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Fermat number - Simple English Wikipedia, the free encyclopedia
A Fermat number is a special positive number. Fermat numbers are named after Pierre de Fermat. The formula that generates them is
{\displaystyle F_{n}=2^{2^{\overset {n}{}}}+1}
where n is a nonnegative integer. The first nine Fermat numbers are (sequence A000215 in the OEIS):
F8 = 2256 + 1 = 115792089237316195423570985008687907853269984665640564039457584007913129639937 = 1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321
As of 2007, only the first 12 Fermat numbers have been completely factored. (written as a product of prime numbers) These factorizations can be found at Prime Factors of Fermat Numbers.
If 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. Every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0,...,F4.
1 Interesting things about Fermat numbers
2 What they are used for
3 Fermat's conjecture
Interesting things about Fermat numbers[change | change source]
No two Fermat numbers have common divisors.
Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.
What they are used for[change | change source]
Today, Fermat numbers can be used to generate random numbers, between 0 and some value N, which is a power of 2.
Fermat's conjecture[change | change source]
Fermat, when he was studying these numbers, conjectured that all Fermat numbers were prime. This was proven to be wrong by Leonhard Euler, who factorised
{\displaystyle F_{5}}
Sequence of Fermat numbers Archived 2001-07-16 at the Wayback Machine
Prime Glossary Page on Fermat Numbers
History of Fermat Numbers Archived 2007-09-28 at the Wayback Machine
Unification of Mersenne and Fermat Numbers Archived 2006-10-02 at the Wayback Machine
Prime Factors of Fermat Numbers Archived 2016-02-10 at the Wayback Machine
Fermat Number at MathWorld
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Fermat_number&oldid=8027699"
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2018 Solution of Sequential Hadamard Fractional Differential Equations by Variation of Parameter Technique
Mohammed M. Matar
We obtain in this article a solution of sequential differential equation involving the Hadamard fractional derivative and focusing the orders in the intervals
\left(\mathrm{1,2}\right)
\left(\mathrm{2,3}\right)
. Firstly, we obtain the solution of the linear equations using variation of parameter technique, and next we investigate the existence theorems of the corresponding nonlinear types using some fixed-point theorems. Finally, some examples are given to explain the theorems.
Mohammed M. Matar. "Solution of Sequential Hadamard Fractional Differential Equations by Variation of Parameter Technique." Abstr. Appl. Anal. 2018 1 - 7, 2018. https://doi.org/10.1155/2018/9605353
Received: 30 September 2017; Accepted: 4 January 2018; Published: 2018
Mohammed M. Matar "Solution of Sequential Hadamard Fractional Differential Equations by Variation of Parameter Technique," Abstract and Applied Analysis, Abstr. Appl. Anal. 2018(none), 1-7, (2018)
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Chamber with one port and fixed volume of two-phase fluid - MATLAB - MathWorks Italia
Constant Volume Chamber (2P)
Chamber with one port and fixed volume of two-phase fluid
The Constant Volume Chamber (2P) block models the accumulation of mass and energy in a chamber containing a fixed volume of two-phase fluid. The chamber has one inlet, labeled A, through which fluid can flow. The fluid volume can exchange heat with a thermal network, for example one representing the chamber surroundings, through a thermal port labeled H.
The mass of the fluid in the chamber varies with density, a property that in a two-phase fluid is generally a function of pressure and temperature. Fluid enters when the pressure upstream of the inlet rises above that in the chamber and exits when the pressure gradient is reversed. The effect in a model is often to smooth out sudden changes in pressure, much like an electrical capacitor does with voltage.
The flow resistance between the inlet and interior of the chamber is assumed to be negligible. The pressure in the interior is therefore equal to that at the inlet. Similarly, the thermal resistance between the thermal port and interior of the chamber is assumed to be negligible. The temperature in the interior is equal to that at the thermal port.
Mass can enter and exit the chamber through port A. The volume of the chamber is fixed but the compressibility of the fluid means that its mass can change with pressure and temperature. The rate of mass accumulation in the chamber must exactly equal the mass flow rate in through port A:
\left[{\left(\frac{\partial \rho }{\partial p}\right)}_{u}\frac{dp}{dt}+{\left(\frac{\partial \rho }{\partial u}\right)}_{p}\frac{du}{dt}\right]V={\stackrel{˙}{m}}_{\text{A}}+{ϵ}_{M},
\stackrel{˙}{m}
{ϵ}_{M}=\frac{M-V/\nu }{\tau }.
\frac{dM}{dt}={\stackrel{˙}{m}}_{A}.
Energy can enter and exit the chamber in two ways: with fluid flow through port A and with heat flow through port H. No work is done on or by the fluid inside the chamber. The rate of energy accumulation in the internal fluid volume must then equal the sum of the energy flow rates in through ports A and H:
\stackrel{˙}{E}={\varphi }_{\text{A}}+\text{}{Q}_{\text{H}},
E=Mu.
The pressure drop due to viscous friction between port A and the interior of the chamber is assumed to be negligible. Gravity is ignored as are other body forces. The pressure in the internal fluid volume must then equal that at port A:
p={p}_{\text{A}}.
Opening through which fluid enters or exits the chamber.
2-Port Constant Volume Chamber (2P) | 3-Port Constant Volume Chamber (2P) | Reservoir (2P)
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During this chapter, you will use your new solving skills to solve word problems. Think about and use the strategies you already have to answer the questions below.
Andy is
4
years older than Eduardo. If Andy is
x
years old, write an expression to represent Eduardo’s age.
If Eduardo is
4
years younger than Andy, how could his age be represented if Andy is
x
x-4
In Eduardo’s collection, the number of butterflies is
12
more than twice the number of moths. If there are
x
moths, write an expression to represent the number of butterflies he has.
If twice the number of moths,
x
, can be represented as
2x
, how can the number of butterflies be represented?
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8-Word Processing Tools - Maple Help
Home : Support : Online Help : Getting Started : Tutorials : 8-Word Processing Tools
Part 8: Word Processing Tools
In Part 8: Word Processing Tools, you will learn about some of Maple's technical document creation features.
Maple contains numerous word processing tools to help you create professional-looking reports. For your reference, here is a list of some of the more common ones.
Built-in headings styles
Drop-down list on toolbar
Insert > Section, Edit > Remove Section
Font control and ability to define new styles
Toolbar buttons, Format > Styles...
Ability to insert images and other objects
Insert > Image
A spell-checker aware of mathematical terms
Tools > Spellcheck
Insert > Hyperlink
Format > Bookmarks...
Insert > Header Footer...
The fundamental organization of a Maple document is controlled by using sections and tables.
Using Sections in a Document
Use sections in your Maple document to:
Hide distracting code or detailed information
To insert a section (or subsection): Insert > Section.
To expand or collapse sections: View > Sections > Expand All Sections / Collapse All Sections
To expand or collapse one section: Click on the arrow beside the section name.
To organize existing content: Use the Indent and Outdent toolbar icons to shift selected text into a new subsection or out of a subsection level.
Enclose the selection in a section or subsection
Outdent the selection to the next section level, if possible.
To delete a section: Place the cursor in the title and press [Ctrl][Delete] or Edit > Delete Element
Tables can contain text, math, plots, graphics, and embedded components. Use tables to:
Organize your content effectively, increasing readability and reducing wasted space
Control the layout of embedded components
To insert a table: Insert > Table
To change table size/column widths: Click and drag column boundaries.
To add or delete rows and columns: Use the Table menu found in the Format menu or through the context panel for the table.
To switch between navigation and indentation modes for the Tab key: Use Tab Navigation.
Tab Navigation selected
Allows you to move between table cells using [Tab].
Tab Navigation not selected
Allows you to indent using [Tab].
