~.}
For example, if the function is
{\displaystyle f(x)=x^{2}}
, then we may write
{\displaystyle x\mapsto x^{2}}
Domain and Range[edit | edit source]
{\displaystyle f:A\rightarrow B}
{\displaystyle A}
is called the domain o{\displaystyle f}
Figure 1. Domain and range of a function.
The range o{\displaystyle f}
{\displaystyle R(A)=\{f(x)\in B~|~x\in A\}}
{\displaystyle R(A)\subset B}
One-to-one mapping (injection)[edit | edit source]
{\displaystyle f:A\rightarrow B}
is called one-to-one (or an injection) if no two distinct elements of
{\displaystyle A}
are mapped to the same element of
{\displaystyle B}
Onto mapping (surjection)[edit | edit source]
{\displaystyle f:A\rightarrow B}
is called onto if for every
{\displaystyle b\in B}
{\displaystyle a\in A}
{\displaystyle f(a)=b}
If that case,
{\displaystyle R(A)=B}
One-to-one and onto mapping (bijection)[edit | edit source]
When a mapping is both one-to-one and onto it is called a bijection. For example, if
{\displaystyle f:A\rightarrow B}
{\displaystyle f(x)=x^{2}}
{\displaystyle A=\{x\in \mathbb {R} ^{}~|~x\geq 0\}~;~~B=\{x\in \mathbb {R} ^{}~|~x\geq 0\}}
the map is one-to-one and onto.
{\displaystyle A=\{x\in \mathbb {R} ^{}~|~x\geq 0\}~;~~B=\{x\in \mathbb {R} ^{}\}}
the map is one-to-one but not onto.
{\displaystyle A=\{x\in \mathbb {R} ^{}\}~;~~B=\{x\in \mathbb {R} ^{}\}}
the map is neither one-to-one nor onto.
Image, pre-image, and inverse functions[edit | edit source]
Suppose we have a function
{\displaystyle f:A\rightarrow B}
{\displaystyle D}
{\displaystyle A}
(see Figure~1). Let us define
{\displaystyle f(D):=\{f(d)\in B~|~d\in D\}~.}
{\displaystyle f(D)}
{\displaystyle D}
{\displaystyle C}
{\displaystyle B}
and we define
{\displaystyle f^{-1}(C):=\{a\in A~|~f(a)\in C\}~.}
{\displaystyle f^{-1}(C)}
is called the inverse image or pre-image of
{\displaystyle C}
{\displaystyle f:A\rightarrow B}
is one-to-one and onto, then there is a unique function
{\displaystyle f^{-1}:B\rightarrow A}
{\displaystyle f(f^{-1}(b))=b~{\text{for all}}~b\in B~~~~~~{\text{and}}~~~~f^{-1}(f(a))=a~{\text{for all}}~a\in A~.}
{\displaystyle f^{-1}}
is called an inverse function o{\displaystyle f}
Identity map[edit | edit source]
{\displaystyle f:A\rightarrow A}
{\displaystyle f(x)=x}
{\displaystyle x\in A}
is called the identity map. This map is one-to-one and onto.
A notation that you will commonly see in papers on nonlinear solid mechanics is the composition of two functions. See Figure 2.
Figure 2. Composition of functions.
{\displaystyle f}
{\displaystyle g}
be two functions such that
{\displaystyle f:A\rightarrow B}
{\displaystyle g:B\rightarrow C}
. The composition
{\displaystyle g\circ f:A\rightarrow C}
{\displaystyle g\circ f(a):=g(f(a))}
Let us consider a stretch (
{\displaystyle f}
) followed by a translation (
{\displaystyle g}
). Then we can write
{\displaystyle f:x\mapsto \lambda x}
{\displaystyle g:y\mapsto y+c}
{\displaystyle g\circ f}
{\displaystyle g\circ f:x\mapsto \lambda x+c}
The inverse composition
{\displaystyle f\circ g}
{\displaystyle f\circ g:x\mapsto \lambda (x+c)}
Isomorphism and Homeomorphism[edit | edit source]
You will also come across the terms isomorphism and homeomorphism in the literature on nonlinear solid mechanics.
Isomorphism is a very general concept that appears in several areas of mathematics. The word means, roughly, "equal shapes". It usually refers to one-to-one and onto maps that preserve relations among elements.
A homeomorphism is a continuous transformation between two geometric figures that is continuous in both directions. The map has to be one-to-one to be homeomorphic. It also has to satisfy the requirements on an equivalence relation.
Continuously differentiable functions[edit | edit source]
{\displaystyle f:\Omega \rightarrow \mathbb {R} ^{}}
{\displaystyle k}
-times continuous differentiable or of class
{\displaystyle C^{k}}
if its derivatives of order
{\displaystyle j}
{\displaystyle 0\leq j\leq k}
) exist and are continuous functions.
Figure 3 shows three functions (
{\displaystyle f(x)}
{\displaystyle g(x)}
{\displaystyle h(x)}
) and their derivatives.
Figure 3: Continuity of functions.
C-1 Functions[edit | edit source]
{\displaystyle f(x)}
is called the Heaviside step function (usually written
{\displaystyle H(x)}
) which is defined as
{\displaystyle H(x)={\begin{cases}0&~{\rm {{if}~x<0}}\\1&~{\rm {{if}~x\geq 0}}\end{cases}}}
The derivative of the Heaviside function is the Dirac delta function (written
{\displaystyle \delta (x)}
) which has the defining property that
{\displaystyle \int _{-\infty }^{\infty }F(x)\delta (x-c)~dx=F(c)}
{\displaystyle F(x)}
{\displaystyle c}
. The delta function is singular and discontinuous. Hence, the Heaviside function is not continuously differentiable. Sometimes the Heaviside function is said to belong to the class of
{\displaystyle C^{-1}}
C0 Functions[edit | edit source]
{\displaystyle g(x)}
in Figure 3 (also called a hat function) is continuous but has discontinuous derivatives. In this particular case, the function has the form
{\displaystyle g(x)={\begin{cases}{\cfrac {x}{a}}+1&~{\text{if}}~-a\leq x\leq 0\\1-{\cfrac {x}{a}}&~{\text{if}}~~0\leq x\leq a\end{cases}}}
Such functions that are differentiable only once are called
{\displaystyle C^{0}}
C∞ Functions[edit | edit source]
{\displaystyle h(x)}
in Figure 3 is infinitely differentiable and has continuous derivatives every time it is differentiable. Such functions are called
{\displaystyle C^{\infty }}
functions. Since the function can be differentiated once to give a continuous derivative, it also falls into the category of
{\displaystyle C^{1}}
Sobolev spaces of functions[edit | edit source]
You will find Sobolev spaces being mentioned when you read the finite element literature. A clear understanding of these function spaces needs a knowledge of functional analysis. The book Introduction to Functional Analysis with Applications to Boundary Value Problems and Finite Elements by B. Daya Reddy is a good starting point that is just right for engineers. We will not get into the details here.
Of particular interest in finite element analysis are Sobolev spaces of functions such as
{\displaystyle {\mathcal {H}}^{k}=\{w~|~w\in {\mathcal {L}}_{2},{\frac {\partial w}{\partial x}}\in {\mathcal {L}}_{2},\dots ,{\cfrac {\partial ^{k}w}{\partial x^{k}}}\in {\mathcal {L}}_{2}\}}
{\displaystyle {\mathcal {L}}_{2}=\{w~|~\int _{0}^{1}w^{2}~dx<\infty \}~.}
The function space
{\displaystyle {\mathcal {L}}_{2}}
is the space of square integrable functions.
Of interest to us is an outcome of Sobolev's theorem which says that if a function is of class
{\displaystyle {\mathcal {H}}^{k+1}}
then it is actually a bounded
{\displaystyle C^{k}}
function. If we choose our basis functions from the set of square integrable functions with continuous derivatives, certain singularities are automatically precluded.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Continuum_mechanics/Functions&oldid=1711582" |
Aerial_(album) Knowpia
Aerial (album)
Aerial is the eighth album by English singer-songwriter and musician Kate Bush. It was released as a double album in 2005, twelve years after her 1993 album The Red Shoes.
Kate Bush chronology
(1994) Aerial
(2005) Director's Cut
Kate Bush studio album chronology
Singles from Aerial
Aerial was Bush's first double album, was released after a twelve-year absence from the music industry during which Bush devoted her time to family and the raising of her son, Bertie. The anticipation leading up to the album's release was immense, with press articles devoted to Bush being printed months, even years before.[1] Like Bush's previous album, The Red Shoes, Aerial does not feature a cover photograph of Bush, but rather one that is emblematic of the album's celebration of sky, sea, and birdsong. The cover image, which seems to show a mountain range at sunset reflected on the sea is in fact a waveform of a blackbird song superimposed over a glowing photograph.
Aerial is one of Bush's most critically acclaimed albums.[2] Musically, the album is a multi-layered work, incorporating elements of folk, Renaissance, classical, reggae, flamenco, and rock. As with 1985's Hounds of Love, the album is divided into two thematically distinct collections. The first disc, subtitled A Sea of Honey, features a set of unrelated songs including the hit single "King of the Mountain", a Renaissance-style ode to her son Albert "Bertie" McIntosh performed with period instruments, and a song based on the story of Joan of Arc named "Joanni". In the song "
{\displaystyle \pi }
", Bush sings the number to its 78th decimal place, then from its 101st to its 137th decimal place.[3][4] The piano and vocal piece "A Coral Room", dealing with the loss of Bush's mother and the passage of time, was hailed by critics as "stunning" in its simplicity,[5] "profoundly moving"[6] and as "one of the most beautiful" pieces Bush has ever recorded.[6]
The second disc, subtitled A Sky of Honey, consists of a single piece of music reveling in the experience of outdoor adventures on a single summer day, beginning in the morning and ending twenty-four hours later with the next sunrise.[7] The songs are saturated with the presence of birdsong, and all refer to the sky and sunlight, with the sea also featuring as an important element. Beginning with blackbirds singing in the dawn chorus, a woodpigeon cooing, solo piano, and Bush's son saying, "Mummy, Daddy, the day is full of birds," the piece begins with an early morning awakening to a beautiful day of sun shining "like the light in Italy"; it proceeds through a visit with a painter who is working on a new piece of pavement art ("An Architect's Dream" and "The Painter's Link") and then passes on to a crimson "Sunset". The interlude "Aerial Tal", consists of Bush imitating various samples of birdsong, while "Somewhere in Between" celebrates the ambiguous nature of dusk. "Nocturn", features a pair of lovers bathing in the sea after dark under a star-studded "diamond sky". The song cycle ends with "Aerial" and its euphoric welcome of the following morning's sunrise with the refrain "I need to get up on the roof...in the sun."
In the album's initial release, A Sky of Honey features Rolf Harris playing the didgeridoo and providing vocals on "An Architect's Dream" and "The Painter's Link". Following Harris' 2014 conviction for indecent assault, his vocals were replaced on the 2018 remaster with new recordings by McIntosh.[8][9] Other guest artists include Peter Erskine, Eberhard Weber, Lol Creme and Procol Harum's Gary Brooker. In one of his final projects before his death in 2003, long-time Bush collaborator Michael Kamen arranged the string sections, performed by the London Metropolitan Orchestra.[citation needed]
In the 2014 series of concerts in London, Before the Dawn, Bush performed "King of the Mountain," "Joanni" and the whole Sky of Honey song cycle live for the first time.
On 13 November 2005, Aerial entered the UK Albums Chart at number three, selling more than 90,000 copies in its first week on release. In Canada, the album was certified Platinum (100,000 copies sold). On 10 January 2006, Bush was nominated for two BRIT Awards for Best British Female Solo Artist and Best British Album for Aerial.[21] On 27 January 2006, the album went up against Demon Days by Gorillaz and Coles Corner by Richard Hawley in the pop category of the South Bank Show's Annual Arts Awards, but was beaten by Hawley. UK music magazine Mojo named it their third best album of 2005, behind I Am a Bird Now by Antony and the Johnsons and Funeral by Arcade Fire. Rob Chapman, writing in The Times stated that "...its closing triptych, Somewhere in Between, Nocturn, and Aerial, represents the most joyous and euphoric finale to an album that you will hear all year."[20]
In an article for Stylus Magazine, Marcello Carlin reflected that "Aerial was a triumph, a towering dual masterpiece arriving like a huge galleon into the shallow pool of enforced worthiness and happiness which defined that era’s pop. It sought to give new life to dead souls—whether Elvis or her own mother or even the number Pi—and found that renewed life in young Bertie."[22]
The only single from the album was "King of the Mountain". The song makes references to Elvis Presley and the film Citizen Kane. The track was played for the first time on BBC Radio 2 on 21 September 2005, and was made available for download on 27 September. The B-side (or second track) of the single was a Marvin Gaye cover, "Sexual Healing", recorded in 1994, and was not available on any of her albums until the release of the compilation The Other Sides in 2018. The single entered (and peaked in) the UK singles chart at No. 4, and gave Bush her first top-five hit in twenty years and her third-highest singles chart placing. The song also peaked at No. 6 on the UK download chart.
As of mid-May 2010, Aerial was released for the first time on iTunes. The second disc, A Sky of Honey, now runs as one continual track, and its title was changed to An Endless Sky of Honey, with each track titles merged altogether on the sleeve. In August 2010, the CD version was reissued by Sony Legacy in the United States.[23] The following year Kate Bush re-released Aerial alongside others of her albums on her own label Fish People, where they appeared again in 2018 in remastered versions.
In the 2018 remastered edition, A Sky of Honey was returned to its original nine tracks. The spoken parts of "An Architect's Dream" and "The Painter's Link" originally performed by Rolf Harris were removed in light of the sexual assault allegations against him and replaced by Albert "Bertie" McIntosh, Kate Bush's son.[24]
On 17 May 2015, a sequence from the song "π" was featured on The Simpsons' twenty-sixth-season finale, "Mathlete's Feat".[25]
All tracks are written by Kate Bush.
"King of the Mountain" 4:53
"π" 6:09
"Bertie" 4:18
"Mrs. Bartolozzi" 5:58
"How to Be Invisible" 5:32
"Joanni" 4:56
"A Coral Room" 6:12
Disc two: A Sky of Honey (original release and 2018 remaster)
"Prelude" 1:26
"An Architect's Dream" 4:50
"The Painter's Link" 1:35
"Aerial Tal" 1:01
"Somewhere in Between" 5:00
"Nocturn" 8:34
"Aerial" 7:52
Disc two: An Endless Sky of Honey (2010 release)
"An Endless Sky of Honey" 42:00
Kate Bush – vocals, keyboards (1 –3, 5, 6, 8, 10, 13 –16), piano (4, 7, 9, 12)
Dan McIntosh – electric and acoustic guitar (1, 2, 5, 6, 10, 12, 14 –16)
Del Palmer – bass guitar (1, 5, 6, 14, 16)
Paddy Bush – backing vocals (1)
Steve Sanger – drums (1, 16)
Stuart Elliott – drums (2, 5, 12, 14)
Eberhard Weber – electric upright bass (2, 9)
Lol Creme – backing vocals (2, 15)
Eligio Quinteiro – renaissance guitar (3)
Richard Campbell and Susanna Pell – viol (3)
Bill Thorp – string arrangement (3)
Robin Jeffrey – renaissance percussion (3)
Chris Hall – accordion (5)
Michael Wood – male vocal (7)
Albert McIntosh (Bertie – Kate Bush's son) – "The Sun" (8), "The Painter" (10, 11, in 2018 edition)
Peter Erskine – drums (9, 10, 15)
London Metropolitan Orchestra strings (9, 11)
Michael Kamen – orchestral arranger and conductor
John Giblin – bass guitar (10, 12, 15)
Rolf Harris – "The Painter" (10, 11), didgeridoo (11) (first release only)
Gary Brooker – backing vocals (12, 14), hammond organ (14, 15)
Bosco D'Oliveira – percussion (15, 16)
Del Palmer – recording and mixing engineer
Simon Rhodes – engineer (Abbey Road Studios)
Chris Bolster – assistant engineer (Abbey Road Studios)
Joseph Southall – "Fishermen" painting
Weekly chart performance for Aerial
2005 year-end chart performance for Aerial
Certifications and sales for Aerial
^ "Kate Bush: The Sequel". The Independent. 2 September 2005. Retrieved 3 April 2007.
^ "Aerial". MetaCritic.com. Archived from the original on 16 April 2007. Retrieved 3 April 2007.
^ Singh, Simon (20 December 2005). "Curse of a festive pedant". The Telegraph. Retrieved 17 January 2017.
^ Hilton, Boyd (5 November 2005). "Aerial review". Heat. Retrieved 4 April 2007.
^ a b Thompson, Ben (5 November 2006). "Ben Thompson reviews an album of two-halves". The Sunday Telegraph. Retrieved 3 April 2007.
^ J. Cowley notes A Sky of Honey covers a single day through to next morning.
^ Sinclair, Paul (7 November 2018). "Kate Bush removes Rolf Harris from the 2018 remaster of her album Aerial". SuperDeluxeEdition. Retrieved 25 February 2022.
^ Cashmere, Paul (13 November 2018). "Kate Bush removes Rolf Harris contribution from Aerial reissue". Noise11. Retrieved 25 February 2022.
^ "Reviews for Aerial by Kate Bush". Metacritic. Retrieved 14 August 2012.
^ Jurek, Thom. "Aerial – Kate Bush". AllMusic. Retrieved 14 August 2012.
^ Weingarten, Marc (14 November 2005). "Aerial". Entertainment Weekly. Archived from the original on 2 February 2019. Retrieved 14 August 2012.
^ Petridis, Alexis (4 November 2005). "Kate Bush, Aerial". The Guardian. Retrieved 14 August 2012.
^ Gill, Andy (4 November 2005). "Album: Kate Bush". The Independent. Archived from the original on 6 November 2005. Retrieved 14 August 2012.
^ Irvin, Jim (December 2005). "And is there honey still for tea?". Mojo (145): 96.
^ "Kate Bush: Aerial". NME: 45. 12 November 2005.
^ Leone, Dominique (9 November 2005). "Kate Bush: Aerial". Pitchfork. Retrieved 14 August 2012.
^ Blake, Mark (December 2005). "Mother Superior". Q (233): 142.
^ Walters, Barry (3 November 2005). "Aerial : Kate Bush". Rolling Stone. Archived from the original on 6 November 2007. Retrieved 14 August 2012.
^ a b Chapman, Rob (29 October 2005). "The Big CD". The Times. Retrieved 14 August 2012.
^ "Search results for Kate Bush". The BRIT Awards. Archived from the original on 28 September 2007. Retrieved 15 February 2007.
^ Carlin, Marcello (16 April 2007). "This Sentinent Soul, This Living Breath". Stylus Magazine. Archived from the original on 30 March 2014. Retrieved 6 October 2020.
^ "Aerial: Kate Bush: Music". Retrieved 2 November 2011.
^ "Kate Bush Deletes Rolf Harris From 'Ariel' Re-Issue". Nova News. 14 November 2018. Retrieved 18 December 2021.
^ "Simpsons, Episode 574, clip feat. Kate Bush's "π"". 18 May 2015. Archived from the original on 21 December 2021. Retrieved 19 May 2015.
^ "Australiancharts.com – Kate Bush – Aerial". Hung Medien. Retrieved 12 May 2013.
^ "Austriancharts.at – Kate Bush – Aerial" (in German). Hung Medien. Retrieved 12 May 2013.
^ "Ultratop.be – Kate Bush – Aerial" (in Dutch). Hung Medien. Retrieved 12 May 2013.
^ "Ultratop.be – Kate Bush – Aerial" (in French). Hung Medien. Retrieved 12 May 2013.
^ "Czech Albums – Top 100". ČNS IFPI. Note: On the chart page, select 200610 on the field besides the word "Zobrazit", and then click over the word to retrieve the correct chart data. Retrieved 13 February 2022.
^ "Danishcharts.dk – Kate Bush – Aerial". Hung Medien. Retrieved 24 April 2020.
^ "Dutchcharts.nl – Kate Bush – Aerial" (in Dutch). Hung Medien. Retrieved 12 May 2013.
^ "Hits of the World". Billboard. Vol. 117, no. 48. 26 November 2005. p. 81. ISSN 0006-2510 – via Google Books.
^ "Kate Bush: Aerial" (in Finnish). Musiikkituottajat – IFPI Finland. Retrieved 24 April 2020.
^ "Lescharts.com – Kate Bush – Aerial". Hung Medien. Retrieved 24 April 2020.
^ "Offiziellecharts.de – Kate Bush – Aerial" (in German). GfK Entertainment Charts. Retrieved 24 April 2020.
^ "Top 50 Ελληνικών και Ξένων Άλμπουμ" [Top 50 Greek and Foreign Albums] (in Greek). IFPI Greece. 11 December 2005. Archived from the original on 20 December 2005. Retrieved 24 April 2020.
^ "Irish-charts.com – Discography Kate Bush". Hung Medien. Retrieved 12 May 2013.
^ "Italiancharts.com – Kate Bush – Aerial". Hung Medien. Retrieved 12 May 2013.
^ エアリアル/ケイト・ブッシュ [Aerial / Kate Bush] (in Japanese). Oricon. Archived from the original on 5 March 2014. Retrieved 12 May 2013.
^ "Charts.nz – Kate Bush – Aerial". Hung Medien. Retrieved 30 August 2019.
^ "Norwegiancharts.com – Kate Bush – Aerial". Hung Medien. Retrieved 12 May 2013.
^ "Official Scottish Albums Chart Top 100". Official Charts Company. Retrieved 24 April 2020.
^ "Swedishcharts.com – Kate Bush – Aerial". Hung Medien. Retrieved 12 May 2013.
^ "Swisscharts.com – Kate Bush – Aerial". Hung Medien. Retrieved 24 April 2020.
^ "Kate Bush Chart History (Billboard 200)". Billboard. Retrieved 24 April 2020.
^ "Jaaroverzichten 2005 – Albums" (in Dutch). Ultratop. Retrieved 25 January 2014.
^ "Jaaroverzichten – Album 2005" (in Dutch). Dutch Charts. Retrieved 26 January 2014.
^ "Top Albums annuel (physique + téléchargement + streaming) – 2005" (in French). Syndicat National de l'Édition Phonographique. Archived from the original on 9 August 2018. Retrieved 8 August 2018.
^ "End of Year Album Chart Top 100 – 2005". Official Charts Company. Retrieved 24 April 2020.
^ "Jaaroverzichten – Album 2006" (in Dutch). Dutch Charts. Retrieved 8 August 2018.
^ "Canadian album certifications – Kate Bush – Aerial". Music Canada.
^ a b "Kate Bush" (in Finnish). Musiikkituottajat – IFPI Finland.
^ "French album certifications – Kate Bush – Aerial" (in French). InfoDisc. Retrieved 26 November 2021. Select KATE BUSH and click OK.
^ "Gold-/Platin-Datenbank (Kate Bush; 'Aerial')" (in German). Bundesverband Musikindustrie.
^ "Wyróżnienia – Złote płyty CD - Archiwum - Przyznane w 2006 roku" (in Polish). Polish Society of the Phonographic Industry. Retrieved 18 April 2021.
^ "British album certifications – Kate Bush – Aerial". British Phonographic Industry.
^ Cashmere, Paul (13 January 2007). "EMI Share Price Drops on Restructure Announcement". Undercover.fm. Archived from the original on 16 June 2013. Retrieved 21 March 2011.
'I'm not some weirdo recluse' (The Guardian 28 October 2005)
This Bush's mission finally gets accomplished
Aerial by Kate Bush: Music review (National Post 22 December 2005) |
The graph below represents the temperature of a cup of hot water that is left on a kitchen table.
What does the horizontal line (asymptote) represent?
The line appears to be the lowest temperature that the water can reach.
In a room, the lowest temperature that something can reach is the temperature of the room.
How would the graph be different if the cup of hot water was left outside during winter instead?
If a cup were outside in the cold, the lowest temperature that the liquid could reach would still be the temperature of its surroundings.
The asymptote would be lower, but still parallel to the
x
If the temperature outside was below zero, the asymptote would be below the
x |
In some cases, the Lie group parameter introduced by SymmetryTransformation appears embedded into a subexpression, for example as in
{ⅇ}^{\mathrm{_ε}}
, and only appears through functions of that subexpression. To have these cases returned with
\mathrm{_ε}
instead of - say -
{ⅇ}^{\mathrm{_ε}}
, use the option redefinegroupparameter.
\mathrm{with}\left(\mathrm{PDEtools},\mathrm{SymmetryTransformation},\mathrm{ChangeSymmetry},\mathrm{InfinitesimalGenerator}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{SymmetryTransformation}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ChangeSymmetry}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{InfinitesimalGenerator}}]
u\left(x,t\right)
, and consider the list of infinitesimals of a symmetry group
S≔[\mathrm{_ξ}[x]=x,\mathrm{_ξ}[t]=1,\mathrm{_η}[u]=u]
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_η}}}_{\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{u}]
S
[x,1,u]
G≔\mathrm{InfinitesimalGenerator}\left(S,u\left(x,t\right)\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\right)
\mathrm{PDESYS}
is invariant under the symmetry transformation generated by
G
G\left(\mathrm{PDESYS}\right)=0
, where, in this formula,
G
\mathrm{PDESYS}
The actual form of this finite, one-parameter, symmetry transformation relating the original variables
{t,x,u\left(x,t\right)}
to new variables,
{r,s,v\left(r,s\right)}
, that leaves invariant any PDE system admitting the symmetry represented by
G
\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right),v\left(r,s\right)\right)
{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{_ε}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)}
\mathrm{_ε}
is a (Lie group) transformation parameter. To express this transformation using jetnotation use
\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right),v\left(r,s\right),\mathrm{jetnotation}\right)
{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{_ε}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}}
\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right),v\left(r,s\right),\mathrm{jetnotation}=\mathrm{jetnumbers}\right)
{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{_ε}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}[]\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}[]}
That this transformation leaves invariant any PDE system invariant under
G
above is visible in the fact that it also leaves invariant the infinitesimals
S
\mathrm{TR},\mathrm{NewVars}≔\mathrm{solve}\left(,{t,x,u\left(x,t\right)}\right),\mathrm{map}\left(\mathrm{lhs},\right)
\textcolor[rgb]{0,0,1}{\mathrm{TR}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{NewVars}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{_ε}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{r}}{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\right)}{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\right)}
\mathrm{ChangeSymmetry}\left(\mathrm{TR},S,u\left(x,t\right),\mathrm{NewVars}\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\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{_ξ}}}_{\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}{\mathrm{_η}}}_{\textcolor[rgb]{0,0,1}{v}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}]
S
(but written in terms of
v\left(r,s\right)
u\left(x,t\right)
). So to this list of infinitesimals corresponds, written in terms of
v\left(r,s\right)
\mathrm{InfinitesimalGenerator}\left(,v\left(r,s\right)\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{r}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{s}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{v}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\right)
which is also equal to
G
, only written in terms of
v\left(r,s\right)
If the new variables,
v\left(r,s\right)
, are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced
\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right)\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_t1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{_ε}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_u1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_t1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t2}}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)}
An example where the Lie group parameter
\mathrm{_ε}
appears only through the subexpression
{ⅇ}^{\mathrm{_ε}}
\mathrm{SymmetryTransformation}\left([0,0,z,0,0],[u\left(x,y,z,t\right)]\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_t1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t3}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ε}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t4}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_u1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_t1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t4}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{u}\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}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)}
\mathrm{SymmetryTransformation}\left([0,0,z,0,0],[u\left(x,y,z,t\right)],'\mathrm{redefinegroupparameter}'\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_t1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t3}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{_ε}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t4}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_u1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_t1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_t4}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{u}\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}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)} |
To W. D. Fox [24 April 1845]1
It is some time since we have had any communication. I write now chiefly to say, that I heard some little time ago from Shrewsbury, in which they said they had wished to have asked you to have come to the Shrewsbury Agricult. Meeting,2 but some invited & more self-offerers have filled the house, more in my opinion than ought to have been allowed. I shall keep out of the way. Mr & Mrs. Wilmot of Nott:3 will be there & Mr & Mrs. Miss Gifford that was G. Holland & E. Holland.4 By the way, was it not an odd & friendly thing, Mrs. Darwin & young Mr D. of Elston5 called on my Brother a few weeks ago; & it seems the young man, whose appearance my Brother liked, has called several times formerly at Grt. Marlborough St. They would not let my Brother return the call.— I have forgotten to add that they desire me to say that they shall be particularly glad to see you at Shrewsbury, if you are inclined to go there any other time either before or after July. My Father has been pretty well lately; but yesterday we heard that his leg has suddenly inflamed & was very painful I hope it will not last; I intend going there for a week very shortly.6
We have had Ellen Tollet staying with us, & heard indirectly much of you: the Tolletts & you seem to have many acquaintances in common.
Our children are very well, notwithstanding this most cold-catching weather: poor Emma is as bad as she always is, when she is, as she is. (this last sentence is quite Shakespearian)7
As for myself, my most important news is that I have agreed with Murray for a second Edition of my Journal in the Colonial Library in three numbers; & thanks to the Geological fates, I have written my S. American volume the first time over.8
The only other piece of news about myself is, that I am turned into a Lincolnshire squire! my Father having invested for me in a Farm of 324 acres of good land near Alford.9
Have you read that strange unphilosophical, but capitally-written book, the Vestiges,10 it has made more talk than any work of late, & has been by some attributed to me.—at which I ought to be much flattered & unflattered.
Ever yours My dear Fox.— | C. Darwin
The date is based on CD’s completion of the first draft of South America (see n. 8, below).
In 1845 the annual meeting of the Royal Agricultural Society took place in Shrewsbury. The principal day of the show was 17 July.
Possibly Edward Woollett Wilmot of Worksop Manor, Nottinghamshire, a governor of the Society, and his wife, Emma Elizabeth née Darwin.
George Henry Holland, his wife Charlotte Dorothy née Gifford, and Edward Holland.
Elizabeth de St Croix Darwin and her son Robert Alvey Darwin.
According to his ‘Journal’ (Correspondence vol. 3, Appendix II), CD left for Shrewsbury on 29 April, returning on 10 May.
Emma Darwin was pregnant with her fifth child.
CD completed the first draft of South America on 24 April (‘Journal’; Correspondence vol. 3, Appendix II). Regarding the agreement with John Murray, see letter to John Murray, 17 [April 1845].
The Beesby farm. CD’s Investment Book (Down House MS) shows that he paid £213 13s. 8d. interest to his father on 10 August 1846. The total amount advanced by his father was £13,592 0s. 7
\frac{1}{2}
d. See also Keith 1955, p. 222.
[Chambers] 1844.
Keith, Arthur. 1955. Darwin revalued. London: Watts. |
Computational fluid dynamics - Wikiversity
Part of the Wikiversity Division of Fluid Mechanics, Division of Applied Mechanics, School of Engineering and the Engineering and Technology Portal
CFD Simulation of the fairing aerodynamics of an éco-mobile (velomobile).
1.1 Computational Fluid Dynamics(CFD) Applications
2.2 CFD Lessons
2.2.1 Lesson 1
All those CFD enthusiasts out there.. Lets start a new CFD course which will meet every requirement that has to be met...
Computational Fluid Dynamics(CFD) Applications[edit | edit source]
Automotive Drag for Fuel Efficient Design
Fire Modeling and Prediction
Atmospheric Modeling (Weather/Climate Change Prediction)
Wake Vortex Study at Wallops Island The air flow from the wing of this agricultural plane is made visible by a technique that uses colored smoke rising from the ground. The swirl at the wingtip traces the aircraft's wake vortex, which exerts a powerful influence on the flow field behind the plane. Because of wake vortex, the Federal Aviation Administration (FAA) requires aircraft to maintain set distances behind each other when they land. A joint NASA-FAA program aimed at boosting airport capacity, however, is aimed at determining conditions under which planes may fly closer together. NASA researchers are studying wake vortex with a variety of tools, from supercomputers to wind tunnels to actual flight tests in research aircraft. Their goal is to fully understand the phenomenon, then use that knowledge to create an automated system that could predict changing wake vortex conditions at airports. Pilots already know, for example, that they have to worry less about wake vortex in rough weather because windy conditions cause them to dissipate more rapidly.
CFD Lessons[edit | edit source]
Lesson 1[edit | edit source]
EXAMPLE For a source contaminant concentration
{\displaystyle \ C_{0}}
entering a flow of velocity
{\displaystyle {\vec {U}}}
{\displaystyle \ x}
upstream from a point, the downstream concentration
{\displaystyle \ C}
at that point is determined by the ratio...
{\displaystyle {\frac {C}{C_{0}}}=e^{\frac {x*{\vec {U}}}{D}}}
{\displaystyle \ D}
is the local dispersion coefficient determined by
{\displaystyle \ D=0.067*depth*V_{f}}
and where friction velocity is
{\displaystyle V_{f}={\sqrt {g*depth*ChannelSlope}}}
Wikipedia article:Computational Fluid Dynamics
Wikipedia article:xxx
cfd-online.com CFD Wiki
US/Cal Govt. Water Modeling Tools
MIT OCW - Transport Processes in the Environment course
NPTEL, Chemical Engineering, Computational Fluid Dynamics
NPTEL, Mechanical Engineering, Computational Fluid Dynamics
OpenFVM
Retrieved from "https://en.wikiversity.org/w/index.php?title=Computational_fluid_dynamics&oldid=1988690" |
EUDML | On Lyapunov inequality in stability theory for Hill's equation on time scales. EuDML | On Lyapunov inequality in stability theory for Hill's equation on time scales.
On Lyapunov inequality in stability theory for Hill's equation on time scales.
Merdivenci Atici, F.; Guseinov, G.Sh.; Kaymakçalan, B.
Merdivenci Atici, F., Guseinov, G.Sh., and Kaymakçalan, B.. "On Lyapunov inequality in stability theory for Hill's equation on time scales.." Journal of Inequalities and Applications [electronic only] 5.6 (2000): 603-620. <http://eudml.org/doc/121338>.
@article{MerdivenciAtici2000,
author = {Merdivenci Atici, F., Guseinov, G.Sh., Kaymakçalan, B.},
keywords = {Lyapunov inequality; Hill's equation; time scale; bounded solution; unbounded solution; Lyapunov stability; linear -differential equation; linear -differential equation},
title = {On Lyapunov inequality in stability theory for Hill's equation on time scales.},
AU - Merdivenci Atici, F.
AU - Guseinov, G.Sh.
AU - Kaymakçalan, B.
TI - On Lyapunov inequality in stability theory for Hill's equation on time scales.
KW - Lyapunov inequality; Hill's equation; time scale; bounded solution; unbounded solution; Lyapunov stability; linear -differential equation; linear -differential equation
Lyapunov inequality, Hill's equation, time scale, bounded solution, unbounded solution, Lyapunov stability, linear
\text{Δ}
-differential equation, linear
{\Delta }
-differential equation
Articles by Merdivenci Atici
Articles by Guseinov
Articles by Kaymakçalan |
EUDML | Splitting of Gysin extensions. EuDML | Splitting of Gysin extensions.
Splitting of Gysin extensions.
Berrick, A.J.; Davydov, A.A.
Berrick, A.J., and Davydov, A.A.. "Splitting of Gysin extensions.." Algebraic & Geometric Topology 1 (2001): 743-762. <http://eudml.org/doc/121833>.
@article{Berrick2001,
author = {Berrick, A.J., Davydov, A.A.},
keywords = {Gysin sequence; Hochschild homology; differential graded algebra; formal space; -structure; Massey triple product; -structure},
title = {Splitting of Gysin extensions.},
AU - Berrick, A.J.
AU - Davydov, A.A.
TI - Splitting of Gysin extensions.
KW - Gysin sequence; Hochschild homology; differential graded algebra; formal space; -structure; Massey triple product; -structure
Gysin sequence, Hochschild homology, differential graded algebra, formal space,
{A}_{\infty }
-structure, Massey triple product,
{A}_{\infty }
Spectral sequences and homology of fiber spaces
Articles by Berrick
Articles by Davydov |
Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Lie Symmetry Method : liesymm : &^
`&^`(a, b, c)
It computes the wedge product of differential forms relative to the coordinates defined by setup().
All 1-forms are generated by applying d() to the coordinates.
All wedge products are automatically simplified to a wedge product of n 1-forms by extracting coefficients of wedge degree 0.
All results of a wedge product are reported using an address ordering of the 1-forms to facilitate simplifications. Thus
d\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(x\right)
may simplify to
-d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right)
and if so will do so consistently within a given session.
The ordering used for simplifications of the products of 1-forms is available as
{\mathrm{wedgeset}\left(1\right)}
\mathrm{with}\left(\mathrm{liesymm}\right):
\mathrm{setup}\left(x,y,z\right)
[\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}{z}]
d\left(t\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(x\right)
\textcolor[rgb]{0,0,1}{0}
d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\left(\left(5d\left(x\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(3d\left(z\right)\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{&^}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)
\mathrm{`&^`}\left(ad\left(x\right),bd\left(y\right),cd\left(z\right)\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{&^}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right) |
Helmholtz–Kohlrausch effect - Wikipedia
The Helmholtz–Kohlrausch effect (after Hermann von Helmholtz and V. A. Kohlrausch[1]) is a perceptual phenomenon wherein the intense saturation of spectral hue is perceived as part of the color's luminance. This brightness increase by saturation, which grows stronger as saturation increases, might better be called chromatic luminance, since "white" or achromatic luminance is the standard of comparison. It appears in both self-luminous and surface colors, although it is most pronounced in spectral lights.
Each color on top has approximately the same luminance level and yet they do not appear equally bright or dark. The yellow (second from the left) appears to be much darker than the magenta (right-most). However, when the top image is converted to grayscale, we have the image on the bottom--a single shade of gray.
3 Helmholtz color coordinates
4 Effects on industry
Even when they have the same luminance, colored lights seem brighter to human observers than white light does. The way humans perceive the brightness of the lights is different for everyone. When the colors are more saturated, our eyes interpret it as the color's luminance and chroma. This makes us believe that the colors are actually brighter. An exception to this is when the human observer is red-green colorblind, they cannot distinguish the differences between the lightness of the colors. Certain colors do not have significant effect, however, any hue of colored lights still seem brighter than white light that has the same luminance. Two colors that do not have as great of an Helmholtz–Kohlrausch effect as the others are green and yellow.[2]
The Helmholtz–Kohlrausch effect is affected by the viewing environment. This includes the surroundings of the object and the lighting that the object is being viewed under. The Helmholtz–Kohlrausch effect works best in darker environments where there are not any other outside factors influencing the colors. For example, this is why theaters are all dark environments.[2]
An example of this lightness factor would be if there were different colors on a grey background that all are of the same lightness. Obviously the colors look different because they are different colors not just grey, but if the image were converted all to grey scale, all of the colors would match the grey background because they all have the same lightness.[2]
Brightness is affected most by what is surrounding the object. In other words, the object can look lighter or darker depending on what is around it. In addition, the brightness can also appear different depending on the color of the object. For example, an object that is more saturated will look brighter than the same object that is less saturated even when they have the same luminance.[3]
The difference between brightness and lightness is that the brightness is the intensity of the object independent of the light source. Lightness is the brightness of the object in respect to the light reflecting on it. This is important because the Helmholtz–Kohlrausch effect is a measure of the ratio between the two.[3]
Helmholtz color coordinates[edit]
Similar to the Munsell color system, Helmholtz designed a coordinate system. He used the principals of wavelength and purity (chroma) of the color for each hue to describe the location of when high saturation indicates a small amount of white.[4]
The percentage of purity for each wavelength can be determined by the equation below:[4]
{\displaystyle \%P=100\cdot (S-N)/(DW-N),}
where %P is the percent of purity, S is the point being assessed, N is the position of the white point, and DW the dominant wavelength.[4]
Effects on industry[edit]
It is essential for lighting users to be aware of the Helmholtz–Kohlrausch effect when working in theaters or in other venues where lighting is often used. In order to get the greatest effect to illuminate their stage or theater, the lighting users need to understand that color has an effect on brightness. For example, one color may appear brighter than another but really they have the same brightness. On stage, lighting users have the ability to make a white light appear much brighter by adding a color gel. This occurs even though gels can only absorb some of the light.[2] When lighting a stage, the lighting users tend to choose reds, pinks, and blues because they are highly saturated colors and are really very dim. However, we perceive them as being brighter than the other colors because they are most affected by the Helmholtz–Kohlrausch effect. We perceive that the color white does not look any brighter to us than individual colors. LED lights are a good example of this.
The Helmholtz–Kohlrausch effect influences the use of LED lights in different technological practices. Aviation is one field that relies upon the results of the Helmholtz–Kohlrausch effect. A comparison of runway LED lamps and filtered and unfiltered incandescent lights all at the same luminance shows that in order to accomplish the same brightness, the white reference incandescent lamp needs to have twice the luminance of the red LED lamp, therefore suggesting that the LED lights do appear to have a greater brightness than the traditional incandescent lights. One condition that affects this theory is the presence of fog.[4]
Another field that uses this is the automotive industry. LEDs in the dashboard and instrument lighting are designed for use in mesopic luminance. In studies, it has been found that red LEDs appear brighter than green LEDs, which means that a driver would be able to see red light more intense thus more alerting than green lights when driving at night.[4]
^ Kohlrausch, V. A. (1935). "Zur photometrie farbiger lichtern". Das Licht. 6: 259–279.
^ a b c d Wood, Mike (2012). "Lightness – The Helmholtz-Kohlrausch effect" (PDF). Out of the Wood. Retrieved 11 November 2015.
^ a b Corney, D; Haynes, JD; Rees, G; Lotto, RB (2009). "The Brightness of Colour". PLOS ONE. 4 (3): e5091. Bibcode:2009PLoSO...4.5091C. doi:10.1371/journal.pone.0005091. PMC 2659800. PMID 19333398.
^ a b c d e Donofrio, Robert L. (2011). "Review Paper: The Helmholtz-Kohlrausch Effect". Journal of the Society for Information Display. 19 (10): 658. doi:10.1889/JSID19.10.658.
Yoshinobu, Nayatani (February 1998). "A colorimetric Explanation of the Helmholtz-Kohlrausch Effect". Color Research & Application. 23 (6): 374–378. doi:10.1002/(SICI)1520-6378(199812)23:6<374::AID-COL5>3.0.CO;2-W.