To format a table: Format > Table > Properties
This dialog box allows you to adjust table properties. Options include:
Table Size Mode: There are two options:
Fixed percentage of page width
The width of the table is adjusted whenever the width of the worksheet changes. This option is useful for ensuring that the entire content of the table always fits onto the screen or printed page.
Scale with zoom factor
Preserves table appearance regardless of the size of the worksheet window or the zoom factor by using horizontal scroll when needed. Ensures line breaks occur in fixed locations, allowing you to control the breaks in long expressions. With this option, the table could be truncated when printing if it is too wide.
Set visibility of borders: As an example, in the previous bullet item a table is used to create two columns with the borders set to None.
Add table captions: Use to add a table numbering or titles. Then, elsewhere in the document, you can easily refer to a table using a table cross-reference.
Set order of cell execution: Control the order in which math in the table is executed.
Note: To avoid confusion, ensure execution order is visibly obvious to reader.
For information on tables, see Overview of Tables.
For information on embedded components, see 9-Dynamic Applications.
You can create annotations that will pop up when a user hovers over the annotated text or math.
Select the text or math to be annotated.
Click on Format > Annotations > Annotate Selection and enter the note.
Click outside the entry box to finish entering text. The selection will have a colored background indicating the annotation was added.
Move your mouse over the selection to see the annotation.
Type the math equation,
y'={y}^{2}+y
Select the equation, and from the Format menu, choose Annotations > Annotate Selection. Enter the note "Ref. p. 127, "Introductory Differential Equations.""
y'={y}^{2}+y
y'={y}^{2}+y
To edit or remove an annotation, place your cursor in the annotated text and from the Format menu, choose Annotations > Edit Annotation or Delete Annotation.
Drawing Tools and Plot Annotations
Using the drawing tools, you can sketch an idea in a canvas, draw on a plot, or draw on an image.
To draw on a plot, click on the plot, then click on the button on the toolbar. You can add additional information to plots such as text, 2-D math, lines, arrows, and shapes. See 4-Plotting for an example using plot annotations.
To create a new drawing, insert a canvas from the Insert menu.
From the Drawing toolbar, you can use the following tools: selection tool, pencil (free style drawing), eraser, text insert, straight line, rectangle, rounded rectangle, oval, diamond, alignment, drawing outline, drawing fill, drawing linestyle, and drawing canvas properties.
You can hide the gridlines through the drawing canvas properties.
For more information on using the drawing tools, including how to rotate objects and how to fill an object with an image, see Drawing Tools.
Hyperlinks in your document can link to:
Different location in current document
Other Maple documents
Highlight the text that you want to make a hyperlink.
From the Format menu, select Convert To > Hyperlink. This option is also available in the Context Panel.
Specify the hyperlink Type and Target. To link to a Maple document, choose Worksheet for the Type.
To insert an image hyperlink:
1. From the Insert menu, select Hyperlink.
2. In the Hyperlink Properties dialog box, select the Image check box and click Choose Image for the file. The image appears as the link. You can resize the image as necessary. Click and drag from the corners of the image to resize.
3. Specify the hyperlink Type and Target.
Create a link to a help topic.
1. Type "For more information, see hyperlinks."
2. Highlight hyperlinks. From the Format menu, select Convert To > Hyperlink.
3. In the Hyperlink Properties dialog, set the target to worksheet,managing,linking.
4. Click OK. Now, test the link.
For more information, see hyperlinks.
To view or edit the properties of an existing hyperlink: With your cursor in the hyperlink, select Format > Hyperlinks > Hyperlink Properties.
You can link to a specific location in a Maple document. To do this, you will need to create a bookmark.
To display bookmark formatting icons, activate the Marker feature: View > Markers.
1. Place the cursor at the location at which to place the bookmark. For example, place the cursor in a section title.
2. From the Format menu, select Bookmarks. The Bookmark dialog opens, listing existing bookmarks in the document.
3. Click New. The Create Bookmark dialog opens. Enter a bookmark name and click Create.
Note: You can also rename and delete bookmarks from the Bookmark dialog.
To link to a bookmark: Follow the steps to create a hyperlink. Enter the bookmark target under Bookmark in the Hyperlink Properties.
Using Startup Code
To enter startup code:
From the Edit menu and select Startup Code. Alternatively, click the Edit startup code icon in the toolbar. A dialog appears in which you can enter Maple commands.
Enter the desired Maple code.
When you are finished defining your startup code, and you have checked the code for syntax errors, save the code and exit, either select Save from the File menu, or click the Save button.
You can hide document blocks, designate regions of the document for automatic execution, and use embedded components to further enhance your document. You can use slideshow mode to share your document. For more information, see Overview of Document Processing or use the Help System to find more information on a specific topic.
annotations, document blocks, document processing, startup code
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Visualizing Energy Landscapes through Manifold Learning
Benjamin W. B. Shires, Chris J. Pickard
Energy landscapes provide a conceptual framework for structure prediction, and a detailed understanding of their topological features is necessary to develop efficient methods for their exploration. The ability to visualize these surfaces is essential, but the high dimensionality of the corresponding configuration spaces makes this visualization difficult. Here, we present stochastic hyperspace embedding and projection (SHEAP), a method for energy landscape visualization inspired by state-of-the-art algorithms for dimensionality reduction through manifold learning, such as
t
-SNE and UMAP. The performance of SHEAP is demonstrated through its application to the energy landscapes of Lennard-Jones clusters, solid-state carbon, and the quaternary system
\mathrm{C}+\mathrm{H}+\mathrm{N}+\mathrm{O}
. It produces meaningful and interpretable low-dimensional representations of these landscapes, reproducing well-known topological features such as funnels and providing fresh insight into their layouts. In particular, an intrinsic low dimensionality in the distribution of local minima across configuration space is revealed.
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Detect Closely Spaced Sinusoids - MATLAB & Simulink - MathWorks India
Consider a sinusoid,
f\left(x\right)={e}^{j2\pi \nu x}
, windowed with a Gaussian window,
g\left(t\right)={e}^{-\pi {t}^{2}}
. The short-time transform is
{V}_{g}f\left(t,\eta \right)={e}^{j2\pi \nu t}{\int }_{-\infty }^{\infty }{e}^{-\pi \left(x-t{\right)}^{2}}{e}^{-j2\pi \left(x-t\right)\left(\eta -\nu \right)}\phantom{\rule{0.16666666666666666em}{0ex}}dx={e}^{-\pi \left(\eta -\nu {\right)}^{2}}{e}^{j2\pi \nu t}.
When viewed as a function of frequency, the transform combines a constant (in time) oscillation at
\nu
with Gaussian decay away from it. The synchrosqueezing estimate of the instantaneous frequency,
{\Omega }_{g}f\left(t,\eta \right)=\frac{1}{j2\pi }\frac{{e}^{-\pi \left(\eta -\nu {\right)}^{2}}\frac{\partial }{\partial t}{e}^{j2\pi \nu t}}{{e}^{-\pi \left(\eta -\nu {\right)}^{2}}{e}^{j2\pi \nu t}}=\nu ,
equals the value obtained by using the standard definition,
\left(2\pi {\right)}^{-1}d\mathrm{arg}f\left(x\right)/dx
. For a superposition of sinusoids,
f\left(x\right)=\sum _{k=1}^{K}{A}_{k}{e}^{j2\pi {\nu }_{k}x},
the short-time transform becomes
{V}_{g}f\left(t,\eta \right)=\sum _{k=1}^{K}{A}_{k}{e}^{-\pi \left(\eta -{\nu }_{k}{\right)}^{2}}{e}^{j2\pi {\nu }_{k}t}.
Create 1024 samples of a signal consisting of two sinusoids. One sinusoid has a normalized frequency of
{\omega }_{0}=\pi /5
rad/sample. The other sinusoid has three times the frequency and three times the amplitude.
x = exp(1j*w0*n)+3*exp(1j*3*w0*n);
Compute the short-time Fourier transform of the signal. Use a 256-sample Gaussian window with
\alpha =20
, 255 samples of overlap between adjoining sections, and 1024 DFT points. Plot the absolute value of the transform.