Yoshinobu, Nayatani. (June 1997). "Simple Estimation Methods for the Helmholtz-Kohlrausch Effect". Color Research & Application. 22 (6): 385–401. doi:10.1002/(SICI)1520-6378(199712)22:6<385::AID-COL6>3.0.CO;2-R.
Quantification of the Helmholtz-Kohlrausch effect for CRT color monitors
LED Projection Enters the Mainstream
Retrieved from "https://en.wikipedia.org/w/index.php?title=Helmholtz–Kohlrausch_effect&oldid=1082534809"
Color appearance phenomena |
Almost_everywhere Knowpia
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of almost surely in probability theory.
A simple example measure assigns to a subregion of the rectangle the fraction of the geometrical area it occupies. Then, the rectangle's boundary has measure 0, while its interior has measure 1. Almost every point of the rectangle is an interior point, yet the interior has a nonempty complement.
More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero,[1][2] or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated.
The term almost everywhere is abbreviated a.e.;[3] in older literature p.p. is used, to stand for the equivalent French language phrase presque partout.[4]
A set with full measure is one whose complement is of measure zero. In probability theory, the terms almost surely, almost certain and almost always refer to events with probability 1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space.
Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for almost all elements (though the term almost all can also have other meanings).
{\displaystyle (X,\Sigma ,\mu )}
is a measure space, a property
{\displaystyle P}
is said to hold almost everywhere in
{\displaystyle X}
if there exists a set
{\displaystyle N\in \Sigma }
{\displaystyle \mu (N)=0}
{\displaystyle x\in X\setminus N}
have the property
{\displaystyle P}
.[5] Another common way of expressing the same thing is to say that "almost every point satisfies
{\displaystyle P\,}
", or that "for almost every
{\displaystyle x}
{\displaystyle P(x)}
holds".
It is not required that the set
{\displaystyle \{x\in X:\neg P(x)\}}
has measure 0; it may not belong to
{\displaystyle \Sigma }
. By the above definition, it is sufficient that
{\displaystyle \{x\in X:\neg P(x)\}}
be contained in some set
{\displaystyle N}
that is measurable and has measure 0.
If property
{\displaystyle P}
holds almost everywhere and implies property
{\displaystyle Q}
, then property
{\displaystyle Q}
holds almost everywhere. This follows from the monotonicity of measures.
{\displaystyle (P_{n})}
is a finite or a countable sequence of properties, each of which holds almost everywhere, then their conjunction
{\displaystyle \forall nP_{n}}
holds almost everywhere. This follows from the countable sub-additivity of measures.
By contrast, if
{\displaystyle (P_{x})_{x\in \mathbf {R} }}
is an uncountable family of properties, each of which holds almost everywhere, then their conjunction
{\displaystyle \forall xP_{x}}
does not necessarily hold almost everywhere. For example, if
{\displaystyle \mu }
is Lebesgue measure on
{\displaystyle X=\mathbf {R} }
{\displaystyle P_{x}}
is the property of not being equal to
{\displaystyle x}
{\displaystyle P_{x}(y)}
is true if and only if
{\displaystyle y\neq x}
), then each
{\displaystyle P_{x}}
holds almost everywhere, but the conjunction
{\displaystyle \forall xP_{x}}
does not hold anywhere.
As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction.[citation needed] This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point".
If f : R → R is a Lebesgue integrable function and
{\displaystyle f(x)\geq 0}
almost everywhere, then
{\displaystyle \int _{a}^{b}f(x)\,dx\geq 0}
{\displaystyle a<b}
{\displaystyle f(x)=0}
If f : [a, b] → R is a monotonic function, then f is differentiable almost everywhere.
If f : R → R is Lebesgue measurable and
{\displaystyle \int _{a}^{b}|f(x)|\,dx<\infty }
{\displaystyle a<b}
, then there exists a set E (depending on f) such that, if x is in E, the Lebesgue mean
{\displaystyle {\frac {1}{2\varepsilon }}\int _{x-\varepsilon }^{x+\varepsilon }f(t)\,dt}
converges to f(x) as
{\displaystyle \epsilon }
decreases to zero. The set E is called the Lebesgue set of f. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of f converges to f almost everywhere.
A bounded function f : [a, b] → R is Riemann integrable if and only if it is continuous almost everywhere.
As a curiosity, the decimal expansion of almost every real number in the interval [0, 1] contains the complete text of Shakespeare's plays, encoded in ASCII; similarly for every other finite digit sequence, see Normal number.
Definition using ultrafiltersEdit
Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection F of subsets of X such that:
If U ∈ F and U ⊆ V then V ∈ F
The intersection of any two sets in F is in F
The empty set is not in F
A property P of points in X holds almost everywhere, relative to an ultrafilter F, if the set of points for which P holds is in F.
For example, one construction of the hyperreal number system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.
The definition of almost everywhere in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter.
Dirichlet's function, a function that is equal to 0 almost everywhere.
^ Weisstein, Eric W. "Almost Everywhere". mathworld.wolfram.com. Retrieved 2019-11-19.
^ Halmos, Paul R. (1974). Measure theory. New York: Springer-Verlag. ISBN 0-387-90088-8.
^ "Definition of almost everywhere | Dictionary.com". www.dictionary.com. Retrieved 2019-11-19.
^ Ursell, H. D. (1932-01-01). "On the Convergence Almost Everywhere of Rademacher's Series and of the Bochnerfejér Sums of a Function almost Periodic in the Sense of Stepanoff". Proceedings of the London Mathematical Society. s2-33 (1): 457–466. doi:10.1112/plms/s2-33.1.457. ISSN 0024-6115.
^ "Properties That Hold Almost Everywhere - Mathonline". mathonline.wikidot.com. Retrieved 2019-11-19.
Billingsley, Patrick (1995). Probability and measure (3rd ed.). New York: John Wiley & Sons. ISBN 0-471-00710-2. |
Administratum debris - The RuneScape Wiki
The remnants of housing for imperial officers.
The administratum debris are excavation hotspots at Kharid-et - Barracks excavation site, found in the Kharid-et Dig Site, that players can excavate after reaching level 25 Archaeology.
The hotspots initially appear as ancient gravel, requiring uncovering to become usable. Uncovering the hotspots yields a one-time reward of 65 Archaeology experience.
3.4 Journal pages
'Solem in Umbra' painting 664.1 3 N/A • Art Critic Jacques (Imperial Impressionism)
• Soran, Emissary of Zaros (Zarosian I)
• Velucia (Museum - Zarosian I) N/A
Legatus Maximus figurine 664.1 4 N/A • Chief Tess (Showy Fings)
• Wise Old Man (Magic Man)
Tyrian purple 1 3/10 5,105 150
Legatus Maximus figurine (damaged) 1 1/2 Not sold Not alchemisable
'Solem in Umbra' painting (damaged) 1 1/2 Not sold Not alchemisable
Journal pages[edit | edit source]
Custodian's log page 1 1 Uncommon Not sold Not alchemisable
{\displaystyle {\frac {L+E}{250{,}000}}}
{\displaystyle L}
{\displaystyle E}
{\displaystyle {\frac {1}{125{,}000}}}
{\displaystyle {\frac {1}{1{,}042}}}
template = Archaeology Hotspot Calculator Lite/Results form = Administratum_debrisArchForm result = ArchResult param=name|Hotspot Name|Administratum debris|combobox|Acropolis debris,Administratum debris,Aetherium forge,Amphitheatre debris,Ancient magick munitions,Animal trophies,Armarium debris,Aughra remains,Autopsy table,Bandos sanctum debris,Bibliotheke debris,Big High War God shrine,Byzroth remains,Carcerem debris,Castra debris,Ceramics studio debris,Chthonian trophies,Crucible stands debris,Culinarum debris,Cultist footlocker,Destroyed golem,Dis dungeon debris,Dis overspill,Dominion Games podium,Dragonkin coffin,Dragonkin reliquary,Experiment workbench,Flight research debris,Gladitorial goblin remains,Goblin dorm debris,Goblin trainee remains,Gravitron research debris,Hellfire forge,Howls workshop debris,Icyene weapon rack,Ikovian memorial,Infernal art,Keshik ger,Keshik weapon rack,Kharid-et chapel debris,Kyzaj champions boudoir,Legionary remains,Lodge art storage,Lodge bar storage,Makeshift pie oven,Moksha device,Monoceros remains,Oikos fishing hut remnants,Oikos studio debris,Optimatoi remains,Orcus altar,Pontifex remains,Praesidio remains,Praetorian remains,Prodromoi remains,Sacrificial altar,Saurthen debris,Stadio debris,Stockpiled art,Tailory debris,Tsutsaroth remains,Venator remains,Varanusaur remains,War table debris,Warforge scrap pile,Warforge weapon rack,Weapons research debris,Xolo mine,Xolo remains,Yubiusk animal pen param=method|Method|High intensity|buttonselect|High intensity,Medium intensity,AFK param=mattock|Mattock|Necronium|select|Bronze,Iron,Steel,Mithril,Adamant,Rune,Orikalkum,Dragon,Necronium,Crystal,Bane,Imcando,Elder Rune,Time and Space,Guildmaster Tony
Retrieved from ‘https://runescape.wiki/w/Administratum_debris?oldid=35700442’ |
Uniform convergence of the Lie–Dyson expansion with respect to the Planck constant | EMS Press
Uniform convergence of the Lie–Dyson expansion with respect to the Planck constant
Mirko Degli Espositi
We prove that the Lie-Dyson expansion for the Heisenberg observables has a nonzero convergence radius in the variable
\ep t
which does not depend on the Planck constant
\hbar
. Here the quantum evolution
U_{\hbar,\ep}(t)
is generated by the \Sc\ operator defined by the maximal action in
L^2(\R^n)
-\hbar^2\Delta+\Q+\ep V
\Q
is a positive definite quadratic form on
\R^n
; the observables and
V
belong to a suitable class of pseudodifferential operators with analytic symbols. It is furthermore proved that, up to an error of order
\ep
, the time required for an exchange of energy between the unperturbed oscillator modes is exponentially long time independently of
\hbar
Sandro Graffi, Dario Bambusi, Mirko Degli Espositi, Uniform convergence of the Lie–Dyson expansion with respect to the Planck constant. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 2, pp. 153–162 |
Index (group theory) - zxc.wiki
Index (group theory)
In the mathematical branch of group theory , the index of a subgroup is a measure of the relative size to the entire group .
Let it be a group and a subgroup. Then the set of left secondary classes and the set of right secondary classes are equal. Their thickness is the index of in and is sometimes referred to as or .
{\ displaystyle G}
{\ displaystyle U}
{\ displaystyle G / U}
{\ displaystyle U \ backslash G}
{\ displaystyle U}
{\ displaystyle G}
{\ displaystyle (G \ colon U)}
{\ displaystyle [G \ colon U]}
{\ displaystyle | G \ colon U |}
It applies . (Here denotes the order of .)
{\ displaystyle (G \ colon 1) = | G |}
{\ displaystyle | G |}
{\ displaystyle G}
The index is multiplicative, i. H. is a subset of , and a subset of , the following applies
{\ displaystyle U}
{\ displaystyle G}
{\ displaystyle V}
{\ displaystyle U}
{\ displaystyle (G \ colon V) = (G \ colon U) \ cdot (U \ colon V).}
The special case is often referred to as Lagrange's theorem (after J.-L. Lagrange ):
{\ displaystyle V = 1}
The following applies to a group and a subgroup :
{\ displaystyle G}
{\ displaystyle U}
{\ displaystyle | G | = (G \ colon U) \ cdot | U |.}
In the case of finite groups, the index of a subgroup can be written as
{\ displaystyle (G \ colon U) = {\ frac {| G |} {| U |}}}
If a normal divisor , then the index of in is just the order of the factor group , that is
{\ displaystyle N \ vartriangleleft G}
{\ displaystyle N}
{\ displaystyle G}
{\ displaystyle G / N}
{\ displaystyle (G \ colon N) = \ left | G / N \ right |}
A subgroup of index 2 is a normal divisor, because of the two (left) secondary classes, one is the subgroup itself and the other is its complement.
More general: If is a subgroup of and its index, which is also the smallest divisor of the order , then in is a normal divisor .
{\ displaystyle U}
{\ displaystyle G}
{\ displaystyle p> 1}
{\ displaystyle | G |}
{\ displaystyle U}
{\ displaystyle G}
In the context of topological groups subgroups are of finite index a special role:
A subset of finite index is open if and only if it is closed . (Open subgroups are always closed.)
Every open subgroup of a compact group has a finite index.
The index of the centralizer of a group element corresponds to the power of its conjugation class.
In Galois theory , the Galois correspondence gives a connection between the relative indices of subgroups of the Galois group and the relative degrees of body extensions.
Index in group theory:
Thomas W. Hungerford: Algebra . 5th edition. Springer, New York 1989, ISBN 0-387-90518-9 , pp. 38 ff .
In topological groups:
Lev Pontryagin : Topological Groups . Teubner, Leipzig 1957 (Russian: Nepreryvnye gruppy . Translated by Viktor Ziegler).
↑ Hungerford (1989), p. 89
↑ Hungerford (1989), p. 247
This page is based on the copyrighted Wikipedia article "Index_%28Gruppentheorie%29" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
Rotation matrix for rotations around y-axis - MATLAB roty - MathWorks España
Rotation matrix for rotations around y-axis
R = roty(ang)
R = roty(ang) creates a 3-by-3 matrix used to rotate a 3-by-1 vector or 3-by-N matrix of vectors around the y-axis by ang degrees. When acting on a matrix, each column of the matrix represents a different vector. For the rotation matrix R and vector v, the rotated vector is given by R*v.
Construct the matrix for a rotation of a vector around the y-axis by 45°. Then let the matrix operate on a vector.
R = roty(45)
Under a rotation around the y-axis, the y-component of a vector is invariant.
Rotation angle specified as a real-valued scalar. The rotation angle is positive if the rotation is in the counter-clockwise direction when viewed by an observer looking along the y-axis towards the origin. Angle units are in degrees.
{R}_{y}\left(\beta \right)=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]
for a rotation angle β.
{v}^{\prime }=Av={R}_{z}\left(\gamma \right){R}_{y}\left(\beta \right){R}_{x}\left(\alpha \right)v
{R}_{x}\left(\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]
{R}_{y}\left(\beta \right)=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]
{R}_{z}\left(\gamma \right)=\left[\begin{array}{ccc}\mathrm{cos}\gamma & -\mathrm{sin}\gamma & 0\\ \mathrm{sin}\gamma & \mathrm{cos}\gamma & 0\\ 0& 0& 1\end{array}\right]
{A}^{-1}A=1
{R}_{x}^{-1}\left(\alpha \right)={R}_{x}\left(-\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & \mathrm{sin}\alpha \\ 0& -\mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]={R}_{x}^{\prime }\left(\alpha \right)
i,j,k
{i}^{\prime },j{,}^{\prime }{k}^{\prime }
\begin{array}{ll}{i}^{\prime }\hfill & =Ai\hfill \\ {j}^{\prime }\hfill & =Aj\hfill \\ {k}^{\prime }\hfill & =Ak\hfill \end{array}
\left[\begin{array}{c}{i}^{\prime }\\ {j}^{\prime }\\ {k}^{\prime }\end{array}\right]={A}^{\prime }\left[\begin{array}{c}i\\ j\\ k\end{array}\right]
v={v}_{x}i+{v}_{y}j+{v}_{z}k={{v}^{\prime }}_{x}{i}^{\prime }+{{v}^{\prime }}_{y}{j}^{\prime }+{{v}^{\prime }}_{z}{k}^{\prime }
\left[\begin{array}{c}{{v}^{\prime }}_{x}\\ {{v}^{\prime }}_{y}\\ {{v}^{\prime }}_{z}\end{array}\right]={A}^{-1}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]={A}^{\prime }\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]
rotx | rotz |
EUDML | Identifying codes with small radius in some infinite regular graphs. EuDML | Identifying codes with small radius in some infinite regular graphs.
Identifying codes with small radius in some infinite regular graphs.
Charon, Irène; Hudry, Olivier; Lobstein, Antoine
Charon, Irène, Hudry, Olivier, and Lobstein, Antoine. "Identifying codes with small radius in some infinite regular graphs.." The Electronic Journal of Combinatorics [electronic only] 9.1 (2002): Research paper R11, 25 p.-Research paper R11, 25 p.. <http://eudml.org/doc/121793>.
author = {Charon, Irène, Hudry, Olivier, Lobstein, Antoine},
title = {Identifying codes with small radius in some infinite regular graphs.},
TI - Identifying codes with small radius in some infinite regular graphs.
r
r |
EUDML | On -convergence for problems of jumping type. EuDML | On -convergence for problems of jumping type.
\text{Γ}
-convergence for problems of jumping type.
Groli, Alessandro. "On -convergence for problems of jumping type.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2003 (2003): Paper No. 60, 16 p., electronic only-Paper No. 60, 16 p., electronic only. <http://eudml.org/doc/123575>.
@article{Groli2003,
author = {Groli, Alessandro},
keywords = {-convergence; jumping problem; nonsmooth critical point theory; Gamma-convergence; -convergence},
title = {On -convergence for problems of jumping type.},
AU - Groli, Alessandro
TI - On -convergence for problems of jumping type.
KW - -convergence; jumping problem; nonsmooth critical point theory; Gamma-convergence; -convergence
\text{Γ}
-convergence, jumping problem, nonsmooth critical point theory, Gamma-convergence,
{\Gamma }
Articles by Groli |
EUDML | Stationary solutions for a Schrödinger-Poisson system in . EuDML | Stationary solutions for a Schrödinger-Poisson system in .
Stationary solutions for a Schrödinger-Poisson system in
{ℝ}^{3}
Benmlih, Khalid. "Stationary solutions for a Schrödinger-Poisson system in .." Electronic Journal of Differential Equations (EJDE) [electronic only] 2002 (2002): 65-76. <http://eudml.org/doc/127139>.
@article{Benmlih2002,
author = {Benmlih, Khalid},
keywords = {Schrödinger equation; Poisson equation; standing wave solutions; variational methods},
title = {Stationary solutions for a Schrödinger-Poisson system in .},
AU - Benmlih, Khalid
TI - Stationary solutions for a Schrödinger-Poisson system in .
KW - Schrödinger equation; Poisson equation; standing wave solutions; variational methods
Khalid Benmlih, Otared Kavian, Existence and asymptotic behaviour of standing waves for quasilinear Schrödinger–Poisson systems in
{R}^{3}
Schrödinger equation, Poisson equation, standing wave solutions, variational methods
Articles by Benmlih |
A k -Dimensional System of Fractional Finite Difference Equations
k
-Dimensional System of Fractional Finite Difference Equations
Dumitru Baleanu, Shahram Rezapour, Saeid Salehi
We investigate the existence of solutions for a
k
-dimensional system of fractional finite difference equations by using the Kranoselskii’s fixed point theorem. We present an example in order to illustrate our results.
Dumitru Baleanu. Shahram Rezapour. Saeid Salehi. "A
k
-Dimensional System of Fractional Finite Difference Equations." Abstr. Appl. Anal. 2014 (SI62) 1 - 8, 2014. https://doi.org/10.1155/2014/312578
Dumitru Baleanu, Shahram Rezapour, Saeid Salehi "A
k
-Dimensional System of Fractional Finite Difference Equations," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI62), 1-8, (2014) |
On the solutions of certain Lebesgue–Ramanujan–Nagell equations
April 2021 On the solutions of certain Lebesgue–Ramanujan–Nagell equations
Kalyan Chakraborty, Azizul Hoque, Richa Sharma
We completely solve the Diophantine equation
{x}^{2}+{2}^{k}1{1}^{\ell }1{9}^{m}={y}^{n}
x,y\ge 1
k,\ell ,m\ge 0
n\ge 3
gcd\left(x,y\right)=1
, except the case
2|k
2\nmid \ell m
5|n
. We use this result to recover some earlier results in the same direction.
Kalyan Chakraborty. Azizul Hoque. Richa Sharma. "On the solutions of certain Lebesgue–Ramanujan–Nagell equations." Rocky Mountain J. Math. 51 (2) 459 - 471, April 2021. https://doi.org/10.1216/rmj.2021.51.459
Received: 12 October 2020; Revised: 23 October 2020; Accepted: 24 October 2020; Published: April 2021
Keywords: Diophantine equation , integer solution , Lucas sequences , primitive divisors , S-integers
Kalyan Chakraborty, Azizul Hoque, Richa Sharma "On the solutions of certain Lebesgue–Ramanujan–Nagell equations," Rocky Mountain Journal of Mathematics, Rocky Mountain J. Math. 51(2), 459-471, (April 2021) |
EUDML | On signatures and a subgroup of a central extension to the mapping class group. EuDML | On signatures and a subgroup of a central extension to the mapping class group.
On signatures and a subgroup of a central extension to the mapping class group.
Natov, Jonathan. "On signatures and a subgroup of a central extension to the mapping class group.." Homology, Homotopy and Applications 5.1 (2003): 251-260. <http://eudml.org/doc/51134>.
@article{Natov2003,
author = {Natov, Jonathan},
keywords = {homology classes; multiplication},
title = {On signatures and a subgroup of a central extension to the mapping class group.},
AU - Natov, Jonathan
TI - On signatures and a subgroup of a central extension to the mapping class group.
KW - homology classes; multiplication
homology classes, multiplication
{E}^{2}
2
{E}^{4}
4
{S}^{3}
Articles by Natov |
return the form of the geometric object
The routine returns the form of the given object supported by the geometry package. It returns point2d, segment2d, dsegment2d, line2d, triangle2d, square2d, circle2d, ellipse2d, parabola2d, hyperbola2d, or FAIL if it is unable to recognize the object.
The command with(geometry,form) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,1,1\right):
\mathrm{form}\left(A\right)
\textcolor[rgb]{0,0,1}{\mathrm{point2d}}
\mathrm{line}\left(l,[A,B]\right)
\textcolor[rgb]{0,0,1}{l}
\mathrm{form}\left(l\right)
\textcolor[rgb]{0,0,1}{\mathrm{line2d}} |
Class 9 Science Questions & Answers | NCERT/CBSE/ICSE Answers from flipClass
Class 9 Science - Questions & Answers CBSE ICSE
Class 9 Science has a total of 15 topics including Atoms and Molecules, Structure of the Atom and others. You can choose a topic of your choice to practice your understanding/ knowledge in these.
According to the law of conservation of mass, during any chemical change, the total mass of the _________.
Mercuric oxide
\to
mercury + oxygen
\to
92.6 g + ______
What will be the amount of oxygen obtained in this reaction?
In water, hydrogen and oxygen are present in the ratio of _________.
An example of a triatomic molecule is _________.
The quantity of matter present in an object is called its _________.
Indivisibility of an atom was proposed by _________.
All samples of carbon dioxide contain carbon and oxygen in the mass ratio of 3 : 8. This is in agreement with the law of ____________.
The atomic mass of sodium is 23. The number of moles in 46g of sodium is __________.
The molecular formula of potassium nitrate is __________.
Kalium is the Latin name of _________. |
PLOToptions - Maple Help
Home : Support : Online Help : Programming : Data Types : Conversion : PLOToptions
convert/PLOToptions
convert user level plot options into internal structures
convert( p, PLOToptions )
This routine converts user-level plot options to internal form. The result is a list of plot options in the internal structures. These structures can be directly inserted into a PLOT data structure. For a complete list of user-level options, see plot/options.
Some function evaluation dependent options such as resolution, numpoints, color function, etc. cannot be converted.
\mathrm{convert}\left(\mathrm{style}=\mathrm{point},\mathrm{PLOToptions}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{STYLE}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{POINT}}\right)]
\mathrm{convert}\left([\mathrm{thickness}=3,\mathrm{color}=\mathrm{red},\mathrm{title}="My Beautiful Plot"],\mathrm{PLOToptions}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{COLOUR}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{RGB}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.00000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{THICKNESS}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{TITLE}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"My Beautiful Plot"}\right)] |
How the initialization affects the stability of the $k$-means algorithm
How the initialization affects the stability of the
k
-means algorithm
Bubeck, Sébastien ; Meilă, Marina ; von Luxburg, Ulrike
We investigate the role of the initialization for the stability of the қ-means clustering algorithm. As opposed to other papers, we consider the actual қ-means algorithm (also known as Lloyd algorithm). In particular we leverage on the property that this algorithm can get stuck in local optima of the қ-means objective function. We are interested in the actual clustering, not only in the costs of the solution. We analyze when different initializations lead to the same local optimum, and when they lead to different local optima. This enables us to prove that it is reasonable to select the number of clusters based on stability scores.
Mots clés : clustering, қ-means, stability, model selection
author = {Bubeck, S\'ebastien and Meil\u{a}, Marina and von Luxburg, Ulrike},
title = {How the initialization affects the stability of the $k$-means algorithm},
AU - Bubeck, Sébastien
AU - Meilă, Marina
AU - von Luxburg, Ulrike
TI - How the initialization affects the stability of the $k$-means algorithm
Bubeck, Sébastien; Meilă, Marina; von Luxburg, Ulrike. How the initialization affects the stability of the $k$-means algorithm. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 436-452. doi : 10.1051/ps/2012013. http://archive.numdam.org/articles/10.1051/ps/2012013/
[1] D. Arthur and S. Vassilvitskii, қ-means++ : the advantages of careful seeding, in Proc. of SODA (2007). | Zbl 1302.68273
[2] S. Ben-David and U. Von Luxburg, Relating clustering stability to properties of cluster boundaries, in Proc. of COLT (2008).
[3] S. Ben-David, U. Von Luxburg and D. Pál, A sober look on clustering stability, in Proc. of COLT (2006). | Zbl 1143.68520
[4] S. Ben-David, D. Pál and H.-U. Simon, Stability of қ-means clustering, in Proc. of COLT (2007). | Zbl 1203.68138
[5] L. Bottou and Y. Bengio, Convergence properties of the қ-means algorithm, in Proc. of NIPS (1995).
[6] S. Dasgupta and L. Schulman, A probabilistic analysis of EM for mixtures of separated, spherical Gaussians. J. Mach. Learn. Res. 8 (2007) 203-226. | MR 2320668 | Zbl 1222.62142
[7] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions. Springer (2000). | MR 1764176 | Zbl 0951.60003
[8] D. Hochbaum and D. Shmoys, A best possible heuristic for the -center problem. Math. Operat. Res. 10 (1985) 180-184. | MR 793876 | Zbl 0565.90015
[9] T. Lange, V. Roth, M. Braun and J. Buhmann, Stability-based validation of clustering solutions. Neural Comput. 16 (2004) 1299-1323. | Zbl 1089.68100
[10] R. Ostrovsky, Y. Rabani, L.J. Schulman and C. Swamy, The effectiveness of Lloyd-type methods for the қ-means problem, in Proc. of FOCS (2006). | Zbl 1281.68229
[11] O. Shamir and N. Tishby, Cluster stability for finite samples, in Proc. of NIPS (2008).
[12] O. Shamir and N. Tishby, Model selection and stability in қ-means clustering, in Proc. of COLT (2008).
[13] O. Shamir and N. Tishby, On the reliability of clustering stability in the large sample regime, in Proc. of NIPS (2008).
[14] N. Srebro, G. Shakhnarovich and S. Roweis, An investigation of computational and informational limits in Gaussian mixture clustering, in Proc. of ICML (2006).
[15] Z. Zhang, B. Dai and A. Tung, Estimating local optimums in EM algorithm over Gaussian mixture model, in Proc. of ICML (2008). |
Catch Mine Drift 2018: Rules | wbrent.lievers.net
The Catch Mine Drift project requires student teams to build a device that can move independently through an enclosed passage. The device must be self-contained and fit within the two doors that form the starting area. Once the inner door is slid open, the device must move independently through the passageway without any user intervention or assistance.
Teams will be given up to 4 minutes to set up their device and make as much progress as possible. Marks will be assigned based on forward progress.
Each device must be designed and constructed according to the following rules:
Device must operate independently without student assistance or intervention.
The device must fit between the two doors that form the starting area.
The following power sources are not permissible: compressed gas canisters, chemicals, explosives. Electrical power is permitted through the use of batteries only.
ALL valid parts must be sourced from an approved list of Canadian suppliers and be accompanied by a link to the Canadian website which lists the price in Canadian dollars. The inclusion of any non-standard parts will result in a 15% deduction from your combined score. Be careful that the American version of a website is not used accidentally (e.g. http://www.lowes.com instead of http://www.lowes.ca). The approved list of websites includes:
Best Buy <https://www.bestbuy.ca>
Canadian Tire <https://www.canadiantire.ca>
Digi-Key Electronics <https://www.digikey.ca>
Great Hobbies <https://www.greathobbies.com>
Home Depot <https://www.homedepot.ca>
Home Hardware <https://www.homehardware.ca>
Les Jeux <https://lesjeux.ca>
Lowes <https://www.lowes.ca>
Newegg <https://newegg.ca>
Princess Auto <https://www.princessauto.com/>
The Robot Shop <https://www.robotshop.com/ca>
Rona <https://www.rona.ca>
Staples <https://www.staples.ca>
The Source <https://www.thesource.ca>
Walmart <ttps://www.walmart.ca>
Items need not be purchased from the supplier listed; however, they must have a Canadian supplier as described above.
Performance competition guidelines
All performance testing will proceed according to the following steps:
Prior to testing, each team will be required to bring a printed copy of their itemized cost analysis of all the parts used in their device. There will be a 5% deduction to your cost score for failing to bring a physical copy of the analysis sheet.
The TA will review your cost sheet against your design and request additions or changes as needed. No changes may be made to the design after this point.
The device will be photographed.
Students will load their device into the starting area with the inner door closed.
The device must be able to fit in the enclosed starting area. The outer door may be closed to verify that this size constraint is met.
Once loading is completed, and sizing is confirmed, the outer door may be left open.
The inner door will be slid opened and the device allowed to progress through the passageway. One team member may turn the device on, if needed, with a single hand. If so, that same hand must be withdrawn and used to raise the inner door.
The device will receive one point when the front-most portion of the device passes a progress line. It will also receive a point for when the rear-most portion of the device passes a progress line. Because the front and rear are expected to progress at an equivalent rate, the front of the device can only score up to one more point than the rear.
Scores will determined based on maximum forward progress in the event that the device slides or moves backward.
Teams may complete up to a maximum of three attempts in the allotted time. Only the maximum score will be used. If time expires in the midst of an attempt, only points earned to that moment will be counted.
Teams are responsible for uploading an electronic version of the cost analysis sheet. There will be a 10% deduction to your cost score for not submitting the document by the specified deadline.
Problems or issues not expressly covered by these rules may be treated or penalized at the discretion of the instructor.
The cost of the device is determined as follows:
Costs must be calculated using the list price on an official website, regardless of the actual purchase price. Nothing can be free. Everything must have a cost.
Item costs and calculations should be reported rounded to the nearest penny (i.e., two decimal places).
All taxes should be excluded from calculations.
Where items are sold in packages of two or more, the item cost may be calculated as a fraction of the total. For example, 3⁄4” #10 wood screws are sold in packages of 75 for $4.57. Therefore, the cost of three screws is:
\text{cost}=\left(\frac{\text{3}}{\text{75}}\right)$4.57=$0.18
after rounding to the nearest penny).
The cost of wood or other material may be scaled based on the number of usable parts that could be machined from a given raw piece of material. This is different that the area or volume of material used.
Costs of consumables such as tape or glue may also be prorated based on the length or mass of material used.
The costs of tools should NOT be included in the analysis.
Cost analyses not submitted according to these rules are subject to a 15% deduction to your cost score.
Determining testing grades
Two rounds of testing are performed using two iterations of the design: a prototype round, and a final round. Scoring during the prototype round is relative, whereas the final round testing is an absolute scoring system determined based on the prototype results. In both rounds the total score is based on a weighting of 70% for performance and 30% for cost.
During the prototype round, grades will be calculated using the following relative system:
\text{performance grade}=50%\left(1+\frac{\text{[your score]}–\text{[lowest score]}}{\text{[highest score]}–\text{[lowest score]}}\right)
\text{cost grade}=50%\left(2–\frac{\text{[your cost]}–\text{[lowest cost]}}{\text{[highest cost]}–\text{[lowest cost]}}\right)
Based on the results of the prototype testing, the following absolute scoring system was determined for the final round:
\text{performance grade}=100%×\text{min}\left(1,\frac{\text{[your score]}}{\text{36}}+0.25\right)
\text{cost score}=50%\left[1–\text{erf}\left(\frac{\text{[your cost]-$50}}{\sqrt{\text{2}{\left(\text{$15}\right)}^{2}}}\right)\right] |
Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters
March 2021 Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters
Paul Dario, Chenlin Gu
Paul Dario,1 Chenlin Gu2
1School of Mathematical Sciences, Tel Aviv University
2DMA, École Normale Supérieure
We study the heat kernel and the Green’s function on the infinite supercritical percolation cluster in dimension
\mathit{d}\ge 2
and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 1–27). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. Math. 71 (2018) 1717–1849), Gaussian bounds on the heat kernel established by Barlow in (Ann. Probab. 32 (2004) 3024–3084) and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a
{\mathit{C}}^{0,1}
-large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.
We would like to thank Jean-Christophe Mourrat and Scott Armstrong for helpful discussions and comments. PD is supported by the Israel Science Foundation grants 861/15 and 1971/19 and by the European Research Council starting grant 678520 (LocalOrder). CG is supported by the PhD scholarship from École Polytechnique.
Paul Dario. Chenlin Gu. "Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters." Ann. Probab. 49 (2) 556 - 636, March 2021. https://doi.org/10.1214/20-AOP1456
Received: 1 September 2019; Revised: 1 April 2020; Published: March 2021
Keywords: large-scale regularity , local central limit theorem , parabolic equation , Stochastic homogenization , Supercritical percolation
Paul Dario, Chenlin Gu "Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters," The Annals of Probability, Ann. Probab. 49(2), 556-636, (March 2021) |
Time to nuclear Armageddon - Everything Wiki
File:Graves-JSM2019-08-01.webm
"Time to nuclear Armageddon" presentation by Spencer Graves at the Joint Statistical Meetings, 2019-08-01.
This article is the narrative basis for the accompanying video of presentation at the Joint Statistical Meetings 2019-08-01. It is on Wikiversity to invite further discussion, expansion, correction, and revision of the narrative presented here subject to the standard Wikimedia rules of writing from a neutral point of view citing credible sources.
This work was inspired by Daniel Ellsberg's 2017 book, The Doomsday Machine. In this book Ellsberg says that as long as the world maintains large nuclear arsenals, it is only a matter of time before there is a nuclear war, which he claims will almost certainly lead to a nuclear winter that lasts over a decade, during which 98 percent of humanity will starve to death if they do not die of something else sooner.[1][2]
Ellsberg's claims suggest statistical questions regarding the probability distribution of the time to a nuclear war and the severity of the consequences.
The following outlines a methodology for addressing these statistical questions. Previous estimates of the probability of a nuclear war in the next year range from 1 chance in a million to 7 percent, with 0.7 percent being offered by the Good Judgment Project, which arguably uses the best known methodology for making such estimates. If that rate is assumed to have been constant over the 70 years since the first test of a nuclear weapon by the Soviet Union in 1949, these estimates of the probability of a nuclear war in 70 years range from 70 chances in a million to 99 percent. The Good Judgment answer translates into a 40 percent chance of such a war in 70 years, past or future, or equivalently 20 chances in a million that the next 24 hours might see the initiation of a crisis that leads to a nuclear war.
Moreover, nuclear proliferation is continuing. This suggests that the probability of a nuclear war and winter is likely increasing and will continue to increase until something happens to make it effectively impossible for anyone to make more nuclear weapons for a very long time. Two possible scenarios might produce such a nuclear disarmament:
A nuclear war and winter ending civilization.
An unprecedented international movement that strengthens international law to the point that the poor and disfranchised have effective nonviolent means for pursuing a redress of grievances.
This article ends with an outline of possible future research in this area.
3 Other leading figures supporting Ellsberg's claims
We suggest here the following methodology:
1. Select a list of incidents.
2. Model the time between such incidents.
3. Estimate subjective probabilities for (a) an essentially equivalent repetition of the same incident leading to a nuclear war, and (b) the distribution of the severity of the consequences of the war. And
4. Combine “2” and “3” into compelling communications.
Someone attacked item number “3” saying, “You, Spencer Graves, are willing to speculate. That's just a rank speculation. I am not willing to speculate.”
My response is an unwillingness to speculate is essentially equivalent to saying that the probability is zero, and I think that is an unrealistic speculation.
A prototype use of this methodology considers only two incidents:
(1) The 1962 Cuban Missile Crisis, and
(2) The 1983 Soviet nuclear false alarm incident.
President Kennedy, who was the US President during the Cuban Missile Crisis, said that there was a probability of between a third and a half that that incident would have gone to a nuclear war. He died before learning that Soviet nuclear weapons were in Cuba at that time. The crisis ended less than 48 hours before a planned invasion by the US, predicated on the belief that there were no such weapons in Cuba at that time.[3] At a 30th anniversary conference in 1992, Fidel Castro (Cuban head of state in 1962) told Robert McNamara (US Secretary of State in 1962) that if the US had invaded, those nuclear weapons would have been used, even though Castro knew that not one person on Cuba would survive.[4]
The 1983 Soviet nuclear false alarm incident occurred when US President Ronald Reagan was building up the US military and challenging the Soviets. Andropov, the Soviet Premier, and his inner circle believed that the US was preparing for a nuclear first attack.
This gives us one observation of
{\displaystyle t_{1}}
= 21 years of the time between the 1962 Cuban Missile Crisis and the 1983 Soviet nuclear false alarm incident. In addition, the time to the next incident of a similar magnitude is censored at the
{\displaystyle t_{2}}
= 36 years between the 1983 Soviet nuclear false alarm incident and 2019, as this is being written. Standard statistical theory says that the likelihood for these two observations is the product of the density at
{\displaystyle t_{1}}
and the survival function at
{\displaystyle t_{2}}
{\displaystyle L_{e}=f(t_{1})S(t_{2})}
It seems reasonable to assume, at least for an initial demonstration of this methodology, an exponential distribution. This means the likelihood is as follows:
{\displaystyle L_{e}=\exp[-(21+36)/\tau ]/\tau }
To the extent that this is accurate, it says that the maximum likelihood estimate of the mean time to the next comparable nuclear crisis is [(21 + 36) divided by 1] = 57 years.
{\displaystyle {\hat {\tau }}}
We can get an equivalent answer by exploiting the well-known duality between exponential and Poisson distributions by considering this history as consisting one Poisson distributed observation on the number of such incidents in each of the 57 years between 1962 and this writing in 2019: We have one such incident in 1983 and 0 in the other 56 years. The likelihood for this formulation is as follows:
{\displaystyle L_{p}=\lambda \exp(-57\lambda )}
This is maximized with
{\displaystyle {\hat {\lambda }}}
= 1/57 = 0.018 such incidents per year.
The Poisson formulation is useful, because it is easier to consider non-constant hazard. The glm function in the R (programming language) can easily model a liner relationship between
{\displaystyle \log(\lambda )}
and the time since the very first test of a nuclear weapon by the United States in 1949. Moreover, the bssm package for R can model a normal random walk of log(Poisson mean). These options will will not be pursued here but might be useful in future work, either with a larger list of incidents or with nuclear proliferation, discussed below.
Simon Beard[5] shared the following literature review of studies estimating something like the probability of a nuclear war in the next year, which he compiled in joint with Tom Rowe of Virginia Tech, and James Fox at the University of Oxford.[6] Beard's analysis is augmented here with the probability of a nuclear war in the 70 years between the first test of a nuclear weapon by the Soviet Union (now Russia) in 1949 and the time that this is being written in 2019. It is augmented also by columns translating the annual probabilities in to the number of chances in a million (parts per million, ppm) that a crises leading to a nuclear war will begin on any given day.
The 70-year numbers use the fact that if there is a constant probability
{\displaystyle p}
of a nuclear war in a given year, the probability of at least one nuclear war in 70 years is
{\displaystyle [1-(1-p)^{70}]}
. The upper limit of 7% for the probability of a nuclear war in the next year (Barrett et al., 2013) is clearly not plausible as a constant probability of a nuclear war each year during that period: Otherwise the probability that we would already have had one is 99%.
Probability of a nuclear war
daily (ppm)
Hellman (2008) 0.02% 0.5% 1% 30% 0.5 14
Barrett et al (2013) 0.0001% 7% 0.007% 99% 0.003 200
Lundgren (2013) 1.4% 60% 40
Project for the Study of the 21st Century (2015) 0.3% 20% 8
Good Judgment Project (2018) 0.7% 40% 20
Turchin (2010) 0.5% 30% 14
Pamlin and Armstrong (2015)[7] 0.1% 7% 3
Sandberg and Bostrom (2008)[8] 0.4% 24% 11
Similarly, the ppm numbers can be interpreted as equivalent to suggesting that each new day the leaders of the nuclear-weapon states roll the cylinder and pull the trigger in a game of Russian Roulette with the indicated chance of the result being a nuclear war.
It seems useful to highlight the Good Judgment Project (2018), because it uses a methodology developed by a 20-year project funded in part by the Intelligence Advanced Research Projects Agency and documented in Tetlock and Gardner (2015). Their methodology produced 30% better forecasts than intelligence agents with access to classified information. It is as follows:
Recruit volunteers and ask them a series of forecasting questions, like estimating the probability of a certain event in a specific time period (typically 1, 2 or 3 years).
Identify the volunteers with the best forecasts.
Organize them into teams.
Study what the best teams did.
The result is documented in Tetlock and Gardner (2015). This methodology might potentially be crowdsourced on a platform like Wikidata, Wikiversity and Wikipedia.
The 0.7 percent chance of a nuclear war starting in the coming year estimated by the Good Judgment Project is equivalent to a 40 percent chance in 70 years and 20 chances in a million that it will start in the next 24 hours.
Other leading figures supporting Ellsberg's claims[edit]
Ellsberg is not alone in his concern about this. Robert McNamara also said that as long as the world has large nuclear arsenals, it's only a question of time before there is a nuclear war.[9] Similar concerns led former US Senator Sam Nunn and media executive Ted Turner to found the Nuclear Threat Initiative, also supported by former US Secretary of Defense William J. Perry, and former US Secretaries of State Henry Kissinger and George Shultz. Perry wrote, "The threat of Russia intentionally launching a nuclear attack against the United States today is vanishingly small", but we are much more likely to stumble into a nuclear war because of a cybersecurity breach, similar to Stuxnet, which reportedly destroyed a fifth of Iran's weapons-grade nuclear enrichment capabilities in 2010.[10] In 2020 the US federal government was the recipient of a similar attack. New York Times cybersecurity and digital espionage expert Nicole Perlroth wrote, "This is how they tell me the world ends."[11]
Atmospheric scientists Owen Toon, Alan Robock et al. (2017) have estimated that a relatively minor nuclear war between India and Pakistan could involve at least 100 nuclear weapons, leading to a nuclear autumn during which two billion people -- just over a quarter of humanity -- not involved in the nuclear exchange would starve to death.[12]
A hundred nuclear weapons is only about 2 percent of the US nuclear arsenal. A nuclear war involving the US would likely be closer to Ellsberg's doomsday scenario than the two billion dead mentioned by Toon, Robock et al. (2017).
File:NuclearProliferation.svg
NuclearProliferation: Time between "first tests" of a nuclear weapon by succeeding nuclear-weapon states. The US is not on this plot, because it had no predecessors. RU = Russia (the USSR in 1949). GB = United Kingdom. FR = France. CN = China. IN = India. IL = Israel, recorded here with the date of the Vela Incident. PK = Pakistan. KP = North Korea.
The fact that nuclear proliferation is continuing suggests that any model that assumes that the risk of a nuclear war is constant or declining is probably wrong. When the Nonproliferation Treaty treaty took effect in 1970, there were 5 nuclear weapon states. When US President George W. Bush announced an “axis of evil” consisting of North Korea, Iran and Iraq on 2002-01-28, there were 8. As this is being written in 2019, there are 9. As long as nuclear weapon states continue to threaten countries without them, the pressure for nuclear proliferation will continue, and the risks of a nuclear war will likely grow.
number of nuclear-weapon states
1970 5 Nonproliferation Treaty
2002 8 Axis of evil speech by US President Bush condemning North Korea, Iran and Iraq
2006 9 First test of a nuclear weapon by North Korea
It is relatively easy to use the glm function in the R (programming language) to model a random walk in the log(Poisson mean) of the number of first-tests of new nuclear-weapon states each year.
Beyond this, it could be useful to try to expand the present study to consider larger lists of incidents threatening nuclear war. For this purpose, it might be useful to try to recruit volunteers to use Wikimedia Foundation projects, especially Wikipedia, Wikiversity, and Wikidata to produce estimates like this using the methodology of the Good Judgment Project (2018) described in Tetlock and Gardner (2017). Wikipedia already does something like this: Peter Binkley in an invited 2006 article for a Canadian Library Association journal said that on controversial topics "the two sides actually engaged each other and negotiated a version of the [Wikipedia] article that both can more or less live with. This is a rare sight indeed in today’s polarized political atmosphere, where most online forums are echo chambers for one side or the other”.[13]
Another potentially useful project could be to write an R function to convert probability distributions generated by models like those discussed here estimates of the probability that a person of any age, especially a child born today, would die prematurely from a nuclear war. Stanford Engineering Professor Emeritus Martin Hellman has estimated that the probability is at least 10 percent that a child born today would die prematurely from a nuclear war.[14] These kinds of analyses might help a broader audience understand the seriousness of this issue.
Time to extinction of civilization, which provides more detail behind part of this present discussion.
Forecasting nuclear proliferation, which predicts continuing increases in the number of nuclear weapon states, thereby seemingly increasing the threat of nuclear war and Armageddon.
{{#invoke:Cite Q|cite_q|qid=Q66151664 }}[15]
Good Judgment Project (2018), nonpublic data, cited in comment 2018-08-06 by Carl Shulman to {{#invoke:Cite Q|cite_q|qid=Q66730943 }}.
{{#invoke:Cite Q|cite_q|qid=Q66156663 }}
↑ {{#invoke:Cite Q|cite_q|qid=Q64226035 }}
↑ A version of this article is scheduled to appear in {{#invoke:Cite Q|cite_q|qid=Q66918248 }}. More on this appears in Time to extinction of civilization.
↑ {{#invoke:Cite Q|cite_q|qid=Q63862699 }}, p. 206.
↑ {{#invoke:Cite Q|cite_q|qid=Q64736611 }}, pp. 189-190.
↑ cited from private communication from Simon Beard. The numbers here correct minor errors in the corresponding slide in the accompanying video, commons:File:Graves-JSM2019-08-01.webm
↑ p. 16/212: “The likelihood of a full-scale nuclear war between the USA and Russia has probably decreased. Still, the potential for deliberate or accidental nuclear conflict has not been removed, with some estimates putting the risk in the next century or so at around 10%”. This makes the risk in 1 year of [1-.9^(1/100)] = 0.001053, and the risk in 70 years = 0.071, ignoring their comment that “The likelihood ... has probably decreased” and ignoring the chances of a nuclear war involving other nuclear weapons diads. Later, they write, “Based on available assessments the best current estimate for nuclear war within the next 100 years is 5% for infinite threshold [and] 0.005% for infinite impact” (p. 148).
↑ p. # 1 (p. 2 of 6 in the pdf): 30% chance of “at least 1 million dead” “total killed in all nuclear wars” by 2100 from 2008.
↑ {{#invoke:Cite Q|cite_q|qid=Q102046116 }}.
↑ {{#invoke:Cite Q|cite_q|qid=Q105623657 }}. For a review, see {{#invoke:Cite Q|cite_q|qid=Q105623758 }}.
↑ See also {{#invoke:Cite Q|cite_q|qid=Q63256454 }}. This could further spell the end of civilization, according to {{#invoke:Cite Q|cite_q|qid=Q106659147 }}. See also {{#invoke:Cite Q|cite_q|qid=Q106657529 }}.
↑ 15.0 15.1 cited from {{#invoke:Cite Q|cite_q|qid=Q66147141 }}
Retrieved from "https://everything.wiki/index.php?title=Time_to_nuclear_Armageddon&oldid=3754401" |
Stability margin requirement for control system tuning - MATLAB - MathWorks América Latina
TuningGoal.Margins class
ScalingOrder
SISO Margin Requirement Evaluated with Additional Loop Opening
MIMO Margin Requirement in Frequency Band
Stability margin requirement for control system tuning
Use TuningGoal.Margins to specify a tuning goal for the gain and phase margins of a SISO or MIMO feedback loop. You can use this tuning goal for validating a tuned control system with viewGoal. You can also use the tuning goal for control system tuning with tuning commands such as systune or looptune.
After you create a tuning goal, you can configure it further by setting Properties of the object.
After using the tuning goal to tune a control system, you can visualize the tuning goal and the tuned value using the viewGoal command. For information about interpreting the margins goal, see Stability Margins in Control System Tuning.
Req = TuningGoal.Margins(location,gainmargin,phasemargin) creates a tuning goal that specifies the minimum gain and phase margins at the specified location in the control system.
Location in the control system at which the minimum gain and phase margins apply, specified as a character vector or cell array of character vectors that identify one or more locations in the control system to tune. What locations are available depends on what kind of system you are tuning:
The margin requirements apply to the point-to-point, open-loop transfer function at the specified loop-opening location. That transfer function is the open-loop response obtained by injecting signals at the specified location, and measuring the return signals at the same point.
If location is a cell array, then the margin requirement applies to the MIMO open-loop transfer function.
Required minimum gain margin for the feedback loop, specified as a scalar value in dB. TuningGoal.Margins uses disk-based gain and phase margins, which provide a stronger guarantee of stability than the classical gain and phase margins. (For details about disk margins, see Stability Analysis Using Disk Margins (Robust Control Toolbox).)
The gain margin indicates how much the gain of the open-loop response can increase or decrease without loss of stability. For instance,
For a SISO system, setting gainmargin = 3 specifies a requirement that the closed-loop system remain stable for changes in the open-loop gain of up to ±3 dB.
For a MIMO system, setting gainmargin = 3 specifies a requirement that the closed-system remain stable for gain changes up to ±3 dB in each feedback channel. The gain can change in all channels simultaneously, and by a different amount in each channel.
Required minimum phase margin for the feedback loop, specified as a scalar value in degrees. TuningGoal.Margins uses disk-based gain and phase margins, which provide a stronger guarantee of stability than the classical gain and phase margins. (For details about disk margins, see Stability Analysis Using Disk Margins (Robust Control Toolbox).)
The phase margin indicates how much the phase of the open-loop response can increase or decrease without loss of stability. For instance,
For a SISO system, setting phasemargin = 45 specifies a requirement that the closed-loop system remain stable for changes of up to ±45° in the phase of the open-loop response.
For a MIMO system, setting phasemargin = 45 specifies a requirement that the closed-system remain stable for phase changes up to ±45° in each feedback channel. The phase can change in all channels simultaneously, and by a different amount in each channel.
Required minimum gain margin for the feedback loop, specified as a scalar value in decibels (dB).
The value of the GainMargin property is set by the gainmargin input argument when you create the tuning goal.
Required minimum phase margin for the feedback loop, specified as a scalar value in degrees.
The value of the PhaseMargin property is set by the phasemargin input argument when you create the tuning goal.
Controls the order (number of states) of the scalings involved in computing MIMO stability margins. Static scalings (ScalingOrder = 0) are used by default. Increasing the order may improve results at the expense of increased computations. Use viewGoal to assess the gap between optimized and actual margins. If this gap is too large, consider increasing the scaling order. See Stability Margins in Control System Tuning.
Default: 0 (static scaling)
Set the Focus property to limit enforcement of the tuning goal to a particular frequency band. For best results with stability margin requirements, pick a frequency band extending about one decade on each side of the gain crossover frequencies. For example, suppose Req is a TuningGoal.Margins object that you are using to tune a system with approximately 10 rad/s bandwidth. To limit the enforcement of the tuning goal, use the following command:
Location at which the minimum gain and phase margins apply, specified as a cell array of character vectors that identify one or more analysis points in the control system to tune. For example, if Location = {'u'}, the tuning goal enforces the minimum gain and phase margins at an analysis point 'u'.
Create a margin requirement for the inner loop of the following control system. The requirement imposes a minimum gain margin of 5 dB and a minimum phase margin of 40 degrees.
Create a tuning requirement object.
Req = TuningGoal.Margins('AP2',5,40);
This requirement imposes the specified stability margins on the feedback loop identified by the AnalysisPoint channel 'AP2', which is the inner loop.
Specify that these margins are evaluated with the outer loop of the control system open.
Adding 'AP1' to the Openings property of the tuning requirements object ensures that systune evaluates the requirement with the loop open at that location.
Use systune to tune the free parameters of T to meet the tuning requirement specified by Req. You can then use viewGoal to validate the tuned control system against the requirement.
Create a requirement that sets minimum gain and phase margins for the loop defined by three loop-opening locations in a control system to tune. Because this loop is defined by three loop-opening locations, it is a MIMO loop.
The requirement sets a minimum gain margin of 10 dB and a minimum phase margin of 40 degrees, within the band between 0.1 and 10 rad/s.
Req = TuningGoal.Margins({'r','theta','phi'},10,40);
The names 'r', 'theta', and 'phi' must specify valid loop-opening locations in the control system that you are tuning.
Limit the requirement to the frequency band between 0.1 and 10 rad/s.
Req.Focus = [0.1 10];
For TuningGoal.Margins, f(x) is given by:
f\left(x\right)={‖2\alpha S-\alpha I‖}_{\infty }.
S = D–1[I – L(s,x)]–1D is the scaled sensitivity function.
L(s,x) is the open-loop response being shaped.
D is an automatically-computed loop scaling factor. For more information about D, see Stability Margins in Control System Tuning.
α is a scalar parameter computed from the specified gain and phase margin. For more information about α, see Stability Analysis Using Disk Margins (Robust Control Toolbox).
looptune | systune | systune (for slTuner) (Simulink Control Design) | looptune (for slTuner) (Simulink Control Design) | viewGoal | evalGoal |
EUDML | On extensions of lax monads. EuDML | On extensions of lax monads.
On extensions of lax monads.
Clementino, Maria Manuel, and Hofmann, Dirk. "On extensions of lax monads.." Theory and Applications of Categories [electronic only] 13 (2004): 41-60. <http://eudml.org/doc/125873>.
author = {Clementino, Maria Manuel, Hofmann, Dirk},
keywords = {relation; V-matrix; lax monad; lax algebra},
title = {On extensions of lax monads.},
TI - On extensions of lax monads.
KW - relation; V-matrix; lax monad; lax algebra
relation, V-matrix, lax monad, lax algebra
2 |
EUDML | The Linear Groups of Injective Factors and of Matroid C*Algebras are Contractible to a Point. EuDML | The Linear Groups of Injective Factors and of Matroid C*Algebras are Contractible to a Point.
The Linear Groups of Injective Factors and of Matroid C*Algebras are Contractible to a Point.
F.A. Sukochev; V.E. Sheremetyev
Sukochev, F.A., and Sheremetyev, V.E.. "The Linear Groups of Injective Factors and of Matroid C*Algebras are Contractible to a Point.." Mathematica Scandinavica 77.1 (1995): 119-128. <http://eudml.org/doc/167354>.
@article{Sukochev1995,
author = {Sukochev, F.A., Sheremetyev, V.E.},
keywords = {injective factors; matroid -algebras; contractibility; group of all linear continuous invertible operators; topology induced by the operator norm; general linear groups; von Neumann algebras; -algebras},
title = {The Linear Groups of Injective Factors and of Matroid C*Algebras are Contractible to a Point.},
AU - Sukochev, F.A.
AU - Sheremetyev, V.E.
TI - The Linear Groups of Injective Factors and of Matroid C*Algebras are Contractible to a Point.
KW - injective factors; matroid -algebras; contractibility; group of all linear continuous invertible operators; topology induced by the operator norm; general linear groups; von Neumann algebras; -algebras
injective factors, matroid
{C}^{*}
-algebras, contractibility, group of all linear continuous invertible operators, topology induced by the operator norm, general linear groups, von Neumann algebras,
{C}^{*}
{C}^{*}
{W}^{*}
{C}^{*}
{C}^{*}
Articles by F.A. Sukochev
Articles by V.E. Sheremetyev |
Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Modular Subpackage : Transpose
compute the transpose of a mod m Matrix or Vector
Transpose(m, A, inplace)
(optional) keyword for inplace operation on square Matrix
The Transpose function returns the transpose of the input mod m Matrix or Vector.
It uses the more flexible mod m Copy function to compute the transpose.
If A is a square Matrix, the transpose can be performed in-place.
This command is part of the LinearAlgebra[Modular] package, so it can be used in the form Transpose(..) only after executing the command with(LinearAlgebra[Modular]). However, it can always be used in the form LinearAlgebra[Modular][Transpose](..).
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Modular}]\right):
p≔97
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{97}
M≔\mathrm{Mod}\left(p,\mathrm{Matrix}\left(3,4,\left(i,j\right)↦\mathrm{rand}\left(\right)\right),\mathrm{integer}[]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{77}& \textcolor[rgb]{0,0,1}{96}& \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{86}\\ \textcolor[rgb]{0,0,1}{58}& \textcolor[rgb]{0,0,1}{36}& \textcolor[rgb]{0,0,1}{80}& \textcolor[rgb]{0,0,1}{22}\\ \textcolor[rgb]{0,0,1}{44}& \textcolor[rgb]{0,0,1}{39}& \textcolor[rgb]{0,0,1}{60}& \textcolor[rgb]{0,0,1}{39}\end{array}]
\mathrm{Transpose}\left(p,M\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{77}& \textcolor[rgb]{0,0,1}{58}& \textcolor[rgb]{0,0,1}{44}\\ \textcolor[rgb]{0,0,1}{96}& \textcolor[rgb]{0,0,1}{36}& \textcolor[rgb]{0,0,1}{39}\\ \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{80}& \textcolor[rgb]{0,0,1}{60}\\ \textcolor[rgb]{0,0,1}{86}& \textcolor[rgb]{0,0,1}{22}& \textcolor[rgb]{0,0,1}{39}\end{array}]
\mathrm{V1}≔\mathrm{Mod}\left(p,\mathrm{Vector}[\mathrm{row}]\left(3,i↦\mathrm{rand}\left(\right)\right),\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{\mathrm{V1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{43.}& \textcolor[rgb]{0,0,1}{12.}& \textcolor[rgb]{0,0,1}{55.}\end{array}]
\mathrm{Transpose}\left(p,\mathrm{V1}\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{43.}\\ \textcolor[rgb]{0,0,1}{12.}\\ \textcolor[rgb]{0,0,1}{55.}\end{array}]
\mathrm{V2}≔\mathrm{Mod}\left(p,\mathrm{Vector}[\mathrm{column}]\left(3,i↦\mathrm{rand}\left(\right)\right),\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{\mathrm{V2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{24.}\\ \textcolor[rgb]{0,0,1}{71.}\end{array}]
\mathrm{Transpose}\left(p,\mathrm{V2}\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{24.}& \textcolor[rgb]{0,0,1}{71.}\end{array}]
\mathrm{Copy}\left(p,\mathrm{V2},'\mathrm{transpose}'\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{24.}& \textcolor[rgb]{0,0,1}{71.}\end{array}]
\mathrm{Multiply}\left(p,\mathrm{Transpose}\left(p,\mathrm{V2}\right),\mathrm{V2}\right)
\textcolor[rgb]{0,0,1}{92.}
M≔\mathrm{Create}\left(p,3,3,\mathrm{random},\mathrm{integer}\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{43}& \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{76}\\ \textcolor[rgb]{0,0,1}{58}& \textcolor[rgb]{0,0,1}{15}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{69}& \textcolor[rgb]{0,0,1}{76}& \textcolor[rgb]{0,0,1}{38}\end{array}]
\mathrm{Transpose}\left(p,M,\mathrm{inplace}\right):
M
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{43}& \textcolor[rgb]{0,0,1}{58}& \textcolor[rgb]{0,0,1}{69}\\ \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{15}& \textcolor[rgb]{0,0,1}{76}\\ \textcolor[rgb]{0,0,1}{76}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{38}\end{array}] |
Loop Shape and Stability Margin Specifications - MATLAB & Simulink - MathWorks América Latina
Loop shape requirements are constraints on the open-loop response
L
. For tuning purposes, they are converted into closed-loop gain constraints on the sensitivity function
S=1/\left(1+L\right)
and complementary sensitivity function
T=L/\left(1+L\right)
. Use viewGoal to visualize the target loop shape and corresponding gain bounds on
S
(green) and
T
(red).
The TuningGoal.MaxLoopGain requirement rests on the fact that the open- and closed-loop gains are comparable when the loop gain is small (
|L|\ll 1
). As a result, it can be ineffective at keeping the loop gain below some value close to 1. For example, suppose that flexible modes cause gain spikes beyond the crossover frequency and that you need to keep these spikes below 0.5 (-6 dB). Instead of using TuningGoal.MaxLoopGain, you can directly constrain the gain of
L
using TuningGoal.Gain with a loop opening at "u".
The TuningGoal.Margins requirement uses the notion of disk margin to enforce minimum amounts of gain and phase margins at the specified loop opening site(s). For MIMO feedback loops, this requirement guarantees stability for gain or phase variations in each feedback channel. The gain or phase can change in all channels simultaneously, and by a different amount in each channel. See Stability Margins in Control System Tuning for details. For example,the following code enforces
±6
dB of gain margin and 45 degrees of phase margin at a location "u". |
On Convergence in L-Valued Fuzzy Topological Spaces
2015 On Convergence in
L
-Valued Fuzzy Topological Spaces
A. A. Ramadan, M. El-Dardery, Hu Zhao
We introduce the concept of L-fuzzy neighborhood systems using complete MV-algebras and present important links with the theory of L-fuzzy topological spaces. We investigate the relationships among the degrees of L-fuzzy r-adherent points (r-convergent, r-cluster, and r-limit, resp.) in an L-fuzzy topological spaces. Also, we investigate the concept of LF-continuous functions and their properties.
A. A. Ramadan. M. El-Dardery. Hu Zhao. "On Convergence in
L
-Valued Fuzzy Topological Spaces." Abstr. Appl. Anal. 2015 1 - 10, 2015. https://doi.org/10.1155/2015/730940
A. A. Ramadan, M. El-Dardery, Hu Zhao "On Convergence in
L
-Valued Fuzzy Topological Spaces," Abstract and Applied Analysis, Abstr. Appl. Anal. 2015(none), 1-10, (2015) |
(Redirected from Commendation points)
Experience in any combat skill that is equal to or greater than level 25. The formula is the following:
{\displaystyle {\frac {L^{2}}{N}}}
{\displaystyle L}
equals level;
{\displaystyle N}
equals approximately 5.195 for magic and range, approximately 19.23 for summoning and prayer, and 5 for all others. The experience is always rounded down to a whole number. |
Speed of light - Simple English Wikipedia, the free encyclopedia
The speed of light, in any medium,which is usually denoted by c, is a physical constant important in many areas of physics.It is denoted by 'c^0' especially in vacuum medium, although the symbol 'c' can be used to refer to that in any medium. It is exactly 299,792,458 metres per second (983,571,056 feet per second) by definition.[1][2] A photon (particle of light) travels at this speed in a vacuum.
According to special relativity, c is the maximum speed at which all energy, matter, and physical information in the universe can travel. It is the speed of all massless particles such as photons, and associated fields—including electromagnetic radiation such as light—in a vacuum.
It is predicted by the current theory to be the speed of gravity (that is, gravitational waves). Such particles and waves travel at c regardless of the motion of the source or the inertial frame of reference of the observer. In the theory of relativity, c interrelates space and time, and appears in the famous equation of mass–energy equivalence E = mc2.[3]
The special theory of relativity is based on the prediction, so far upheld by observations, that the measured speed of light in a vacuum is the same whether or not the source of the light and the person doing the measuring are moving relative to each other. This is sometimes expressed as "the speed of light is independent of the reference frame."
1 Example explaining how speed does not depend on reference frame
2 Relation to fundamental electric and magnetic properties of space
3.1 Rømer
Example explaining how speed does not depend on reference frame[change | change source]
This behavior is different from our common ideas about motion as shown by this example:
George is standing on the ground next to some train tracks (railroad). There is a train rushing by at 30 mph (48 km/h). George throws a baseball at 90 mph (140 km/h) in the direction the train is moving. Tom, a passenger on the train, has a device (like a radar gun) to measure throwing speeds. Because he is on the train, Tom is already moving at 30 mph (48 km/h) in the direction of the throw, so Tom measures the speed of the ball as only 60 mph (97 km/h).
In other words, the speed of the baseball, as measured by Tom on the train, depends on the speed of the train.
In the example above, the train was moving at 1/3 the speed of the ball, and the speed of the ball as measured on the train was 2/3 of the throwing speed as measured on the ground.
Now, repeat the experiment with light instead of a baseball; that is, George has a flashlight instead of throwing a baseball. George and Tom both have devices that are the same to measure the speed of light (instead of the radar gun in the baseball example).
George is standing on the ground next to some train tracks. There is a train rushing by at 1/3 the speed of light. George flashes a light beam in the direction the train is moving. George measures the speed of light as 186,282 miles per second (299,792 kilometres per second). Tom, a passenger on the train, measures the speed of the light beam. What speed does Tom measure?
Intuitively, one may think that the speed of the light from the flashlight as measured on the train should be 2/3 the speed measured on the ground, just like the speed of the baseball was 2/3. But in fact, the speed measured on the train is the full value, 186,282 miles per second (299,792 kilometres per second), not 124,188 miles per second (199,861 kilometres per second).
It sounds impossible, but that is what one measures.
A consequence of this fact that the speed of light is the same for all observers, is that nothing can go faster than the speed of light. Another consequence is that for objects that have mass, no matter how much energy is used to increase the speed of an object, it will get closer and closer, but it will never reach the speed of light. These ideas were discovered in the early 1900s by Albert Einstein, whose work completely changed our understanding of light.
Relation to fundamental electric and magnetic properties of space[change | change source]
Maxwell's equations predicted the speed of light and confirmed Michael Faraday's idea that light was an electromagnetic wave (a way that energy moves). From these equations, we find that the speed of light is related to the inverse of the square root of the permittivity of free space, ε0, and the permeability of free space, μ0:
{\displaystyle c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}\ .}
The index of refraction of a clear material is the ratio between the speed of light in a vacuum and the speed of light in that material.
Measurement[change | change source]
Rømer[change | change source]
Ole Christensen Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light.[4][5] When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it. The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the diameter of the Earth's orbit around the Sun. The observed change in the moon's orbital period is actually the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect for Jupiter's innermost moon Io, and he deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.
Bradley[change | change source]
Another method is to use the aberration of light, discovered and explained by James Bradley in the 18th century.[6] This effect results from the vector addition of the velocity of light arriving from a distant source (such as a star) and the velocity of its observer (see diagram on the right). A moving observer thus sees the light coming from a slightly different direction and consequently sees the source at a position shifted from its original position. Since the direction of the Earth's velocity changes continuously as the Earth orbits the Sun, this effect causes the apparent position of stars to move around. From the angular difference in the position of stars,[7] it is possible to express the speed of light in terms of the Earth's velocity around the Sun. This, with the known length of a year, can be easily converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled 10,210 times faster than the Earth in its orbit (the modern figure is 10,066 times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.[6]
Nowadays, the "light time for unit distance"—the inverse of c (1/c), expressed in seconds per astronomical unit—is measured by comparing the time for radio signals to reach different spacecraft in the Solar System. The position of spacecraft is calculated from the gravitational effects of the Sun and various planets. By combining many such measurements, a best fit value for the light time per unit distance is obtained. As of 2009[update], the best estimate, as approved by the International Astronomical Union (IAU), is:[8][9]
light time for unit distance: 499.004783836(10) s
c = 0.00200398880410(4) AU/s
c = 173.144632674(3) AU/day.
The relative uncertainty in these measurements is 0.02 parts per billion (2×10−11), as equivalent to the uncertainty in Earth-based measurements of length by interferometry.[10] Since the metre is defined to be the length travelled by light in a certain time interval, the measurement of the light time for unit distance can also be interpreted as measuring the length of an AU in metres. The metre is considered to be a unit of proper length, whereas the AU is often used as a unit of observed length in a given frame of reference.
Practical effects[change | change source]
The finite speed of light is a major constraint on long-distance space travel. Supposing a journey to the other side of the Milky Way, the total time for a message and its reply would be about 200,000 years. Even more seriously, no spacecraft could travel faster than light, so all galactic-scale transport would be effectively one-way, and would take much longer than than any modern civilisation has existed.
The speed of light can also be of concern over very short distances. In supercomputers, the speed of light imposes a limit on how quickly data can be sent between processors.[11] If a processor operates at 1 gigahertz, a signal can only travel a maximum of about 30 centimetres (1 ft) in a single cycle. Processors must therefore be placed close to each other to minimize communication latencies; this can cause difficulty with cooling. If clock frequencies continue to increase, the speed of light will eventually become a limiting factor for the internal design of single chips.[11]
↑ SI Brochure: The International System of Units (SI) (PDF) (9 ed.), BIPM, 2019, p. 128, retrieved 2020-01-12
↑ Cox, Brian; Forshaw, Jeff (2010). Why does E=mc2?: (and why should we care?). Da Capo. p. 2. ISBN 978-0-306-81911-7.
↑ Uzan, J-P; Leclercq, B (2008). The natural laws of the universe: understanding fundamental constants. Springer. pp. 43–4. ISBN 978-0387734545.
↑ Cohen, IB (1940). "Roemya and the first determination of the velocity of light (1676)". Isis. 31 (2): 327–79. doi:10.1086/347594. hdl:2027/uc1.b4375710. S2CID 145428377.
↑ "Touchant le mouvement de la lumiere trouvé par M. Rŏmer de l'Académie Royale des Sciences" (PDF). Journal des sçavans (in French): 233–36. 1676.
Translated in "On the motion of light by M. Romer". Philosophical Transactions of the Royal Society. 12 (136): 893–95. 1677. doi:10.1098/rstl.1677.0024. S2CID 186210345. (As reproduced in Hutton, C; Shaw, G (1809). "On the Motion of Light by M. Romer". In Pearson, R (ed.). The Philosophical Transactions of the Royal Society of London, from Their Commencement in 1665, in the Year 1800: Abridged. Vol. 2. London: C. & R. Baldwin. pp. 397–98. )
↑ 6.0 6.1 Bradley, J (1729). "Account of a new discoved Motion of the Fix'd Stars". Philosophical Transactions. 35: 637–660.
↑ at most 20.5 arcseconds Duffett-Smith, P (1988). Practical astronomy with your calculator. Cambridge University Press. p. 62. ISBN 0521356997.
↑ Pitjeva, EV; Standish, EM (2009). "Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit". Celestial Mechanics and Dynamical Astronomy. 103 (4): 365–372. Bibcode:2009CeMDA.103..365P. doi:10.1007/s10569-009-9203-8. S2CID 121374703.
↑ IAU Working Group on Numerical Standards for Fundamental Astronomy. "IAU WG on NSFA Current Best Estimates". US Naval Observatory. Archived from the original on 2009-12-08. Retrieved 2009-09-25.
↑ "NPL's Beginner's Guide to Length". UK National Physical Laboratory. Archived from the original on 2010-08-31. Retrieved 2009-10-28.
↑ 11.0 11.1 Parhami, B (1999). Introduction to parallel processing: algorithms and architectures. Plenum Press. p. 5. ISBN 9780306459702. and (2009) "Software transactional memories: an approach for multicore programming" in 10th International Conference, PaCT 2009, Novosibirsk, Russia, August 31-September 4, 2009. Malyshkin, V Springer.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Speed_of_light&oldid=8066868" |
Introduction - Kickflow Documentation
What is Kickflow?
Kickflow is a decentralized platform that enables community funding for projects on Tezos. Through the concept of Quadratic Funding, it gives the community the power to take the best projects forward!
Besides being a general crowdfunding platform, Kickflow conducts periodic match funding rounds, where entries receive an optimal share from a sponsor-funded pool.
Lifecycle of a Matching Round
Kickflow is governed by a DAO that manages the funding rounds and operates on a token voting mechanism using the Kickflow Governance Token ($KFL).
The need for Quadratic Funding
Match funding schemes for public goods suffer from a fundamental weakness- the match amount for a project is proportional to the sum of contributions it receives during a stipulated period of time. This has its benefits, but it entirely rules out individual preferences.
Think of a situation where two projects A and B have received a contribution of $1000 from the community. A has received it from 10 unique contributors, whereas B from just 2 unique contributors. Even though it signifies A has a better reach and preference than B, both of them end up receiving an equal amount of match from the funding pool.
To solve this problem, we bring in CLR* matching. Here, instead of being linear, the match is calculated as the square of the sum of the square roots of the individual contributions.
M_{i} = ( \sum_{j = 0}^n \sqrt{C_{j}} )^{2}
Let's say A received $100 each from 10 contributors whereas B received $500 from 2 contributors. If we use CLR matching, A's match would be $10,000 whereas B's match would be $2000. Here clearly the individual preferences have been factored in.
For a more in-depth study of Quadratic Funding, view the original paper by Vitalik Buterin.
*CLR refers to Capital-constrained Liberal Radicalism. It brings about a democratic essence to capital distribution amongst public goods. |
Mapped solid axle suspension - Simulink - MathWorks Switzerland
Solid Axle Suspension - Mapped
Axle and wheels lumped principal moments of inertia about longitudinal axis, AxlIxx
Axle and wheels lumped mass, AxlM
Track hardpoint coordinates relative to axle center, TrackCoords
Suspension hardpoint coordinates relative to axle center, SuspCoords
Wheel and axle interface compliance constant, KzWhlAxl
Wheel and axle interface compliance preload, F0zWhlAxl
Wheel and axle interface damping constant, CzWhlAxl
Mapped solid axle suspension
The Solid Axle Suspension - Mapped block implements a mapped solid axle suspension for multiple axles with multiple tracks per axle.
The block models the suspension compliance, damping, and geometric effects as functions of the track positions and velocities, with axle-specific compliance and damping parameters. Using the track position and velocity, the block calculates the vertical track position and suspension forces on the vehicle and wheel. The block uses the Z-down (defined in SAE J670) and a solid axle coordinate system. The solid axle coordinate system, shown here, is aligned with the Z-down vehicle coordinate system, with the x-axis in the direction of forward vehicle motion.
Suspension parameters.
The block contains energy-storing spring elements and energy-dissipating damper elements. The block also stores energy via the axle roll angular acceleration and axle center of mass vertical and lateral acceleration.
Two axles.
Two tracks per axle.
Steering angle input for both tracks on the front axle.
The block uses a lookup table that relates the vertical damping and compliance to the suspension height, suspension height rate of change, and steering angle. You can calibrate the wheel force lookup table so that steering angle changes from the nominal center position generate a force that increases the vehicle height. Specifically, the block:
Longitudinal and lateral displacement and velocity of the vehicle.
Longitudinal and lateral displacement and velocity of the track.
Vertical wheel forces applied to the vehicle.
Suspension forces applied to the axle center.
Vertical displacements and velocities of the vehicle and track.
Longitudinal, lateral, and vertical suspension forces and moments applied to the vehicle.
Longitudinal, lateral, and vertical suspension forces and moments applied to the wheel.
To calculate the dynamics of the axle, the block implements these equations. The block neglects the effects of:
Lateral and longitudinal translational velocity.
Angular velocity about the vertical and lateral axes.
\begin{array}{l}\left[\begin{array}{c}{\stackrel{¨}{x}}_{a}\\ {\stackrel{¨}{y}}_{a}\\ {\stackrel{¨}{z}}_{a}\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}{F}_{xa}\\ {F}_{ya}\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}{\stackrel{˙}{x}}_{a}\\ {\stackrel{˙}{y}}_{a}\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}0\\ 0\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ g\end{array}\right]=\left[\begin{array}{c}0\\ p{\stackrel{˙}{z}}_{a}\\ \frac{{F}_{za}}{{M}_{a}}+g\end{array}\right]\\ \\ \left[\begin{array}{c}\stackrel{˙}{p}\\ \stackrel{˙}{q}\\ \stackrel{˙}{r}\end{array}\right]=\left[\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]-\left[\begin{array}{c}p\\ q\\ r\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}\\ =\left[\left[\begin{array}{c}{M}_{x}\\ 0\\ 0\end{array}\right]-\left[\begin{array}{c}p\\ q\\ 0\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}=\left[\begin{array}{c}\frac{{M}_{x}}{{I}_{xx}}\\ 0\\ 0\end{array}\right]\end{array}
For the forces and moments, the block uses lookup tables.
\begin{array}{l}{F}_{w{z}_{a,t}}=f\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)\\ {M}_{v{z}_{a,t}}=f\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)\end{array}
The suspension forces and moments applied to the vehicle are equal to the suspension forces and moments applied to the wheel.
\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=-{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}
\left[\begin{array}{ccc}{\xi }_{a,t}& {\eta }_{a,t}& {\zeta }_{a,t}\end{array}\right]={G}_{alookup}f\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{\delta }_{whlstee{r}_{a,t}}={\delta }_{stee{r}_{a,t}}+{G}_{alookup}f\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{H}_{a,t}=-\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}}-\mathrm{median}\left(f_susp_dz_bp\right)\right)
{z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}
\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlVz}={\stackrel{˙}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{˙}{z}}_{{w}_{1,1}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]
\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]
\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]
\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]
\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{v}\\ {\stackrel{˙}{y}}_{v}\\ {\stackrel{˙}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{v}_{1,1}}& {\stackrel{˙}{x}}_{{v}_{1,2}}& {\stackrel{˙}{x}}_{{v}_{2,1}}& {\stackrel{˙}{x}}_{{v}_{2,2}}\\ {\stackrel{˙}{y}}_{{v}_{1,1}}& {\stackrel{˙}{y}}_{{v}_{1,2}}& {\stackrel{˙}{y}}_{{v}_{2,1}}& {\stackrel{˙}{y}}_{{v}_{2,2}}\\ {\stackrel{˙}{z}}_{{v}_{1,1}}& {\stackrel{˙}{z}}_{{v}_{1,2}}& {\stackrel{˙}{z}}_{{v}_{2,1}}& {\stackrel{˙}{z}}_{{v}_{2,2}}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
Wheel angles according to the axle.
\mathrm{WhlAng}\left[1,...\right]=\xi =\left[{\xi }_{a,t}\right]
\mathrm{WhlAng}\left[2,...\right]=\eta =\left[{\eta }_{a,t}\right]
\mathrm{WhlAng}\left[3,...\right]=\zeta =\left[{\zeta }_{a,t}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
[1 0]—For a two-axle vehicle, enables axle 1 steering and disables axle 2 steering
[1 1]—For a two-axle vehicle, enables axle 1 and axle 2 steering
Setting an element of the Steered axle enable by axle, StrgEnByAxl vector to 1:
Creates input port StrgAng.
Creates these parameters
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
Axle and wheels lumped principal moments of inertia about longitudinal axis, AxlIxx — Inertia
Axle and wheels lumped principal moments of inertia about longitudinal axis, AxleIxx a, in kg*m^2.
Axle and wheels lumped mass, AxlM — Mass
Axle and wheels lumped mass, a, in kg.
Track hardpoint coordinates relative to axle center, TrackCoords — Point
[0 0 0 0;-1 1 -1 1;0 0 0 0] (default) | array
Track hardpoint coordinates, Tct, along the solid axle x, y, and z-axes, in m.
For example, for a two-axle vehicle with two tracks per axle, the TrackCoords array:
Contains four track hardpoint coordinates according to their axle and track locations.
T{c}_{t}=\left[\begin{array}{cccc}{x}_{{w}_{1,1}}& {x}_{{w}_{1,2}}& {x}_{{w}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{{w}_{1,1}}& {y}_{{w}_{1,2}}& {y}_{{w}_{2,1}}& {y}_{{w}_{2,2}}\\ {z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
TrackCoords(1,1) 1 1 Solid axle x-axis
TrackCoords(1,2) 1 2
TrackCoords(2,1) 1 1 Solid axle y-axis
TrackCoords(3,1) 1 1 Solid axle z-axis
Suspension hardpoint coordinates relative to axle center, SuspCoords — Point
Suspension hardpoint coordinates, Sct, along the solid axle x-, y-, and z-axes, in m.
For example, for a two-axle vehicle with two tracks per axle, the SuspCoords array:
S{c}_{t}=\left[\begin{array}{cccc}{x}_{{s}_{1,1}}& {x}_{{s}_{1,2}}& {x}_{{s}_{2,1}}& {x}_{{s}_{2,2}}\\ {y}_{{s}_{1,1}}& {y}_{{s}_{1,2}}& {y}_{{s}_{2,1}}& {y}_{{s}_{2,2}}\\ {z}_{{s}_{1,1}}& {z}_{{s}_{1,2}}& {z}_{{s}_{2,1}}& {z}_{{s}_{2,2}}\end{array}\right]
SuspCoords(1,1) 1 1 Solid axle x-axis
SuspCoords(1,2) 1 2
SuspCoords(2,1) 1 1 Solid axle y-axis
SuspCoords(3,1) 1 1 Solid axle z-axis
Wheel and axle interface compliance constant, KzWhlAxl — Spring rate
Wheel and axle interface compliance constant, Kz, in N/m.
Wheel and axle interface compliance preload, F0zWhlAxl — Spring rate
Wheel and axle interface compliance preload, F0z, in N.
Wheel and axle interface damping constant, CzWhlAxl — Damping
Wheel and axle interface damping constant, Cz, in m.
Solid Axle Suspension | Solid Axle Suspension - Coil Spring | Solid Axle Suspension - Leaf Spring |
Isopentenyl-diphosphate delta isomerase - Wikipedia
Isopentenyl-diphosphate delta isomerase octamer, Thermus thermophilus
Isopentenyl-pyrophosphate delta isomerase 1
Isopentenyl pyrophosphate isomerase (IPP isomerase), also known as Isopentenyl-diphosphate delta isomerase,[1] is an isomerase that catalyzes the conversion of the relatively un-reactive isopentenyl pyrophosphate (IPP) to the more-reactive electrophile dimethylallyl pyrophosphate (DMAPP). This isomerization is a key step in the biosynthesis of isoprenoids through the mevalonate pathway and the MEP pathway.
isopentenyl diphosphate
{\displaystyle \rightleftharpoons }
dimethylallyl diphosphate
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is isopentenyl-diphosphate Delta3-Delta2-isomerase. Other names in common use include isopentenylpyrophosphate Delta-isomerase, methylbutenylpyrophosphate isomerase, and isopentenylpyrophosphate isomerase.[2][3][4]
2.1 Structural studies
IPP isomerase catalyzes the isomerization of IPP to DMAPP by an antarafacial transposition of hydrogen.[5][6] The empirical evidence suggests that this reaction proceeds by a protonation/deprotonation mechanism, with the addition of a proton to the re-face of the inactivated C3-C4 double bond resulting in a transient carbocation intermediate.[7][8] The removal of the pro-R proton from C2 forms the C2-C3 double bond of DMAPP.
The mechanism for the isomerization between IPP and DMAPP. Generic proton donors and acceptors are shown because the identities of the amino acids that carry out these functions have not conclusively been established.
A cartoon diagram of human IPP isomerase with the catalytic cysteine residue (Cys87) in red and the catalytic glutamic acid residue (Glu149) in blue (RCSB Protein Data Bank accession number 2ICJ).
Crystallographic studies have observed that the active form of IPP isomerase is a monomer with alternating α-helices and β-sheets.[9][10] The active site of IPP isomerase is deeply buried within the enzyme and consists of a glutamic acid residue and a cysteine residue that interact with opposite sides of the IPP substrate, consistent with the antarafacial stereochemistry of isomerization.[9][11] The origin of the initial protonation step has not been conclusively established. Recent evidence suggests that the glutamic acid residue is involved in the protonating step despite the observation that its carboxylic acid side-chain is stabilized in its carboxylate form.[12] This discrepancy has been addressed by the discovery of a water molecule in the active site of human IPP isomerase, suggesting a mechanism where the glutamine residue polarizes the double bond of IPP and makes it more susceptible to protonation by water.[13]
IPP isomerase also requires a divalent cation to fold into its active conformation. The enzyme contains several amino acids, including the catalytic glutamate, that are involved in coordinating with Mg2+ or Mn2+.[9][14] The coordination of the metal cation to the glutamate residue stabilizes the carbiocation intermediate after protonation.