[s,w,t] = spectrogram(x,gausswin(Nw,alpha),Nw-1,nfft,'centered');
surf(t,w/pi,abs(s),'EdgeColor','none')
ylabel('Normalized Frequency (\times\pi rad/sample)')
The Fourier synchrosqueezed transform results in a sharper, better localized estimate of the spectrum.
[ss,sw,st] = fsst(x,[],gausswin(Nw,alpha));
fsst(x,'yaxis')
The sinusoids are visible as constant oscillations at the expected frequency values. To see that the decay away from the ridges is Gaussian, plot an instantaneous value of the transform and overlay two instances of a Gaussian. Express the Gaussian amplitude and standard deviation in terms of
\alpha
and the window length. Recall that the standard deviation of the frequency-domain window is the reciprocal of the standard deviation of the time-domain window.
rstdev = (Nw-1)/(2*alpha);
amp = rstdev*sqrt(2*pi);
instransf = abs(s(:,128));
plot(w/pi,instransf)
plot(w/pi,[1 3]*amp.*exp(-rstdev^2/2*(w-[1 3]*w0).^2),'--')
lg = legend('Transform','First sinusoid','Second sinusoid');
lg.Location = 'best';
The Fourier synchrosqueezed transform concentrates the energy content of the signal at the estimated instantaneous frequencies.
stem(sw/pi,abs(ss(:,128)))
The synchrosqueezed estimates of instantaneous frequency are valid only if the sinusoids are separated by more than
2\Delta
\Delta =\frac{1}{\sigma }\sqrt{2\mathrm{log}2}
for a Gaussian window and
\sigma
Repeat the previous calculation, but now specify that the second sinusoid has a normalized frequency of
{\omega }_{0}+1.9\Delta
D = sqrt(2*log(2))/rstdev;
w1 = w0+1.9*D;
x = exp(1j*w0*n)+3*exp(1j*w1*n);
instransf = abs(s(:,20));
plot(w/pi,[1 3]*amp.*exp(-rstdev^2/2*(w-[w0 w1]).^2),'--')
The Fourier synchrosqueezed transform cannot resolve the sinusoids well because
|{\omega }_{1}-{\omega }_{0}|<2\Delta
. The spectral estimates decrease significantly in value.
fsst | gausswin | ifsst | spectrogram
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Solve stiff differential equations — trapezoidal rule + backward differentiation formula - MATLAB ode23tb - MathWorks Switzerland
Solve stiff differential equations — trapezoidal rule + backward differentiation formula
[t,y] = ode23tb(odefun,tspan,y0)
[t,y] = ode23tb(odefun,tspan,y0,options)
[t,y,te,ye,ie] = ode23tb(odefun,tspan,y0,options)
sol = ode23tb(___)
[t,y] = ode23tb(odefun,tspan,y0), where tspan = [t0 tf], integrates the system of differential equations
y\text{'}=f\left(t,y\right)
y\text{'}=f\left(t,y\right)
M\left(t,y\right)y\text{'}=f\left(t,y\right)
[t,y] = ode23tb(odefun,tspan,y0,options) also uses the integration settings defined by options, which is an argument created using the odeset function. For example, use the AbsTol and RelTol options to specify absolute and relative error tolerances, or the Mass option to provide a mass matrix.
[t,y,te,ye,ie] = ode23tb(odefun,tspan,y0,options) additionally finds where functions of (t,y), called event functions, are zero. In the output, te is the time of the event, ye is the solution at the time of the event, and ie is the index of the triggered event.
sol = ode23tb(___) returns a structure that you can use with deval to evaluate the solution at any point on the interval [t0 tf]. You can use any of the input argument combinations in previous syntaxes.
{y}^{\prime }=-10t.
[t,y] = ode23tb(@(t,y) -10*t, tspan, y0);
Solve the stiff system using the ode23tb solver, and then plot the first column of the solution y against the time points t. The ode23tb solver passes through stiff areas with far fewer steps than ode45.
[t,y] = ode23tb(@vdp1000,[0 3000],[2 0]);
Solve the ODE using ode23tb. Specify the function handle such that it passes in the predefined values for A and B to odefcn.
[t,y] = ode23tb(@(t,y) odefcn(t,y,A,B), tspan, y0);
{y}^{\prime }=-\lambda y.
\lambda
\lambda =1×1{0}^{9}
y\left(0\right)=1
J=\frac{\partial f}{\partial y}=-\lambda
f\left(t,y\right)
y\text{'}=5y-3
\begin{array}{l}y{\text{'}}_{1}={y}_{1}+2{y}_{2}\\ y{\text{'}}_{2}=3{y}_{1}+2{y}_{2}\end{array}
ode23tb is an implementation of TR-BDF2, an implicit Runge-Kutta formula with a trapezoidal rule step as its first stage and a backward differentiation formula of order two as its second stage. By construction, the same iteration matrix is used in evaluating both stages. Like ode23s and ode23t, this solver may be more efficient than ode15s for problems with crude tolerances [1], [2].
[1] Bank, R. E., W. C. Coughran, Jr., W. Fichtner, E. Grosse, D. Rose, and R. Smith, “Transient Simulation of Silicon Devices and Circuits,” IEEE Trans. CAD, 4 (1985), pp. 436–451.
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Perturb the solver's solution of a system's states to better satisfy time-invariant solution relationships - MATLAB Projection - MathWorks Deutschland
Perturb System States Using a Solution Invariant
Perturb the solver's solution of a system's states to better satisfy time-invariant solution relationships
This method is intended for use with S-functions that model dynamic systems whose states satisfy time-invariant relationships, such as those resulting from mass or energy conservation or other physical laws. The Simulink® engine invokes this method at each time step after the model's solver has computed the S-function's states for that time step. Typically, slight errors in the numerical solution of the states cause the solutions to fail to satisfy solution invariants exactly. Your Projection method can compensate for the errors by perturbing the states so that they more closely approximate solution invariants at the current time step. As a result, the numerical solution adheres more closely to the ideal solution as the simulation progresses, producing a more accurate overall simulation of the system modeled by your S-function.
Your Projection method's perturbations of system states must fall within the solution error tolerances specified by the model in which the S-function is embedded. Otherwise, the perturbations may invalidate the solver's solution. It is up to your Projection method to ensure that the perturbations meet the error tolerances specified by the model. See Perturb System States Using a Solution Invariant for a simple method for perturbing a system's states. The following articles describe more sophisticated perturbation methods that your mdlProjection method can use.
C.W. Gear, “Maintaining Solution Invariants in the Numerical Solution of ODEs,” Journal on Scientific and Statistical Computing, Vol. 7, No. 3, July 1986.
L.F. Shampine, “Conservation Laws and the Numerical Solution of ODEs I,” Computers and Mathematics with Applications, Vol. 12B, 1986, pp. 1287–1296.
L.F. Shampine, “Conservation Laws and the Numerical Solution of ODEs II,” Computers and Mathematics with Applications, Vol. 38, 1999, pp. 61–72.
Here is a simple, Taylor-series-based approach to perturbing a system's states. Suppose your S-function models a dynamic system having a solution invariant,
g\left(X,t\right)
g
is a continuous, differentiable function of the system states,
X
t
, whose value is constant with time. Then
{X}_{n}\cong {X}_{n}^{*}+{J}_{n}^{T}{\left({J}_{n}{J}_{n}^{T}\right)}^{-1}{R}_{n}
{X}_{n}
is the system's ideal state vector at the solver's current time step
{X}_{n}^{*}
is the approximate state vector computed by the solver at the current time step
{J}_{n}
is the Jacobian of the invariant function evaluated at the point in state space specified by the approximate state vector at the current time step:
{J}_{n}=\frac{\partial g}{\partial X}\left({X}_{n}^{*},{t}_{n}\right)
{t}_{n}
is the time at the current time step
{R}_{n}
is the residual (difference) between the invariant function evaluated at
{X}_{n}
{X}_{n}^{*}
at the current time step:
{R}_{n}=g\left({X}_{n},{t}_{n}\right)-g\left({X}_{n}^{*},{t}_{n}\right)
g\left({X}_{n},{t}_{n}\right)
is the same at each time step and is known by definition.