As of late 2007, 25 structures have been solved for this class of enzymes, with PDB accession codes 1HX3, 1HZT, 1I9A, 1NFS, 1NFZ, 1OW2, 1P0K, 1P0N, 1PPV, 1PPW, 1PVF, 1Q54, 1R67, 1VCF, 1VCG, 1X83, 1X84, 2B2K, 2DHO, 2G73, 2G74, 2I6K, 2ICJ, 2ICK, and 2PNY.
The protonation of an inactivated double bond is rarely seen in nature, highlighting the unique catalytic mechanism of IPP isomerase. The isomerization of IPP to DMAPP is a crucial step in the synthesis of isoprenoids and isoprenoid-derivatives, compounds that play vital roles in the biosynthetic pathways of all living organisms.[15] Because of the importance of the melavonate pathway in isoprenoid biosynthesis, IPP isomerase is found in a variety of different cellular compartments, including plastids and mammalian mitochondria.[16]
Mutations in IDI1, the gene that codes for IPP isomerase 1, have been implicated in decreased viability in a number of organisms, including the yeast Saccharomyces cerevisiae, the nematode Caenorhabditis elegans and the plant Arabidopsis thaliana.[17][18][19] While there have been no evidence directly implicating IDI1 mutations in human disease, genomic analysis has identified a copy-number gain near two IPP isomerase genes in a substantial proportion of patients with sporadic amyotrophic lateral sclerosis, suggesting that the isomerase may play a role in this disease.[20]
^ "IDI1 - Isopentenyl-diphosphate Delta-isomerase - Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (Baker's yeast) - IDI1 gene & protein". UniProt. Retrieved 6 June 2016.
^ Kaneda K, Kuzuyama T, Takagi M, Hayakawa Y, Seto H (2001). "An unusual isopentenyl diphosphate isomerase found in the mevalonate pathway gene cluster from Streptomyces sp. strain CL190". Proc. Natl. Acad. Sci. U.S.A. 98 (3): 932–7. doi:10.1073/pnas.020472198. PMC 14687. PMID 11158573.
^ Bishop JM (1983). "Cellular oncogenes and retroviruses". Annu. Rev. Biochem. 52: 301–54. doi:10.1146/annurev.bi.52.070183.001505. PMID 6351725.
^ Agranoff BW, Eggerer H, Henning U, Lynen F (1960). "Biosynthesis of terpenes. VII. Isopentenyl pyrophosphate isomerase". J. Biol. Chem. 235 (2): 326–32. doi:10.1016/S0021-9258(18)69523-7. PMID 13792054.
^ Cornforth JW, Cornforth RH, Popják G, Yengoyan L (Sep 1966). "Studies on the biosynthesis of cholesterol. XX. Steric course of decarboxylation of 5-pyrophosphomevalonate and of the carbon to carbon bond formation in the biosynthesis of farnesyl pyrophosphate". The Journal of Biological Chemistry. 241 (17): 3970–3987. doi:10.1016/S0021-9258(18)99800-5. PMID 4288360.
^ Cornforth RH, Popják G (1969). "Chemical syntheses of substrates of sterol biosynthesis". In Raymond BC (ed.). Methods in Enzymology. Vol. 15. Academic Press. pp. 359–390.
^ Reardon JE, Abeles RH (Sep 1986). "Mechanism of action of isopentenyl pyrophosphate isomerase: evidence for a carbonium ion intermediate". Biochemistry. 25 (19): 5609–5616. doi:10.1021/bi00367a040. PMID 3022798.
^ Street IP, Christensen DJ, Poulter CD (1990). "Hydrogen exchange during the enzyme-catalyzed isomerization of isopentenyl diphosphate and dimethylallyl diphosphate". Journal of the American Chemical Society. 112 (23): 8577–8578. doi:10.1021/ja00179a049.
^ a b c Hall NR, Fish DE, Hunt N, Goldin RD, Guillou PJ, Monson JR (Jun 1992). "Is the relationship between angiogenesis and metastasis in breast cancer real?". Surgical Oncology. 1 (3): 223–229. doi:10.1016/0960-7404(92)90068-v. PMID 1285217.
^ Zheng W, Sun F, Bartlam M, Li X, Li R, Rao Z (Mar 2007). "The crystal structure of human isopentenyl diphosphate isomerase at 1.7 A resolution reveals its catalytic mechanism in isoprenoid biosynthesis". Journal of Molecular Biology. 366 (5): 1447–1458. doi:10.1016/j.jmb.2006.12.055. PMID 17250851.
^ Street IP, Coffman HR, Baker JA, Poulter CD (Apr 1994). "Identification of Cys139 and Glu207 as catalytically important groups in the active site of isopentenyl diphosphate:dimethylallyl diphosphate isomerase". Biochemistry. 33 (14): 4212–4217. doi:10.1021/bi00180a014. PMID 7908830.
^ Wouters J, Oudjama Y, Barkley SJ, Tricot C, Stalon V, Droogmans L, Poulter CD (Apr 2003). "Catalytic mechanism of Escherichia coli isopentenyl diphosphate isomerase involves Cys-67, Glu-116, and Tyr-104 as suggested by crystal structures of complexes with transition state analogues and irreversible inhibitors". The Journal of Biological Chemistry. 278 (14): 11903–11908. doi:10.1074/jbc.M212823200. PMID 12540835.
^ Zhang C, Liu L, Xu H, Wei Z, Wang Y, Lin Y, Gong W (Mar 2007). "Crystal structures of human IPP isomerase: new insights into the catalytic mechanism". Journal of Molecular Biology. 366 (5): 1437–1446. doi:10.1016/j.jmb.2006.10.092. PMID 17137593.
^ Bonanno JB, Edo C, Eswar N, Pieper U, Romanowski MJ, Ilyin V, Gerchman SE, Kycia H, Studier FW, Sali A, Burley SK (Nov 2001). "Structural genomics of enzymes involved in sterol/isoprenoid biosynthesis". Proceedings of the National Academy of Sciences of the United States of America. 98 (23): 12896–12901. Bibcode:2001PNAS...9812896B. doi:10.1073/pnas.181466998. PMC 60796. PMID 11698677.
^ Bach TJ (Mar 1995). "Some new aspects of isoprenoid biosynthesis in plants--a review". Lipids. 30 (3): 191–202. doi:10.1007/BF02537822. PMID 7791527. S2CID 3999323.
^ Ramos-Valdivia AC, van der Heijden R, Verpoorte R (Dec 1997). "Isopentenyl diphosphate isomerase: a core enzyme in isoprenoid biosynthesis. A review of its biochemistry and function". Natural Product Reports. 14 (6): 591–603. doi:10.1039/np9971400591. PMID 9418296.
^ Mayer MP, Hahn FM, Stillman DJ, Poulter CD (Sep 1992). "Disruption and mapping of IDI1, the gene for isopentenyl diphosphate isomerase in Saccharomyces cerevisiae". Yeast. 8 (9): 743–748. doi:10.1002/yea.320080907. PMID 1441751. S2CID 19430360.
^ Yochem J, Hall DH, Bell LR, Hedgecock EM, Herman RK (Apr 2005). "Isopentenyl-diphosphate isomerase is essential for viability of Caenorhabditis elegans". Molecular Genetics and Genomics. 273 (2): 158–166. doi:10.1007/s00438-004-1101-x. PMID 15765206. S2CID 1637634.
^ Okada K, Kasahara H, Yamaguchi S, Kawaide H, Kamiya Y, Nojiri H, Yamane H (Apr 2008). "Genetic evidence for the role of isopentenyl diphosphate isomerases in the mevalonate pathway and plant development in Arabidopsis". Plant & Cell Physiology. 49 (4): 604–616. doi:10.1093/pcp/pcn032. PMID 18303110.
^ Kato T, Emi M, Sato H, Arawaka S, Wada M, Kawanami T, Katagiri T, Tsuburaya K, Toyoshima I, Tanaka F, Sobue G, Matsubara K (Nov 2010). "Segmental copy-number gain within the region of isopentenyl diphosphate isomerase genes in sporadic amyotrophic lateral sclerosis". Biochemical and Biophysical Research Communications. 402 (2): 438–442. doi:10.1016/j.bbrc.2010.10.056. PMID 20955688.
isopentenyldiphosphate+delta-isomerase at the US National Library of Medicine Medical Subject Headings (MeSH)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Isopentenyl-diphosphate_delta_isomerase&oldid=1042196773" |
How to Calculate Gross Profit Margin: 8 Steps (with Pictures)
1 Calculating Gross Profit Margin
Gross profit is a way to compare the cost of the goods your company sells and the income derived from those goods. All you need for the gross profit formula is your total revenue, and the cost of goods sold (COGS). You can use your gross profit margin to quickly and meaningfully compare your company to your competitors, the industry as a whole, or even your own past performance. Our how-to guide breaks it down for you, including examples.
Calculating Gross Profit Margin Download Article
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/59\/Calculate-Gross-Profit-Margin-Step-1-Version-4.jpg\/v4-460px-Calculate-Gross-Profit-Margin-Step-1-Version-4.jpg","bigUrl":"\/images\/thumb\/5\/59\/Calculate-Gross-Profit-Margin-Step-1-Version-4.jpg\/aid1481899-v4-728px-Calculate-Gross-Profit-Margin-Step-1-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Look up Net Sales and Cost of Goods Sold. The company's income statement lists both values.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/ea\/Calculate-Gross-Profit-Margin-Step-2-Version-4.jpg\/v4-460px-Calculate-Gross-Profit-Margin-Step-2-Version-4.jpg","bigUrl":"\/images\/thumb\/e\/ea\/Calculate-Gross-Profit-Margin-Step-2-Version-4.jpg\/aid1481899-v4-728px-Calculate-Gross-Profit-Margin-Step-2-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Gross Profit Margin = (Net Sales - Cost of Goods Sold) ÷ Net Sales.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/f\/f3\/Calculate-Gross-Profit-Margin-Step-3-Version-4.jpg\/v4-460px-Calculate-Gross-Profit-Margin-Step-3-Version-4.jpg","bigUrl":"\/images\/thumb\/f\/f3\/Calculate-Gross-Profit-Margin-Step-3-Version-4.jpg\/aid1481899-v4-728px-Calculate-Gross-Profit-Margin-Step-3-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Example. A company makes $4,000 selling goods that cost $3,000 to produce. Its gross profit margin is
{\displaystyle {\frac {4000-3000}{4000}}={\frac {1}{4}}}
Understand Gross Profit Margin. The Gross Profit Margin (GPM) is the percentage of revenue a company has left over after paying direct costs of producing goods.[1] X Research source All other expenditures (including shareholder dividends) must come out of this percentage. This makes the GPM a good indicator of profitability.
Define Net Sales. A company's net sales equal its total sales minus returns, allowances for damaged merchandise, and discounts.[2] X Research source This is a more accurate measure of incoming money than total sales alone.
Measure Costs of Goods Sold. Abbreviated COGS, this figure includes the cost of materials, labor, and other expenses directly related to the production of goods or services.[3] X Research source It does not include costs of distribution, labor that does not go into goods production, or other indirect costs.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/e3\/Calculate-Gross-Profit-Margin-Step-7.jpg\/v4-460px-Calculate-Gross-Profit-Margin-Step-7.jpg","bigUrl":"\/images\/thumb\/e\/e3\/Calculate-Gross-Profit-Margin-Step-7.jpg\/aid1481899-v4-728px-Calculate-Gross-Profit-Margin-Step-7.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Avoid confusing Gross Profit with GPM. The Gross Profit equals the Net Sales minus the Cost of Goods Sold. This is expressed in dollars or other units of currency. The formula above converts Gross Profit to GPM, a percentage, for easy comparison with other companies.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/1\/1b\/Calculate-Gross-Profit-Margin-Step-8.jpg\/v4-460px-Calculate-Gross-Profit-Margin-Step-8.jpg","bigUrl":"\/images\/thumb\/1\/1b\/Calculate-Gross-Profit-Margin-Step-8.jpg\/aid1481899-v4-728px-Calculate-Gross-Profit-Margin-Step-8.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Understand why these figures are important. Investors look at Gross Profit Margin to see how efficiently a company can use its resources. If one company has a GPM of 10% and a second company has a GPM of 20%, the second company is making twice as much money per dollar spent on goods. Assuming other costs are roughly equal between the two companies, the second company is probably the better investment opportunity.
It's best to compare companies in the same sector. Some goods and services have a lower average profit margin than others.
How do I calculate the profit, expressed as a percentage, that the vendor makes when selling a 180 airtime voucher?
Subtract the cost of the voucher from the price received from its sale. the difference is gross profit. To calculate the Gross Profit Margin percentage, divide the price received for the sale by the gross profit and convert the decimals into a percentage. For example, 0.01 equals 1%, 0.1 equals 10 percent, and 1.0 equals 100 percent.
How do I calculate the gross as a percentage if I have the GP in dollars and the net sales in dollars?
Net Sales is Gross Sales less returns, discounts, and allowances for damaged or missing goods. Calculate Gross Sales by adding back those amounts if known to Net Sales, then divide your Gross Profit by Gross Sales. This will provide the answers in decimals that can be converted into a percentage.
To calculate gross profit margin, start by subtracting the cost of goods sold from the net sales. Then, divide the difference by the net sales to find the gross profit margin. If you're not sure what the net sales and cost of goods sold are, you can look them up on the company's income statement. For more tips from our Financial co-author, like how to interpret gross profit margin, scroll down!
Italiano:Calcolare il Margine Lordo
Bashir Qabaha
"I have a certificate in accounting field, this article is really helped me by understanding how GPM is important, using it for comparison by several companies."..." more
"Thanks to the wikiHow team for a wonderful article regarding basic calculation of business profit. It is very easy to understand with examples."..." more
Leah Creed
"I always look up wikiHow. It has images and makes explanations clear with easy directions to follow."
"The formulas helped me understand GPM and revise for my accounting principles test."
Miso T.
"As a novice, it helped me focus and understand the terms." |
Recover symbol timing phase using fourth-order nonlinearity method - Simulink - MathWorks France
MSK-Type Signal Timing Recovery
Timing Phase Recovery sublibrary of Synchronization
The MSK-Type Signal Timing Recovery block recovers the symbol timing phase of the input signal using a fourth-order nonlinearity method. This block implements a general non-data-aided feedback method that is independent of carrier phase recovery but requires prior compensation for the carrier frequency offset. This block is suitable for systems that use baseband minimum shift keying (MSK) modulation or Gaussian minimum shift keying (GMSK) modulation.
By default, the block has one input port. The input signal could be (but is not required to be) the output of a receive filter that is matched to the transmitting pulse shape, or the output of a lowpass filter that limits the amount of noise entering this block.
This block accepts a scalar-valued or column vector input signal. The input uses N samples to represent each symbol, where N > 1 is the Samples per symbol parameter.
For a column vector input signal, the block operates in single-rate processing mode. In this mode, the output signal inherits its sample rate from the input signal. The input length must be a multiple of N.
For a scalar input signal, the block operates in multirate processing mode. In this mode, the input and output signals have different sample rates. The output sample rate equals N multiplied by the input sample rate.
This block accepts input signals of type Double or Single
If you set the Reset parameter to On nonzero input via port, then the block has a second input port, labeled Rst. The Rst input determines when the timing estimation process restarts, and must be a scalar.
If the input signal is a scalar value, the sample time of the Rst input equals the symbol period
If the input signal is a column vector, the sample time of the Rst input equals the input port sample time
This block accepts reset signals of type Double or Boolean
The block has two output ports, labeled Sym and Ph:
The Sym output is the result of applying the estimated phase correction to the input signal. This output is the signal value for each symbol, which can be used for decision purposes. The values in the Sym output occur at the symbol rate:
For a column vector input signal of length N*R, the Sym output is a column vector of length R having the same sample rate as the input signal.
For a scalar input signal, the sample rate of the Sym output equals N multiplied by the input sample rate.
The Ph output gives the phase estimate for each symbol in the input.
The Ph output contains nonnegative real numbers less than N. Noninteger values for the phase estimate correspond to interpolated values that lie between two values of the input signal. The sample time of the Ph output is the same as that of the Sym output.
If the Ph output is very close to either zero or Samples per symbol, or if the actual timing phase offset in your input signal is very close to zero, then the block's accuracy might be compromised by small amounts of noise or jitter. The block works well when the timing phase offset is significant rather than very close to zero.
The output signal inherits its data type from the input signal.
When the input signal is a vector, this block incurs a delay of two symbols. When the input signal is a scalar, this block incurs a delay of three symbols.
The type of modulation in the system. Choices are MSK and GMSK.
The number of samples, N, that represent each symbol in the input signal. This must be greater than 1.
Error update gain
A positive real number representing the step size that the block uses for updating successive phase estimates. Typically, this number is less than 1/N, which corresponds to a slowly varying phase.
This parameter is tunable in normal mode, Accelerator mode and Rapid Accelerator mode. If you use the Simulink® Coder™ rapid simulation (RSIM) target to build an RSIM executable, then you can tune the parameter without recompiling the model. For more information, see Tunable Parameters (Simulink).
Determines whether and under what circumstances the block restarts the phase estimation process. Choices are None, Every frame, and On nonzero input via port. The last option causes the block to have a second input port, labeled Rst.
This block's algorithm extracts timing information by passing the sampled baseband signal through a fourth-order nonlinearity followed by a digital differentiator whose output is smoothed to yield an error signal. The algorithm then uses the error signal to make the sampling adjustments.
More specifically, this block uses a timing error detector whose result for the kth symbol is e(k), given in [2] by
\begin{array}{c}e\left(k\right)={\left(}^{-}\mathrm{Re}\left\{{r}^{2}\left(kT-{T}_{s}+{d}_{k-1}\right){r}^{*2}\left(\left(k-1\right)T-{T}_{s}+{d}_{k-2}\right)\right\}\\ -{\left(}^{-}\mathrm{Re}\left\{{r}^{2}\left(kT+{T}_{s}+{d}_{k-1}\right){r}^{*2}\left(\left(k-1\right)T+{T}_{s}+{d}_{k-1}\right)\right\}\end{array}
r is the block's input signal
T is the symbol period
Ts is the sampling period
* means complex conjugate
dk is the phase estimate for the kth symbol
D is 1 for MSK and 2 for Gaussian MSK modulation
[1] D'Andrea, A. N., U. Mengali, and R. Reggiannini, "A Digital Approach to Clock Recovery in Generalized Minimum Shift Keying," IEEE Transactions on Vehicular Technology, Vol. 39, No. 3, August 1990, pp. 227-234.
[2] Mengali, Umberto and Aldo N. D'Andrea, Synchronization Techniques for Digital Receivers, New York, Plenum Press, 1997. |
A\mathrm{sin}\left(x\right)
A
x
t
n
p
p
p
The trace=n option specifies that a number of previous frames of the animation be kept visible. When
n
n+1
n=5
When
is a list of integers, then the frames in those positions are the frames that remain visible. Each integer in
n=0
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=5,\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=[30,35,40,45,50],\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot3d},[A\left({x}^{2}+{y}^{2}\right),x=-3..3,y=-3..3],A=-2..2,\mathrm{style}=\mathrm{patchcontour}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+{y}^{2}={r}^{2},x=-3..3,y=-3..3],r=1..3,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+Axy-{y}^{2}=1,x=-2..2,y=-3..3],A=-2..2,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{sin}\left(t\right),\mathrm{sin}\left(t\right)\mathrm{exp}\left(-\frac{t}{5}\right)],t=0..x],x=0..6\mathrm{\pi },\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t=0..A]],A=0..2\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\frac{1-{t}^{2}}{1+{t}^{2}},\frac{2t}{1+{t}^{2}},t=-10..A]],A=-10..10,\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50,\mathrm{view}=[-1..1,-1..1]\right)
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{numpoints}=100,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),\left(2+\mathrm{sin}\left(A\right)\right)t],t=0..20,\mathrm{opts}],A=0..2\mathrm{\pi }\right)
B≔\mathrm{plot3d}\left(1-{x}^{2}-{y}^{2},x=-1..1,y=-1..1,\mathrm{style}=\mathrm{patchcontour}\right):
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[t,t,1-2{t}^{2}],t=-1..A,\mathrm{opts}],A=-1..1,\mathrm{frames}=11,\mathrm{background}=B\right)
\mathrm{animate}\left(\mathrm{ball},[0,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=100\right)
\mathrm{sinewave}≔\mathrm{plot}\left(\mathrm{sin}\left(x\right),x=0..4\mathrm{\pi }\right):
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{background}=\mathrm{sinewave},\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{trace}=10,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(F,[\mathrm{\theta }],\mathrm{\theta }=0..2\mathrm{\pi },\mathrm{background}=\mathrm{plot}\left([\mathrm{cos}\left(t\right)-2,\mathrm{sin}\left(t\right),t=0..2\mathrm{\pi }]\right),\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{none}\right) |
Turney, ACL 2002 - Cohen Courses
Turney, ACL 2002
Turney, P. D. 2002. Thumbs up or thumbs down? Semantic orientation applied to unsupervised classification of reviews. In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, 417–424.
This is an early and influential paper presenting an unsupervised approach to review classification. There are three basic ideas introduced here.
One key idea is to score the polarity of a review based on the total polarity of the phrases in it.
A second idea is to use patterns of part of speech tags to pick out phrases that are likely to be meaningful and unambiguous with respect to semantic orientation (e.g. ADJ NOUN might pick out "good service" or "delicious desserts").
Finally, these potentially-meaningful phrases are then scored using pointwise mutual information (PMI) to seed words on known polarity. Specifically, Turney uses PMI to compare each phrase to the words "excellent" or "poor", and then uses these distances to give an overall score for the polarity to each phrase, based on the difference of its PMI with "excellent" to the PMI with "poor". A very large corpus was used here (the Web, via queries to a search engine), which appears to be important in making this simple technique work.
The algorithm takes a written review as an input. First it assigns a POS tag to each word in the review to identify adjective or adverb phrases in the input review. They have used PMI-IR algorithm to estimate the semantic orientation of a phrase. The Pointwise Mutual Information (PMI) between two words
{\displaystyle w_{1}}
{\displaystyle w_{2}}
{\displaystyle PMI(w_{1},w_{2})=log_{2}(p(w_{1}\ and\ w_{2})/p(w_{1})p(w_{2}))}
{\displaystyle p(w_{1},w_{2})}
{\displaystyle w_{1}}
{\displaystyle w_{2}}
co-occur. They have defined the semantic orientation of a phrase as follow:
{\displaystyle SO(phrase)=PMI(phrase,'excellent')-PMI(phrase,'poor')}
We can modify the above definition to obtain the following formula:
{\displaystyle SO(phrase)=log_{2}\left({\frac {hits(phrase\ NEAR\ 'excellent')hits('poor')}{hits(phrase\ NEAR\ 'poor')hits('excellent')}}\right)}
where operator NEAR means that the two phrases should be appeared close to each other in the corpus. Using the above formula they have calculated the average semantic orientation for a review. They have shown that the value of average semantic orientation for phrases in the items that are tagged as "recommended" by the users are usually positive and those that are tagged as "not recommended" are usually negative.
This approach was fairly successful on a range of review-classification tasks: it achieved accuracy of between 65% and 85% in predicting an author-assigned "recommended" flag for Epinions ratings for eight diverse products, ranging from cars to movies. Many later writers used several key ideas from the paper, including: treating polarity prediction as a document-classification problem; classifying documents based on likely-to-be-informative phrases; and using unsupervised or semi-supervised learning methods.
The widely cited Pang et al EMNLP 2002 paper was influenced by this paper - but considers supervised learning techniques. The choice of movie reviews as the domain was suggested by the (relatively) poor performance of Turney's method on movies.
An interesting follow-up paper is Turney and Littman, TOIS 2003 which focuses on evaluation of the technique of using PMI for predicting the semantic orientation of words.
Retrieved from "http://curtis.ml.cmu.edu/w/courses/index.php?title=Turney,_ACL_2002&oldid=16525" |
Basic Science for Class 8 Science Chapter 15 - Natural Resources
Basic Science Solutions for Class 8 Science Chapter 15 Natural Resources are provided here with simple step-by-step explanations. These solutions for Natural Resources are extremely popular among Class 8 students for Science Natural Resources Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Basic Science Book of Class 8 Science Chapter 15 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Basic Science Solutions. All Basic Science Solutions for class Class 8 Science are prepared by experts and are 100% accurate.
The natural resources that cannot be renewed or replenished by natural processes are called non-renewable resources. Examples: Coal and petroleum
Plantations cannot make up for the loss of primary forests because these forests have evolved over centuries. Moreover, forests consist of a vast variety of organisms that are not present in planted forests.
Natural causes of deforestation are:
(c) Storms
Human activities that cause deforestation are:
(a) Extraction of timber
(b) Production of paper
Forests protect soil as the leaves of plants cover the soil and protect it from direct impact of rain. Also, the roots of the plants keep soil in place. Thus, soil erosion is prevented.
Consequences of deforestation are:
(a) It causes soil erosion.
(b) It can lead to floods and droughts.
(c) It can alter the climate of a place.
Coal, petroleum and natural gas are called fossil fuels because they are formed by the fossilisation of the living organisms or remains of plants and animals that are trapped between layers of rocks.
Commercial energy refers to the energy that is bought or sold. It includes energy obtained from coal, petroleum and natural gas.
The products of destructive distillation of coal are coal tar, coal gas, coke and ammoniacal liquor. Ammoniacal liquor is used for production of fertilisers.
Alternative sources of energy that can be used for generation of electricity are biomass energy, solar energy, wind energy, geothermal energy and hydroelectricity.
Following measures can be taken to bring down the number of trees being cut annually for the production of paper:
(a) Reducing wasteful consumption of paper
(b) Recycling of paper, i.e., using the waste paper to make a new paper
Affects of deforestation on climate are:
(a) It can cause global changes in the weather pattern by increasing the amount of carbon dioxide in the air.
(b) It can cause a worldwide increase in temperature, leading to global warming.
Petroleum and natural gas are formed by the decomposition of the remains of marine organisms by the bacteria and get deposited under the layers of sediments. Under the earth, a part of the product of decomposition got liquefied (petroleum) and other got converted into gas (natural gas) due to high temperature and pressure.
Shifting cultivation is a method of agriculture in which a forest is converted into crop land and pastures. In this method, a part of the forest is cleared by cutting the vegetation and burning it for growing crops on the cleared land. After the soil gets exhausted, the farmers move onto the other parts of the forest.
Shifting cultivation has destroyed forests in South and Central America. This also causes the soil to get exhausted and due to increasing population, the soil does not get adequate time (20
-
25 years) to restore and gain its fertility.
Large-scale logging for timber destroys forests in the following ways:
(a) Timber is used for making houses, furniture, crates, etc. With increase in population, the demand for timber has also increased. This has destroyed large parts of forests.
(b) Trees are cut down with electrical machines for industrial uses. This has destroyed forests in various ways. For every cubic metre of timber removed, about double that quantity of forest is destroyed.
Deforestation of mountains, slopes and uplands causes water to rush down to the rivers. This causes the rivers to overflow and lower lands get flooded. Example: Deforestation of Himalayas has been causing floods in India, Bangladesh and Pakistan every year.
Forests hold water and deforestation causes water to rush down very fast. Because of this, the uplands get deprived of water, soon after the rains. This leads to droughts in the uplands.
Example: Deforestation of Himalayas in India has changed the perennial streams into seasonal streams that run out of water soon after the monsoon.
Following stages are involved in the formation of coal:
(a) Peat: It is the first stage and has the lowest content of carbon. It is the most inferior type of coal and is formed by the decomposition of plant remains, buried under swamps, by anaerobic bacteria.
(b) Carbonisation: It is a process by which the carbon content of coal increases. Peat got pushed under the ground by earthquakes and volcanic eruptions. It experienced a high pressure and temperature as it sank, and this drove out gaseous product from the remains, increasing the carbon content. The deeper the remains sank, the more is their carbon content.
Anthracite is the best quality of coal with highest carbon content.
1. For the rural poor in developing countries, fuelwood is the primary source of energy.
2. The process by which the carbon content of coal increases is called carbonisation.
3. Energy derived from animal excreta and plant waste is called biomass energy.
4. A natural fountain of hot water and steam is called a geyser.
5. A worldwide increase in temperature is called global warming.
6. Petrol made from sources other than petroleum is called synthetic petrol.
(a) stabilised
In recent years, the total forest cover in our country has stabilised.
(a) north-eastern India
Shifting cultivation is practised in north-eastern India.
Coal fulfils about 65 percent of the energy requirement of our country.
Carbon black is a product of natural gas.
(c) animal fat and vegetable oil
Biodiesel is made from animal fat and vegetable oil. |
EUDML | Adjointability of operators on Hilbert -modules. EuDML | Adjointability of operators on Hilbert -modules.
Adjointability of operators on Hilbert
{C}^{*}
Manuilov, V.M.
Manuilov, V.M.. "Adjointability of operators on Hilbert -modules.." Acta Mathematica Universitatis Comenianae. New Series 65.2 (1996): 161-169. <http://eudml.org/doc/120024>.
@article{Manuilov1996,
author = {Manuilov, V.M.},
keywords = {non-self-dual Hilbert module; -algebra; inner products; dual module; compactness of operators; representability of functionals; -algebra},
title = {Adjointability of operators on Hilbert -modules.},
AU - Manuilov, V.M.
TI - Adjointability of operators on Hilbert -modules.
KW - non-self-dual Hilbert module; -algebra; inner products; dual module; compactness of operators; representability of functionals; -algebra
non-self-dual Hilbert module,
{C}^{*}
-algebra, inner products, dual module, compactness of operators, representability of functionals,
{C}^{*}
Individual linear operators as elements of algebraic systems
Operators in
{C}^{*}
- or von Neumann algebras
{C}^{*}
{W}^{*}
{C}^{*}
Articles by Manuilov |
EUDML | Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative. EuDML | Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative.
Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative.
Girg, Petr. "Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2000 (2000): Paper No. 63, 28 p., electronic only-Paper No. 63, 28 p., electronic only. <http://eudml.org/doc/121410>.
@article{Girg2000,
author = {Girg, Petr},
keywords = {-Laplacian; Leray-Schauder degree; Landesman-Lazer condition; -Laplacian},
title = {Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative.},
AU - Girg, Petr
TI - Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative.
KW - -Laplacian; Leray-Schauder degree; Landesman-Lazer condition; -Laplacian
David Arcoya, Naira del Toro, Semilinear elliptic problems with nonlinearities depending on the derivative
p
-Laplacian, Leray-Schauder degree, Landesman-Lazer condition,
p
Articles by Girg |
EUDML | New prime gaps between and . EuDML | New prime gaps between and .
New prime gaps between
{10}^{15}
5×{10}^{16}
Nyman, Bertil; Nicely, Thomas R.
Nyman, Bertil, and Nicely, Thomas R.. "New prime gaps between and .." Journal of Integer Sequences [electronic only] 6.3 (2003): Art. 03.3.1, 6 p., electronic only-Art. 03.3.1, 6 p., electronic only. <http://eudml.org/doc/50800>.
@article{Nyman2003,
author = {Nyman, Bertil, Nicely, Thomas R.},
keywords = {prime gaps; maximal gaps; first occurrences; prime numbers; kilogaps; maximal prime gaps},
title = {New prime gaps between and .},
AU - Nyman, Bertil
AU - Nicely, Thomas R.
TI - New prime gaps between and .
KW - prime gaps; maximal gaps; first occurrences; prime numbers; kilogaps; maximal prime gaps
prime gaps, maximal gaps, first occurrences, prime numbers, kilogaps, maximal prime gaps
Articles by Nyman
Articles by Nicely |
Positive Volume Index (PVI) Definition
What Is the Positive Volume Index (PVI)?
The positive volume index (PVI) is an indicator used in technical analysis that provides signals for price changes based on positive increases in trading volume. The PVI helps in assessing trend strength and potentially confirming price reversals and can be calculated for popular market indexes as well as used to analyze movements in individual securities.
The positive volume index (PVI) is based on price moves depending on whether the current volume is higher than the previous period.
If volume doesn't increase from one period to the next, the PVI stays the same.
The PVI is often shown as a moving average (to help smooth out its movements) and compared to a one-year average (255 days).
Traders watch the relationship of a nine-period PVI moving average (or other MA length) relative to the 255-period PVI moving average.
When the PVI is above the one-year average, it helps confirm a price rise. When the PVI drops below the one-year average, it helps confirm a price drop.
The Formula for the Positive Volume Index (PVI)
If today’s volume is greater than yesterday’s volume, then:
\begin{aligned} &\text{PVI} = PPVI + \frac{(TCP - YCP)}{YCP}\times PPVI \\ &\textbf{where:}\\ &PVI=\text{positive volume index}\\ &PPVI=\text{previous positive volume index}\\ &TCP=\text{today's closing price}\\ &YCP=\text{yesterday's closing price}\\ \end{aligned}
PVI=PPVI+YCP(TCP−YCP)×PPVIwhere:PVI=positive volume indexPPVI=previous positive volume indexTCP=today’s closing priceYCP=yesterday’s closing price
If volume today is less than or equal to volume yesterday:
PVI = \text{Previous PVI}
PVI=Previous PVI
How to Calculate the Positive Volume Index (PVI)
If volume today is greater than volume yesterday, then use the PVI formula.
Input price data for today and yesterday, along with the previous PVI calculation.
If there is no previous PVI calculation, use the price calculation from today as the previous PVI as well.
If volume today is not greater than volume yesterday, then the PVI stays the same for that day.
Understanding the Positive Volume Index (PVI)
The PVI is typically followed in conjunction with a negative volume index (NVI) calculation. Together they are known as price accumulation volume indicators.
The PVI and NVI were first developed in the 1930s by Paul Dysart using market breadth indicators such as the advance-decline line. The PVI and NVI indicators gained popularity following their inclusion in a 1976 book titled Stock Market Logic by Norman Fosback, who expanded their application to individual securities.
Fosback's research, which encompassed the period from 1941 to 1975, suggested that when the PVI is below its one-year average, there is a 67% chance of a bear market. If the PVI is above its one-year average, the chance of a bear market drops to 21%.
Generally, traders will watch both the PVI and NVI indicators to get a sense of the market’s trend in terms of volume. The PVI will be more volatile when volume is rising and the NVI will be more volatile when volume is decreasing.
Since the primary factor of the PVI is price, traders will see the PVI increasing when volume is high and prices are increasing. The PVI will decrease when volume is high but prices are decreasing. Therefore, the PVI can be a signal for bullish and bearish trends.
The general belief is that high-volume days are associated with the crowd. When the PVI is above its one-year moving average (about 255 trading days), it shows that the crowd is optimistic, which helps fuel further price increases. If the PVI falls below the one-year average, that signals the crowd is turning pessimistic, and a price decline is forthcoming or is already underway.
Traders will often plot a nine-period moving average (MA) of PVI and compare it to a 255-period MA of PVI. They will then watch for the relationships as described above. Crossovers signal potential trend changes in price. For example, if the PVI rises above the 255-period MA from below, that could signal a new uptrend is underway. That uptrend is confirmed as long as the PVI stays above the one-year average.
Keep in the mind the probabilities mentioned above. The PVI signals are not 100% accurate. Generally, the PVI compared to a one-year MA helps confirm trends and reversals, but it won't be correct all the time.
Some traders prefer the NVI over the PVI, or they use them together to help confirm each other. The NVI looks at lower volume days, which are associated with professional trader activity, and not the crowd. Therefore, the NVI shows what the "smart money" is doing.
The Positive Volume Index (PVI) vs. On Balance Volume (OBV)
Positive volume is a price calculation based on whether volume rose in the current session relative to the prior. On balance volume (OBV) is a running total of positive and negative volume based on whether the price today was higher or lower than the price yesterday, respectively.
In other words, both indicators are factoring in volume and price, but do it in very different ways. Due to their different calculations, the PVI and OBV will provide different trade signals and information to traders.
Limitations of Using the Positive Volume Index (PVI)
The PVI is tracking the crowd, whose activity is typically associated with higher volume days. The crowd typically loses money, or fairs less well than professional traders. Therefore, the PVI is tracking the "not-smart money." For better quality signals, and for a better context of what a particular market or stock is doing, the PVI is used in conjunction with the NVI.
In the historical tests, the PVI did a decent job of highlighting the bull and bear markets in price. Although it is not 100% accurate...nothing is.
The indicator can be prone to whipsaws, which is when multiple crossovers occur in quick succession, making it hard to determine the true trend direction based on the indicator alone. The PVI is also prone to some anomalies. For example, it may continually move lower, even if the price is rising aggressively.
For these reasons, it is recommended traders use the PVI along with price action analysis, other technical indicators, and fundamental analysis if looking at longer-term trading opportunities.
Norman G. Fosback. "Stock Market Logic," Pages 120-124. Dearborn Financial Publishing, 1993.
Norman Fosback. "Stock Market Logic," Page 123. Dearborn Financial Publishing, 1993.
Up volume generally refers to an increase in the volume of shares traded in either a market or security that leads to an increase in value. |
Neutron activation - Wikipedia
Induction of radioactivity by neutron radiation
Neutron activation is the process in which neutron radiation induces radioactivity in materials, and occurs when atomic nuclei capture free neutrons, becoming heavier and entering excited states. The excited nucleus decays immediately by emitting gamma rays, or particles such as beta particles, alpha particles, fission products, and neutrons (in nuclear fission). Thus, the process of neutron capture, even after any intermediate decay, often results in the formation of an unstable activation product. Such radioactive nuclei can exhibit half-lives ranging from small fractions of a second to many years.
Neutron activation is the only common way that a stable material can be induced into becoming intrinsically radioactive. All naturally occurring materials, including air, water, and soil, can be induced (activated) by neutron capture into some amount of radioactivity in varying degrees, as a result of the production of neutron-rich radioisotopes.[citation needed] Some atoms require more than one neutron to become unstable, which makes them harder to activate because the probability of a double or triple capture by a nucleus is below that of single capture. Water, for example, is made up of hydrogen and oxygen. Hydrogen requires a double capture to attain instability as tritium (hydrogen-3), while natural oxygen (oxygen-16) requires three captures to become unstable oxygen-19. Thus water is relatively difficult to activate, as compared to sodium chloride (NaCl), in which both the sodium and chlorine atoms become unstable with a single capture each. These facts were realized first-hand at the Operation Crossroads atomic test series in 1946.
3 Effects on materials over time
4.1.1 Neutron detection
4.1.2 Materials analysis
4.2 Semiconductor production
Main article: Neutron capture
An example of this kind of a nuclear reaction occurs in the production of cobalt-60 within a nuclear reactor: The cobalt-60 then decays by the emission of a beta particle plus gamma rays into nickel-60. This reaction has a half-life of about 5.27 years, and due to the availability of cobalt-59 (100% of its natural abundance), this neutron bombarded isotope of cobalt is a valuable source of nuclear radiation (namely gamma radiation) for radiotherapy.[1]
In other cases, and depending on the kinetic energy of the neutron, the capture of a neutron can cause nuclear fission—the splitting of the atomic nucleus into two smaller nuclei. If the fission requires an input of energy, that comes from the kinetic energy of the neutron. An example of this kind of fission in a light element can occur when the stable isotope of lithium, lithium-7, is bombarded with fast neutrons and undergoes the following nuclear reaction:
+ gamma rays + kinetic energy
In other words, the capture of a neutron by lithium-7 causes it to split into an energetic helium nucleus (alpha particle), a hydrogen-3 (tritium) nucleus and a free neutron. The Castle Bravo accident, in which the thermonuclear bomb test at Enewetak Atoll in 1954 exploded with 2.5 times the expected yield, was caused by the unexpectedly high probability of this reaction.
In the areas around a pressurized water reactors or boiling water reactors during normal operation, a significant amount of radiation is produced due to the fast neutron activation of coolant water oxygen via a (n,p) reaction. The activated oxygen-16 nucleus emits a proton (hydrogen nucleus), and transmutes to nitrogen-16, which has a very short life (7.13 seconds) before decaying back to oxygen-16 (emitting 6.13 MeV beta particles).[2]
(Decays rapidly)
This activation of the coolant water requires extra biological shielding around the nuclear reactor plant. It is the high energy gamma ray in the second reaction that causes the major concern. This is why water that has recently been inside a nuclear reactor core must be shielded until this radiation subsides. One to two minutes is generally sufficient.
In facilities that housed a cyclotron, the reinforced concrete foundation can become radioactive due to neutron activation. Six important long-lived radioactive isotopes (54Mn, 55Fe, 60Co, 65Zn, 133Ba, and 152Eu) can be found within concrete nuclei affected by neutrons.[3] The residual radioactivity is predominantly due to trace elements present, and thus the amount of radioactivity derived from cyclotron activation is minuscule, i.e., pCi/g or Bq/g. The release limit for facilities with residual radioactivity is 25 mrem/year.[4] An example of 55Fe production from iron rebar activation is shown below:
Neutron activation is the only common way that a stable material can be induced into becoming intrinsically radioactive. Neutrons are only free in quantity in the microseconds of a nuclear weapon's explosion, in an active nuclear reactor, or in a spallation neutron source.
In an atomic weapon neutrons are only generated for from 1 to 50 microseconds, but in huge numbers. Most are absorbed by the metallic bomb casing, which is only just starting to be affected by the explosion within it. The neutron activation of the soon-to-be vaporized metal is responsible for a significant portion of the nuclear fallout in nuclear bursts high in the atmosphere. In other types of activation, neutrons may irradiate soil that is dispersed in a mushroom cloud at or near the Earth's surface, resulting in fallout from activation of soil chemical elements.
Effects on materials over time[edit]
In any location with high neutron fluxes, such as within the cores of nuclear reactors, neutron activation contributes to material erosion; periodically the lining materials themselves must be disposed of, as low-level radioactive waste. Some materials are more subject to neutron activation than others, so a suitably chosen low-activation material can significantly reduce this problem (see International Fusion Materials Irradiation Facility). For example, Chromium-51 will form by neutron activation in chrome steel (which contains Cr-50) that is exposed to a typical reactor neutron flux.[5]
Carbon-14, most frequently but not solely, generated by the neutron activation of atmospheric nitrogen-14 with a thermal neutron, is (together with its dominant natural production pathway from cosmic ray-air interactions and historical production from atmospheric nuclear testing) also generated in comparatively minute amounts inside many designs of nuclear reactors which contain nitrogen gas impurities in their fuel cladding, coolant water and by neutron activation of the oxygen contained in the water itself. Fast breeder reactors (FBR) produce about an order of magnitude less C-14 than the most common reactor type, the pressurized water reactor, as FBRs do not use water as a primary coolant.[6]
For physicians and radiation safety officers, activation of sodium in the human body to sodium-24, and phosphorus to phosphorus-32, can give a good immediate estimate of acute accidental neutron exposure.[7]
Neutron detection[edit]
One way to demonstrate that nuclear fusion has occurred inside a fusor device is to use a Geiger counter to measure the gamma ray radioactivity that is produced from a sheet of aluminium foil.
In the ICF fusion approach, the fusion yield of the experiment (directly proportional to neutron production) is usually determined by measuring the gamma-ray emissions of aluminium or copper neutron activation targets.[8] Aluminium can capture a neutron and generate radioactive sodium-24, which has a half life of 15 hours[9][10] and a beta decay energy of 5.514 MeV.[11]
The activation of a number of test target elements such as sulfur, copper, tantalum, and gold have been used to determine the yield of both pure fission[12][13] and thermonuclear weapons.[14]
Materials analysis[edit]
Main article: Neutron activation analysis
Neutron activation analysis is one of the most sensitive and precise methods of trace element analysis. It requires no sample preparation or solubilization and can therefore be applied to objects that need to be kept intact such as a valuable piece of art. Although the activation induces radioactivity in the object, its level is typically low and its lifetime may be short, so that its effects soon disappear. In this sense, neutron activation is a non-destructive analysis method.
Neutron activation analysis can be done in situ. For example, aluminium (Al-27) can be activated by capturing relatively low-energy neutrons to produce the isotope Al-28, which decays with a half-life of 2.3 minutes with a decay energy of 4.642 MeV.[15] This activated isotope is used in oil drilling to determine the clay content (clay is generally an alumino-silicate) of the underground area under exploration.[16]
Historians can use accidental neutron activation to authenticate atomic artifacts and materials subjected to neutron fluxes from fission incidents. For example, one of the fairly unique isotopes found in trinitite, and therefore with its absence likely signifying a fake sample of the mineral, is a barium neutron activation product, the barium in the Trinity device coming from the slow explosive lens employed in the device, known as Baratol.[17]
Semiconductor production[edit]
Neutron irradiation may be used for float-zone silicon slices (wafers) to trigger fractional transmutation of Si atoms into phosphorus (P) and therefore doping it into n-type silicon [18]: 366
{\displaystyle {\ce {Si14^{30}}}+neutron\xrightarrow {} {\ce {Si14^{31}}}+\gamma -ray\xrightarrow {2.62hr} {\ce {P31^{15}}}+\beta -ray}
Phosphorus-32 produced when sulfur captures a neutron.
Salted bomb
^ Neeb, Karl Heinz (1997). The Radiochemistry of Nuclear Power Plants with Light Water Reactors. Berlin-New York: Walter de Gruyter. p. 227. ISBN 3-11-013242-7.
^ Vichi, Sara (2016). "Efficiency calibration of a portable CZT detector for". Radiation Effects and Defects in Solids. 171: 705–713. doi:10.1080/10420150.2016.1244675. S2CID 99556734.
^ Nuclear Regulatory Commission 10 CFR 20.1402. "Standards for Protection Against Radiation".
^ "IAEA Technical report series no.421, Management of Waste Containing Tritium and Carbon-14" (PDF).
^ ORNL Report Archived 2013-10-01 at the Wayback Machine on determination of dose from criticality accidents
^ Stephen Padalino; Heather Oliver & Joel Nyquist. "DT neutron yield measurements using neutron activation of aluminum". LLE Collaborators: Vladimir Smalyukand, Nancy Rogers.
^ "4 Identified radioactive isotopes". Aanda.org. 1998-03-02. Retrieved 2019-11-14.
^ "Wayback Machine". November 29, 2014. Archived from the original on 2014-11-29.
^ Kerr, George D.; Young, Robert W.; Cullings, Harry M.; Christy, Robert F. (2005). "Bomb Parameters" (PDF). In Robert W. Young, George D. Kerr (ed.). Reassessment of the Atomic Bomb Radiation Dosimetry for Hiroshima and Nagasaki – Dosimetry System 2002. The Radiation Effects Research Foundation. pp. 42–43. Archived from the original (PDF) on 2015-08-10. Retrieved 2014-03-13.
^ "Search Results - Schlumberger Oilfield Glossary". www.glossary.oilfield.slb.com.
^ Parekh, PP; Semkow, TM; Torres, MA; Haines, DK; Cooper, JM; Rosenberga, PM; Kittoa, ME (2006). "Radioactivity in Trinitite six decades later" (PDF). Journal of Environmental Radioactivity. 85 (1): 103–120. CiteSeerX 10.1.1.494.5179. doi:10.1016/j.jenvrad.2005.01.017. PMID 16102878.
^ Sze, S. M. (2012). Semiconductor devices : physics and technology. M. K. Lee (3 ed.). New York, NY: Wiley. ISBN 0-470-53794-9. OCLC 869833419.
Neutron Activation Analysis web
Handbook on Nuclear Activation Cross-Sections, IAEA, 1974
Decay Data in MIRD Format from the National Nuclear Data Center at Brookhaven National Laboratory
Neutron capture as it relates to nucleosynthesis
Neutron capture and the Chart of the nuclides
The chart of the Nuclides
Discovery of the Chromium isotopes, Chromium-55 by Cr-54 neutron capture
ORILL : 1D transmutation, fuel depletion, and radiological protection code
US Army (1952). Operation Ivy Final Report Joint Task Force 132 (PDF).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Neutron_activation&oldid=1089067764" |
Improving Knowledge-Based Weakly Supervised Information Extraction - Cohen Courses
Improving Knowledge-Based Weakly Supervised Information Extraction
6 Our Big Idea
8 Comments from William
9 More Comments from William
Wangshu Pang
Our first idea was to better populate wikipedia infoboxes using semi-supervised techniques. After looking a little more closely at that problem, we discovered that it was somewhat ill-posed, and that producing a baseline would be difficult, as recent systems that address that problem do not have freely available code.
So instead we switched to a related problem, that of using a knowledge base (like Freebase, or DBpedia) to weakly supervise an information extractor. The system we will start from is that of Hoffmann et al., ACL 2011. They were generous enough to provide both code and data, giving us a dataset and a baseline.
We will address the problem of Relation Extraction, and we will address it using a set of known relations between entities in a knowledge base as our only supervision (in addition to a parser and a named-entity extractor). Specifically, given a corpus of text and a set of relations
{\displaystyle R}
, return a set of entities
{\displaystyle E}
in the text along with instances
{\displaystyle r(e_{1},e_{2})}
of those relations between entities (the entities could be inputs instead of outputs, as well, or some entities could be input, with additional entities as output).
We will use as our corpus of text the New York Times Annotated Corpus, as provided by LDC. We will use Freebase as a knowledge source to provide weak supervision to our system. Details of the dataset are described in Hoffmann et al., ACL 2011.
We will use the system of Hoffmann et al., ACL 2011 as our baseline. Their method is described in the linked page, so we will not repeat the description here. We have their code, so we can reproduce the results in their paper exactly.
A key motivation for Hoffmann et al. was that their method was simpler than previous systems and it also allowed for multiple relations between pairs of entities (e.g., both CEOof(Jobs, Apple) and Founder(Jobs, Apple) - previous systems would have to pick one or the other). Our idea is the Hoffmann et al. did not simplify it enough, and they do not allow multiple relations between pairs of entities in a single sentence - we go farther than Hoffmann et al. in both of their points. For instance, the sentence "Steve Jobs, the founder and CEO of Apple, Inc., ..." clearly contains two relations between Apple and Steve Jobs, but the system of Hoffmann et al. would have to pick one or the other. However, it is likely that such sentences are not incredibly common, so a model that only improved that point would not be incredibly interesting. The reason we think this is an interesting experiment is that it should also simplify inference, giving decreased training and test time, because we will essentially be turning a multi-class classifier into a set of independent binary classifiers, reducing some coupling in the inference.
[1] D. Lange, C. Böhm, F. Naumann, "Extracting Structured Information from Wikipedia Articles to Populate Infoboxes", CIKM, Oct 2010.
Comments from William
This is a nice problem. Semi-supervised learning won't be covered will later in the class, though, so you guys will have to be proactive about finding the appropriate papers for this. One nice paper that might get you started is: http://dl.acm.org/citation.cfm?id=1870675
You guys should also look into the Wu and Weld papers on Infobox extraction, which are quite nice.
--Wcohen 20:59, 22 September 2011 (UTC)
More Comments from William
What baseline method and dataset are you using? --Wcohen 14:37, 11 October 2011 (UTC)
Retrieved from "http://curtis.ml.cmu.edu/w/courses/index.php?title=Improving_Knowledge-Based_Weakly_Supervised_Information_Extraction&oldid=9113" |
XOR gate — Wikipedia Republished // WIKI 2
This article is about XOR digital logic gate (e.g. SN7486 or CD4030B). For XOR logical operation, see Exclusive or. For other uses, see XOR (disambiguation).
XOR gate (sometimes EOR, or EXOR and pronounced as Exclusive OR) is a digital logic gate that gives a true (1 or HIGH) output when the number of true inputs is odd. An XOR gate implements an exclusive or (
{\displaystyle \nleftrightarrow }
) from mathematical logic; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "must have one or the other but not both".
XOR can also be viewed as addition modulo 2. As a result, XOR gates are used to implement binary addition in computers. A half adder consists of an XOR gate and an AND gate. Other uses include subtractors, comparators, and controlled inverters.[1]
The algebraic expressions
{\displaystyle A\cdot {\overline {B}}+{\overline {A}}\cdot B}
{\displaystyle (A+B)\cdot ({\overline {A}}+{\overline {B}})}
{\displaystyle A\oplus B}
all represent the XOR gate with inputs A and B. The behavior of XOR is summarized in the truth table shown on the right.
XOR and XNOR Logic Gates
Logical Operators − Exclusive OR
1 Symbols
2 Pass-gate-logic wiring
2.1 Analytical representation
4 More than two inputs
5.1 Uses in addition
5.2 Pseudo-random number generator
5.3 Correlation and sequence detection
There are three schematic symbols for XOR gates: the traditional ANSI and DIN symbols and the IEC symbol. In some cases, the DIN symbol is used with ⊕ instead of ≢. For more information see Logic Gate Symbols.
ANSI XOR Schematic Symbol IEC XOR Schematic Symbol DIN XOR Schematic Symbol
The logic symbols ⊕, Jpq, and ⊻ can be used to denote an XOR operation in algebraic expressions.
C-like languages use the caret symbol ^ to denote bitwise XOR. (Note that the caret does not denote logical conjunction (AND) in these languages, despite the similarity of symbol.)
Pass-gate-logic wiring
An XOR gate can be constructed using MOSFETs. Here is a diagram of a pass transistor logic implementation of an XOR gate.[2][3][4][5][6]
Transmission Gate Logic wiring of an XOR gate
Note: The "Rss" resistor prevents shunting current directly from "A" and "B" to the output. Without it, if the circuit that provides inputs A and B does not have the proper driving capability, the output might not swing rail to rail or be severely slew-rate limited. The "Rss" resistor also limits the current from Vdd to ground which protects the transistors and saves energy when the transistors are transitioning between states.
{\displaystyle f(a,b)=a+b-2ab}
is an analytical representation of XOR gate:
{\displaystyle f(0,0)=0+0-2\cdot 0\cdot 0=0}
{\displaystyle f(0,1)=0+1-2\cdot 0\cdot 1=1}
{\displaystyle f(1,0)=1+0-2\cdot 1\cdot 0=1}
{\displaystyle f(1,1)=1+1-2\cdot 1\cdot 1=0}
{\displaystyle f(a,b)=|a-b|}
is an alternative analytical representation.
XOR gate circuit using three mixed gates
If a specific type of gate is not available, a circuit that implements the same function can be constructed from other available gates. A circuit implementing an XOR function can be trivially constructed from an XNOR gate followed by a NOT gate. If we consider the expression
{\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)}
, we can construct an XOR gate circuit directly using AND, OR and NOT gates. However, this approach requires five gates of three different kinds.
As alternative, if different gates are available we can apply Boolean algebra to transform
{\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)\equiv (A+B)\cdot ({\overline {A}}+{\overline {B}})}
as stated above, and apply de Morgan's Law to the last term to get
{\displaystyle (A+B)\cdot {\overline {(A\cdot B)}}}
which can be implemented using only three gates as shown on the right.
An XOR gate circuit can be made from four NAND gates. In fact, both NAND and NOR gates are so-called "universal gates" and any logical function can be constructed from either NAND logic or NOR logic alone. If the four NAND gates are replaced by NOR gates, this results in an XNOR gate, which can be converted to an XOR gate by inverting the output or one of the inputs (e.g. with a fifth NOR gate).
An alternative arrangement is of five NOR gates in a topology that emphasizes the construction of the function from
{\displaystyle (A+B)\cdot ({\overline {A}}+{\overline {B}})}
, noting from de Morgan's Law that a NOR gate is an inverted-input AND gate. Another alternative arrangement is of five NAND gates in a topology that emphasizes the construction of the function from
{\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)}
, noting from de Morgan's Law that a NAND gate is an inverted-input OR gate.
For the NAND constructions, the upper arrangement requires fewer gates. For the NOR constructions, the lower arrangement offers the advantage of a shorter propagation delay (the time delay between an input changing and the output changing).
Literal interpretation of the name "exclusive or", or observation of the IEC rectangular symbol, raises the question of correct behaviour with additional inputs. If a logic gate were to accept three or more inputs and produce a true output if exactly one of those inputs were true, then it would in effect be a one-hot detector (and indeed this is the case for only two inputs). However, it is rarely implemented this way in practice.
It is most common to regard subsequent inputs as being applied through a cascade of binary exclusive-or operations: the first two signals are fed into an XOR gate, then the output of that gate is fed into a second XOR gate together with the third signal, and so on for any remaining signals. The result is a circuit that outputs a 1 when the number of 1s at its inputs is odd, and a 0 when the number of incoming 1s is even. This makes it practically useful as a parity generator or a modulo-2 adder.
For example, the 74LVC1G386 microchip is advertised as a three-input logic gate, and implements a parity generator.[7]
Example half adder circuit diagram
Example full adder circuit diagram
XOR gates and AND gates are the two most-used structures in VLSI applications.[8]
Uses in addition
The XOR logic gate can be used as a one-bit adder that adds any two bits together to output one bit. For example, if we add 1 plus 1 in binary, we expect a two-bit answer, 10 (i.e. 2 in decimal). Since the trailing sum bit in this output is achieved with XOR, the preceding carry bit is calculated with an AND gate. This is the main principle in Half Adders. A slightly larger Full Adder circuit may be chained together in order to add longer binary numbers.
Pseudo-random number (PRN) generators, specifically linear-feedback shift registers (LFSR), are defined in terms of the exclusive-or operation. Hence, a suitable setup of XOR gates can model a linear-feedback shift register, in order to generate random numbers.
Correlation and sequence detection
XOR gates produce a 0 when both inputs match. When searching for a specific bit pattern or PRN sequence in a very long data sequence, a series of XOR gates can be used to compare a string of bits from the data sequence against the target sequence in parallel. The number of 0 outputs can then be counted to determine how well the data sequence matches the target sequence. Correlators are used in many communications devices such as CDMA receivers and decoders for error correction and channel codes. In a CDMA receiver, correlators are used to extract the polarity of a specific PRN sequence out of a combined collection of PRN sequences.
A correlator looking for 11010 in the data sequence 1110100101 would compare the incoming data bits against the target sequence at every possible offset while counting the number of matches (zeros):
1110100101 (data)
11010 (target)
00111 (XOR) 2 zero bits
00000 5 zero bits
01110 2 zero bits
10011 2 zero bits
01000 4 zero bits
11111 0 zero bits
Matches by offset:
: :
: : : : :
In this example, the best match occurs when the target sequence is offset by 1 bit and all five bits match. When offset by 5 bits, the sequence exactly matches its inverse. By looking at the difference between the number of ones and zeros that come out of the bank of XOR gates, it is easy to see where the sequence occurs and whether or not it is inverted. Longer sequences are easier to detect than short sequences.
Exclusive or
AND gate
OR gate
Inverter (NOT gate)
NAND gate
NOR gate
XNOR gate
IMPLY gate
Logic gate
Wikimedia Commons has media related to XOR gates.
^ Fletcher, William (1980). An engineering approach to digital design. Prentice-Hall. p. 98. ISBN 0-13-277699-5.
^ "Designing combinational logic gates in CMOS". p. 233
^ "Transmission Gate XOR".
^ "transmission-gate XOR (tiny XOR)" (via [1])
^ "Figure 3, Exclusive OR and XNOR gate".
^ "Pass-Transistor Logic: Transmission Gate XOR" (p. 11)
^ 74LVC1G386 data sheet
^ "Comparison of different design techniques of XOR & AND gate using EDA simulation tool". XOR & AND gates are most important basic building blocks of any VLSI applications.
{\displaystyle \top }
Alternative denial (NAND gate)
{\displaystyle \uparrow }
Converse implication
{\displaystyle \leftarrow }
Implication (IMPLY gate)
{\displaystyle \rightarrow }
Disjunction (OR gate)
{\displaystyle \lor }
Negation (NOT gate)
{\displaystyle \neg }
Exclusive or (XOR gate)
{\displaystyle \not \leftrightarrow }
Biconditional (XNOR gate)
{\displaystyle \leftrightarrow }
Statement (Digital buffer)
Joint denial (NOR gate)
{\displaystyle \downarrow }
Nonimplication (NIMPLY gate)
{\displaystyle \nrightarrow }
Converse nonimplication
{\displaystyle \nleftarrow }
Conjunction (AND gate)
{\displaystyle \land }
{\displaystyle \bot } |
Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct - Maple Help
Home : Support : Online Help : Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct
Error, empty plot
Note: For users of Maple 13 and earlier versions, this warning occurs when you create a Maple plot without points, curves, or surfaces. In the warning message, "function" refers to the function being plotted (expressed as an algebraic expression or procedure) and "region" refers to the plotting domain.
Example 1: 3-D plot of a function
h≔\left(x,y\right)→{x}^{2} \mathrm{cos}\left(y\right)
\textcolor[rgb]{0,0,1}{h}\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}{→}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)
\mathrm{plot3d}\left(h,x=-2..2,y=-2 \mathrm{π}..2 \mathrm{π}\right)
Solution 1: Specify
h\left(x,y\right)
h
in the plot3d calling sequence. This solution uses an expression (
{x}^{2} \mathrm{cos}\left(y\right)
) as the argument to the plot3d command.
\mathrm{plot3d}\left(h\left(x,y\right),x=-2..2,y=-2 \mathrm{π}..2 \mathrm{π}\right)
Solution 2: To plot the function
h
, leave
h
in operator form and use the second calling sequence listed on the plot3d help page.
\mathrm{plot3d}\left(h,-2..2,-2 \mathrm{\pi }..2 \mathrm{\pi }\right)
Example 2: Expression with non-real values
\mathrm{plot}\left(\sqrt{x}\mathit{,}\mathit{ }x\mathit{=}\mathit{-}\mathit{10}\mathit{..}\mathit{-}\mathit{1}\right)\mathit{;}
In the range given, the solutions for the function
\sqrt{x}
are all complex values that cannot be plotted with the plot command.
Solution: Change the range to one for which the expression is real-valued.
\mathrm{plot}\left(\sqrt{x}\mathit{,}\mathit{ }x\mathit{=}\mathit{0}\mathit{..}\mathit{10}\right)\mathit{;}
Example 3: Additional parameter not given numerical value
\mathrm{plot}\left(x\mathit{+}y\mathit{,}\mathit{ }x\mathit{=}\mathit{-}\mathit{1}\mathit{..}\mathit{1}\right)\mathit{;}
The expression to be plotted includes two unknowns, but plot expects only one unknown..
Solution 1: To plot this expression, replace
y
with a numeric value.
\mathrm{plot}\left(x\mathit{+}\mathit{1}\mathit{,}\mathit{ }x\mathit{=}\mathit{-}\mathit{1}\mathit{..}\mathit{1}\right)\mathit{;}
Solution 2: To create a 3-D plot of this expression, use plot3d and specify a range for
y
\mathrm{plot3d}\left(x\mathit{+}y\mathit{,}\mathit{ }x\mathit{=}\mathit{-}\mathit{1}\mathit{..}\mathit{1}\mathit{,}y\mathit{=}\mathit{-}\mathit{1}\mathit{..}\mathit{1}\right)\mathit{;}
Example 4: Independent variable not specified
\mathrm{plot}\left(x\mathit{+}\mathit{1}\mathit{,}\mathit{ }\mathit{-}\mathit{1}\mathit{..}\mathit{1}\right)
Solution: To plot this function, explicitly assign the range to
x
\mathrm{plot}\left(x\mathit{+}\mathit{1}\mathit{,}\mathit{ }x\mathit{=}\mathit{-}\mathit{1}\mathit{..}\mathit{1}\right)
Important: Previously in Maple 10, you had to specify a range when entering a plot.
Example 5 (Maple 10): Function has not been assigned to anything
\mathrm{plot}\mathit{}\left(f\right)
Here, plot(f) is interpreted as operator-form and a default range of -10..10 is assumed. The error occurs because
f
has not been assigned to anything.
Example 5 (Maple 11 and later): You can enter a plot without having to specify a range.
f
is interpreted as the independent variable. A default range of -10..10 is assumed.
\mathrm{plot}\left(f\right)
plot, plot3d, plots[complexplot] |
Taskar et al. 2004. Max-margin Parsing - Cohen Courses
Taskar et al. 2004. Max-margin Parsing
Max-margin parsing, by Ben Taskar, Taskar, B. and Klein, D. and Collins, M. and Koller, D. and Manning, C.. In Proc. EMNLP, 2004.
This paper presents a novel approach to Parsing by maximizing separating margins using Support Vector Machines. They show how we can reformulate the parsing problem as a discriminative task, which allow an arbitrary number of features to be used. Also, such a formulation allows them to incorporate a loss function that directly penalizes incorrect parse trees appropriately.
Instead of a probabilistic interpretation for parse trees, we seek to find:
{\displaystyle y_{i}=\arg \max _{y\in \mathbf {G} (x_{i})}\langle \mathbf {w} ,\Phi (x_{i},y)\rangle }
for all sentences
{\displaystyle x_{i}}
in the training data,
{\displaystyle y_{i}}
being the parse tree,
{\displaystyle \mathbf {G} (x_{i})}
the set of possible parses for
{\displaystyle x_{i}}
Formulating it as an optimization problem,
{\displaystyle \max _{\gamma }\{\langle \mathbf {w} ,\Phi _{i,y_{i}}-\Phi _{i,y}\rangle \geq \gamma L_{i,y}\}\forall y\in \mathbf {G} (x_{i}),||\mathbf {w} ||^{2}\leq 1}
Using SVM, we can find the dual of the above program
{\displaystyle \max C\sum _{i,y}\alpha _{i,y}L_{i,y}-{\dfrac {1}{2}}||C\sum _{i,y}(I_{i,y}-\alpha _{i,y})\Phi _{i,y}||}
{\displaystyle \sum _{y}\alpha _{i,y}=1,\forall i;\alpha _{i,y}\geq 0,\forall i,y}
{\displaystyle I_{i,y}}
indicates whether
{\displaystyle y}
is the true parse for sentence
{\displaystyle i}
For each sentence, we need to enumerate all possible parse trees, which is exponential in size. However, we can make use of local substructures similar to chart parsing dynamic programming algorithm to factor these trees into parts like
{\displaystyle \langle A,s,e,i\rangle }
{\displaystyle \langle A\rightarrow BC,s,m,e,i\rangle }
{\displaystyle s,m,e,i}
refers to start, split, end points and sentence number respectively.
{\displaystyle \Phi (x,y)=\sum _{r\in R(x,y)}\phi (x,r)}
{\displaystyle R(x,y)}
is the set of all possible parts.
{\displaystyle \phi }
can be any function that maps a rule production part to some feature vector representation. In addition, the loss function can also be decomposed into sum of parts similar to above. In the paper, the loss function used was the number of constituent errors made in a parse.
By incorporating parts, the factored dual objective can be expressed in polynomial number of variables, which is in fact cubic in the length of the sentence.
Experiments on the Penn Treebank dataset with lexical features achieved 0.43 f-score over the Collins 99 parser.
McDonald_et_al,_ACL_2005:_Non-projective_dependency_parsing_using_spanning_tree_algorithms Margin learning for dependency parsing
Tsochantaridis,_Joachims_,_Support_vector_machine_learning_for_interdependent_and_structured_output_spaces_2004 Using SVMs for structured output space.
Retrieved from "http://curtis.ml.cmu.edu/w/courses/index.php?title=Taskar_et_al._2004._Max-margin_Parsing&oldid=9505" |
Truncated tetrahedron - Wikipedia
Elements F = 8, E = 18, V = 12 (χ = 2)
Conway notation tT
Schläfli symbols t{3,3} = h2{4,3}
Symmetry group Td, A3, [3,3], (*332), order 24
6-6: 70°31′44″
3D model of a truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.
A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.[1]
A truncated tetrahedron is the Goldberg polyhedron GIII(1,1), containing triangular and hexagonal faces.
A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.
2 Densest packing
4 Orthogonal projection
5.1 Friauf polyhedron
7 Truncated tetrahedral graph
The area A and the volume V of a truncated tetrahedron of edge length a are:
{\displaystyle {\begin{aligned}A&=7{\sqrt {3}}a^{2}&&\approx 12.124\,355\,65a^{2}\\V&={\tfrac {23}{12}}{\sqrt {2}}a^{3}&&\approx 2.710\,575\,995a^{3}.\end{aligned}}}
Densest packing[edit]
The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using Monte Carlo methods.[2][3] Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space.[2]
Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:
(+3,−1,−1), (+1,−3,−1), (+1,−1,−3)
Orthogonal projection showing Cartesian coordinates inside it bounding box: (±3,±3,±3). The hexagonal faces of the truncated tetrahedra can be divided into 6 coplanar equilateral triangles. The 4 new vertices have Cartesian coordinates:
(−1,−1,−1), (−1,+1,+1),
(+1,−1,+1), (+1,+1,−1). As solid this can represent a 3D dissection making 4 red octahedra and 6 yellow tetrahedra. The set of vertex permutations (±1,±1,±3) with an odd number of minus signs forms a complementary truncated tetrahedron, and combined they form a uniform compound polyhedron.
Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of:
Orthogonal projection[edit]
The truncated tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
hexagon-centered
Friauf polyhedron[edit]
A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D2d symmetry) is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges.[4] It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2".[5]
Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in Expo 67. They were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times.[6]
The Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a truncated tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations.
Truncated tetrahedral graph[edit]
Truncated tetrahedral graph
24 (S4)[7]
Hamiltonian, regular, 3-vertex-connected, planar graph
In the mathematical field of graph theory, a truncated tetrahedral graph is an Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges.[8] It is a connected cubic graph,[9] and connected cubic transitive graph.[10]
It is also a part of a sequence of cantic polyhedra and tilings with vertex configuration 3.6.n.6. In this wythoff construction the edges between the hexagons represent degenerate digons.
*n33 orbifold symmetries of cantic tilings: 3.6.n.6
Cantic figure
3.6.2.6 3.6.3.6 3.6.4.6 3.6.5.6 3.6.6.6 3.6.∞.6
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
Truncated tetrahedron in rotation
Truncated tetrahedron (Matemateca IME-USP)
Truncated 4-sided die
Quarter cubic honeycomb – Fills space using truncated tetrahedra and smaller tetrahedra
Truncated 5-cell – Similar uniform polytope in 4-dimensions
Triakis truncated tetrahedron
Octahedron – a rectified tetrahedron
^ Chisholm, Matt; Avnet, Jeremy (1997). "Truncated Trickery: Truncatering". theory.org. Retrieved 2013-09-02.
^ a b Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces". ACS Nano. 6 (2012): 609–614. arXiv:1109.1323. doi:10.1021/nn204012y. PMID 22098586. S2CID 12785227.
^ Jiao, Yang; Torquato, Sal (Sep 2011). "A Packing of Truncated Tetrahedra that Nearly Fills All of Space". arXiv:1107.2300 [cond-mat.soft].
^ http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf[bare URL PDF]
^ Friauf, J. B. (1927). "The crystal structure of the intermetallic compound MgCu2". J. Am. Chem. Soc. 49: 3107–3114. doi:10.1021/ja01411a017.
^ "Expo 67 - Man the Producer - page 1".
^ a b c d e f An Atlas of Graphs, page=172, C105
^ An Atlas of Graphs, page 267, truncated tetrahedral graph
^ An Atlas of Graphs, page 130, connected cubic graphs, 12 vertices, C105
^ An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11
Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press
Wikimedia Commons has media related to Truncated tetrahedron.
Eric W. Weisstein, Truncated tetrahedron (Archimedean solid) at MathWorld.
Weisstein, Eric W. "Truncated tetrahedral graph". MathWorld.
Klitzing, Richard. "3D convex uniform polyhedra x3x3o - tut".
Editable printable net of a truncated tetrahedron with interactive 3D view
Retrieved from "https://en.wikipedia.org/w/index.php?title=Truncated_tetrahedron&oldid=1082970420" |
Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties
1 December 2010 Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties
Duke Math. J. 155(3): 483-518 (1 December 2010). DOI: 10.1215/00127094-2010-061
G
be a unitary group over a totally real field, and let
X
be a Shimura variety associated to
G
. For certain primes
p
of good reduction for
X
, we construct cycles
{X}_{{\tau }_{0},i}
on the characteristic
p
fiber of
X
. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on
X
. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group
G\prime
G
at all finite places but not isomorphic to
G
at archimedean places. More precisely, each cycle
{X}_{{\tau }_{0},i}
has a natural desingularization
{\stackrel{~}{X}}_{{\tau }_{0},i}
, which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety
X\prime
G\prime
. We exploit this relationship to construct an injection of the étale cohomology of
X\prime
into that of
X
. This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of
G\prime
to automorphic representations of
G
David Helm. "Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties." Duke Math. J. 155 (3) 483 - 518, 1 December 2010. https://doi.org/10.1215/00127094-2010-061
David Helm "Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties," Duke Mathematical Journal, Duke Math. J. 155(3), 483-518, (1 December 2010) |
Dictionary:Backus filter - SEG Wiki
(bok’ ∂s) An inverse filter that removes the effects of reverberation involving a simple water bottom. The filter's z-transform expression is
{\displaystyle 1+2kz^{q}+k^{2}z^{2q},\ }
where k is the water-bottom reflection coefficient and qts is the two-way traveltime through the water layer if ts is the sample interval. See Backus (1959)[1].
↑ Backus, M. M., 1959, Water reverberations: their nature and elimination: Geophysics, 24, 233–261.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Backus_filter&oldid=38900" |
Predict state and state estimation error covariance at next time step using extended or unscented Kalman filter, or particle filter - MATLAB predict
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k-1|k-1\right]
x\left[k\right]=\sqrt{x\left[k-1\right]+u\left[k-1\right]}+w\left[k-1\right]
y\left[k\right]=x\left[k\right]+2*u\left[k\right]+v\left[k{\right]}^{2}
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k-1|k-1\right]
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k\right] |
Current Directory in libname
Orientation for 3-D Plots
Parametric Infinite Sums
Maple Spreadsheets
Tabulate Return Value
Prevent Automatic Evaluation of Pi in Float Expressions
By default, the Import command now returns a DataFrame when importing from Excel (XLS and XLSX), CSV, DIF, and TSV file formats, and returns a DataSeries when importing from ODS and SXC file formats. Use the output option to specify a different format. For example, Import(origin, output=Matrix).
The libname variable stores the paths to repository files where variables and packages will be implicitly loaded. Previous versions of Maple automatically put . as an entry in libname. This meant that Maple would look in the current directory for .mla files in order to load variables and data. Now Workbook files can be used interchangeably with .mla files, and .maple files give you a place to store variables associated with specific worksheets. As a result, . is not automatically added to libname.
The default orientation for 3-D plots has changed. The new default orientation is [55, 75, 0]. In previous versions of Maple, the default orientation was [45, 45, 0].
In previous versions of Maple, the zoom level for worksheets has been stored in the worksheet itself. In Maple 2016, the zoom level is no longer stored in the Maple worksheet. The default zoom for any Maple worksheet is now set to use the zoom level as set in the Tools > Options > Interface tab > Default zoom. More details, see the Interface page.
In Maple 2016, the default behavior for geometric, hypergeometric, polylog and Zeta type parametric infinite sums has changed. While in earlier Maple versions some of these sums used to return a formal answer even if the sum diverges for some parameter values, now the default behavior is to return an unevaluated sum. Consider, for example, the geometric sum:
sum(x^n, n=0..infinity);
\textcolor[rgb]{0,0,1}{\sum }_{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{n}}
This behavior can be changed by one of the following means:
Supply appropriate assumptions on the parameter(s)
sum(x^n, n=0..infinity) assuming abs(x)<1;
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
sum(x^n, n=0..infinity) assuming x>=1;
\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
Use option parametric
sum(x^n, n=0..infinity, parametric);
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}& |\textcolor[rgb]{0,0,1}{x}|\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{\mathrm{\infty }}& \textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{x}\\ \textcolor[rgb]{0,0,1}{\mathrm{undefined}}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
Use the _EnvFormal environment variable or option formal
_EnvFormal := true:
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
_EnvFormal := '_EnvFormal':
\textcolor[rgb]{0,0,1}{\sum }_{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{n}}
sum(x^n, n=0..infinity, formal);
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
See Advanced Math for more details.
Similarly, the default behavior for convert/hypergeom has changed as well.
S := Sum(x^n/n,n=1..infinity);
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}\frac{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{n}}}{\textcolor[rgb]{0,0,1}{n}}
convert(S, hypergeom);
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}\frac{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{n}}}{\textcolor[rgb]{0,0,1}{n}}
convert(S, hypergeom) assuming abs(x)<1;
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)
_EnvFormal := true;
\textcolor[rgb]{0,0,1}{\mathrm{_EnvFormal}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{true}}
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)
The Spreadsheet menu has been removed from the standard menu bar, and the Insert > Spreadsheet menu item has been removed from the Insert menu.
To access the commands that were in the Spreadsheet menu, right-click (Control-click for Macintosh) on the spreadsheet to bring up the context-sensitive menu. For more information, see Overview of Spreadsheets.
To insert a new Maple spreadsheet, use the Spread:-CreateSpreadsheet(); command. For more information, see Spread.
The Typesetting rules can no longer be set to the value query with the Typesetting:-Settings command. Similarly, the query value is no longer available in the Typesetting Rule Assistant.
The default values for the functionassign and numericalderivs rules are now true in both cases.
A consequence of these changes is that the Clarify Expression dialog no longer appears when any of these rules are invoked. For example, f(x) := 4 in 1-D input will add an entry to the remember table of f, while
f\left(x\right)≔4
in 2-D input will assign the operator x->4 to f.
The return value for the DocumentTools:-Tabulate command is now the identity of the inserted parent Table.
The display of the returned value may be suppressed by terminating the calling statement with a full colon.
The returned identity, which is a string, may be used to modify properties of the Table following insertion. See the DocumentTools:-Tabulate help page for examples.
Maple 2015 introduced a change in the handling of Pi. Whenever Pi is adjacent to a floating-point number in a sum, product, or power, Pi is automatically converted to a floating-point number. See Compatibility Issues in Maple 2015.
In order to keep the symbol Pi in such cases, set the kernelopts option floatPi to false.
The hashmset package has been deprecated. Use the superseding data structure MultiSet instead.
The NumberTheory package updates and replaces the numtheory package.
The list of supported video player file formats and associated codecs can be found here: https://docs.oracle.com/javafx/2/api/javafx/scene/media/package-summary.html#SupportedMediaTypes. |
Decision Making - Wikiversity
1 What is a Decision?
4 Programs and Algorithms
What is a Decision?[edit | edit source]
Decision making is a process of choosing an option from alternatives. Achieving the choice can be either risky (include some uncertainties) or certain. Usually decision process concentrates on choosing the best alternative. The best alternative can be determined by calculating and comparing the utility values and the probabilities of the choices.
Types of Decisions[edit | edit source]
In the context of Artificial Intelligence, computer scientists, software engineers and programmers deal with the ways machines make decisions. The simplest desision has a multiplicity of one-to-one (1::1), where only two possible choices exist: true or false; yes or no; one or zero; etc.. The two results are said to be mutually exclusive, thought of in human terms as "either, but not both" and "one or the other". Some terms for this type of desision are:
mutex (short for mutual exclusion)
boolean as in Boolean Algebra
In Topic:Web Design, a common means to allow a Human to enter the answer to a yes or no question is the checkbox.
Color-coded regions of the world based on the seven commonly-recognised continents
A simple form showing most of the html form elements
The next type of decision has a multiplicity of one-to-many (1::n), where one of several choices exist:
Color: red, yellow, blue, ...
yes, no, no preference (mutex with a "doesn't matter" choice)
Time Zone range: (UTC +12 ... UTC ... UTC-12)
Continent: Africa, Antarctica, Asia, Australia, Europe, N. America, S. America
Again, these choice are exclusive in that only one of the several choices is allowed. These are sometimes processes by a program or algorithm as conditional variables or set as parameters. Some terms for this type of desicion are:
Selection - choose only one please
condvar (short for conditional variable)
parameter - setting a value for a variable
For processing dynamic websites with PHP, a choice can be entered by way of a radio button or drop-down list to allow the user to make his or her choice.
(Logic and Fuzzy Logic - Inference System) Logical rules (combinations of AND, OR, NOT, ...) can be used to expressed a logical structure. Compare the classical logic and its inference with Fuzzy Logic and its inference system.
What are the difference and joint properties of classical Logic and Fuzyy Logic?
What are use-cases in which you would use classical logic for decision making and what are use cases in which you would use Fuzzy Logic?
(Spatial Decision Support System) What are Spatial Decision Support Systems and identify uses cases in which you would consider e.g. a spatial distribution of risk and a spatial distribution of the availability of resources?
Assume you have a property
{\displaystyle P}
that can be evaluated at a specific geolocation
{\displaystyle (x,y)}
(e.g. coordinates of the location (latitude, longitude)). How can you combine a logic property spatially? What is different if you use Fuzzy Logic for that spatial property
{\displaystyle P(x,y)\in [0,1]}
Transfer decision making to a spatial context of Risk Management. Use your own application scenario and take as decision task: "Allocate the limited available resources according to the risk".
Programs and Algorithms[edit | edit source]
Programmes and algorithms are used to support decision makers in finding the best decision option according to the criteria the institution of organisation defines.
Analyse the concept of Dynamic Document Generation to create up-to-data reports for decision making.
What is the role of Geographic Information Systems to support spatial decision making?
Analyse e.g. existing OpenSource R packages that can be used for an algorithmic implementation of decision support.
What are decisions that can be performed by an algorithm or software and what are decision in which the decision maker should remain a human decision maker? Discuss also legal aspects of responsibility for decisions making, that was performed by a piece of software.
Spatial Decision Support Layer
Spatial Decision Support Systems/Fuzzy Controller
Dynamic Document Generation as a decision support product
Retrieved from "https://en.wikiversity.org/w/index.php?title=Decision_Making&oldid=2237488" |
EUDML | Property T for discrete groups in terms of their regular representation. EuDML | Property T for discrete groups in terms of their regular representation.
Property T for discrete groups in terms of their regular representation.
Jolissaint, Paul. "Property T for discrete groups in terms of their regular representation.." Mathematische Annalen 297.3 (1993): 539-552. <http://eudml.org/doc/165145>.
@article{Jolissaint1993,
author = {Jolissaint, Paul},
keywords = {Kazhdan’s property ; discrete groups; von Neumann algebra; Fourier-Stieltjes algebra; positive definite functions},
title = {Property T for discrete groups in terms of their regular representation.},
AU - Jolissaint, Paul
TI - Property T for discrete groups in terms of their regular representation.
KW - Kazhdan’s property ; discrete groups; von Neumann algebra; Fourier-Stieltjes algebra; positive definite functions
Kazhdan’s property
T
, discrete groups, von Neumann algebra, Fourier-Stieltjes algebra, positive definite functions
Positive definite functions on groups, semigroups, etc.
{C}^{*}
{W}^{*}
{C}^{*}
{W}^{*}
Articles by Paul Jolissaint |
EUDML | Discrete group actions and the minimal primal ideal space. EuDML | Discrete group actions and the minimal primal ideal space.
Discrete group actions and the minimal primal ideal space.
Ferdinand Beckhoff
Beckhoff, Ferdinand. "Discrete group actions and the minimal primal ideal space.." Mathematica Scandinavica 80.2 (1997): 289-309. <http://eudml.org/doc/167422>.
@article{Beckhoff1997,
author = {Beckhoff, Ferdinand},
keywords = {minimal primal ideal; -algebra; crossed product -algebra; Fell topology; essentially inner unitaries},
title = {Discrete group actions and the minimal primal ideal space.},
AU - Beckhoff, Ferdinand
TI - Discrete group actions and the minimal primal ideal space.
KW - minimal primal ideal; -algebra; crossed product -algebra; Fell topology; essentially inner unitaries
minimal primal ideal,
{C}^{*}
-algebra, crossed product
{C}^{*}
-algebra, Fell topology, essentially inner unitaries
{C}^{*}
{W}^{*}
{C}^{*}
Articles by Ferdinand Beckhoff |
To Charles Lyell 22 January [1865]1
I thank you for your very interesting letter.2 I have the true English instinctive reverence for rank & therefore liked to hear about the Princess Royal.3 You ask what I think of the Duke’s address4 & I shall be glad to tell you.