Given a continuous, differentiable invariant function for the system that your S-function models, this formula allows your S-function's mdlProjection method to compute a perturbation
{J}_{n}^{T}{\left({J}_{n}{J}_{n}^{T}\right)}^{-1}{R}_{n}
of the solver's numerical solution,
{X}_{n}^{*}
, that more closely matches the ideal solution,
{X}_{n}
, keeping the S-function's solution from drifting from the ideal solution as the simulation progresses.
This example illustrates how the perturbation method outlined in the previous section can keep a model's numerical solution from drifting from the ideal solution as a simulation progresses. Consider the following model,mdlProjectionEx1:
The PredPrey block references an S-function, predprey_noproj.m, that uses the Lotka-Volterra equations
\begin{array}{l}\stackrel{˙}{x}=ax\left(1-y\right)\\ \stackrel{˙}{y}=-cy\left(1-x\right)\end{array}
to model predator-prey population dynamics, where
x\left(t\right)
is the population density of the predators and
y\left(t\right)
is the population density of prey. The ideal solution to the predator-prey ODEs satisfies the time-invariant function
{x}^{-c}{e}^{cx}{y}^{-a}{e}^{ay}=d
a
c
, and
are constants. The S-function assumes a = 1, c = 2, and d = 121.85.
The Invariant Residual block in this model computes the residual between the invariant function evaluated along the system's ideal trajectory through state space and its simulated trajectory:
{R}_{n}=d-{x}_{n}^{-c}{e}^{c{x}_{n}}{y}_{n}^{-a}{e}^{a{y}_{n}}
{x}_{n}
{y}_{n}
are the values computed by the model's solver for the predator and prey population densities, respectively, at the current time step. Ideally, the residual should be zero throughout simulation of the model, but simulating the model reveals that the residual actually strays considerably from zero:
Now consider the following model, mdlProjectionEx2:
This model is the same as the previous model, except that its S-function, predprey.m, includes a mdlProjection method that uses the perturbation approach outlined in Perturb System States Using a Solution Invariant to compensate for numerical drift. As a result, the numerical solution more closely tracks the ideal solution as the simulation progresses as demonstrated by the residual signal, which remains near or at zero throughout the simulation:
Simulink.MSFcnRunTimeBlock, mdlProjection,
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Comparative Technology | Economic Evaluation Of Four Biomass To Electricity Systems – Engineeringness
Comparative Technology | Economic Evaluation Of Four Biomass To Electricity Systems
Harris Khan October 1, 2020, 6:16 pm
The following article analyses the different techniques involved in producing electricity from biomass systems. This article will outline 4 different biomass systems using 4 different scenarios for each system to calculate and determine the overall efficiency, specific capital cost, and the cost of electricity. Each scenario will be assessed for the individual systems over a specified range of rated power which will be 2.5MWe to 25MWe. PyrEng, GasEng, IGCC and Combust are the four different technologies outlined within this report.
Biomass systems are an alternative to conventional fossil fuels as energy sources due to the increasing scarcity of fossil fuels as well as being less environmentally damaging. With the introduction of pellets in the early 2000’s which have high energy densities and can be easily fed within heating systems, increasing government regulation and legislation requiring companies to reduce and even meet targets on greenhouse gas emissions coupled with increasing knowledge and information affecting public opinions on companies all leading to better Press for companies who are opting to ‘go green’ this which has led to a large increase in the popularity, use and development of biomass systems by companies as a viable long term option over fossil fuels as energy sources (ATech Electronics, 2019). For example in 2019 there are around 3800 active biomass power plants worldwide with an expected 5600 commissioned by the end of 2027 (ecoprog, 2019).
PyrEng: is the fast pyrolysis (fluidised bed) with compression-ignition engine, including intermediate liquids storage.
GasEng: atmospheric gasification (fluidised bed) with spark ignition engine, including Tar cracker and gas clean up.
Integrated Gasification Combined Cycle or IGCC: Pressurized gasification (fluidised bed) with gas turbine combined cycle, including hot as clean up.
Combust: Combustion (moving grate) with boiler and Rankine cycle.
The overall efficiency is the annual net electricity delivered from the plant to the grid
Measured as a percentage and can be calculated using Equation 1.11 below:
\frac{Net power output from plant \left(GJ {y}^{–1}\right)}{Total fuel input to the plant \left(GJ {y}^{–1}\right)} = Overall efficieny \left(%\right)
Equation 1.11 Equation to determine the overall efficiency of the plant used for each biomass system.
The specified range of rated power is split into 10 ranges. Rated power can be defined as the net power exported by the power plant if it is running at full load.
For each of the rated power ranges an initial biomass wet feed (mB) is assumed which then allows the chemical energy in (Ec) to be calculated using the Heating value (H), which is of the biomass as shown in equation 1.12:
{E}_{c} = {m}_{B}. H
Ec = Chemical Energy in (GJ y-1),
MB = Initial Biomass wet feed (tonnes/ year),
H = Heating Value of biomass (GJ/ Tonne)
Equation 1.12 Equation to calculate the chemical energy in of the plant for each biomass system.
Once the chemical energy in has been determined the electrical energy out (EE) can be calculated using the chemical energy in (Ec) and the overall electrical efficiency (e) as shown in equation 1.13 below:
{E}_{E}={E}_{c}.{\eta }_{e}
EE = Electrical Energy out (GJ y-1),
e = Heating Value of biomass (GJ/ Tonne)
Equation 1.13 Equation to calculate the electrical energy out of the plant for each biomass system.
Conversion Efficiency is defined as efficiency in converting the raw biomass material energy to energy within the intermediary product for example with the PyrEng the energy this means the pyrolyser to the bio oil intermediary bio oil liquid storage. This Is calculated by using a function of the given values.
Generation efficiency is defined as the efficiency with which the conversion of energy within the intermediary product is converted into electrical energy which can be used and transported to the grid. This is also calculated by using a function of the given values.
Rated power (PE) for each scenario and system can then be worked out using the electrical energy out (EE) as seen in equation 1.14 below:
{P}_{E}=C.{E}_{E}/f
PE = Rated Power (MWe),
C = Units constant,
f = Capacity Factor
Equation 1.14 Equation to calculate the Rated power for each biomass system.
Capacity factor is calculated using equation 1.15:
\frac{Actual amount of energy out per year}{Amount of energy our per year if power station ran continously at rated power} = Capacity Factor \left(f\right)
Equation 1.15 Equation to calculate capacity factor used in the calculation for equation 1.14.
Once the above has been worked out for each biomass systems and each of its 4 scenarios the economic measures can be calculated.
The specific capital costs (SCC) consist of four different elements known as preparation, drying, conversion and generation.
SCC can be worked out using equation 1.16 shown below:
\frac{Total Capital Costs \left(£\right) }{Rated Power \left(kWe\right)} = Specific Capital C\mathrm{os}ts \left(£/kWe\right)
Equation 1.16 equation to calculate the specific capital costs at each rated power range for each biomass system.
Cost of Electricity (CoE) is then worked out following the determination of the specific capital costs (SCC) and can be calculated using the equations 1.17:
\frac{\sum _{all system operating \mathrm{cos}ts per annum \left(£\right)}}{amount of electricity exported per annum \left(£\right)} = \mathrm{cos}t of electricity \left(£/kWh\right)
Equation 1.17 equation to determine the cost of electricity at each rated power range for each biomass system.
The total of all system annual operating costs consist of seven elements known as feed production, feed transport, labour, utilities, overheads, maintenance, and annual cost of capital.
Feed preparation includes the reception, handling, screening, grinding and storage.
Overheads are a percentage of the total capital cost.
Feed production is calculated using equation 1.18 below:
Price per tonne of Dry feed (£k) x annual supply (kt/year) = Feed production cost (£/kt/year)
Equation 1.18 equation to calculate the feed production costs per year.
When calculating the feed production cost, it is done on a dry basis. The feed rate is given as a wet basis. Wet basis is defined as the mass of moisture divided by the total mass on the product. To gain the dry basis mass the wet basis mass is divided by 2 as an assumption is made that the product is 50% wet basis.
Annual capital costs are the annual cost equivalent spread over the lifetime of the process. As stated above annual cost of capital (ACC) is used in the calculation to determine the sum of annual operating costs and can be evaluated using equation 1.19:
\frac{C.r.{\left(1+r\right)}^{n}}{{\left(1+r\right)}^{n}–1} = Annual capital \mathrm{cos}ts \left(£k/y\right)
C = Total capital cost (£k),
r = interest rate (%),
Equation 1.19 equation to calculate the annual capital costs included in the operating costs per annum outlined in equation 1.17.
Learning factors are the reflection in the learning process of new plants and can be defined as the amount capital cost reduces every time the number of plants in operation doubles.
The learning factors can be factored into calculation using equation 1.20 below:
{C}_{n}={C}_{m} . {\left(1–l\right)}^{\frac{\mathrm{ln}\left(n\right) – \mathrm{ln}\left(m\right)}{\mathrm{ln}\left(2\right)}}
Cn = Capital cost for n amount of plants,
Cm = Capital cost for m amount of plants,
l = learning factor
Equation 1.20 equation to calculate the new capital cost including the learning factor.
Wherever a biomass system is described as ‘mature’ the learning factor, l, is set to 0.
Graph 1.11 Graph to show the overall efficiency compared to the rated power.
Graph 1.11 above compares the overall efficiency (%) between all four systems over the rated power range (MWe).
IGCC is the most efficient of all the biomass systems over all the rated power ranges. For example, at a rated Power range of 5 MWe the IGCC had an overall efficiency of 0.36% compared to the second-highest of 0.28% for GasEng followed by PyrEng at 0.25% and lastly the Combust at 0.18%. This trend does not change over the rated power range.
The IGCC system works by utilizing gasification followed by the use of gas clean-up and lastly a gas and steam turbine to create a combined system to allow for a more environmentally friendly, efficient and clean way of creating electricity especially compared to classic combustion systems as seen in graph 1.11 above (Wang, 2016).
Gasification is the process in which carbon-based material such as biomass is turned into fuel/ energy without the use of combustion. This is done by incomplete combustion of the biomass feed in an oxygen-deficient area. This leads to less heat being released as compared to a combustion method as gasification packs energy into the bonds whereas combustion releases the energy from the bonds by oxidizing the feed giving off heat (Basu, 2010).
In the gasification process, syngas and ash amongst other intermediary products are converted into a gas.
Once the Gasification process is done gas is cleaned up by removing contaminants including CO2, if carbon capture is used before the fuel is combusted, in the gas turbine phase meaning there is greater heating value in the combustion of the fuel as compared to combustion biomass systems. Moreover, it also leads to less harmful and polluting gases such as mercury and sulphur being combusted with sulphur being one of the lead causes of acid rain. Due to IGCC plants being run at high pressure the efficiency of the removal of the contaminants increases with the capital costs being reduced due to the lower volumetric flow rate of the pre combusted fuel as compared to cleaning the gas at ambient temperature post combustion at a higher volumetric flow rate. This also contains more gases as compared to just fuel which is pre combusted (Wang, 2016).
Figure 1.11 Diagram demonstrating the IGCC biomass system (Mitsubishi Hitachi Power Systems, 2019).
Combust is the oldest of the four systems and the least efficient. Combustion systems such as the Rankine System work by utilizing an external heat source such as the combustion of biomass material to provide heat to a closed system containing an operating fluid.
A pump is used to pump high pressurized operating liquid into a boiler where it is heated using the external heat source, in this case combusted biomass material, where the operating liquid undergoes Isobaric heat transfer reaching its saturation temperature and further heated until it evaporates and is fully converted into saturated steam.
This saturated steam then travels to the turbine upstream of the boiler within the closed system and here it undergoes isentropic expansion. The saturated steam here expands, working on the surroundings producing electricity by spinning the turbine. This expansion, however, is limited by factors such as the temperature of the cooling medium, erosion of the turbine blades and the liquid entrainment when the medium reaches a two-phase region which all affect its efficiency.
This vapour liquid mixture leaves the turbine undergoing isobaric heat rejection in which the vapour liquid mixture enters a surface condenser. Here it is cooled and condensed using a cooling medium such as cooling water.
This condensed liquid is then recycled and sent back to the pump at the beginning where it begins the cycle once again (Muller-Steinhagen, 2011). NOx exhaust must be controlled more in combust than other systems.
There are though, inefficiencies in the combust process such as the inefficiencies of the boiler. The boiler does not convert 100% of the fuel energy into steam energy due to the high temperature difference between the combusting fuel and the vapour temperature. This high-temperature difference leads to greater entropy leading to greater energy dispersion which, compared to the IGCC system which does not fully combust its fuel, has less energy lost between the gas turbine and the incompletely combusted fuel (Muller-Steinhagen, 2011).
Moreover, there are losses of energy through the turbine due to a number of reasons such as erosion of the turbine blades and as the steam cools vapour entrainment occurs (Muller-Steinhagen, 2011). Vapour entrainment is when liquid droplets are trapped within vapour. This can cause mechanical damage to turbine blades as well as reducing the efficiency of separation in the evaporation stage (R K BAGUL, 2013). Furthermore, the build-up of dirt/fouling reduces the efficiency of the condensing within the condenser (Muller-Steinhagen, 2011).
Figure 1.12 Diagram to show the Rankine Cycle and example of combust (Muller-Steinhagen, 2011).
Fast Pyrolysis is used in the PyrEng biomass system. Fast pyrolysis involves the thermal decomposition of carbon-based material in this case biomass in the absence of oxygen at low pressure and around 500oC. This results in the biomass being vaporized leaving a residue of char and ash. This fast pyrolysis in the PyrEng system produces bio oil as an intermediary liquid which can be compressed and ignited as it is an energy-dense liquid feedstock that can be utilized within a compression engine and used as a fuel to produce electricity (A.V. Bridgwater, 2002).
The bio-oil is created by first drying the feedstock making sure most of the moisture is released. It is then heated to further release moisture and some gas and char. This is done around 100 – 300oC. The biomass is then cooled to form ash, char, permanent gases and the desired bio-oil intermediary liquid. The remaining vapours are further vaporized at a temperature above 600oC to become secondary char and more permanent gases.
However, bio-oil if stored for too long can start to separate out into different phases. Moreover, it can become increasingly thick leading to complications in use as a fuel for compression engines. To combat the separation bio-oil must be kept in storage conditions with less than 30% water content to stop the bio-oil mixture from separating from an aqueous phase and a gummy-like phase as seen in figure 1.13. Furthermore, bio-oil can age leading to a change in properties reducing usability. To protect against ageing, it is important to keep low temperatures and reduce sheer stress and both can lead to accelerated ageing. This leads to storage, handling and transportation differences compared to other biomass systems.