It seems to me extremely clever like every thing that I have read of his; but I am not shaken; perhaps you will say that neither gods nor men could shake me. I demur to the Duke reiterating his objection that the brilliant plumage of the male humming bird could not have been acquired through selection, at the same time entirely ignoring my discussion (p. 93 3rd Edition) on beautiful plumage being acquired thro’ sexual selection.5 The Duke may think this insufficient, but that is another question. All analogy makes me quite disagree with the Duke that the differences in the beak, wing & tail are not of importance to the several species.6 In the only two species which I have watched, the difference in flight & in the use of the tail was conspicuously great.7
The Duke who knows my orchis book so well might have learnt a lesson of caution from it, with respect to his doctrine of differences for mere variety or beauty.8 It may be confidently said that no tribe of plants presents such grotesque & beautiful differences which no one until lately conjectured were of any use; but now in almost every case, I have been able to shew their important service.
It should be remembered that with humming birds or orchids a modification in one part will cause correlated changes in other parts.9 I agree with what you say about beauty. I formerly thought a good deal on the subject & was led quite to repudiate the doctrine of beauty being created for beauty’s sake.10 I demur also to the Duke’s expression of “new births”:11 that may be a very good theory but it is not mine,—unless indeed he calls a bird born with a beak
\frac{1}{100}
th of an inch longer than usual “a new birth”; but this is not the sense in which the term wd usually be understood.
The more I work the more I feel convinced that it is by the accumulation of such extremely slight variations that new species arise. I do not plead guilty to the Duke’s charge that I forget that natural selection means only the preservation of variations which independently arise.12 I have expressed this in as strong language as I could use; but it wd have been infinitely tedious had I on every occasion thus guarded myself. I will cry “peccavi”13 when I hear of the Duke or you attacking Breeders for saying that man has made his improved Shorthorns or Pouter-pigeons or Bantams. And I cd quote still stronger expressions used by agricuturalists. Man does make his artificial breeds, for his selective power is of such importance relatively to that of the slight spontaneous variations. But no one will attack Breeders for using such expressions, & the rising generation will not blame me.14
Many thanks for your offer of sending me the Elements;15 I hope to read it all, but unfortunately reading makes my head whiz more than any thing else. I am able most days to work for 2 or 3 hours & this makes all the difference in my happiness.16 I have resolved not to be tempted astray, & to publish nothing till my Vol. on Variation is completed.17
You gave me excellent advice about the foot-notes in my Dog chapter,18 but their alteration gave me infinite trouble, & I often wished all the Dogs & I fear sometimes you yourself in the nether regions.
We (dictater & writer) send our best love to Lady Lyell19 | yours affectionately | Charles Darwin
If ever you shd. speak with the Duke on the subject please say how much interested I was with his Address & tell him about Sexual Selection.
The year is established by the relationship between this letter and the letter from Charles Lyell, 16 January 1865.
Letter from Charles Lyell, 16 January 1865.
Victoria Adelaide Mary Louise, Queen Victoria’s daughter. Lyell had written about her interest in CD’s theory in his letter of 16 January 1865.
Lyell had discussed the opening address delivered to the Royal Society of Edinburgh on 5 December 1864 by George Douglas Campbell, eighth duke of Argyll (G. D. Campbell 1864); see letter from Charles Lyell, 16 January 1865.
See letter from Charles Lyell, 16 January 1865 and nn. 10 and 11. CD’s discussion of sexual selection in Origin 3d ed., pp. 92–5, was unchanged from that of the first edition. CD addressed Campbell’s views on humming-birds in Descent 2: 151–3. CD’s notes on G. D. Campbell 1864, dated December 1864, are in DAR 47: 20.
Campbell argued that the variations in bills, wings, and tails of humming-birds were not such as to give competitive advantage in the struggle for existence (G. D. Campbell 1864, p. 281): ‘It seems rather to have been a rule having for its object the mere multiplying of life, and the fitting of new forms for new spheres of enjoyment, according as these might arise out of corresponding changes in other departments of the organic world’.
CD described the manner of flight of two species of humming-birds, Trochilus forficatus and T. gigas, which he saw in Chile, in Zoology 3: 110–12. See also Journal of researches, pp. 330–2, in which Trochilus forficatus is identified as Mellisuga Kingii, and R. D. Keynes ed. 2000, pp. 235–6, 245–6, 279.
Against the view that natural structures were created for the sake of beauty or variety, CD argued in Orchids, pp. 346–60, that the unusual or beautiful forms of the orchid flowers facilitated pollination by insects. Campbell had reviewed Orchids in the October 1862 issue of the Edinburgh Review ([G. D. Campbell] 1862).
CD discussed the principle of correlation of growth in Origin, pp. 11–12 and 143–50. According to this principle, when slight variations occur in one part of an organism, and are accumulated through natural selection, other parts of the organism are modified.
In Origin, p. 199, CD briefly discussed the question of beauty in relation to selection, stating: ‘[some naturalists] believe that very many structures have been created for beauty in the eyes of man, or for mere variety. This doctrine, if true, would be absolutely fatal to my theory.’ The discussion of beauty is expanded in Origin 4th ed., pp. 238–41. CD began to make notes on differing ideals of beauty in humans and on mate selection between 1838 and 1840 (see Notebooks, Notebook D, 99; Notebook M, 32; Notebook N, 26–9; and Old and useless notes, 8, 14, 22–4; see also Barrett 1980). See also Correspondence vol. 8, letter to G. H. K. Thwaites, 21 March [1860], and Correspondence vol. 12, letter to A. R. Wallace, 28 [May 1864] and n. 18.
According to Campbell, CD’s theory implied ‘the possibility of new births being the means of introducing new species’. Campbell emphasised, however, that CD offered no explanation of such births (G. D. Campbell 1864, p. 286).
See letter from Charles Lyell, 16 January 1865 and n. 6.
Peccavi: ‘I have sinned’.
In Origin, p. 30, CD discussed the predominance of human selection in the production of domestic breeds, relative to the direct action of external conditions of life, habit, and simple variability: ‘We cannot suppose that all the breeds were suddenly produced as perfect and as useful as we now see them. … The key is man’s power of accumulative selection: nature gives successive variations; man adds them up in certain directions useful to him. In this sense he may be said to make for himself useful breeds.’ He also gave numerous examples of animal breeders who habitually spoke of an animal’s organisation as something they could ‘model almost as they please’ (ibid., p. 31).
C. Lyell 1865. See letter from Charles Lyell, 16 January 1865 and n. 2.
After a period of improvement in the spring and summer of 1864, CD had a return to ill health toward the end of 1864 (see Correspondence vol. 12). See letter to J. D. Hooker, 7 January [1865].
CD had begun Variation in 1860; after several interruptions, he most recently resumed work in September 1864 (see Correspondence vol. 12). Variation was published in 1868.
CD sent Lyell the draft of his chapter on dogs for Variation in August or September 1860 (see Correspondence vol. 8, letter to Charles Lyell, 11 August [1860], and letter from Charles Lyell, 18 September 1860). Lyell had had difficulties accepting CD’s view, first expressed in Origin, p. 17, that the various breeds of dogs had descended from several distinct species rather than from a single progenitor (see Correspondence vol. 7). Lyell returned CD’s manuscript at the end of September, and remarked in his letter of 25 September 1860 (Correspondence vol. 8): ‘The case you make out seems very strong’. In his letter of 26 [September 1860] (Correspondence vol. 8), CD wrote that he was ‘grieved’ to hear from Lyell that material from the footnotes should be worked into the text, but would ‘obey’. The manuscript has not been found. See Variation 1: 15–45.
Mary Elizabeth Lyell. The letter is in Emma Darwin’s hand.
Criticises Duke of Argyll’s address [to the Royal Society of Edinburgh (1864)] and demurs on Argyll’s "new birth" theory.
Agrees with CL on beauty.
Enjoyed hearing of Princess Royal’s discussion [on Darwinism].
CD’s illness.
CL’s advice on chapter [of Variation] on dogs was excellent. |
EUDML | Three classical results on representations of a number. EuDML | Three classical results on representations of a number.
Three classical results on representations of a number.
Hirschhorn, Michael D.
Volume: 42, page B42f, 8 p.-B42f, 8 p.
Hirschhorn, Michael D.. "Three classical results on representations of a number.." Séminaire Lotharingien de Combinatoire [electronic only] 42 (1999): B42f, 8 p.-B42f, 8 p.. <http://eudml.org/doc/120032>.
@article{Hirschhorn1999,
author = {Hirschhorn, Michael D.},
keywords = {number of representations; number of divisors; -series; -series},
pages = {B42f, 8 p.-B42f, 8 p.},
title = {Three classical results on representations of a number.},
AU - Hirschhorn, Michael D.
TI - Three classical results on representations of a number.
SP - B42f, 8 p.
EP - B42f, 8 p.
KW - number of representations; number of divisors; -series; -series
number of representations, number of divisors,
q
q
Articles by Hirschhorn |
Transform position and velocity components from discontinued Standard Besselian Epoch (B1950) to Standard Julian Epoch (J2000) - Simulink - MathWorks España
Transform position and velocity components from discontinued Standard Besselian Epoch (B1950) to Standard Julian Epoch (J2000)
The Besselian Epoch to Julian Epoch block transforms two 3-by-1 vectors of Besselian Epoch position
\left({\overline{r}}_{B1950}\right)
\left({\overline{v}}_{B1950}\right)
into Julian Epoch position
\left({\overline{r}}_{J2000}\right)
\left({\overline{v}}_{J2000}\right)
Position in Standard Besselian Epoch (B1950), specified as a 3-by-1 vector.
Velocity in Standard Besselian Epoch (B1950), specified as a 3-by-1 vector.
Position in Standard Julian Epoch (J2000), returned as a 3-by-1 vector.
Velocity in Standard Julian Epoch (J2000), returned as a 3-by-1 vector.
\left[\begin{array}{l}{\overline{r}}_{J2000}\\ {\overline{v}}_{J2000}\end{array}\right]=\left[\begin{array}{cc}{\overline{M}}_{rr}& {\overline{M}}_{vr}\\ {\overline{M}}_{rv}& {\overline{M}}_{vv}\end{array}\right]\left[\begin{array}{l}{\overline{r}}_{B1950}\\ {\overline{v}}_{B1950}\end{array}\right]
\left({\overline{M}}_{rr},{\overline{M}}_{vr},{\overline{M}}_{rv},{M}_{vv}\right)
{\overline{M}}_{rr}\left[\begin{array}{ccc}0.9999256782& -0.0111820611& -0.0048579477\\ 0.0111820610& 0.9999374784& -\text{0}\text{.0000271765}\\ 0.0048579479& -\text{0}\text{.0000271474}& 0.9999881997\end{array}\right]
{\overline{M}}_{vr}=\left[\begin{array}{ccc}0.00000242395018& -\text{0}\text{.00000002710663}& -0.00000001177656\\ 0.00000002710663& 0.00000242397878& -0.00000000006587\\ 0.00000001177656& -0.00000000006582& 0.00000242410173\end{array}\right]
{\overline{M}}_{rv}=\left[\begin{array}{ccc}-0.000551& -0.238565& 0.435739\\ 0.238514& -0.002667& -0.008541\\ -0.435623& 0.012254& 0.002117\end{array}\right]
{\overline{M}}_{vv}=\left[\begin{array}{ccc}0.99994704& -0.01118251& -0.00485767\\ 0.01118251& 0.99995883& -0.00002718\\ 0.00485767& -0.00002714& 1.00000956\end{array}\right] |
Lead(II,IV)_oxide Knowpia
Lead(II,IV) oxide, also called red lead or minium, is the inorganic compound with the formula
{\displaystyle {\ce {Pb3O4}}}
. A bright red or orange solid, it is used as pigment, in the manufacture of batteries, and rustproof primer paints. It is an example of a mixed valence compound, being composed of both Pb(II) and Pb(IV) in the ratio of two to one.[2]
Lead tetroxide [1]
Minium, red lead, triplumbic tetroxide
A4G39L7HN2 Y
InChI=1S/4O.3Pb Y
Key: XMFOQHDPRMAJNU-UHFFFAOYSA-N Y
Appearance Vivid orange crystals
Vapor pressure 1.3 kPa (at 0 °C)
Tetragonal, tP28
P42/mbc, No. 135
Related lead oxides
Lead(II,IV) oxide has a tetragonal crystal structure at room temperature, which then transforms to an orthorhombic (Pearson symbol oP28, Space group Pbam, No. 55) form at temperature 170 K (−103 °C). This phase transition only changes the symmetry of the crystal and slightly modifies the interatomic distances and angles.[3]
Unit cell of tetragonal Pb3O4
(Key: Pb O)
Part of tetragonal red lead's crystal structure
Lead(II,IV) oxide is prepared by calcination of lead(II) oxide (
{\displaystyle {\ce {PbO}}}
; also called litharge) in air at about 450–480 °C:[4]
{\displaystyle {\ce {6 PbO + O2 -> 2 Pb3O4}}}
{\displaystyle {\ce {PbO + KOH + H2O -> K[Pb(OH)3]}}}
Another method of preparation relies on annealing of lead(II) carbonate (cerussite) in air:
{\displaystyle {\ce {6PbCO3 + O2 -> 2Pb3O4 + 6CO2}}}
Yet another method is oxidative annealing of white lead:
{\displaystyle {\ce {3Pb2CO3(OH)2 + O2 -> 2Pb3O4 + 3CO2 + 3H2O}}}
In solution, lead(II,IV) oxide can be prepared by reaction of potassium plumbate with lead(II) acetate, yielding yellow insoluble lead(II,IV) oxide monohydrate
{\displaystyle {\ce {Pb3O4.H2O}}}
, which can be turned into the anhydrous form by gentle heating:
{\displaystyle {\ce {K2PbO3 + 2Pb(OCOCH3)2 + H2O -> Pb3O4 + 2KOCOCH3 + 2 CH3COOH}}}
Natural minium is uncommon, forming only in extreme oxidizing conditions of lead ore bodies. The best known natural specimens come from Broken Hill, New South Wales, Australia, where they formed as the result of a mine fire.[5]
Red lead is virtually insoluble in water and in ethanol. However, it is soluble in hydrochloric acid present in the stomach, and is therefore toxic when ingested. It also dissolves in glacial acetic acid and a diluted mixture of nitric acid and hydrogen peroxide.
{\displaystyle {\ce {2Pb3O4 -> 6PbO + O2}}}
{\displaystyle {\ce {Pb3O4 + 4HNO3 -> PbO2 + 2Pb(NO3)2 + 2H2O}}}
With iron oxides and with elemental iron, lead(II,IV) oxide forms insoluble iron(II) and iron(III) plumbates, which is the basis of the anticorrosive properties of lead-based paints applied to iron objects.
Red lead has been used as a pigment for primer paints for iron objects. Due to its toxicity, its use is being limited. It finds limited use in some amateur pyrotechnics as a delay charge and was used in the past in the manufacture of dragon's egg pyrotechnic stars.
Red lead is used as a curing agent in some polychloroprene rubber compounds. It is used in place of magnesium oxide to provide better water resistance properties.
Red lead was used for engineer's scraping, before being supplanted by engineer's blue.
It is also used as an adultering agent in turmeric powder.
When inhaled, lead(II,IV) oxide irritates lungs. In case of high dose, the victim experiences a metallic taste, chest pain, and abdominal pain. When ingested, it is dissolved in the gastric acid and absorbed, leading to lead poisoning. High concentrations can be absorbed through skin as well, and it is important to follow safety precautions when working with lead-based paint.
Long-term contact with lead(II,IV) oxide may lead to accumulation of lead compounds in organisms, with development of symptoms of acute lead poisoning. Chronic poisoning displays as agitation, irritability, vision disorders, hypertension, and a grayish facial hue.
Lead(II,IV) oxide was shown to be carcinogenic for laboratory animals. Its carcinogenicity for humans was not proven.
Minium from a mine fire at Broken Hill, Australia
This compound's Latin name minium originates from the Minius, a river in northwest Iberia where it was first mined.
Lead(II,IV) oxide was used as a red pigment in ancient Rome, where it was prepared by calcination of white lead. In the ancient and medieval periods it was used as a pigment in the production of illuminated manuscripts, and gave its name to the minium or miniature, a style of picture painted with the colour.
Made into a paint with linseed oil, red lead was used as a durable paint to protect exterior ironwork. In 1504 the portcullis at Stirling Castle in Scotland was painted with red lead, as were cannons including Mons Meg.[6]
As a finely divided powder, it was also sprinkled on dielectric surfaces to study Lichtenberg figures.
In traditional Chinese medicine, red lead is used to treat ringworms and ulcerations, though the practice is limited due to its toxicity. Also, azarcón, a Mexican folk remedy for gastrointestinal disorders, contains up to 95% lead(II,IV) oxide.[7]
It was also used before the 18th century as medicine.[8]
Lead(II) oxide,
{\displaystyle {\ce {PbO}}}
Lead(IV) oxide,
{\displaystyle {\ce {PbO2}}}
Minium (pigment)
^ "VOLUNTARY RISK ASSESSMENT REPORT ON LEAD AND SOME INORGANIC LEAD COMPOUNDS". Retrieved 2012-12-25.
^ Gavarri, J; Weigel, Dominique; Hewat, A. W. (1978). "Oxydes de plomb. IV. Évolution structurale de l'oxyde Pb3O4 entre 240 et 5 °K et mécanisme de la transition" [Lead oxides. IV. Structural evolution of the oxide Pb3O4 between 240 and 5 K and mechanism of transition]. Journal of Solid State Chemistry. 23 (3–4): 327. doi:10.1016/0022-4596(78)90081-6.
^ Carr, Dodd S. "Lead Compounds". Ullmann's Encyclopedia of Industrial Chemistry. Weinheim: Wiley-VCH. doi:10.1002/14356007.a15_249.
^ Minium
^ James Balfour Paul, Accounts of the Treasurer of Scotland, vol. 2 (Edinburgh, 1900), p. 277.
^ Bose, A.; Vashistha, K; O'Loughlin, B. J. (1983). "Azarcón por empacho – another cause of lead toxicity". Pediatrics. 72: 108–118.
^ "The London Lancet: A Journal of British and Foreign Medicine, Physiology, Surgery, Chemistry, Criticism, Literature and News". 1853.
National Pollutant Inventory - Lead and Lead Compounds Fact Sheet
Minium mineral data |
K
-Theory Groups of Certain Crossed Product
C*
-A1gebras
Michael OberguggenbergerYa-Guang Wang
On the Behaviour of Solutions to the Dirichiet Problem for Second Order Elliptic Equations near Edges and Polyhedral Vertices with Critical Angles
Vladimir G. Maz'yaJürgen Rossmann
Approximation by Linear Combinations of Fundamental Solution of Elliptic Systems of Partial Differential Operators
Uniqueness Theorems in Linear Theory of Microporous Solids
Tengiz G. GegeliaLothar Jentsch
General Approximation-Solvability of Nonlinear Equations Involving
\mathcal A
-regular Operators
On Spectral Factorization and Generalized Nehari Problems
Tangential Carathéodory-Fejer Interpolation for Stieltjes Functions at Real Points
Extending Chains of Factorizations and Minimal Negative Signatures
Tiberiu ConstantinescuAurelian Gheondea
Modular Convergence Theorems in Fractional Musielak-Orlicz Spaces
Carlo BardaroGianluca Vinti
Estimating Remainder Functionals by the Moduli of Smoothness |
Spectrographs - Everything Wiki
This problem set is designed for astronomy to help the student, teacher, and researcher understand the inner workings of a spectrograph. Template:One Box Section
{\displaystyle n_{0}}
{\displaystyle n_{1}}
{\displaystyle n_{2}}
{\displaystyle \theta '}
{\displaystyle {\begin{aligned}\theta '_{0}&=\,{\text{arcsin}}{\Big (}{\frac {n_{0}}{n_{1}}}\,\sin \theta _{0}{\Big )}\\\theta _{1}&=\alpha -\theta '_{0}\\\theta '_{1}&=\,{\text{arcsin}}{\Big (}{\frac {n_{1}}{n_{2}}}\,\sin \theta _{1}{\Big )}\\\theta _{2}&=\theta '_{1}-\alpha \end{aligned}}}
{\displaystyle n_{0}=n_{2}\simeq 1}
{\displaystyle n=n_{1}}
{\displaystyle \delta }
{\displaystyle \delta =\theta _{0}+\theta _{2}=\theta _{0}+{\text{arcsin}}{\Big (}n\,\sin {\Big [}\alpha -{\text{arcsin}}{\Big (}{\frac {1}{n}}\,\sin \theta _{0}{\Big )}{\Big ]}{\Big )}-\alpha }
{\displaystyle \theta _{0}}
{\displaystyle \alpha }
{\displaystyle \sin \theta \approx \theta }
{\displaystyle {\text{arcsin}}x\approx x}
{\displaystyle \delta }
{\displaystyle \delta \approx \theta _{0}-\alpha +{\Big (}n\,{\Big [}{\Big (}\alpha -{\frac {1}{n}}\,\theta _{0}{\Big )}{\Big ]}{\Big )}=\theta _{0}-\alpha +n\alpha -\theta _{0}=(n-1)\alpha \ .}
{\displaystyle \delta (\lambda )\approx [n(\lambda )-1]\alpha }
{{Charge ontology}}{{Flight resources}}{{Principles of radiation astronomy}}Template:Radiation astronomy resources{{Repellor vehicle}}{{Technology resources}}
Retrieved from "https://everything.wiki/index.php?title=Spectrographs&oldid=3738319" |
Your last account,1 some months ago, was so little satisfactory, that I have often been thinking of you, & should be really obliged if you would fly me a few lines, & tell me how your voice & chest are. I most sincerely hope that your report will be good; this wonderfully mild winter must be in your favour.
As for myself I really have no news: just lately my stomach has been a little extra ailing.2 All other members of the family are flourishing. My eldest Boy is now home from Rugby: he is a thoroughily steady, industrious & good boy; I fancy, (though perhaps it is fancy) that I see the contracting effects on his mind of his very steady attention to classics: formerly I think he had more extended interests, & cared more for the causes & reasons of things.3 Our second lad Georgie, has a strong mechanical turn: & we think of making him an engineer: I shall try & find out for him some less classical school,—perhaps Bruce Castle.4 I certainly shd. like to see more diversity in Education, than there is any ordinary school: no exercise of the observing or reasoning faculties,—no general knowledge acquired,—I must think it a wretched system: on the other hand a Boy who has learnt to stick at Latin & conquer its difficulties, ought to be able to stick at any labour.— I shd. always be glad to hear anything about schools or education from you.
I am at my old, never-ending subject, but trust I shall really go to press in a few months with my second volume on Cirripedes:5 I have been much pleased by finding some odd facts in my 1st. vol. believed by Owen, & a few others, whose good opinion I regard as final.—6 I have this morning been dissecting a most abnormal cirripede, which after a good meal has to vomit forth the residuum, for there is no other exit!7
I heard yesterday from Dr. Hooker, who married Henslow’s eldest daughter, of the birth of a son8 under Chloroform, at Hitcham.
I wonder when we shall see you here again: it wd. give Emma & myself no common pleasure. Do write pretty soon & tell me all you can about yourself—& family & I trust your Report of yourself may be much better than your last.
Catherine & Susan are at present staying with Erasmus in London, & perhaps I shall go up & see them next week.9 I have been very little in London of late, & have not seen Lyell since his return from America: how lucky he was to exhume with his own hand parts of 3 skeletons of Reptiles out of the Carboniferous strata, & out of the inside of a fossil tree, which had been hollow within!10
Farewell | My dear Fox | Your’s affectionately | Charles Darwin
See letter to W. D. Fox, 24 [October 1852], in which CD mentions Fox’s chest ailment. See also Correspondence vol. 1, letters from W. D. Fox, 30 June 1832 and 29 August – 28 September 1832, for the onset of the illness affecting his lungs.
See letter to G. R. Waterhouse, 18 January [1853], n. 3. At the end of January, CD summed up that he had had 11 days on which he felt very well. This compared with 24 such days in December 1852.
After much consideration of the effects of ‘the old stereotyped stupid classical education’, CD had chosen to send William Erasmus Darwin to Rugby School rather than to the educationally innovative Bruce Castle School (letter to W. D. Fox, 7 March [1852], and Correspondence vol. 4, letter to W. D. Fox, 10 October [1850]).
George Howard Darwin, then 7
\frac{1}{2}
years old, was not sent to Bruce Castle either. In August 1856 he went to Clapham Grammar School, where science and mathematics had a more prominent place in the curriculum than at more traditional schools. The school was run by Charles Pritchard, who later became Savilian Professor of Astronomy at Oxford (Moore 1977 p. 53).
The final proofs of Living Cirripedia (1854) were not sent to the printer until July 1854, and the proofs of Fossil Cirripedia (1854) were not ready until mid-September.
The sexual relations of Ibla and Scalpellum (Living Cirripedia (1851): 281–93). See letter to Richard Owen, 17 July [1852].
Alcippe lampas has no rectum or anus (Living Cirripedia (1854): 546–7).
William Henslow Hooker, born 24 January 1853.
CD recorded the expenses of a trip to London on 3 February 1853 in his Account book (Down House MS). His Health diary (Down House MS) indicates that the visit was from 1 to 3 February.
For Charles Lyell’s description of some of the reptile bones, see K. M. Lyell ed. 1881, 2: 183, 186. Later in the year, he published an account of them (C. Lyell 1853a).
Moore, James Richard. 1977. On the education of Darwin’s sons: the correspondence between Charles Darwin and the Reverend G. V. Reed, 1857–1864. Notes and Records of the Royal Society 32 (1977–8): 51–70.
Discusses education of his sons. Would like to see more diversity.
He is pleased that Richard Owen and others had a good opinion of his first volume [on Living Cirripedia]. |
To J. D. Hooker 3 January [1863]
I am burning with indignation & must exhale. If you have not already read, do read the first part of Falconer’s paper on Elephants in N. H. Review & mark Owen’s whole conduct.—1 I could not get to sleep till past 3 last night from indignation. Thinking over his conduct in this case, in the Brain-case2 & towards Mantell3 Nasmyth,4 Huxley,5 you & self,6 & review on Lyell,7 the Terlepeton case8 &c &c, I declare I think every honest man of science is almost bound to show his sense of Owen’s character. I have made up my mind, as far as I can at this distance of time, to attend when next Council of Royal is elected, & if no else does, vote for some other man;9 & if Owen were to come & speak to me I would tell him for what I came. But possibly he may answer Falconer, & explain. I have read only about
\frac{1}{4}
of Falconer’s paper.—10 The Reviews seem good in this number.—11
Now for pleasanter subjects; we were all amused at your defence of stamp collecting & collecting generally.12 Henrietta13 had audacity to say “well I think Dr. Hooker shows that it leads all sorts of vice; yet I shall go on collecting plants, for I love to look at them.” I ought to say nothing against collecting for
\frac{6}{7}
th of my children collect, & I collected seals, franks, coins minerals, shells insects & God knows what else. But by Jove I can hardly stomach a grown man collecting stamps. Who would ever have thought of your collecting Wedgwood ware!14 but that is wholly different like engravings or pictures. We are degenerate descendants of old Josiah W. for we have not a bit of pretty ware in the house.—15
When you see Mr Oldfield pray thank him; my questions were foolish; but not rarely foolish questions, lead, I find, to good results.—16
Notwithstanding the very pleasant reason you give for our not enjoying a holiday, namely that we have no vices, it is a horrid bore.—17 I have been trying for health sake to be idle with no success. What I shall soon have to do, will be to erect a tablet in Down church “sacred to the memory &c” & officially die, & then publish books “by the late Charles Darwin”; for I cannot think what has come over me of late; I always suffered from the excitement of talking, but now it has become ludicrous. I talked lately for 1
\frac{1}{2}
hours (broken by tea by myself) with my nephew18 & I was shaking & vomiting half the night— It is a fearful evil for self & family.
Goodnight | Ever yours | C. Darwin
My children’s dried flowers get a little mouldy; is it not good to wash them with corrosive Sublimate19 (how much?) in spirits? or in water? Sometime tell me.—
In his paper on fossil elephants, which was published in the January 1863 number of the Natural History Review, Hugh Falconer claimed (Falconer 1863a, pp. 43–9) that Richard Owen and his protégé, Charles Carter Blake, had abused the law of priority in zoological nomenclature. He explained that Owen and Blake had supplanted the species name Falconer had first given to the fossil elephant, Elephas columbi (Falconer 1857a, p. 319), with the name E. texianus (Owen 1858 and 1861b, and Blake 1861). In 1862, Blake argued that, according to the rules of zoological nomenclature as defined by the British Association for the Advancement of Science, the name E. columbi could be changed even without ‘published priority’, on the grounds that it was not clearly defined and was likely to propagate errors (Blake 1862, p. 58). Falconer refuted the claims of both Owen and Blake for the name E. texianus, dismissing their reasons as ‘light and trivial’ (Falconer 1863a, p. 49).
CD refers to Owen’s long-running dispute with Thomas Henry Huxley, George Rolleston, and William Henry Flower on the comparative anatomy of human and simian brains, now often referred to as the hippocampus controversy (see Correspondence vols. 8–10, and Rupke 1994, pp. 270–86).
CD refers to the geologist and palaeontologist Gideon Algernon Mantell. In addition to disputes between the two men regarding the nature of particular fossils, in 1850 Owen attempted to reproduce without permission illustrations of fossil reptiles from a publication by Mantell. Owen was also thought to be the author of an obituary of Mantell that appeared in the Literary Gazette, 13 November 1852, p. 842; the article discussed Mantell’s ‘weaknesses’, dismissing him as an enthusiast with an ‘overweaning estimate’ of the value of his own work (see A. Desmond 1982, p. 208 n. 13). On Owen’s relationship with Mantell, see Spokes 1927, pp. 204–8, 221–7; Curwen ed. 1940, pp. 245–7, 260–2; Benton 1982; A. Desmond 1982, pp. 24, 208 n. 13; and Rupke 1994, pp. 6–8, 125–6. See also n. 8, below.
In 1839, a priority dispute developed between Owen and the dentist-surgeon Alexander Nasmyth; both men claimed to have been the first to outline a new theory of the ossific transformation of the cells of the pulp into dental ivory, that is, the development of teeth (see Owen 1839 and Nasmyth 1841). The debate was conducted in the Lancet between 6 June and 4 July 1840, and in the London Medical Gazette between 5 June and 17 July 1840. These exchanges culminated with Owen ridiculing Nasmyth’s efforts in the scientific arena (London Medical Gazette, 17 July 1840, pp. 657*–673*); Nasmyth subsequently claimed that Owen had prevented the publication of his research in the official report of the British Association’s 1839 meeting (see Nasmyth 1841, pp. iii–xvi). See also letter from J. D. Hooker, [6 March 1863] and n. 7.
T. H. Huxley and Owen had maintained a deep-seated antipathy since the early 1850s (see A. Desmond 1982, pp. 19–55, A. Desmond 1994–7, and Rupke 1994; see also n. 2, above).
CD refers in part to Owen’s anonymous review of Origin ([Owen] 1860a). CD thought the review ‘malignant’, and that it misrepresented and misquoted his work; in his letter to Charles Lyell of 10 April [1860] (Correspondence vol. 8), CD referred to the review: ‘It is painful to be hated in the intense degree with which Owen hates me’. For accounts of the additional differences between CD and Owen, see Hull 1973, pp. 171–215, and Rupke 1994. CD also strongly objected to the ‘slighting way’ Owen alluded to J. D. Hooker 1859, in which Hooker announced his support for the theory of natural selection (see Correspondence vol. 8, letter to Asa Gray, 18 May [1860]). See also Correspondence vol. 8, letter to T. H. Huxley, 9 April [1860]. There is an annotated copy of [Owen] 1860a in the Darwin Pamphlet Collection–CUL.
In an anonymous review ([Owen] 1851) of the eighth edition of Charles Lyell’s Principles of geology (C. Lyell 1850), and of C. Lyell 1851a and 1851b, Owen attacked Lyell’s anti-progressionism and uniformitarianism.
Telerpeton (Leptopleuron) was a fossil reptile discovered in Scotland in 1851, which Owen and Mantell may both have been asked to describe; a priority dispute followed (Benton 1982).
Hooker and CD had discussed the possibility of ‘organising an opposition’ to Owen’s election to the council of the Royal Society of London in November 1862 (see Correspondence vol. 10, letter from J. D. Hooker, [15 and] 20 November [1862], and letter to J. D. Hooker, 24 [November 1862] and nn. 9 and 10). Owen was not re-elected at the 30 November 1863 anniversary meeting (see Proceedings of the Royal Society 13: 39), although CD did not attend the meeting (see ‘Journal’ (Correspondence vol. 11, Appendix II)).
Falconer 1863a.
CD refers to the January 1863 number of the Natural History Review, which included a review of the first part of volume 1 of Bentham and Hooker 1862–83. CD’s annotated copy of this number of the journal is in the Darwin Library–CUL. See also following letter and n. 5.
See Correspondence vol. 10, letter from J. D. Hooker, [27 or 28 December 1862].
CD’s daughter, Henrietta Emma Darwin, was 19 years old.
Hooker had written that he was collecting Wedgwood ware ‘solely because they are pretty & I love them’ (Correspondence vol. 10, letter from J. D. Hooker, [27 or 28 December 1862]).
Both CD and Emma Darwin were grandchildren of the master-potter, Josiah Wedgwood I.
The botanist Augustus Frederick Oldfield, who had travelled widely in Australia and Tasmania, was a frequent visitor to the Royal Botanic Gardens at Kew; Hooker offered to convey any questions that CD might have for him (see Correspondence vol. 10, letter from J. D. Hooker, [21 December 1862] and n. 5). In his letter to Hooker of 24 December [1862] (ibid.), CD asked several questions about the diet and behaviour of Australian aborigines. Hooker sent Oldfield’s replies to CD’s queries with his letter of [27 or 28 December 1862] (ibid.).
See Correspondence vol. 10, letter from J. D. Hooker, [31 December 1862].
Henry Parker, the son of CD’s sister, Marianne Parker, visited Down House on 29 December 1862 (see Correspondence vol. 10, letter to J. D. Hooker, 29 [December 1862]).
Corrosive sublimate (mercuric chloride) was used in herbaria to provide protection for dried specimens against fungal and insect attack (EB).
Benton, Michael J. 1982. Progessionism in the 1850s: Lyell, Owen, Mantell and the Elgin fossil reptile Leptopleuron (Telerpeton). Archives of Natural History 11: 123–36.
Nasmyth, Alexander. 1841. Three memoirs on the development and structure of the teeth and epithelium, read at the ninth annual meeting of the British Association for the Encouragement of Science, held at Birmingham, in August, 1839; with diagrams in illustration of them. London: John Churchill.
[Owen, Richard.] 1851b. [Review of Charles Lyell’s anniversary address to the Geological Society of London (1851), Principles of geology (8th edition), & other works.] Quarterly Review 89: 412–51.
Spokes, Sidney. 1927. Gideon Algernon Mantell, LLD, FRCS, FRS, surgeon and geologist. London: John Bale, Sons & Danielsson. [Vols. 4,11]
Indignant over Owen’s conduct as described in Hugh Falconer’s article on elephants ["On the American fossil elephant of the regions bordering the Gulf of Mexico", Nat. Hist. Rev. (1863): 43–114]. |
pH - Citizendium
(CC) Drawing: Milton Beychok
The pH scale measures the acidity or alkalinity of an aqueous solution, which is a solution in which water is the solvent. Values for pH range from about 0 (strongly acidic) to about 14 (strongly alkaline or basic). The pH of a neutral solution (neither acid or basic), such as pure water at room temperature and atmospheric pressure is 7, whereas the pH of an acidic solution is less than 7 and the pH of a basic solution is greater than 7. The pH scale is logarithmic which means that a difference of one pH unit is equivalent to a ten-fold difference in hydrogen ion concentration. The notation pH is sometimes referred to as the power of hydrogen or the potential of hydrogen.
The traditional way to determine whether a solution is acidic or basic is by wetting litmus paper with the solution. If the wet litmus paper turns red, the solution has a pH less than 7 and is acidic. If it turns blue, the solution has a pH greater than 7 and is acidic. Measuring the actual pH value of a solution is more accurately done with a pH meter.
The pH scale was originally defined by Danish biochemist Søren Peter Lauritz Sørensen in 1909, who wrote it as PH. It was subsequently changed to the modern notation of pH in 1920 by William Mansfield Clark, an American biochemist, for typographical convenience in his book The Determination of Hydrogen Ions.[1][2]
Definitions and discussion
H+ and OH- ions
molar concentrations vs. pH [3]
H+ concentration
(mole/litre)
OH- concentration
0.01 0.000000000001 2
0.001 0.00000000001 3
0.0001 0.0000000001 4
0.00001 0.000000001 5
0.000001 0.00000001 6
0.000000001 0.00001 9
0.0000000001 0.0001 10
0.00000000001 0.001 11
0.000000000001 0.01 12
0.0000000000001 0.1 13
For more information, see: Activity (chemistry) and Activity coefficient.
A molar concentration of a compound, in moles per litre of solution, is commonly written as the symbol of the compound surrounded by square brackets. Thus, the hydrogen (H+) ion concentration in an aqueous solution is written simply as [H+] or as hydronium [H3O+] and both describe the equivalent entity.[4]
For very dilute solutions, the pH value can be defined by this simple expression:[3][5][6][7]
{\displaystyle {\rm {pH}}=-\log _{10}\left[{\rm {H^{+}}}\right]=\log _{10}{\frac {1}{\left[{\rm {H^{+}}}\right]}}}
and the corresponding expression for the hydroxide (OH-) ions can be expressed as:
{\displaystyle {\rm {pOH}}=-\log _{10}\left[{\rm {OH^{-}}}\right]=\log _{10}{\frac {1}{\left[{\rm {OH^{-}}}\right]}}}
Liquid water molecules undergo the following rapid, reversible dissociation (or ionization) reaction:
This reaction can be called dissociation or self-ionization of water. Since both H3O+ and OH- are simultaneously forming to a slight degree, this makes water both a very weak acid and a very weak base, with the same acid and base dissociation constant, symbolized as Kw. In the expression of this equilibrium constant for an aqueous solution, which can be called the dissociation or ionization constant of water, the molarity of water is omitted by convention. At about 25°C, Kw = 1 x 10-14. The expression for Kw is:
Kw = [H3O+] [OH-] = [H+] [OH-] = 1 x 10-14
Since the product of the concentration of hydrogen ions and the concentration of hydroxide ions is a constant at about 25°C, namely:
{\displaystyle \left[{\rm {H^{+}}}\right]\left[{\rm {OH^{-}}}\right]=1\times 10^{-14}}
taking logarithms gives:
{\displaystyle {\rm {pH+{\rm {pOH=14}}}}}
At about 25°C, the mid-point of 7 in the pH scale indicates ionic neutrality of the solution, namely when [H+] equals [OH−] (see the adjacent table). An aqueous solution is acidic when [H+] > [OH-] and is basic or alkaline when [OH-] > [H+].
As the theory behind chemical reactions became more sophisticated, the definition of pH was reexamined. Specifically, as the role of electrical charge in chemical reactions became better understood, the definition of pH was changed to refer to the active hydrogen ion concentration. The more theoretical definition of pH, while not generally covered in many introductory chemistry textbooks, is the definition adopted by the International Union of Pure and Applied Chemistry (IUPAC):[3][8]
{\displaystyle {\rm {pH}}=-\log _{10}\,(a_{\rm {H^{+}}})\equiv -\log _{10}\,\left(\gamma \,\left[{\rm {H^{+}}}\right]\right)}
{\displaystyle a_{\rm {H^{+}}}}
is the hydrogen ion activity and the factor
{\displaystyle \gamma }
is the hydrogen ion activity coefficient.[3][8]
Only in dilute solutions (about 0.001 moles per litre or less) are all anion and cations so far apart that they are free to be at their maximum activity where:
{\displaystyle \gamma =1\,}
{\displaystyle a_{\rm {H^{+}}}=\left[{\rm {H^{+}}}\right]\,}
At higher acid and base concentrations, the space between cations and anions decreases, so that they begin to obstruct each other and shield each others charge. Thus, the mobility of the any particular ion is impaired by interactions with other ions and their associated electrical fields. These local electric field interactions affect the extent to which the ions can participate in chemical reactions, and give an apparent concentration that is always smaller than the real concentration, that is, the dimensionless parameter γ becomes less than unity. In other words, the ion activity is "slowed down" and [H+] becomes greater than aH+. The coefficient γ (ion activity over concentration) decreases with the increasing acid concentration. Therefore, for acid concentrations greater than about 0.001 moles per liter, it is advisable to use activities instead of concentrations in order to accurately predict pH.[3]
pH of some common substances
The two tables below list the pH ranges for each of a number of fairly common substances:
Human gastric juice 1.0 − 3.0
Car battery acid 1.1 − 1.7
Lime juice 1.8 − 2.0
Soft drinks 2.0 − 4.0
Lemon juice 2.2 − 2.4
Vinegar 2.4 − 3.4
Apple juice 2.9 − 3.3
Wine 3.4 − 3.7
Tomato juice 4.0 − 4.4
Beer 4.0 − 5.0
Coffee 5.0 − 6.5
Rainwater 5.1 − 5.6
Banana juice 4.5 − 4.7
Human urine 4.8 − 8.4
Cow milk 6.3 − 6.6
Human saliva 6.5 − 7.5
Soap suds 7.0 − 10.0
Human blood plasma 7.3 − 7.5
Sea water 7.4 − 8.3
Egg white 7.6 − 8.0
Baking soda solution 8.3 − 8.8
Milk of Magnesia 10.6 − 10.7
Household ammonia 11.0 − 12.0
Household lye 13.6 − 14.0
(1) Car battery acid is an aqueous solution of 65 weight % sulfuric acid, H2SO4
(2) Baking soda solution is an aqueous solution of sodium bicarbonate, NaHC03
(3) Milk of Magnesia is an aqueous solution of magnesium hydroxide, Mg(OH)2
(4) Household ammonia is ammonium hydroxide, NH4OH, a dilute aqueous
(5) Household lye is an aqueous solution of sodium hydroxide, NaOH
↑ William Mansfield Clark (1920). The Determination of Hydrogen Ions, 1st Edition. William & Wilkins Company, page 35. Available in Google Books here
↑ pH: Potenz, The Determination of Hydrogen Ions, History of Analytical Chemistry, Electrochemistry, Past and Present From the JRank Science & Philosophy website
↑ 3.0 3.1 3.2 3.3 3.4 pH Measurement Definitions: The pH Scale
↑ The hydronium notation reflects the physical situation, because the positive point charge H+ binds strongly to H2O. Many textbooks therefore propagate the use of [H3O+].
↑ Darrell D. Ebbing and Mark S. Wrighton (1987). General Chemistry, 2nd Edition. Houghton Mifflin, pp. 103-117. ISBN 0-395-35654-7.
↑ Kenneth W. Whitten and Kenneth D. Gailey (1984). General Chemistry with Qualitative Analysis, 2nd Edition. Saunders College, pp. 263-278. ISBN 0-03-63287-5.
↑ What is pH? Professor Frederick A. Senese, Frostburg State University, Maryland
↑ 8.0 8.1 IUPAC Gold Book: pH
Retrieved from "https://citizendium.org/wiki/index.php?title=PH&oldid=37875" |
To Charles Lyell [8 August 1846]
I was delighted to receive your letter, which was forwarded here to me. I am very glad to hear about the new Edit of the Principles,1 & I most heartily hope you may live to bring out half-a-dozen more editions. There would not have been such books as d’Orbignys S. American Geology2 published, if there had been seven Editions of the Principles distributed in France. I am rather sorry about the small type; but the first Edit, my old true-love,3 which I never deserted for the later editions, was also in small type.— I much fear I shall not be able to give any assistance to Book III;4 I think I formerly gave my few criticisms, but I will read it over again very soon (though I am slaving to finish my S. American Geolog) & see whether I can give you any references.—
I have been thinking over the subject, & can remember no one book of consequence, as all my materials (which are in an absolute chaos on separate bits of paper) have been picked out of books not directly treating of the subjects you have discussed, & which I hope some day to attempt: thus Hooker’s Antarctic Flora,5 I have found eminently useful & yet I declare I do not know what precise facts I could refer you to. Bronn’s Gesichte6 (which you once borrowed) is the only systematic book I have met with on such subjects; & there are no general views in such parts, as I have read, but an immense accumulation of references, very useful to follow up, but not credible in themselves;—thus he gives hybrids from ducks & fowls just as readily as between fowls & pheasants! you can have it again, if you like.— I have no doubt Forbes essay, which is I suppose now fairly out, will be very good under geographical head.7 Koelreuter’s German Book8 is excellent on Hybrids, but it will cost you a good deal of time to work out any conclusions from his numerous details. With respect to variation, I have found nothing, but minute details scattered over scores of volumes.— But I will look over Book III again: What a quantity of work you have in hand! I almost wish you cd have finished America,9 & thus have allowed yourself rather more time for the old Principles, & I am quite surprised that you cd. possibly have worked your own new matter in within six weeks. Your intention of being in Southampton will much strengthen mine & I shall be very glad to hear some of your American Geolog. news.—10
You have pleased me much by saying that you intend looking through my Volcanic volume: it cost me 18 months!!! work & I have heard of very few who has read it; now I shall feel whatever little (& little it is) there is confirmatory of old work or new will work its effect & not be lost. I wish my S. American volume was out for same end, & I daresay you will be heartily glad it is not, for you must with all your work in hand, grudge time for your own new materials. I shd. have liked to have had your opinion on my facts & short discussion regarding the foliation of the metamorphic schists,11 which I am now correcting;—and another on the absence of recent conchiferous deposits & on the tertiary formations having been deposited during subsidence:12 but I will have mercy on you & say no more on my volume, of which I am inexpressibly weary & thank Heavens have now finally corrected
\frac{2}{3}
of, & hope to see published this month.
I return home on Tuesday, having been here for a week to see my Father: Emma & the children have been having colds but are otherwise well.— I hope you found Mr & Mrs Lyell tolerably well.— How I shall enjoy having you for a visit to Down & I believe you cd with quiet & fresh air do more work with us than in that horrid place London.—
I must go to work to proof-sheets: my vol will be about 240 pages, dreadfully dull yet much condensed: I think, whenever you have time to look through it, you will think the collection of facts on the elevation of the land & on the formation of terraces pretty good.
Goodbye with many thanks for your letter & my kindest remembrances to Mrs Lyell. | Ever yours | C. Darwin
The seventh edition of the Principles of geology (C. Lyell 1847).
Orbigny 1835–47, vol. 3, pt 3: Gélogie.
CD had the first edition with him on the Beagle voyage; see Correspondence vol. 1, letter to J. S. Henslow, 18 May – 16 June 1832, and Autobiography, pp. 77, 101, for CD’s adoption of Lyell’s views. CD’s annotated copy is in the Darwin Library–CUL.
The third volume of Lyell’s previous edition (C. Lyell 1840a) contained an extended discussion of the transmutation and first appearance of species (chapters 1–11). CD’s copy is lightly annotated (Darwin Library–CUL).
Bronn 1841–9. CD’s annotated copy is in the Darwin Library–CUL.
Kölreuter 1761–6. CD’s copy is in the Darwin Library–CUL. CD’s frequent citations in the Origin and later works make clear that Joseph Gottlieb Kölreuter’s work was one of CD’s major sources on hybridism and an important influence in the development of his theory. DAR 116 contains CD’s abstracts and notes on thirteen papers by Kölreuter.
An account of Lyell’s second visit to the United States (September 1845 – June 1846) was eventually published as C. Lyell 1849. CD’s annotated copy is in the Darwin Library–CUL.
Lyell presented a short account to the British Association (C. Lyell 1846c).
South America, pp. 140–68.
South America, pp. 135–9.
Lyell, Charles. 1847. Principles of geology; or, the modern changes of the earth and its inhabitants considered as illustrative of geology. 7th ed. London. [Vols. 4,9]
Comments on forthcoming edition [7th (1847)] of CL’s Principles. Mentions other books relevant to CL’s needs by Hooker, H. G. Bronn, Edward Forbes, and J. G. Kölreuter. Discusses his own books on volcanoes and the geology of S. America.
Mentions expected visit to Down by the Lyells. |
How to Figure Cost Per Square Inch of Pizza: 13 Steps
1 Finding the Per Inch Cost of a Round Pizza
2 Finding the Per Inch Cost of a Rectangular Pizza
When deciding where to order a pizza, you likely want to consider value. One way to consider the value of a pizza is to determine how much the pizza costs per square inch. In order to find this cost, you need to determine the area of the pizza, which can be found by using basic formulas. As long as you know the price and dimensions of the pizzas, you can find out which restaurant offers the best value.
Finding the Per Inch Cost of a Round Pizza Download Article
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/26\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-1-Version-2.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/2\/26\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-1-Version-2.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Set up the formula for the area of a circle. The formula is
{\displaystyle A=\pi (r^{2})}
{\displaystyle r}
equals the length of the circle’s radius.[1] X Research source
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/c\/c8\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-2-Version-2.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-2-Version-2.jpg","bigUrl":"\/images\/thumb\/c\/c8\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-2-Version-2.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-2-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Divide the size of the pizza in half. Pizza sizes are measured by their diameter. To find the length of the radius, you have to divide the diameter in half.
For example, if you are ordering a 20 inch pizza, calculate
{\displaystyle 20\div 2=10}
. So the radius of the pizza is 10 inches.
Plug the length of the radius into the formula. Remember to substitute for the variable
{\displaystyle r}
For example, if the radius is 10 inches, your formula will look like this:
{\displaystyle A=\pi (10^{2})}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/ef\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-4-Version-2.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-4-Version-2.jpg","bigUrl":"\/images\/thumb\/e\/ef\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-4-Version-2.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-4-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Square the length of the radius. To square a number, multiply it by itself.
For example, if the radius is 10 inches, you would calculate
{\displaystyle 10\times 10=100}
, so your formula will now look like this:
{\displaystyle A=\pi (100)}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b8\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-5-Version-2.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-5-Version-2.jpg","bigUrl":"\/images\/thumb\/b\/b8\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-5-Version-2.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-5-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
<b class="whb">{\displaystyle \pi }</b>
. You can use a calculator, or use 3.14 for
{\displaystyle \pi }
. The result will give you the area, in square inches, of your pizza.
{\displaystyle A=\pi (100)}
{\displaystyle A=3.14(100)}
{\displaystyle A=314}
So, the area of a 20-inch round pizza is 314 square inches.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/25\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-6.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-6.jpg","bigUrl":"\/images\/thumb\/2\/25\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-6.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-6.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Divide the price of the pizza by the number of square inches. This will give you the cost per square inch of the pizza.
For example, if a pizza costs $32 and has an area of 314 square inches, you would calculate
{\displaystyle 32\div 314=.101}
. So, the cost per square inch of pizza is about .10, or 10 cents.
Compare the value of pizzas. When comparing value, the best value is the pizza with the lowest cost per square inch. This is only true, however, when comparing pizzas with the same toppings. A cheese pizza is likely to be cheaper per square inch than a pizza with several toppings, but that doesn’t necessarily make it the best value.
Finding the Per Inch Cost of a Rectangular Pizza Download Article
Set up the formula for the area of a rectangle. The formula is
{\displaystyle A=l\times w}
{\displaystyle l}
equals the length of the rectangle, and
{\displaystyle w}
equals the width of the rectangle.[2] X Research source
Find out the length and width of the pizza. Usually menus will only give you one measurement. You will have to contact the restaurant and ask if they will provide you with the length and width of the pizza. If it is a square pizza, the length and width will be the same.
For example, you might want to find the value of a square 16-inch pizza. Both the length and the width will be 16 inches.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/0\/06\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-10.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-10.jpg","bigUrl":"\/images\/thumb\/0\/06\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-10.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-10.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Plug the length and width of the pizza into the formula. Due to the commutative property of multiplication, It doesn’t matter which dimension you use for the length and which you use for the width.
For example, for a square 16-inch pizza, your formula will look like this:
{\displaystyle A=16\times 16}
Multiply the length and the width of the pizza. This will give you the area of the pizza in square inches.
{\displaystyle A=16\times 16}
{\displaystyle A=256}
So, the area of a square 16-inch pizza is 256 square inches.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/a6\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-12.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-12.jpg","bigUrl":"\/images\/thumb\/a\/a6\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-12.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-12.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
{\displaystyle 32\div 256=.125}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/c\/c2\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-13.jpg\/v4-460px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-13.jpg","bigUrl":"\/images\/thumb\/c\/c2\/Figure-Cost-Per-Square-Inch-of-Pizza-Step-13.jpg\/aid145595-v4-728px-Figure-Cost-Per-Square-Inch-of-Pizza-Step-13.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Compare the cost per square inch of different pizzas. The pizza with the lowest cost per square inch will have the best value. However, you should only compare similar types of pizzas when looking for the best value (for example, two veggie pizzas). A pizza with fewer toppings will cost less per square inch than a pizza with several toppings, but that doesn’t necessarily make it the best value.
In general, the larger the pizza, the better the value.[3] X Research source
↑ https://www.mathgoodies.com/lessons/vol2/circle_area
↑ http://www.npr.org/sections/money/2014/02/26/282132576/74-476-reasons-you-should-always-get-the-bigger-pizza
Categories: Mathematics | Pizza
Español:calcular el coste por pulgada cuadrada de una pizza |
Physics and Astronomy Labs/Radioactive decay with dice - Wikiversity
Physics and Astronomy Labs/Radioactive decay with dice
< Physics and Astronomy Labs
4 Matlab codes
4.1 Simulates five virtual labs
4.2 Simulates and also graphs
4.3 Collects data from a data file
4.4 Future project(s)
4.5 HTW fall 19
Lede[edit | edit source]
This graph shows (in pink) the amount of a radioactive sample that remains after several half-lives have passed. After one half-life, half the sample is left; after two half-lives, one half of the remainder (or one quarter) is left; and after three half-lives, one half of that (or one eighth) is left. Note that, in reality, the decay of radioactive elements in a rock sample would not cause any visible change in the appearance of the rock; the splashes of color are shown here for conceptual purposes only. [1]
This will be the Lede. It should mention the application to radioactive dating in OpenStax Astronomy. Remove the header Lede when this has been written.
201 dice were rolled and all the "ones" were removed and counted at each throw. The process was simulated five times for comparison with the actual experiment.
Dice half-life decay
Excel Spreadsheet[edit | edit source]
Column roll denotes the number of rolls. We rolled 201 dice 18 times, removing the "one" each time. Column exp denotes the number of dice removed on each roll in our experiment. Columns sim1 through sim5 represent five simulations. Column theor represents the theoretical value one is most likely to obtain: the first value is 201 divided by 6, and each consecutive value is 5/6 times smaller, representing exponential decay.
Column SEexp is the square of the error associated with the experiment. For the first role, this is obtained as follows:
{\displaystyle SEexp(1)=(33-33.5)^{2}=0.25}
Columns SE1 through SE5 represent the squared errors associated with the 5 simulations.
These squared errors are summed at the last row. The values of SoS (Sum of Squares) represent a measure of how well the simulation or the experiment matched the theoretical ideal. If the experimental SoS sufficiently exceeded any of the simulated SoS values, we might wish to question the experimental method.
roll exp sim1 sim2 sim3 sim4 sim5 theo SEexp SE1 SE2 SE3 SE4 SE5
1 33 33 27 29 35 37 33.50 0.25 0.25 42.25 20.25 2.25 12.25
2 25 34 25 28 30 23 27.92 8.51 37.01 8.51 0.01 4.34 24.17
3 21 18 22 27 23 23 23.26 5.13 27.71 1.60 13.96 0.07 0.07
5 17 18 14 21 13 14 16.16 0.71 3.40 4.65 23.47 9.96 4.65
6 14 13 20 13 15 12 13.46 0.29 0.21 42.73 0.21 2.36 2.14
7 11 13 9 15 14 8 11.22 0.05 3.17 4.92 14.30 7.73 10.36
8 13 6 9 12 9 10 9.35 13.33 11.22 0.12 7.03 0.12 0.42
9 12 9 9 6 6 16 7.79 17.72 1.46 1.46 3.21 3.21 67.39
10 9 5 3 3 5 4 6.49 6.29 2.23 12.20 12.20 2.23 6.21
11 5 6 4 4 11 7 5.41 0.17 0.35 1.99 1.99 31.24 2.53
12 4 4 6 6 1 6 4.51 0.26 0.26 2.22 2.22 12.31 2.22
13 2 4 3 5 2 3 3.76 3.09 0.06 0.57 1.54 3.09 0.57
15 2 5 6 3 2 3 2.61 0.37 5.72 11.50 0.15 0.37 0.15
sums 201 67 97 182 131 92 154
Matlab codes[edit | edit source]
To know if the random error is consistent with the laws of probability for such a decay process, we use matlab codes. These codes simulate any number of labs for an arbitrary number of dice.
Simulates five virtual labs[edit | edit source]
this code simulates but we graphed in Excel
Nworms = 5;
Nstart = 201;
Nstop = 18;
data = zeros(Nstop,Nworms);
for nworm=1:Nworms
ncurrent=Nstart; % initiate throws
for count = 1 : Nstop %iterates throws by all the students
n2remove=0;
for diceCount = 1:ncurrent
if rand < 1/6
n2remove=n2remove+1;
end % ends if
end % finish thowing all the dice
ncurrent=ncurrent-n2remove; %remove some dice
data(count,nworm)=n2remove; %record number left
Simulates and also graphs[edit | edit source]
this code graphs in matlab
Y = [42 32 29 25 11 10 14 7 3 6 3 3 2 1 3 1 1]
sizeArray =size(Y);
Nstop = sizeArray(2);
for wormcount = 1:Nworms
x=[1:1:Nstop];
y=data(:,wormcount);
Collects data from a data file[edit | edit source]
This code allows the user to enter classroom data into an Excel file that can be opened by matlab.
matlab code that reads an excel file
Y = xlsread('engineersData.xlsx');
%It is necessary to save an Excel file with this name and extension, and
%then place into "Current folder". This was done on a click and drag the
%only time I have ever tried it.
% The data in that file were:
% 42 32 29 25 11 10 14 7 3 6 3 3 2 1 3 1 1
Future project(s)[edit | edit source]
Matlab codes have been used to make spagetti plots. In the future, it would be nice to also use matlab to perform the sum of squares analysis.This can be of ultimate usefulness because of radioactivity of some elements under the hood.
HTW fall 19[edit | edit source]
↑ https://cnx.org/contents/LnN76Opl@13.120:niEjH03V@3/Dating-Planetary-Surfaces
Retrieved from "https://en.wikiversity.org/w/index.php?title=Physics_and_Astronomy_Labs/Radioactive_decay_with_dice&oldid=2292850" |
EUDML | Hasse invariant and group cohomology. EuDML | Hasse invariant and group cohomology.
Hasse invariant and group cohomology.
Edixhoven, Bas; Khare, Chandrashekhar
Edixhoven, Bas, and Khare, Chandrashekhar. "Hasse invariant and group cohomology.." Documenta Mathematica 8 (2003): 43-50. <http://eudml.org/doc/123728>.
@article{Edixhoven2003,
author = {Edixhoven, Bas, Khare, Chandrashekhar},
title = {Hasse invariant and group cohomology.},
AU - Edixhoven, Bas
AU - Khare, Chandrashekhar
TI - Hasse invariant and group cohomology.
Adam Mohamed, Weight reduction for cohomological mod
p
modular forms over imaginary quadratic fields
Congruences for modular and
p
Automorphic forms on
\mathrm{GL}\left(2\right)
; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Articles by Edixhoven
Articles by Khare |
EUDML | The quantum double of a (locally) compact group. EuDML | The quantum double of a (locally) compact group.
The quantum double of a (locally) compact group.
Koornwinder, T.H.; Muller, N.M.
Koornwinder, T.H., and Muller, N.M.. "The quantum double of a (locally) compact group.." Journal of Lie Theory 7.1 (1997): 101-120. <http://eudml.org/doc/222570>.
@article{Koornwinder1997,
author = {Koornwinder, T.H., Muller, N.M.},
keywords = {locally compact Hausdorff second countable groups; quantum doubles; transformation group -algebras; irreducible representations; transformation group -algebras},
title = {The quantum double of a (locally) compact group.},
AU - Koornwinder, T.H.
AU - Muller, N.M.
TI - The quantum double of a (locally) compact group.
KW - locally compact Hausdorff second countable groups; quantum doubles; transformation group -algebras; irreducible representations; transformation group -algebras
locally compact Hausdorff second countable groups, quantum doubles, transformation group
{C}^{*}
-algebras, irreducible representations, transformation group
{C}^{*}
{C}^{*}
{W}^{*}
Applications of Lie groups to physics; explicit representations
Articles by Koornwinder
Articles by Muller |
2012 Electromagnetic Gyroscopic Motion
A. I. Ismail, T. S. Amer, S. A. El Banna, M. A. El-Ameen
A problem of the gyroscopic motions around a fixed point, under the action of a gyrostatic moment vector, in presence of electromagnetic field and Newtonian one, is considered. The small parameter technique is used to investigate the periodic solutions for the derived equations of such motion problem. A geometric interpretation of motion will be given in terms of Euler’s angles (
\theta ,\psi ,\varphi
). Computer programs are carried out to integrate the attained quasilinear autonomous system using a fourth-order Runge-Kutta method. A comparison between the obtained analytical solutions and the numerical ones is investigated to calculate the errors between them.
A. I. Ismail. T. S. Amer. S. A. El Banna. M. A. El-Ameen. "Electromagnetic Gyroscopic Motion." J. Appl. Math. 2012 1 - 14, 2012. https://doi.org/10.1155/2012/812645
A. I. Ismail, T. S. Amer, S. A. El Banna, M. A. El-Ameen "Electromagnetic Gyroscopic Motion," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-14, (2012) |
Eval and rtable
RegularChains option in RootFinding[Isolate]
By setting interface(helpbrowser=standard), it is possible to view documentation through the Standard GUI's help browser launched from any interface. This gives Command-line and Classic users a nicer interface for viewing documentation including hyperlinks, full-text search, and the table of contents. Additionally, help pages that are written using features specific to Maple's Standard interface and that were previously not viewable in Classic and Command-line help will be accessible. This includes some help pages, example worksheets, and Maple Portal pages. For more information on the helpbrowser variable, see interface.
Eval now works on rtable constructions such as Matrix.
Eval(Matrix([[x^2 + y]]), {x=3, y=2}) mod 5;
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\end{array}]
Users of RootFinding[Isolate] now can choose the method to isolate the real roots of polynomial system. These methods are "RS", which is based on Fabrice Rouillier's RealSolving (RS) C library, and "RC", which is based on the RegularChains package by Marc Moreno Maza et al.
F := [x^2-2, y-1];
\textcolor[rgb]{0,0,1}{F}\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}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}]
Isolate(F, [x,y], method="RS");
[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-1.414213562}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1.000000000}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1.414213562}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1.000000000}]]
Isolate(F, [x,y], method="RC");
[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-1.414213562}\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}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1.414213562}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1.}]]
Programming and Connectivity Changes in Maple 14 |
Potential Outcomes Model | Causal Flows
Hey! This post references terminology and material covered in previous blog posts. If any concept below is new to you, I strongly suggest you check out its corresponding post.
How do we represent causal relationships between the many interconnected processes which comprise our universe?
What is causal inference? Why is it useful? How can you use to amplify your decision-making capabilities?
In my previous post, I discussed a powerful notation for describing hypothesized causal effects between explanatory variables corresponding to a cause, outcome variables corresponding to an effect, and unobserved variables which also correspond to a cause, but are not measured in our dataset. While this notation, structural causal models, are a useful language for describing the existence or absence of causal relationships, it cannot easily describe the quantitative causal hypotheses we are generally interested in analyzing.
In contemporary English, we commonly describe these hypotheses as counterfactual queries, asking questions such as “How much would I earn if I had chosen to attend graduate school?” or “Can I get more of my email subscribers to read a blog post if I include images in its corresponding email alert?”. These queries are an effective way to describe causal effects because they require us to imagine a hypothetical universe which is only different in the absence or presence of a particular cause. For example, when I question what my income would be had I attended graduate school, I am imagining my income in a hypothetical universe in which I attended graduate school, instead of working out of undergrad. Measuring the differential between this hypothetical universe and ours allows us to isolate the effect this cause has on other variables in our universe and to quantify the extent to which the presence of this cause generates particular observed outcomes.
Figure 1: Causal graphs representing the two *hypothetical universes* considered when I ask if my income would be higher had I gone to graduate school.
Counterfactual questions are generally answered by analyzing data with some variation in the explanatory variable which defines two seperate hypothetical universes. For example, If I wanted to estimate the extent to which adding images to my email alerts encourages Causal Flows email subscribers to open my blog posts, I will need to observe data that tracks which of my subscribers opened my blog post during a time in which some subscribers received images in their email alerts and some subscribers did not.
Figure 2: The email alerts for my previous blog post, with and without images. (Although, you should already know what my email alert looks like, and have subscribed to be alerted of every post!)
These counterfactual queries often concern potential outcomes or hypotheses describing the values of outcome variables in the hypothetical universes for which certain explanatory variables have particular values. For example, consider the question: “Can I get one of my email subscribers to read a blog post if I include images in its corresponding email alert?”. Perhaps, I am interested in changing my email alerts to resemble the version displayed on the right of figure 2 in order to get more of my subscribers to click the hyperlink to my latest post. In this example, the explanatory variable is Images In Email Alert (
\textcolor{#7A28CB}{I}
) and the outcome variable is Opens Blog Post (
\textcolor{#EF3E36}{O}
). I am interested in whether or not individual email subscriber
i
opens my blog post after reading my email alert, given that it contains images, and given that it does not. As a reminder, the structural causal model describing the hypothetical effect of the former on the latter is as follows:
\textcolor{#7A28CB}{O} \leftarrow f_i(\textcolor{#A93F55}{I})
And its corresponding causal graph is:
Figure 3: A causal graph representing the causal relationship between Images In Email Alert and Opens Blog Post.
How can we quantify the degree of this hypothetical causal effect? How can we make an inference about the probability that adding an image to my newsletter will encourage an individual subscriber to open an email? Adding images will take a lot of effort, so I wouldn’t want to take the time to make the change if the resultant effect would just make a reader 1% more likely to open my newsletter. Statisticians Jerzey Neuman and Donald Rubin both formalized a model for investigating counterfactual queries commonly referred to as the potential outcomes model. The simplest version of this powerful model consists of four main concepts.
Indicator Variables are mathematical variables used to represent discrete events. Indicator variables take only the value 0 or 1 to indicate the presence or absence of a particular causal event. For example, the event that I include images in the email alert sent to a particular email subscriber
\textcolor{#7A28CB}{i}
can be represented by the following indicator variable.
\color{#7A28CB} I_i = \begin{cases} 1, & \text{if images are included in }i\text{'s email alert} \\ 0, & \text{if images are not included in }i\text{'s email alert} \end{cases}
In the potential outcomes model, an explanatory variable is often referred to as the treatment variable and the state for which an explanatory variable
\color{#7A28CB}X_i = 1
for a particular individual
is commonly referred to as the treatment. This borrows from terminology commonly used in the medical sciences. When an explanatory indicator variable takes the value
\color{#7A28CB} X_i = 1
for individual
i
, we commonly say that individual
is treated, when an explanatory indicator variable takes the value
\color{#7A28CB}X_i = 0
i
, we say that individual
is untreated. For example, let’s suppose I were to send a portion of my email subscribers email alerts with images, and that I were to send another portion email alerts without images. Choosing to send email alerts with images to certain subscribers would be a treatment, those who received said email alerts would be treated, and those who receive email alerts without images would be untreated.
Potential Outcomes are mathematical variables used to represent outcome variables in the two hypothetical universes we consider when asking a counterfactual question. We commonly represent these hypothetical universes using a superscript added to a particular outcome variable, representing the value of an explanatory variable in its corresponding hypothetical universe. With this notation,
\color{#EF3E36}Y_i^1
is the potential outcome observed when its corresponding explanatory variable
\color{#7A28CB}X_i = 1
\color{#EF3E36}Y_i^0
\color{#7A28CB}X_i = 0
For example, when describing our counterfactual question concerning images in email alerts, potential outcomes take on the following values depending on whether or not individual
\textcolor{#EF3E36}{i}
opens a blog post, given the type of email alert they see. Note that the two potential outcomes in this simple version of the potential outcomes model are also indicator variables.
\color{#EF3E36} O_i^0 = \begin{cases} 1, & \text{if }i\text{ opens a blog link in an email \textcolor{#7A28CB}{without images}} \\ 0, & \text{if }i\text{ does not open a blog link in an email \textcolor{#7A28CB}{without images}} \end{cases}
\color{#EF3E36} O_i^1 = \begin{cases} 1, & \text{if }i\text{ opens a blog post link in an email \textcolor{#7A28CB}{with images}} \\ 0, & \text{if }i\text{ does not open a blog post link in an email \textcolor{#7A28CB}{with images}} \end{cases}
The two hypothetical universes described by these potential outcomes are the universe in which individual
i
receives an email alert without images and the universe in which individual
i
receives an email alert with images.
The treatment effect, or causal effect, is the differential between the two “hypothetical universes” we consider when asking our counterfactual question. Concisely, for a particular observed individual
i
, it is the difference
\textcolor{#EF3E36}{\delta_i}
between the two potential outcomes
\textcolor{#EF3E36}{Y_i^1}
(for which individual
\textcolor{#EF3E36}{i}
does receive treatment) and
\textcolor{#EF3E36}{Y_i^0}
\textcolor{#EF3E36}{i}
does not receive treatment) . Utilizations of the potential outcomes model are largely concerned with estimating this crucial value, as this is the quantified level of the causal relationships contemplated endlessly by causal inference practitioners. If I had information about the treatment effect for each of my individual email subscribers I would have a precise measurement of the value of adding images to my email alerts, precisely the data I need to make an informed decision with regard to my proposed intervention.
Observable outcomes are the outcomes which we eventually observe, and depend on whether or not we apply treatment to a particular individual. For a given outcome variable
\textcolor{#EF3E36}{Y}
, and an explanatory indicator variable
\textcolor{#7A28CB}{X}
, the observable outcome
\textcolor{#EF3E36}{Y_i}
is represented by the following switching equation.
\textcolor{#EF3E36}{Y_i} = \textcolor{#7A28CB}{X_i}\textcolor{#EF3E36}{Y_i^1} + (1-\textcolor{#7A28CB}{X_i})\textcolor{#EF3E36}{Y_i^0}
For example, the observable outcome
\textcolor{#EF3E36}{O_i}
I witness when consider the effect that images in email alerts has on my blog post open rate is represented as follows:
\textcolor{#EF3E36}{O_i} = \textcolor{#7A28CB}{I_i}\textcolor{#EF3E36}{O_i^1} + (1-\textcolor{#7A28CB}{I_i})\textcolor{#EF3E36}{O_i^0}
Note that for treated individuals, who receive an email alert with images,
\textcolor{#7A28CB}{I_i} = 1
(1-\textcolor{#7A28CB}{I_i}) = 0
and thus I observe potential outcome
\textcolor{#EF3E36}{O_i} = \textcolor{#EF3E36}{O_i^1}
. Similarly, for untreated individuals, who receive an email alert without images,
(1-\textcolor{#7A28CB}{I_i})
\textcolor{#7A28CB}{I_i}
is 0 and thus I observe potential outcome
\textcolor{#EF3E36}{O_i} = \textcolor{#EF3E36}{O_i^0}
In a typical causal inference setting, our data contains measurements of observable outcomes as well as the indicator explanatory variables describing which individuals were treated. However, the switching equation is just one equation, containing two unknown variables
Y_i^0
Y_i^1
, and is thus underspecified and cannot be solved. This problem is commonly known as the fundamental problem of causal inference. It is impossible to see both potential outcomes
Y_i^0
Y_i^1
at once, to observe the hypothetical universe for which individual
is treated, and that for which individual
is untreated. When I sent out the email alerts for this post, I could not observe a world in which I send an particular subscriber an email alert with images, as well as the world in which I send that same subscriber an alert without images. Thus, the fundamental problem of causal inference is unanswerable. Does that mean we just give up? Of course not!
While the effect of treatment on each observed individual can be valuable, often times analysts are fine with just estimating average treatment effects (ATE) which are the average of all treatment effects identified for all individuals. The formal equation for the ATE of a particular outcome variable
\color{#EF3E36}Y
\textcolor{#EF3E36}{\text{ATE}} = E[\textcolor{#EF3E36}{\delta_i}] =E[\textcolor{#EF3E36}{Y_i^1} - \textcolor{#EF3E36}{Y_i^0}]
E
is the expected value, or arithmetic mean of a given data series.
For example, consider the following (fake) dataset measuring which email subscribers will open my blog post after reading its corresponding email alert, given that there are images in the email alert
\color{#EF3E36}O_i^1
and given that there is none
\color{#EF3E36}O_i^0
. Note that we cannot possibly observe both columns of this data, as they each take place in two different hypothetical universes. However, such an example is useful for analysis of average treatment effects.
\color{#EF3E36}O_i^1
\color{#EF3E36}O_i^0
For these observations:
\text{ATE} = E[\textcolor{#EF3E36}{\delta_i}] =E[\textcolor{#EF3E36}{O_i^1}]- E[\textcolor{#EF3E36}{O_i^0}] = \frac{6}{8} - \frac{2}{8} = \frac{1}{2}
Ok, but our calculation used data of potential outcomes we could not observe, as individual email subscribers cannot both receive an email alert with images and receive one without. Can we calculate average treatment effects without running in to trouble with the fundamental problem of causal inference? Unfortunately not, we still do not have data measuring potential outcomes in the hypothetical universes for which each observed individual generates outcome
\textcolor{#EF3E36}{Y_i^1}
\textcolor{#EF3E36}{Y_i^0}
. We still can only see observable outcomes: the behavior of treated individuals when they are given treatment (which we can see because these are the individuals we treated) and the behavior of untreated individuals when they are not given treatment. With the help of a single calculation we can measure (and a few more which we cannot), we can achieve a robust estimate and even evaluate its statistical significance. Sorry to leave you hanging, but to keep this post at a preferred manageable length, discussion of ATE estimation will have to wait until my next post.
Why Do We Try To Estimate Average Treatment Effects?
Average treatment effect estimation with the potential outcomes model is a crucial tool for quantifying the causal effects of a proposed intervention. ATEs are used in a variety of disciplines to solve a variety of problems, beyond making decisions of whether or not to add images to blog post email alerts. In agile software development they are commonly used in the form of a technique known as A/B testing, or randomized experiments designed to estimate the effect of a particular change to a user interface. In medicine, ATEs are commonly estimated during drug trials in order to identify generalizable estimations of a particular therapy on a patient with a particular medical profile. In economics, ATE estimation is commonly used to discern the effect of a particular public policy decision on target populations, such as the effect that an increase in a high school’s class size can have on a student’s performance.
Figure 3: An example of a small variation that can be evaluated with A/B testing. UI designers of an e-commerce website may use this technique to estimate the best location of an "Add To Cart" button in order to ensure a user is likely to add an item to the site's cart.
What If The “Average” Is Not Enough?
Estimating average treatment effects would be particularly helpful for informing my choice of whether or not to add images to my newsletter emails. As of now, Substack only allows me to send one email alert to all of my subscribers, so I’m not really concerned with how adding images to an alert affects a single subscriber. For causal inference questions concerning costly interventions, such as those regarding the purchase of costly digital ads targeted at a particular demographic, ATEs are not a sufficiently useful measurement to inform the design of a marketing campaign that is optimized for revenue. These decisions require a more granular estimation of how a particular treatment affects different sets of individuals, commonly referred to in causal inference literature as heterogeneous treatment effect estimation. Literature on machine learning techniques for heterogeneous treatment effect estimation provide many examples of practical causal inference tooling that is extremely valuable for effectively managing operating expenses. I will cover HTEs in detail in a future blog post.
A casual introduction to causal inference for business analytics, by Ken Acquah
← Structural Causal Models
Estimating Average Treatment Effects →
About • Newsletter |
Connection technology - zxc.wiki
This article covers joining technology as an assembly method.
For the technology for making an electrical contact, see connection technology (electrical engineering) .
For the creation of connections in scaffolding, see connection technology (scaffolding) .
Connection between the rail and threshold :
positive locking transversely of the rail and frictional connection in the direction of the rail (slips in thermal expansions), traction under the screw heads and the mother (by spring washers improved)
The connection technology describes the constructive methods of assembling technical structures ( machines , equipment , apparatus , equipment and modern buildings ) from their individual parts . The process of connecting is called joining and is a matter of manufacturing technology .
As a rule, these are fixed connections. Connections that only restrict the mobility between two parts are joints (e.g. swivel or sliding joints).
The connections can be detachable if the connection can be detached again without damaging the components ( e.g. screw connection or Velcro fastener ), not detachable if the components have to be destroyed ( e.g. welded connection or adhesive bond ), or conditionally detachable if only the auxiliary joining parts must be destroyed, but not the components (knocking off the rivets for riveted connections ).
The classification according to physical operating principles is: form-fit , force-fit and material-fit.
1 form fit
2 frictional connection
Principle of form and force fit. The upper cube can only be moved a little because it is fitted into a recess. The lower cube is pressed onto the base by a load and can therefore only be moved if the force applied for this exceeds the static friction of the material on the base.
Positive connections are created by the interlocking of at least two connection partners. This means that the connection partners cannot disengage even with or without power transmission. In other words, with a positive connection, one connection partner is in the way of the other . Under normal operating conditions, compressive forces act normally, i.e. at right angles to the surfaces of the connection partners. Such “blocks” occur in at least one direction. If a second homogeneous pair of surfaces is arranged opposite, the opposite direction is also blocked (see figure, principle illustration and bung ). If the pair consists of two mutually coaxial cylinder surfaces, there is a form fit in all directions of the plane perpendicular to the cylinder axis. An example is the pin inserted into a hole that can be removed again. The hole is advantageously a blind hole so that the pin cannot fall through. There is also a one-sided form fit in the axial direction. Pin-like connecting elements are also rivets and screws , screw connections usually being both positive and non-positive.
Primarily only two components have to be connected to one another in a form-fitting manner, but this is often achieved with the help of a third part - the special connecting element. An example is the connection of two overlapping sheet metal edges using rivets or screws. In addition to the plane of the sheet metal, the form fit must also be established perpendicular to it. The metal sheets should be held on top of one another and the connecting elements should not fall out. For this purpose, the rivets have heads on both sides. The screw has its head , and opposite is the nut (if the screw is not screwed into the sheet metal).
Riveted connections of an old railway bridge
With thin sheets or large forces between the sheets in the direction of their planes, there is a risk of plastic deformation or destruction at the hole edges of the sheets (hole embedment ) and the shear in the pins. Sheet metal connections are usually additionally provided with a frictional connection or even exclusively designed so that the frictional connection alone withstands the stress. The rivets and screws are elastically stretched axially, which is done by shrinking after hot riveting or by tightening the screws firmly.
solvable:
Tongue and groove connection
Dovetail connection
Interlocking gears , e.g. B. Gear and rack
not solvable:
Enforcement joining
Non-positive connections require a normal force on the surfaces to be connected. Their mutual displacement is prevented as long as the counter-force caused by the static friction is not exceeded. The frictional connection is lost and the surfaces slip on each other if the tangentially acting load force is greater than the static friction force, for example between wheel and rail or road surface in vehicles with their own drive. In the friction clutch of a car , the frictional connection is interrupted when it is kicked. If it is only partially kicked against the built-in spring causing the normal force, it grinds.
{\ displaystyle F _ {\ text {Last}} \ leq F _ {\ text {Haft}} = \ mu \ cdot F _ {\ text {N}}}
{\ displaystyle F _ {\ text {Last}}}
= tangential load force (blue) = tangential static friction force (red) = static friction coefficient = normal force (green)
{\ displaystyle F _ {\ text {Haft}}}
{\ displaystyle \ mu}
{\ displaystyle F _ {\ text {N}}}
Frictional connection is the cause of the self-locking of loaded wedges or screws . The static friction between the active surfaces prevents the wedge from slipping out or the screw starting to turn. Screws are therefore firmly tightened, even if their preload is not required to create a force fit between the parts they connect (for example in a sheet metal connection, see above). This means that tightly tightened screw connections are also force-fit connections.
The terminal is also a positive connection: something between thumb and forefinger or with a spring clip hold.
Two ropes that are knotted together only transmit a tensile force via a frictional connection, if one neglects that the ropes have a residual stiffness against bending, i.e. that a form fit is also involved to a small extent.
Electrically welded seam (left) and after processing with a slag hammer and wire brush (right)
Cohesive connections are all connections in which the connection partners are held together by atomic or molecular forces. They are also non-detachable connections that can only be separated by destroying the connecting means:
Gottfried W. Ehrenstein: Handbook of plastic connection technology. 1st edition, Hanser Verlag, Munich 2004, ISBN 978-3-446-22668-5 .
Manfred Neitzel, Peter Mitschang, Ulf Breuer (eds.): Manual composite materials. Materials, processing, application, 2nd updated and expanded edition, Hanser Verlag, Munich 2014, ISBN 978-3-446-43696-1 .
Commons : fasteners - collection of images, videos and audio files
DVS study: Connection technology for lightweight construction and renewable energies (accessed on October 15, 2015)
This page is based on the copyrighted Wikipedia article "Verbindungstechnik" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
Comparison of Interfacial Forces During Post-Chemical-Mechanical-Polishing Cleaning | J. Tribol. | ASME Digital Collection
Dedy Ng,
Ng, D., and Liang, H. (May 7, 2008). "Comparison of Interfacial Forces During Post-Chemical-Mechanical-Polishing Cleaning." ASME. J. Tribol. April 2008; 130(2): 021603. https://doi.org/10.1115/1.2908896
This research investigates the interfacial forces involved in tribological interactions while removing nanosized particles during post-chemical-mechanical polishing cleaning. Surface and interfacial forces are discussed to understand the particle adhesion and subsequent removal through physical and chemical interactions. Approaches include theoretical analysis combined with experimental study. The theoretical analysis was focused on the forces that exist between particles and a substrate. Surface interaction consideration includes applied pressure, frictional force, and hydrodynamic drag. The polishing experiments were carried out on silicon wafers with
SiO2
slurry. Cleaning experiments were performed in de-ionized water using a polyvinyl acetal brush to remove particles from a hydrophilic-silicon surface. The fluid-drag force was found to affect the lubricating behavior of cleaning through changing material properties. Values of interfacial forces and their effects on cleaning were discussed along with a lubricating model system.
CMP, post-CMP cleaning, interfacial force, friction, lubrication, Sommerfeld grouping, adhesion, chemical mechanical polishing, elemental semiconductors, friction, interface phenomena, lubrication, silicon, surface cleaning
Adhesion, Drag (Fluid dynamics), Fluids, Friction, Lubrication, Particulate matter, Polishing, Semiconductor wafers, Van der Waals forces, Pressure, Silicon, Bonding, Hydrogen, Water
Chemical Mechanical Planarization of Microelectronics Materials
Trends in Wafer Cleaning
Submicron Particle Removal in Post-Oxide Chemical-Mechanical Planarization (CMP) Cleaning
Particle Adhesion and Removal Mechanisms in Post-CMP Cleaning Processes
Describing Hydrodynamic Particle Removal from Surfaces Using the Particle Reynolds Number
A Theoretical Evaluation of Hydrodynamic and Brush Contact Effects on Particle Removal During Brush Scrubbing
Hydrodynamic Particle Removal From Surfaces
Fransaer
Particle Adhesion and Removal Mechanisms During Brush Scrubber Cleaning
Fundamental Study of the Removal Mechanisms of Nano-Sized Particles Using Brush Scrubber Cleaning
Particle Adhesion Theory and Experiment
Knozinger
Hydrogen Bonds in Systems of Adsorbed Molecules
The Hydrogen Bond: Recent Developments in Theory and Experiments
Sandorfy
Intermolecular and Surface Force
On Hamaker Constants: A Comparison Between Hamaker Constants and Lifshitz–van der Waals Constants
Surface Energy and the Contact of Elastic Solids
The Adhesion of Colloidal Polystyrene Particles to Cellophane as a Function of pH and Ionic Strength
Mutual Coagulation of Colloidal Dispersions
The Potential Energy of Interaction Between Dissimilar Electrical Double Layers
The Effects of Hydrogen Bonds on the Adhesion of Inorganic Oxide Particles on Hydrophilic Silicon Surfaces
The Modeling of Laser Particle Removal From Hydrophilic Silicon Surfaces
Effect of Additives in Post Cu CMP Cleaning Solutions on Particle Adhesion and Removal
Electrochemistry Society Proceedings
Simulation of Particle Adhesion: Implications in Chemical Mechanical Polishing and Post Chemical Mechanical Polishing Cleaning
Analysis of Contact Interactions Between a Rough Deformable Colloid and a Smooth Surface
Effect of Slurry Viscosity Modification on Oxide and Tungsten CMP
Influence of Oxides on Friction During Cu CMP
Lubricating Behavior in Chemical Mechanical Polishing of Copper
Mechanisms of Post-CMP Cleaning
Proceedings of MRS-CMP 2001
Role of Surfactant Molecules in Post-Chemical-Mechanical-Planarization Cleaning
Friction Forces in Post-CMP Cleaning Applications
The Mechanism of Particle Removal and Brush Mechanics in Post-CMP Cleaning Applications
Proceedings of Sixth International Conference on CMP for ULSI Multilevel InterConnection (CMP-MIC)
, Mar. 7–9, pp.