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In recent years, there has been an increasing interest in open-ended language generation thanks to the rise of large transformer-based language models trained on millions of webpages, such as OpenAI's famous GPT2 model. The results on conditioned open-ended language generation are impressive, e.g. GPT2 on unicorns, XLNet, Controlled language with CTRL. Besides the improved transformer architecture and massive unsupervised training data, better decoding methods have also played an important role.
This blog post gives a brief overview of different decoding strategies and more importantly shows how you can implement them with very little effort using the popular transformers library!
All of the following functionalities can be used for auto-regressive language generation (here a refresher). In short, auto-regressive language generation is based on the assumption that the probability distribution of a word sequence can be decomposed into the product of conditional next word distributions:
P(w_{1:T} | W_0 ) = \prod_{t=1}^T P(w_{t} | w_{1: t-1}, W_0) \text{ ,with } w_{1: 0} = \emptyset,
W_0
being the initial context word sequence. The length
T
of the word sequence is usually determined on-the-fly and corresponds to the timestep
t=T
the EOS token is generated from
P(w_{t} | w_{1: t-1}, W_{0})
Auto-regressive language generation is now available for GPT2, XLNet, OpenAi-GPT, CTRL, TransfoXL, XLM, Bart, T5 in both PyTorch and Tensorflow >= 2.0!
We will give a tour of the currently most prominent decoding methods, mainly Greedy search, Beam search, Top-K sampling and Top-p sampling.
Let's quickly install transformers and load the model. We will use GPT2 in Tensorflow 2.1 for demonstration, but the API is 1-to-1 the same for PyTorch.
Greedy search simply selects the word with the highest probability as its next word:
w_t = argmax_{w}P(w | w_{1:t-1})
t
. The following sketch shows greedy search.
Starting from the word
\text{"The"},
the algorithm greedily chooses the next word of highest probability
\text{"nice"}
and so on, so that the final generated word sequence is
(\text{"The"}, \text{"nice"}, \text{"woman"})
having an overall probability of
0.5 \times 0.4 = 0.2
In the following we will generate word sequences using GPT2 on the context
(\text{"I"}, \text{"enjoy"}, \text{"walking"}, \text{"with"}, \text{"my"}, \text{"cute"}, \text{"dog"})
. Let's see how greedy search can be used in transformers:
# encode context the generation is conditioned on
# generate text until the output length (which includes the context length) reaches 50
Alright! We have generated our first short text with GPT2 😊. The generated words following the context are reasonable, but the model quickly starts repeating itself! This is a very common problem in language generation in general and seems to be even more so in greedy and beam search - check out Vijayakumar et al., 2016 and Shao et al., 2017.
The major drawback of greedy search though is that it misses high probability words hidden behind a low probability word as can be seen in our sketch above:
\text{"has"}
with its high conditional probability of
0.9
is hidden behind the word
\text{"dog"}
, which has only the second-highest conditional probability, so that greedy search misses the word sequence
\text{"The"}, \text{"dog"}, \text{"has"}
Thankfully, we have beam search to alleviate this problem!
Beam search reduces the risk of missing hidden high probability word sequences by keeping the most likely num_beams of hypotheses at each time step and eventually choosing the hypothesis that has the overall highest probability. Let's illustrate with num_beams=2:
At time step 1, besides the most likely hypothesis
(\text{"The"}, \text{"nice"})
, beam search also keeps track of the second most likely one
(\text{"The"}, \text{"dog"})
. At time step 2, beam search finds that the word sequence
(\text{"The"}, \text{"dog"}, \text{"has"})
, has with
0.36
a higher probability than
(\text{"The"}, \text{"nice"}, \text{"woman"})
, which has
0.2
. Great, it has found the most likely word sequence in our toy example!
Beam search will always find an output sequence with higher probability than greedy search, but is not guaranteed to find the most likely output.
Let's see how beam search can be used in transformers. We set num_beams > 1 and early_stopping=True so that generation is finished when all beam hypotheses reached the EOS token.
# activate beam search and early_stopping
While the result is arguably more fluent, the output still includes repetitions of the same word sequences.
A simple remedy is to introduce n-grams (a.k.a word sequences of n words) penalties as introduced by Paulus et al. (2017) and Klein et al. (2017). The most common n-grams penalty makes sure that no n-gram appears twice by manually setting the probability of next words that could create an already seen n-gram to 0.
Let's try it out by setting no_repeat_ngram_size=2 so that no 2-gram appears twice:
# set no_repeat_ngram_size to 2
Nice, that looks much better! We can see that the repetition does not appear anymore. Nevertheless, n-gram penalties have to be used with care. An article generated about the city New York should not use a 2-gram penalty or otherwise, the name of the city would only appear once in the whole text!
In transformers, we simply set the parameter num_return_sequences to the number of highest scoring beams that should be returned. Make sure though that num_return_sequences <= num_beams!
As can be seen, the five beam hypotheses are only marginally different to each other - which should not be too surprising when using only 5 beams.
In open-ended generation, a couple of reasons have recently been brought forward why beam search might not be the best possible option:
Beam search can work very well in tasks where the length of the desired generation is more or less predictable as in machine translation or summarization - see Murray et al. (2018) and Yang et al. (2018). But this is not the case for open-ended generation where the desired output length can vary greatly, e.g. dialog and story generation.
We have seen that beam search heavily suffers from repetitive generation. This is especially hard to control with n-gram- or other penalties in story generation since finding a good trade-off between forced "no-repetition" and repeating cycles of identical n-grams requires a lot of finetuning.
In its most basic form, sampling means randomly picking the next word
w_t
according to its conditional probability distribution:
w_t \sim P(w|w_{1:t-1})
Taking the example from above, the following graphic visualizes language generation when sampling.
It becomes obvious that language generation using sampling is not deterministic anymore. The word
(\text{"car"})
is sampled from the conditioned probability distribution
P(w | \text{"The"})
, followed by sampling
(\text{"drives"})
P(w | \text{"The"}, \text{"car"})
In transformers, we set do_sample=True and deactivate Top-K sampling (more on this later) via top_k=0. In the following, we will fix random_seed=0 for illustration purposes. Feel free to change the random_seed to play around with the model.
# activate sampling and deactivate top_k by setting top_k sampling to 0
Interesting! The text seems alright - but when taking a closer look, it is not very coherent. the 3-grams new hand sense and local batte harness are very weird and don't sound like they were written by a human. That is the big problem when sampling word sequences: The models often generate incoherent gibberish, cf. Ari Holtzman et al. (2019).
A trick is to make the distribution
P(w|w_{1:t-1})
sharper (increasing the likelihood of high probability words and decreasing the likelihood of low probability words) by lowering the so-called temperature of the softmax.
An illustration of applying temperature to our example from above could look as follows.
The conditional next word distribution of step
t=1
becomes much sharper leaving almost no chance for word
(\text{"car"})
to be selected.
Let's see how we can cool down the distribution in the library by setting temperature=0.7:
# use temperature to decrease the sensitivity to low probability candidates
OK. There are less weird n-grams and the output is a bit more coherent now! While applying temperature can make a distribution less random, in its limit, when setting temperature
\to 0
, temperature scaled sampling becomes equal to greedy decoding and will suffer from the same problems as before.
Fan et. al (2018) introduced a simple, but very powerful sampling scheme, called Top-K sampling. In Top-K sampling, the K most likely next words are filtered and the probability mass is redistributed among only those K next words. GPT2 adopted this sampling scheme, which was one of the reasons for its success in story generation.
We extend the range of words used for both sampling steps in the example above from 3 words to 10 words to better illustrate Top-K sampling.
Having set
K = 6
, in both sampling steps we limit our sampling pool to 6 words. While the 6 most likely words, defined as
V_{\text{top-K}}
encompass only ca. two-thirds of the whole probability mass in the first step, it includes almost all of the probability mass in the second step. Nevertheless, we see that it successfully eliminates the rather weird candidates
(\text{``not"}, \text{``the"}, \text{``small"}, \text{``told"})
in the second sampling step.
Let's see how Top-K can be used in the library by setting top_k=50:
# set top_k to 50
Not bad at all! The text is arguably the most human-sounding text so far. One concern though with Top-K sampling is that it does not dynamically adapt the number of words that are filtered from the next word probability distribution
P(w|w_{1:t-1})
. This can be problematic as some words might be sampled from a very sharp distribution (distribution on the right in the graph above), whereas others from a much more flat distribution (distribution on the left in the graph above).
t=1
, Top-K eliminates the possibility to sample
(\text{"people"}, \text{"big"}, \text{"house"}, \text{"cat"})
, which seem like reasonable candidates. On the other hand, in step
t=2
the method includes the arguably ill-fitted words
(\text{"down"}, \text{"a"})
in the sample pool of words. Thus, limiting the sample pool to a fixed size K could endanger the model to produce gibberish for sharp distributions and limit the model's creativity for flat distribution. This intuition led Ari Holtzman et al. (2019) to create Top-p- or nucleus-sampling.
Top-p (nucleus) sampling
Instead of sampling only from the most likely K words, in Top-p sampling chooses from the smallest possible set of words whose cumulative probability exceeds the probability p. The probability mass is then redistributed among this set of words. This way, the size of the set of words (a.k.a the number of words in the set) can dynamically increase and decrease according to the next word's probability distribution. Ok, that was very wordy, let's visualize.
p=0.92
, Top-p sampling picks the minimum number of words to exceed together
p=92\%
of the probability mass, defined as
V_{\text{top-p}}
. In the first example, this included the 9 most likely words, whereas it only has to pick the top 3 words in the second example to exceed 92%. Quite simple actually! It can be seen that it keeps a wide range of words where the next word is arguably less predictable, e.g.
P(w | \text{"The''})
, and only a few words when the next word seems more predictable, e.g.
P(w | \text{"The"}, \text{"car"})
Alright, time to check it out in transformers! We activate Top-p sampling by setting 0 < top_p < 1:
# deactivate top_k sampling and sample only from 92% most likely words
Great, that sounds like it could have been written by a human. Well, maybe not quite yet.
While in theory, Top-p seems more elegant than Top-K, both methods work well in practice. Top-p can also be used in combination with Top-K, which can avoid very low ranked words while allowing for some dynamic selection.
Finally, to get multiple independently sampled outputs, we can again set the parameter num_return_sequences > 1:
Cool, now you should have all the tools to let your model write your stories with transformers!
As ad-hoc decoding methods, top-p and top-K sampling seem to produce more fluent text than traditional greedy - and beam search on open-ended language generation. Recently, there has been more evidence though that the apparent flaws of greedy and beam search - mainly generating repetitive word sequences - are caused by the model (especially the way the model is trained), rather than the decoding method, cf. Welleck et al. (2019). Also, as demonstrated in Welleck et al. (2020), it looks as top-K and top-p sampling also suffer from generating repetitive word sequences.
In Welleck et al. (2019), the authors show that according to human evaluations, beam search can generate more fluent text than Top-p sampling, when adapting the model's training objective.
Open-ended language generation is a rapidly evolving field of research and as it is often the case there is no one-size-fits-all method here, so one has to see what works best in one's specific use case.
Good thing, that you can try out all the different decoding methods in transfomers 🤗.
That was a short introduction on how to use different decoding methods in transformers and recent trends in open-ended language generation.
Feedback and questions are very welcome on the Github repository.
For more fun generating stories, please take a look at Writing with Transformers
Thanks to everybody, who has contributed to the blog post: Alexander Rush, Julien Chaumand, Thomas Wolf, Victor Sanh, Sam Shleifer, Clément Delangue, Yacine Jernite, Oliver Åstrand and John de Wasseige.
There are a couple of additional parameters for the generate method that were not mentioned above. We will explain them here briefly!
min_length can be used to force the model to not produce an EOS token (= not finish the sentence) before min_length is reached. This is used quite frequently in summarization, but can be useful in general if the user wants to have longer outputs.
repetition_penalty can be used to penalize words that were already generated or belong to the context. It was first introduced by Keskar et al. (2019) and is also used in the training objective in Welleck et al. (2019). It can be quite effective at preventing repetitions, but seems to be very sensitive to different models and use cases, e.g. see this discussion on Github.
attention_mask can be used to mask padded tokens
pad_token_id, bos_token_id, eos_token_id: If the model does not have those tokens by default, the user can manually choose other token ids to represent them.
For more information please also look into the generate function docstring.
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Modeling and Parametric Study of Torque in Open Clutch Plates | J. Tribol. | ASME Digital Collection
Chinar R. Aphale,
Chinar R. Aphale
e-mail: caphale@umich.edu
Jinhyun Cho,
e-mail: jinhyunc@umich.edu
Steven L. Ceccio,
e-mail: ceccio@umich.edu
Takao Yoshioka,
e-mail: yoshioka-t@mail.dxj.co.jp
Henry Hiraki
e-mail: hiraki-h@mail.dxj.co.jp
Aphale, C. R., Cho, J., Schultz, W. W., Ceccio, S. L., Yoshioka, T., and Hiraki, H. (September 19, 2005). "Modeling and Parametric Study of Torque in Open Clutch Plates." ASME. J. Tribol. April 2006; 128(2): 422–430. https://doi.org/10.1115/1.2162553
The relative motion of the friction and separator plates in wet clutches during the disengaged mode causes viscous shear stresses in the fluid passing through the
100microns
gap. This results in a drag torque on both the disks that wastes energy and decreases fuel economy. The objective of the study is to develop an accurate mathematical model for the above problem with verification using FLUENT and experiments. Initially we two consider flat disks. The mathematical model calculates the drag torque on the disks and the 2D axisymmetric solver verifies the solution. The surface pressure distribution on the plates is also verified. Then, 3D models of one grooved and one flat disk are tested using CFD, experiments and an approximate 3D mathematical model. The number of grooves, depth of groove and clearance between the disks are studied to understand their effect on the torque. The study determines the pressure field that eventually affects aeration incipience (not studied here). The results of the model, computations and experiments corroborate well in the single-phase regime.
clutches, plates (structures), drag, computational fluid dynamics
Clearances (Engineering), Drag (Fluid dynamics), Flow (Dynamics), Plates (structures), Pressure, Simulation, Torque, Modeling, Shear stress, Disks, Fluids, Boundary-value problems
Laminar and Turbulent Reibung
Note on a Class of Solutions of Navier-Stokes Equations Representing Steady Rotationally Symmetric Flow
On the Flow Between Two Rotating Coaxial Discs
Asymptotic Solution for a Film Flow on a Rotating Disk
Effects of Transmission Fluid on Clutch Performance
,” SAE Technical Paper Series No. 910803.
Radial Inflow Between a Rotating and a Stationary Disc
C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers
, Serie II, pp.
Is Laminar Flow in a Cylindrical Container Within a Rotating Cover Batchelor or Stewartson-Type Solution?
Film Flow on a Rotating Disk
,” JSAE Technical Paper No. 20030331.
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Atomic formula — Wikipedia Republished // WIKI 2
This article is about a concept in mathematical logic. For the concept from chemistry, see chemical formula.
In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula, For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formulas are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.[1]
WELL FORMED FORMULAS|WFF|MFCS|DISCRETE MATHEMATICS
How to Read Chemical Formulas
1 Atomic formula in first-order logic
The well-formed terms and propositions of ordinary first-order logic have the following syntax:
{\displaystyle t\equiv c\mid x\mid f(t_{1},\dotsc ,t_{n})}
that is, a term is recursively defined to be a constant c (a named object from the domain of discourse), or a variable x (ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms tk. Functions map tuples of objects to objects.