Characteristics of Plain and Roller Bearing
Zet. Ver. Dent. Ing.
Brooks∕Cole
A Sphere in Contact With a Plane Wall in a Slow Linear Shear Flow
Adhesion and Pull-Off Forces for Polysilicon MEMS Surfaces Using the Sub-Boundary Lubrication Model
Adhesion and Friction Studies of Silicon and Hydrophobic and Low Friction Films and Investigation of Scale Effects
Interfacial Force Analysis and Lubrication Behavior During Post-CMP Cleaning
Effects of Gas Adsorption in Nanotribology and Demonstration of In-Situ Vapor Phase Lubrication of MEMS Devices |
Tridecagon - Simple English Wikipedia, the free encyclopedia
(Redirected from Tridecagram)
Regular tridecagon
A regular tridecagon
A tridecagon or trikaidecagon or triskaidecagon or trisdecagon or 13-gon is a shape with 13 sides and 13 corners.
1 Regular tridecagon
2 Numismatic use
Regular tridecagon[change | change source]
All sides of a regular tridecagon are the same length. Each corner is 147.27°. All corners added together equal 6840°.
The amount of space a regular tridecagon takes up is
{\displaystyle A={\frac {13}{4}}a^{2}\cot {\frac {\pi }{13}}\simeq 13.1858\,a^{2}.}
Numismatic use[change | change source]
The regular tridecagon is used as the shape of the Czech 20 korun coin.[1]
↑ Colin R. Bruce, II, George Cuhaj, and Thomas Michael, 2007 Standard Catalog of World Coins, Krause Publications, 2006, ISBN 0896894290, p. 81.
Eric W. Weisstein, Tridecagon at MathWorld.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Tridecagon&oldid=6560831" |
Analyze electric circuit - MATLAB power_analyze - MathWorks Nordic
Output Arguments: Structure
Output Arguments: Sort
Output Arguments: ss
Output Arguments: Net
Output Arguments: getSwitchStatus
Output Arguments: setSwitchStatus
Analyze electric circuit
sps = power_analyze('sys','structure')
[A,B,C,D,x0,states,inputs,outputs,uss,xss,yss,frequencies,Hlin] =...
power_analyze('sys')
sps = power_analyze('sys','sort')
sps = power_analyze('sys','ss')
power_analyze('sys','net')
SW = power_analyze('sys','getSwitchStatus')
sps = power_analyze('sys','setSwitchStatus',SW)
The power_analyze command computes the equivalent state-space model of the specified electrical model built with Simscape™ Electrical™ Specialized Power Systems software. It evaluates the A, B, C, D standard matrices of the state-space system described by the equations
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}
where the state vector x represents the inductor currents and capacitor voltages, the input vector u represents the voltage and current sources, and the output vector y represents the voltage and current measurements of the model.
Nonlinear elements, such as the motors or machines, are simulated either by current sources driven by the voltages across the nonlinear element terminals, or by voltage sources driven by the currents through the nonlinear element terminals. The nonlinear elements produce additional current and voltage source inputs to the u vector, and additional voltage and current measurements outputs to the y vector.
For the switching devices elements, such as the breakers, diodes, and thyristors, the state-space calculations include only the resistance and capacitance elements of the snubber devices.
The Simulink® blocks of the model, as well as the internal Simulink models of the Simscape Electrical Specialized Power Systems nonlinear elements, are not represented in the state-space matrices.
The A, B, C, D matrices are computed for the particular circuit topology where all the switch devices, if any, are considered to be open status (that is, with infinite impedance).
power_analyze also computes the Aswitch, Bswitch, Cswitch, and Dswitch matrices for the circuit topology that take into account the initial state (open/closed) of the Breaker and Ideal Switch blocks present in the model. The initial state of power electronic devices (Diodes, Thyristors, and so on) is considered to be open, unless a nonzero initial current value is specified in the mask of the device when Lon parameter is different from zero.
For a circuit that contains no switches, the Aswitch, Bswitch, Cswitch, and Dswitch matrices have exactly the same values as the A, B, C, D matrices.
In Simscape Electrical Specialized Power Systems software, each state variable name begins with a prefix Uc_ for capacitor voltages or Il_ for inductor currents, followed by the name of the block in which the element (C or L) is found.
A character vector is added to this prefix for blocks containing more than one inductance or capacitor. For example, the Linear Transformer block is represented with four state variables, one for each of the three leakage inductances, defined with the prefixes Il_winding_x:, where x is the winding number of the transformer, and one state for the magnetization inductance defined with the prefix Il_Lm:.
Each input state variable name begins with a prefix U_ for voltage sources or I_ for current sources, followed by the name of the source block. Text can be added to the prefix for blocks containing more than one source. For example, the Synchronous Machine block produces two current inputs with prefixes I_A: and I_B: (phase A and phase B machine currents).
Each output state variable name begins with a prefix U_ for voltage outputs or I_ for current outputs, followed by the name of the block that produces the output. Text can be added to the prefix for blocks containing more than one output. For example, the Synchronous Machine block produces two voltage outputs with prefixes U_AB: and U_BC: (two machine phase-to-phase voltages).
The following conventions are used for inputs:
Source current flowing in the arrow direction is positive.
Positive source voltage is indicated by a + sign on the icon.
The sign conventions used for voltages and currents of state variables and measurement outputs are described in Measuring Voltages and Currents.
sps = power_analyze('sys','structure') creates a structure array sps with fields and values describing the model sys.
The fields of the structure array are defined in the following order.
char array of state variable names
char array of system input names
char array of system output names
nstates-by-nstates state-space A matrix
nstates-by-ninput state-space B matrix
noutput-by-nstates state-space C matrix
noutput-by-ninput state-space D matrix
nstates-by-1 vector of initial conditions of the state variables listed in states
nstates-by-nfreq steady-state values of states. A set of values is computed for every frequency specified in the frequencies vector.
ninput-by-nfreq steady-state values of inputs. A set of values is computed for every frequency specified in the frequencies vector.
noutput-by-nfreq steady-state values of outputs. A set of values is computed for every frequency specified in the frequencies vector.
1-by-nfreq vector of input source frequencies ordered by increasing values
DependentStates
char array of dependent state variable names. The dependent states are not included in the state-space equations.
x0DependentStates
Vector of initial conditions of dependent states
xssDependentStates
nstates-by-nfreq steady-state values of dependent states
Discrete state-space A matrix. Returns an empty value when the Powergui is in continuous or in phasor mode.
Discrete state-space B matrix. Returns an empty value when the Powergui is in continuous or in phasor mode.
Cdiscrete
Discrete state-space C matrix. Returns an empty value when the Powergui is in continuous or in phasor mode.
Ddiscrete
Discrete state-space D matrix. Returns an empty value when the Powergui is in continuous or in phasor mode.
x0discrete
Vector of discrete initial conditions. Returns an empty value when the Powergui is in continuous or in phasor mode.
Sample time value used to compute discrete state-space matrices
A matrix taking into account the initial status of switch devices
B matrix taking into account the initial status of switch devices
C matrix taking into account the initial status of switch devices
D matrix taking into account the initial status of switch devices
x0switch
Vector of initial values of switch currents
noutput-by-ninput-by-nfreq complex transfer function of impedances of the linear system corresponding to the frequencies contained in the frequencies vector. For a particular frequency, Hlin is defined by
yss(:,i) = Hlin(:,:,i) * uss(:,i)
OscillatoryModes
Display the oscillatory modes of the state-space system
The table uses the following conventions:
nstates is the number of states.
ninput is the number of inputs.
noutput is the number of outputs.
nfreq is the number of input source frequencies.
power_analyze('sys') returns the state-space calculations in separate variables.
sps = power_analyze('sys','sort') returns a structure array sps with the following fields related to the interconnection of Simscape Electrical Specialized Power Systems blocks in a model. The fields are defined in the following order.
Sample time for discrete systems
RlcBranch
rlc matrix in the power_statespace format
RlcBranchNames
List of blocks containing the state variable
SourceBranch
Source matrix in the power_statespace format
SourceBranchNames
Names of the blocks defined as sources
Names of the inputs of the system
Names of the outputs of the system
OutputExpressions
Output expression in the power_statespace format
Output expression in matrix format (internal)
MeasurementBlocks
Names of the voltage and current measurement blocks
sps = power_analyze('sys','ss') creates a continuous state-space model of the model sys with matrices A, B, C, D. You must have Control System Toolbox™ software installed for this option. The output is a state-space object.
power_analyze('sys','net') generates a netlist stored in a file, sys.net. The file contains the node numbers automatically generated by power_analyze, as well as parameter values of all linear elements. See the formats described in the power_statespace reference page.
SW = power_analyze('sys','getSwitchStatus') returns a structure array with switch names and their initial status. You can use the SW structure to specify switch statuses for a particular circuit topology and to compute the corresponding state-space matrices using the command sps = power_analyze('sys','setSwitchStatus',SW). The SW structure contains the following fields.
Names of the switches of the system
Vector of initial states of switches
sps = power_analyze('sys','setSwitchStatus',SW) creates a structure array sps with fields and values describing the state-space matrices of model sys for the switch status defined in SW. Use the command
SW = power_analyze('sys','getSwitchStatus') to obtain the SW structure array.
Obtain the state-space matrices and steady-state voltages and currents for the power_netsim2 circuit.
sps = power_analyze('power_netsim2','structure');
returns the state-space model in the sps structure variable.
sps.A =
sps.uss =
sps.xss =
sps.yss =
sps.inputs =
I_Breaker
U_Source
sps.outputs =
U_Breaker
I_Current Measurement
The inductor current of the 51-ohm, 12-mH block and the capacitor voltage of the 120-ohm, 16-µF block are the two state variables in this circuit. The Breaker block is a nonlinear element that is represented by a current source (the first input) driven by the voltage across its terminals (the first output).
power_statespace | power_init | Powergui |
Estimate frequency response and spectrum using spectral analysis with frequency-dependent resolution - MATLAB spafdr - MathWorks India
y\left(t\right)=G\left(q\right)u\left(t\right)+v\left(t\right)
G\left({e}^{i\omega }\right)
\frac{2\pi }{N{T}_{s}}
\frac{\pi }{{T}_{s}} |
Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents
2018 Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents
Zakariya Chaouai, Abderrahmane El Hachimi
We consider the Dirichlet initial boundary value problem
{\partial }_{t}{u}^{m\left(x\right)}-\mathrm{div}\left({\left|\nabla u\right|}^{p\left(x,t\right)-\mathrm{2}}\nabla u\right)=a\left(x,t\right){u}^{q\left(x,t\right)}
, where the exponents
p\left(x,t\right)>\mathrm{1}
q\left(x,t\right)>\mathrm{0}
m\left(x\right)>\mathrm{0}
are given functions. We assume that
a\left(x,t\right)
is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if
\mathrm{ess}\mathrm{sup}p\left(x,t\right)-\mathrm{1}<\mathrm{ess}\mathrm{inf}m\left(x\right)
, then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when
\mathrm{ess}\mathrm{sup}m\left(x\right)<\mathrm{ess}\mathrm{inf}p\left(x,t\right)-\mathrm{1}
, we get the positivity of solutions for large
t
. In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that
\mathrm{ess}\mathrm{sup}m\left(x\right)<\mathrm{ess}\mathrm{inf}p\left(x,t\right)-\mathrm{1}
q\left(x,t\right)=m\left(x\right)
a\left(x,t\right)\le \mathrm{0}
a.e. Finally, we give a uniform localization of the support of solutions for all
t>\mathrm{0}
, in the case where
a\left(x,t\right)<{a}_{\mathrm{1}}<\mathrm{0}
\mathrm{ess}\mathrm{sup}q\left(x,t\right)<\mathrm{ess}\mathrm{inf}p\left(x,t\right)-\mathrm{1}
Zakariya Chaouai. Abderrahmane El Hachimi. "Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents." Abstr. Appl. Anal. 2018 1 - 14, 2018. https://doi.org/10.1155/2018/3821217
Zakariya Chaouai, Abderrahmane El Hachimi "Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents," Abstract and Applied Analysis, Abstr. Appl. Anal. 2018(none), 1-14, (2018) |
Home : Support : Online Help : Mathematics : Discrete Mathematics : Combinatorics : Iterator : Overview
Overview of the Iterator package
Iterator[command](arguments)
The Iterator package exports constructors of efficient iterators over discrete structures.
Each iterator is an object with a ModuleIterator method. It can be used in for loops and in seq, add, or mul commands.
To reduce memory usage the iterators use a mutable data structure, a one-dimensional Array, as the output.
All constructors provide a compile option that is true by default. When true, the returned iterator is compiled.
See Iterator[Details] for details of an iterator object.
SetPartitionFixedSize
The following subpackages are available.
procedures for converting between permutations and inversion tables
procedures for operating on mixed-radix tuples
procedure for converting between tree representations
subpackage for converting between permutations and inversion tables
subpackage for converting between tree representations
RevolvingDoorCombinations
generate necklaces
generate fixed-size partitions of an integer
generate fixed-size partitions of a set
generate set partitions with restricted growth strings in Gray code order
The Twelve-Fold Way
Richard Stanley, in Enumerative Combinatorics, categorizes common combinatorial selections using the cardinality of unrestricted, injective, and surjective functions between discrete domains in a
4×3
tableau called "The Twelve-fold Way." The following table reproduces this categorization, using the enumeration of ways to place balls into urns, and links to the appropriate iterator.
balls per urn
n
labeled balls,
m
labeled urns
MixedRadixTuples([m$n])
Permute(m,n)
n
unlabeled balls,
m
Multicombination([n$m],n)
Combination(m,n)
Composition(n,parts=m)
n
m
unlabeled urns
SetPartitions(n,maxparts=m)
SetPartitions(n,parts=m)
n
m
PartitionFixedSize(n,maxparts=m)
PartitionFixedSize(n,parts=m)
[1] Partitions of
n
m
ordered parts.
[2] If
n\le m
there is one arrangement that satisfies this, otherwise none.
Index of interesting examples. The link goes to the help page; look in its Examples section for the example.
solve an alphametic puzzle (cryptarithm)
compute number of contingency tables
solve Dudney's century puzzle
Matrix permanent
compute matrix permanent with Ryser's algorithm
compute number of distinct ranks of a poker hand
solve simplified Dudney's century puzzle
Split list of floats
split a list of floats into nearly equal sublists. Demonstrates the creation and use of parallelized iterators.
create a Young rectangle (specialization of a Young tableau)
\mathrm{with}\left(\mathrm{Iterator}\right):
Use Permute to construct an iterator over all permutations of the list [1,2,2,3].
P≔\mathrm{Permute}\left([1,2,2,3]\right):
Use a for-loop to iterate over the permutations.
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}P\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{printf}\left("%d\n",p\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}
The same output is more conveniently generated with the Print method. Here the number of iterations is limited and showrank option is used to display the rank.
\mathrm{Print}\left(P,10,'\mathrm{showrank}'\right):
Use a seq command to create the entire sequence.
\mathrm{seq}\left(p[],p=P\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
Note the use of the square brackets, [], to instantiate the Vector that is assigned to p. Without them, all values in the final expression sequence equal the last value because the p' evaluates to the Vector rather than its content. Here is what happens when the square brackets are omitted.
\mathrm{seq}\left(p,p=P\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
Using hasNext and getNext
Use Combination to generate all triplets of the integers 0 to 4. Extract the two procedures, hasNext and getNext, from the ModuleIterator method of the iterator object and use them in a while-loop.
M≔\mathrm{Combination}\left(5,3\right):
\mathrm{hasNext},\mathrm{getNext}≔\mathrm{ModuleIterator}\left(M\right):
\mathbf{while}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{hasNext}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(\mathrm{getNext}\left(\right)\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
Concurrent iterators
Construct an iterator over the 2-permutations of the list
[1,1,2]
, use Object to create an identical, but independent, second iterator, and use both iterators in a dual-loop.
P≔\mathrm{Permute}\left([1,1,2],2\right):
Q≔\mathrm{Object}\left(P\right):
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}P\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}q\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}Q\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(p,q\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\end{array}]
The Iterator package was introduced in Maple 2016.
Iterator[Details] |
A genericity theorem for algebraic stacks and essential dimension of hypersurfaces
Zinovy ReichsteinAngelo Vistoli
A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds
Stefano VidussiStefan Friedl
The entropy conjecture for diffeomorphisms away from tangencies
Marcelo VianaGang LiaoJiagang Yang
Modular representations of finite groups with trivial restriction to Sylow subgroups
On a magnetic characterization of spectral minimal partitions
Bernard HelfferThomas Hoffmann-Ostenhof
Søren FournaisJan Philip SolovejLászló Erdős
Leonid PotyagailoVictor Gerasimov
Enumeration of real conics and maximal configurations
Erwan BrugalléNicolas Puignau
Mladen BestvinaKevin WortmanAlex Eskin
On the dimension of
p
-harmonic measure in space
Kaj NyströmJohn L. LewisAndrew Vogel
Tensor complexes: multilinear free resolutions constructed from higher tensors
Christine Berkesch ZamaereDaniel ErmanManoj KumminiSteven V Sam
Operations between sets in geometry
Richard J. GardnerDaniel HugWolfgang Weil
A support theorem for Hilbert schemes of planar curves
Luca MiglioriniVivek Shende
The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant
Igor RodnianskiJared Speck
Erratum to “Geometric rigidity of
\times m
invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012)) |
EUDML | Bounded cohomology of subgroups of mapping class groups. EuDML | Bounded cohomology of subgroups of mapping class groups.
Bounded cohomology of subgroups of mapping class groups.
Bestvina, Mladen; Fujiwara, Koji
Bestvina, Mladen, and Fujiwara, Koji. "Bounded cohomology of subgroups of mapping class groups.." Geometry & Topology 6 (2002): 69-89. <http://eudml.org/doc/122903>.
@article{Bestvina2002,
author = {Bestvina, Mladen, Fujiwara, Koji},
keywords = {bounded cohomology; mapping class groups; hyperbolic groups},
title = {Bounded cohomology of subgroups of mapping class groups.},
TI - Bounded cohomology of subgroups of mapping class groups.
KW - bounded cohomology; mapping class groups; hyperbolic groups
bounded cohomology, mapping class groups, hyperbolic groups
{E}^{2}
2
Articles by Bestvina
Articles by Fujiwara |
Now for my day’s annals— In the morning I was baddish, & did hardly any work & was as much overcome by my children, as ever Bishop Coplestone2 was with Duck.3 But the children have been very good all day, & I have grown a good deal better this afternoon, & had a good romp with Baby—4 I see, however, very little of the Blesseds— The day was so thick & wet a fog, that none of them went out, though a thaw & not very cold; I had a long pace in the Kitchen Garden: Lewis5 came up to mend the pipe & paper the W.C. in which apartment there was a considerable crowd for about an hour, when Mr Lewis & his son William, Willy Annie, Baby & Bessy6 were there. Baby insisted on going in, I daresay, greatly to the disturbance of Bessy’s delecacy— Lewis from first dinner to second dinner was a first-rate dispensary, as they never left him— They, also, dined in the Kitchen, and I believe have had a particularly pleasant day.—
I was playing with Baby in the window of the drawing-room this morning, & she was blowing a feeble fly (fry) & blew it on its back, when it kicked so hard, that to my great amusement Baby grew red in the face, looked frightened & pushed away from the window.— The children are growing so quite out of all rule in the drawing-room, jumping on everything & butting like young bulls at every chair & sofa, that I am going to have the dining-room fire lighted tomorrow & keep them out of the drawing-room. I declare a months such wear, wd spoil every thing in the whole drawing-room.—
I read Whately’s Shakspeare7 & very ingenious & interesting it is—and what do you think Mitford’s Greece8 has made me begin, the Iliad by Cowper,9 which we were talking of; & have read 3 books with much more pleasure, than I anticipated.— I have given up acids & gone to puddings again.—
Tuesday morning— I am impatient for your letter this morning to hear how you got on.— I asked Willy how Baby has slept & he answered “she did not cry not one mouthful”. My stomach is baddish again this morning & I almost doubt, whether I will go to London, tomorrow; if I do you won’t hear. Poor Annie has had a baddish knock by Willie’s ball in her eye.—it is swelled a bit, but not otherwise bad.
Your cap cannot ⟨be⟩ found anywhere: Jane says you took one.
\frac{9}{10}
of the snow is gone & the children are going out. Very many thanks for your letter10
Date based on nn. 7 and 8, below, and on Henrietta Litchfield’s statement, before her transcription of parts of this letter, that Emma went to Maer in February 1845 (Emma Darwin (1915) 2: 92). Emma’s diary records that she was away between 31 January and 11 February.
Edward Copleston.
Henrietta Litchfield notes, ‘This must be some family joke. Bishop Copleston had been a friend of Sir James Mackintosh.’ (Emma Darwin (1915) 2: 93).
Henrietta Emma Darwin, born 25 September 1843.
John Lewis was a carpenter in Down village (Post Office directory of the six home counties 1845.)
Elizabeth Harding, nursery maid at Down House (see Emma Darwin (1915) 2: 80–1).
T. Whately 1785. The London Library borrowing list records that CD borrowed Thomas Whately’s book on 30 January and returned it on 27 March 1845 (London Library Archives).
Mitford 1784–1818. Volumes two and three of William Mitford’s History of Greece were borrowed from the London Library on 9 January and returned on 27 March 1845 (London Library Archives).
Cowper 1791.
The final paragraph was written in pencil.
Cowper, William, trans. 1791. The Iliad and Odyssey of Homer translated into English blank verse. 2 vols. London.
Mitford, William. 1784–1818. History of Greece. 5 vols. London: J. Murray and J. Robson.
Whately, Thomas. 1785. Remarks on some of the characters of Shakespeare. London.
News of the children and books he is reading.
Sotheby’s (dealers) (28 March 1983) |
Step-by-step NMO correction - SEG Wiki
This tutorial originally appeared as a featured article in The Leading Edge in February 2017 — see issue.
Open any textbook about seismic data processing and you will inevitably find a section about the normal moveout (NMO) correction. There you'll see that we can correct the measured traveltime of a reflected wave t at a given offset x to obtain the traveltime at normal incidence
{\displaystyle t_{0}}
by applying the following equation:
{\displaystyle t_{0}^{2}=t^{2}-{\frac {x^{2}}{v_{NMO}^{2}}}}
{\displaystyle v_{NMO}}
is the NMO velocity. There are variants of this equation with different degrees of accuracy, but we'll use this one for simplicity.
When applied to a common-midpoint (CMP) section, the equation above is supposed to turn the hyperbola associated with a reflection into a straight horizontal line. What most textbooks won't tell you is how, exactly, to apply this equation to the data.
Read on and I'll explain step-by-step how the algorithm for NMO correction from Yilmaz (2001) works and how to implement it in Python.[1] The accompanying Jupyter notebook (Perez and Granger, 2007) contains the full source code, with documentation and tests for each function.[2] You can download the notebook at https://github.com/seg or at https://github.com/pinga-lab/nmo-tutorial.
1 What the equation doesn't tell us
2 Doing it backwards
3 The code for NMO
What the equation doesn't tell us
Equation 1 relates traveltimes: the one we can measure (t) and the one we want to know (
{\displaystyle t_{0}}
). But the data in our CMP gather are actually a matrix of amplitudes measured as a function of time (t) and offset. Our NMO-corrected gather will also be a matrix of amplitudes as a function of time (
{\displaystyle t_{0}}
) and offset. So what we really have to do is transform one matrix of amplitudes into the other. But the equation has no amplitudes!
This is a major divide between the formula we've all seen before and what actually goes on in the software that implements it. You have probably never thought about it — I certainly hadn't — so let's bridge this divide. Next, I'll explain an algorithm that maps the amplitudes in the CMP to amplitudes in an NMO-corrected gather.
It's surprisingly difficult to find a description of a method for calculating the amplitudes in the NMO correction. The only one I could find is a single paragraph in the book by Yilmaz:
“The idea is to find the amplitude value at A′ on the NMO-corrected gather from the amplitude value at A on the original CMP gather. Given quantities
{\displaystyle t_{0}}
, x,
{\displaystyle v_{NMO}}
, compute t from equation 1. […] The amplitude value at this time can be computed using the amplitudes at the neighboring integer sample values […] This is done by an interpolation scheme that involves the four samples.”
This paragraph is telling us to do the calculation backwards. Instead of mapping where each point in the CMP goes in the NMO-corrected gather, we should map where each point in the NMO gather comes from in the CMP. Figure 1 shows a sketch of the procedure to calculate the amplitude of a point (
{\displaystyle t_{0}}
, x) in the NMO gather.
Figure 1. Sketch of the algorithm for a single trace and
{\displaystyle t_{0}}
. To the left is a trace from the CMP and to the right the corresponding trace from the NMO-corrected gather. The green square in the NMO is the amplitude at
{\displaystyle t_{0}}
that we want to calculate. We apply the equation to find time t in the CMP, then interpolate the amplitude using the four samples around t (orange triangles). This amplitude is then copied over to the NMO.
Start with an NMO gather filled with zeros.
For each point (
{\displaystyle t_{0}}
, x) in the NMO gather:
Calculate the reflection traveltime (t) given
{\displaystyle v_{NMO}}
using the equation in Figure 1.
Go to the trace at offset x in the CMP and find the two samples before and the two samples after time t.
If t is greater than the recording time or if it doesn't have two samples after it, skip the next two steps.
Use the amplitude in these four samples to interpolate the amplitude at time t.
Copy the interpolated amplitude to the NMO gather at (
{\displaystyle t_{0}}
, x).
At the end of this algorithm, we will have a fully populated NMO gather with the amplitudes sampled from the CMP. Notice that we didn't actually use the equation for
{\displaystyle t_{0}}
. Instead we calculate the reflection traveltime (t). Good luck guessing that from the equation alone.
The code for NMO
Now I'll show how to implement the above algorithm in Python using the NumPy and SciPy libraries.[3].
We'll split the algorithm into three functions. This is very important when programming any moderately complex code because it allows us to test each part of our code independently. It also reduces the amount of code we have to search through to find the bug that is messing up our results. Modular code is easier to understand and to reuse.
The first function I'll define performs the NMO correction on a given CMP gather. We'll assume that the CMP gather is a 2D array of amplitudes and that the velocities are a 1D array with
{\displaystyle v_{NMO}}
for each time sample.
def nmo_correction(cmp, dt, offsets, velocities):
nmo = np.zeros_like(cmp)
nsamples = cmp.shape[0]
times = np.arange(0, nsamples*dt, dt)
for i, t0 in enumerate(times):
for j, x in enumerate(offsets):
t = reflection_time(t0, x, velocities[i]) amplitude = sample_trace(cmp[:, j], t, dt) if amplitude is not None:
nmo[i, j] = amplitude
return nmo
This function is essentially the algorithm above translated to Python with some of the details pushed into the reflection_time and sample_trace functions, which we will define below.
First, the function that calculates the reflection traveltime:
def reflection_time(t0, x, vnmo):
t = np.sqrt(t0**2 + x**2/vnmo**2)
For the sample_trace function, we'll use cubic splines from the scipy.interpolate package. For more information on interpolation with scipy, see the tutorial by Matt Hall.[4]
def sample_trace(trace, time, dt):
before = int(np.floor(time/dt))
N = trace.size
samples = np.arange(before − 1, before + 3)
if any(samples < 0) or any(samples >= N):
amplitude = None
times = dt*samples
amps = trace[samples]
interpolator = CubicSpline(times, amps)
amplitude = interpolator(time)
The Jupyter notebook contains the full code for these functions, including documentation through Python documentation strings or “docstrings” and code that tests that the functions work as expected. Also included is an application of our nmo_correction function to a synthetic CMP (Figure 2).
Figure 2. (a) The
{\displaystyle v_{NMO}}
profile passed to nmo_correction. The profile was interpolated on a line using the two picked velocities (black squares). (b) A synthetic CMP gather. (c) The output from our nmo_correction function.
My sincerest thanks to Evan Bianco, Gregorio Kawakami, Jesper Dramsch, and Matt Hall for comments and suggestions and to Öz Yilmaz for generously making the full text of the Seismic Data Analysis book available for free and in the open.
Leonardo Uieda, Universidade do Estado do Rio de Janeiro, Brazil, leouieda gmail.com
Madagascar workflow - reproducible using Madagascar open-source software
↑ Yilmaz, Ö., 2001, Seismic data analysis: Processing, inversion, and interpretation of seismic data: Society of Exploration Geophysicists, http://dx.doi.org/10.1190/1.9781560801580
↑ Perez, F. H., and B. E. Granger, 2007, IPython: A system for interactive scientific computing: Computing in Science & Engineering, 9, no. 3, 21–29, http://dx.doi.org/10.1109/mcse.2007.53.
↑ van der Walt, S., S. C. Colbert, and G. Varoquaux, 2011, The NumPy array: A structure for efficient numerical computation: Computing in Science & Engineering, 13, no. 2, 22–30, http://dx.doi.org/10.1109/mcse.2011.37z
↑ Hall, M., 2016, Geophysical tutorial: The function of interpolation: The Leading Edge, 35, no. 4, 367–369, http://dx.doi.org/10.1190/tle35040367.1
Retrieved from "https://wiki.seg.org/index.php?title=Step-by-step_NMO_correction&oldid=58782" |
3DOF rigid vehicle body to calculate longitudinal, vertical, and pitch motion - Simulink - MathWorks 日本
\begin{array}{l}{F}_{x}={F}_{wF}+{F}_{wR}â{F}_{d,x}â{F}_{sx,F}â{F}_{sx,R}+{F}_{g,x}\\ {F}_{z}={F}_{d,z}â{F}_{sz,F}â{F}_{sz,R}+{F}_{g,z}\\ {M}_{y}=a{F}_{sz,F}âb{F}_{sz,R}+h\left({F}_{wF}+{F}_{wR}+{F}_{sx,F}+{F}_{sx,R}\right)â{M}_{d,y}\end{array}
\begin{array}{l}\stackrel{¨}{x}=\frac{{F}_{x}}{m}âqz\\ \stackrel{¨}{z}=\frac{{F}_{z}}{m}âqx\\ \stackrel{Ë}{q}=\frac{{M}_{y}}{{I}_{yy}}\\ \stackrel{Ë}{\mathrm{θ}}=q\end{array}
\begin{array}{l}F{s}_{F}={N}_{F}\left[F{k}_{F}+F{b}_{F}\right]\\ F{s}_{R}={N}_{R}\left[F{k}_{R}+F{b}_{R}\right]\end{array}
\begin{array}{l}F{k}_{F}=f\left(d{Z}_{F}\right)\\ F{k}_{R}=f\left(d{Z}_{R}\right)\end{array}
\begin{array}{l}F{b}_{F}=f\left(d{\stackrel{Ë}{Z}}_{F}\right)\\ F{b}_{R}=f\left(d{\stackrel{Ë}{Z}}_{R}\right)\end{array}
\begin{array}{l}d{Z}_{F}={Z}_{F}â{\stackrel{¯}{Z}}_{F}\\ d{Z}_{R}={Z}_{R}â{\stackrel{¯}{Z}}_{R}\\ d{\stackrel{Ë}{Z}}_{F}={\stackrel{Ë}{Z}}_{F}â{\stackrel{Ë}{\stackrel{¯}{Z}}}_{F}\\ d{\stackrel{Ë}{Z}}_{R}={\stackrel{Ë}{Z}}_{R}â{\stackrel{Ë}{\stackrel{¯}{Z}}}_{R}\end{array}
{\stackrel{¯}{Z}}_{F},{\stackrel{¯}{Z}}_{R}
{\stackrel{Ë}{\stackrel{¯}{Z}}}_{F},{\stackrel{Ë}{\stackrel{¯}{Z}}}_{R}
\begin{array}{l}{F}_{d,x}=\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{\left(}^{\stackrel{Ë}{x}}\\ {F}_{d,z}=\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{\left(}^{\stackrel{Ë}{x}}\\ {M}_{d,y}=\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{\left(}^{\stackrel{Ë}{x}}\left(a+b\right)\end{array}
{P}_{FxExt}={F}_{xExt}\stackrel{Ë}{x}
{P}_{FzExt}={F}_{zExt}\stackrel{Ë}{z}
{P}_{MzExt}={M}_{zExt}\stackrel{Ë}{\mathrm{θ}}
{P}_{FwFx}={F}_{wF}\stackrel{Ë}{x}
{P}_{FwRx}={F}_{wR}\stackrel{Ë}{x}
{P}_{Fs,F}=â{P}_{FwFx}+{P}_{FsbF}+{P}_{Fsk,F}+{F}_{xF}{\stackrel{Ë}{x}}_{F}+{F}_{zF}{\stackrel{Ë}{z}}_{F}
{P}_{Fs,R}=â{P}_{FwRx}+{P}_{Fsb,R}+{P}_{Fsk,R}+{F}_{xF}{\stackrel{Ë}{x}}_{F}+{F}_{zF}{\stackrel{Ë}{z}}_{F}
{P}_{d,x}={F}_{d,x}\stackrel{Ë}{x}
{P}_{d,z}={F}_{d,z}\stackrel{Ë}{z}
{P}_{d,My}={M}_{d,y}\stackrel{Ë}{\mathrm{θ}}
{P}_{Fsb}=\underset{i=F,R}{\overset{}{â}}{F}_{sb,i}{\stackrel{Ë}{z}}_{i}
{P}_{g}=âmg\stackrel{Ë}{Z}
{P}_{\stackrel{Ë}{x}}=m\stackrel{¨}{x}\stackrel{Ë}{x}
{P}_{\stackrel{Ë}{z}}=m\stackrel{¨}{z}\stackrel{Ë}{z}
{P}_{\stackrel{Ë}{\mathrm{θ}}}={I}_{yy}\stackrel{¨}{\mathrm{θ}}\stackrel{Ë}{\mathrm{θ}}
{P}_{FskF}={F}_{sk,F}{\stackrel{Ë}{z}}_{F}
{P}_{FskF}={F}_{sk,R}{\stackrel{Ë}{z}}_{R}
\stackrel{Ë}{x}
\stackrel{¨}{x}
z\text{, }\stackrel{Ë}{z}\text{, }\stackrel{¨}{z}
{\stackrel{Ë}{Z}}_{F},{\stackrel{Ë}{Z}}_{R}
{\stackrel{¯}{Z}}_{F},{\stackrel{¯}{Z}}_{R}
{\stackrel{Ë}{\stackrel{¯}{Z}}}_{F},{\stackrel{Ë}{\stackrel{¯}{Z}}}_{R}
d{\stackrel{Ë}{Z}}_{F},d{\stackrel{Ë}{Z}}_{R}
External moment about vehicle CG, Mx, My, Mz, in the vehicle-fixed frame, in N·m. Signal vector dimensions are [1x3] or [3x1].
\mathrm{γ}
{\stackrel{¯}{Z}}_{F},{\stackrel{¯}{Z}}_{R}
{\stackrel{Ë}{\stackrel{¯}{Z}}}_{F},{\stackrel{Ë}{\stackrel{¯}{Z}}}_{R}
Ext Fx External moment on vehicle CG about the vehicle-fixed x-axis Computed N·m
Fy External moment on vehicle CG about the vehicle-fixed y-axis Computed N·m
Fz External moment on vehicle CG about the vehicle-fixed z-axis Computed N·m
{\stackrel{Ë}{x}}_{0} |
List of games in game theory - Wikipedia
See also: Category:Game theory game classes
Game theory studies strategic interaction between individuals in situations called games. Classes of these games have been given names. This is a list of the most commonly studied games
Games can have several features, a few of the most common are listed here.
Number of players: Each person who makes a choice in a game or who receives a payoff from the outcome of those choices is a player.
Strategies per player: In a game each player chooses from a set of possible actions, known as pure strategies. If the number is the same for all players, it is listed here.
Number of pure strategy Nash equilibria: A Nash equilibrium is a set of strategies which represents mutual best responses to the other strategies. In other words, if every player is playing their part of a Nash equilibrium, no player has an incentive to unilaterally change his or her strategy. Considering only situations where players play a single strategy without randomizing (a pure strategy) a game can have any number of Nash equilibria.
Sequential game: A game is sequential if one player performs her/his actions after another player; otherwise, the game is a simultaneous move game.
Perfect information: A game has perfect information if it is a sequential game and every player knows the strategies chosen by the players who preceded them.
Constant sum: A game is a constant sum game if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.
Move by nature: A game includes a random move by nature.
No. of pure strategy
Move by nature
Battle of the sexes 2 2 2 No No No No
Blotto games 2 variable variable No No Yes No
Cake cutting N, usually 2 infinite variable[1] Yes Yes Yes No
Centipede game 2 variable 1 Yes Yes No No
Chicken (aka hawk-dove) 2 2 2 No No No No
Coordination game N variable >2 No No No No
Cournot game 2 infinite[2] 1 No No No No
Deadlock 2 2 1 No No No No
Dictator game 2 infinite[2] 1 N/A[3] N/A[3] Yes No
Diner's dilemma N 2 1 No No No No
Dollar auction 2 2 0 Yes Yes No No
El Farol bar N 2 variable No No No No
Game without a value 2 infinite 0 No No Yes No
Gift-exchange game N, usually 2 variable 1 Yes Yes No No
Guess 2/3 of the average N infinite 1 No No Maybe[4] No
Kuhn poker 2 27 & 64 0 Yes No Yes Yes
Matching pennies 2 2 0 No No Yes No
Minimum Effort Game aka Weak-Link Game Infinite 7 7 No No No No
Muddy Children Puzzle N 2 1 Yes No No Yes
Nash bargaining game 2 infinite[2] infinite[2] No No No No
Optional prisoner's dilemma 2 3 1 No No No No
Peace war game N variable >2 Yes No No No
Pirate game N infinite[2] infinite[2] Yes Yes No No
Platonia dilemma N 2
{\displaystyle 2^{N}-1}
No Yes No No
Princess and monster game 2 infinite 0 No No Yes No
Prisoner's dilemma 2 2 1 No No No No
Public goods N infinite 1 No No No No
Rock, paper, scissors 2 3 0 No No Yes No
Screening game 2 variable variable Yes No No Yes
Signaling game N variable variable Yes No No Yes
Stag hunt 2 2 2 No No No No
Traveler's dilemma 2 N >> 1 1 No No No No
Truel 3 1-3 infinite Yes Yes No No
Trust game 2 infinite 1 Yes Yes No No
Ultimatum game 2 infinite[2] infinite[2] Yes Yes No No
Vickrey auction N infinite 1 No No No Yes[5]
Volunteer's dilemma N 2 2 No No No No
War of attrition 2 2 0 No No No No
List of games from gametheory.net
A visual index to common 2x2 games
^ For the cake cutting problem, there is a simple solution if the object to be divided is homogenous; one person cuts, the other chooses who gets which piece (continued for each player). With a non-homogenous object, such as a half chocolate/half vanilla cake or a patch of land with a single source of water, the solutions are far more complex.
^ a b c d e f g h There may be finite strategies depending on how goods are divisible
^ a b Since the dictator game only involves one player actually choosing a strategy (the other does nothing), it cannot really be classified as sequential or perfect information.
^ Potentially zero-sum, provided that the prize is split among all players who make an optimal guess. Otherwise non-zero sum.
^ The real value of the auctioned item is random, as well as the perceived value.
Arthur, W. Brian “Inductive Reasoning and Bounded Rationality”, American Economic Review (Papers and Proceedings), 84,406-411, 1994.
Bolton, Katok, Zwick 1998, "Dictator game giving: Rules of fairness versus acts of kindness" International Journal of Game Theory, Volume 27, Number 2
Gibbons, Robert (1992) A Primer in Game Theory, Harvester Wheatsheaf
Glance, Huberman. (1994) "The dynamics of social dilemmas." Scientific American.
Martin J. Osborne & Ariel Rubinstein: A Course in Game Theory (1994).
McKelvey, R. and T. Palfrey (1992) "An experimental study of the centipede game," Econometrica 60(4), 803-836.
Nash, John (1950) "The Bargaining Problem" Econometrica 18: 155-162.
Ochs, J. and A.E. Roth (1989) "An Experimental Study of Sequential Bargaining" American Economic Review 79: 355-384.
Rapoport, A. (1966) The game of chicken, American Behavioral Scientist 10: 10-14.
Rasmussen, Eric: Games and Information, 2004
Shor, Mikhael. "Battle of the sexes". GameTheory.net. Retrieved September 30, 2006.
Shor, Mikhael. "Deadlock". GameTheory.net. Retrieved September 30, 2006.
Shor, Mikhael. "Matching Pennies". GameTheory.net. Retrieved September 30, 2006.
Shor, Mikhael. "Prisoner's Dilemma". GameTheory.net. Retrieved September 30, 2006.
Shubik, Martin "The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation," The Journal of Conflict Resolution, 15, 1, 1971, 109-111.
Sinervo, B., and Lively, C. (1996). "The Rock-Paper-Scissors Game and the evolution of alternative male strategies". Nature Vol.380, pp. 240–243
Skyrms, Brian. (2003) The stag hunt and Evolution of Social Structure Cambridge: Cambridge University Press.
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_games_in_game_theory&oldid=1088580294" |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.