{\displaystyle A,B,...\equiv P(t_{1},\dotsc ,t_{n})\mid A\wedge B\mid \top \mid A\vee B\mid \bot \mid A\supset B\mid \forall x.\ A\mid \exists x.\ A}
that is, a proposition is recursively defined to be an n-ary predicate P whose arguments are terms tk, or an expression composed of logical connectives (and, or) and quantifiers (for-all, there-exists) used with other propositions.
{\displaystyle P(x)}
{\displaystyle Q(y,f(x))}
{\displaystyle R(z)}
In model theory, structures assign an interpretation to the atomic formulas.
In proof theory, polarity assignment for atomic formulas is an essential component of focusing.
Atomic sentence
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Kt/V - wikidoc
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In medicine, Kt/V is a number used to quantify hemodialysis and peritoneal dialysis treatment adequacy.
K - dialyzer clearance of urea
t - dialysis time
V - patient's total body water
In the context of hemodialysis, Kt/V is a bonafide dimensionless number that can be derived using the Buckingham π theorem. In peritoneal dialysis, it is dimensionless only by definition.
It was developed by Frank Gotch and John Sargent as a way for measuring the dose of dialysis when they analyzed the data from the National Cooperative Dialysis Study.[1] In hemodialysis the US National Kidney Foundation Kt/V target is 1.3, so that one can be sure that the delivered dose is at least 1.2.[2] In peritoneal dialysis the target is 2.0/week.[2]
Despite the name, Kt/V is quite different from standardized Kt/V.
1 Rationale for Kt/V as a marker of dialysis adequacy
2 Relation to URR
2.2 Post-dialysis rebound
4 Reason for adoption
5 Criticisms/disadvantages of Kt/V
5.1 Importance of total weekly dialysis time and frequency
5.2 Kt/V minimums and targets for hemodialysis
5.3 Kt/V minimums and targets for peritoneal dialysis
Rationale for Kt/V as a marker of dialysis adequacy
K (clearance) multiplied by t (time) is a volume (since mL/min x min = mL, or L/hr x hr = L), and (K x t) can be thought of as the mL or L of fluid (blood in this case) cleared of urea (or any other solute) during the course of a single treatment. V also is a volume, expressed in mL or L. So the ratio of K x t / V is a so-called "dimensionless ratio" and can be thought of as a multiple of the volume of plasma cleared of urea divided by the distribution volume of urea. When Kt/V = 1.0, a volume of blood equal to the distribution volume of urea has been completely cleared of urea.
The relationship between Kt/V and the concentration of urea C at the end of dialysis can be derived from the first-order differential equation that describes exponential decay and models the clearance of any substance from the body where the concentration of that substance decreases in an exponential fashion:
{\displaystyle V{\frac {dC}{dt}}=-K\cdot C\qquad (1)}
C is the concentration [mol/m3]
K is the clearance [m3/s]
V is the volume of distribution [m3]
From the above definitions it follows that
{\displaystyle {\frac {dC}{dt}}}
is the first derivative of concentration with respect to time, i.e. the change in concentration with time.
This equation is separable and can be integrated as follows:
{\displaystyle \int {\frac {dC}{C}}=\int -{\frac {K}{V}}\,dt.\qquad (2a)}
{\displaystyle \ln(C)=-{\frac {K\cdot t}{V}}+{\mbox{const}}\qquad (2b)}
const is the constant of integration
If one takes the antilog of Equation 2b the result is:
{\displaystyle C=e^{-{\frac {K\cdot t}{V}}+const}\qquad (2c)}
By integer exponentiation this can be written as:
{\displaystyle C=C_{0}e^{-{\frac {K\cdot t}{V}}}\qquad (3)}
C0 is the concentration at the beginning of dialysis [mmol/L] or [mol/m3].
{\displaystyle {\frac {K\cdot t}{V}}=\ln {\frac {C_{o}}{C}}\qquad (4)}
Normally we measure postdialysis serum urea nitrogen concentration C and compare this with the initial or predialysis level C0. The session length or time is t and this is measured by the clock. The dialyzer clearance K is usually estimated, based on the urea transfer ability of the dialyzer (a function of its size and membrane permeability), the blood flow rate, and the dialysate flow rate. [3] In some dialysis machines, the urea clearance during dialysis is estimated by testing the ability of the dialyzer to remove a small salt load that is added to the dialysate during dialysis.
Relation to URR
The URR is simply the fractional reduction of urea during dialysis. So by definition, URR = 1 -C/C0. So 1-URR = C/C0. So by algebra, substituting into equation (4) above, since ln C/C0 = - ln C0/C, we get:
{\displaystyle {\frac {K\cdot t}{V}}=-\ln(1-URR).\qquad (8)}
Patient has a mass of 70 kg (154 lb) and gets a hemodialysis treatment that lasts 4 hours where the urea clearance 215 ml/min.
K = 215 mL/min
t = 4.0 hours = 240 min
V = 70 kg × 0.6 L of water/kg of body mass = 42 L = 42,000 mL
Kt/V = 1.23
This means that if you dialyze a patient to a Kt/V of 1.23, and measure the postdialysis and predialysis urea nitrogen levels in the blood, then calculate the URR, then -ln (1-URR) should be about 1.23.
The math does not quite work out, and more complicated relationships have been worked-out to account for the fluid removal (ultrafiltration) during dialysis as well as urea generation(see urea reduction ratio). Nevertheless, the URR and Kt/V are so closely related mathematically, that their predictive power has been shown to be no different in terms of prediction of patient outcomes in observational studies.
Post-dialysis rebound
The above analysis assumes that urea is removed from a single compartment during dialysis. In fact, this Kt/V is usually called the "single-pool" Kt/V. Due to the multiple compartments in the human body, a significant concentration rebound occurs following hemodialysis. Usually rebound lowers the Kt/V by about 15%. The amount of rebound depends on the rate of dialysis (K) in relation to the size of the patient (V). Equations have been devised to predict the amount of rebound based on the ratio of K/V, but usually this is not necessary in clinical practice. One can use such equations to calculate an "equilibrated Kt/V" or a "double-pool Kt/V", and some think that this should be used as a measure of dialysis adequacy, but this is not widely done in the United States, and the KDOQI guidelines (see below) recommend using the regular single pool Kt/V for simplicity.
In peritoneal dialysis calculation of Kt/V is easy, because the fluid drained is usually close to 100% saturated with urea. So the daily amount of plasma cleared is simply the drain volume divided by an estimate of the patient's volume of distribution.
As an example, if someone is infusing four 2 liter exchanges a day, and drains out a total of 9 liters per day, then they drain 9 x 7 = 63 liters per week. If the patient has an estimated total body water volume V of about 35 liters, then the weekly Kt/V would be 63/35, or about 1.8.
Calculation is a bit more complicated during Automated PD, when the serum concentration of urea is changing during dialysis. Usually blood samples are measured at some time point in the day thought to represent an average value, and the clearance is determined from this value.
Kt/V has been widely adopted because it was correlated with survival. Before Kt/V nephrologists measured the serum urea concentration (specifically the time-averaged concentration of urea (TAC of urea)), which was found not to be correlated with survival (due to its strong dependence on protein intake) and thus deemed an unreliable marker of dialysis adequacy.
Criticisms/disadvantages of Kt/V
It is complex and tedious to calculate. Many nephrologists have difficulty understanding it.
Urea is not associated with toxicity.[4]
Kt/V only measures a change in the concentration of urea and implicitly assumes the clearance of urea is comparable to other toxins. (It ignores molecules larger than urea having diffusion-limited transport - so called middle molecules).
Kt/V does not take into account the role of ultrafiltration.
It ignores the mass transfer between body compartments and across the plasma membrane (i.e. intracellular to extracellular transport), which has been shown to be important for the clearance of molecules such as phosphate. Practical use of Kt/V requires adjustment for rebound of the urea concentration due to the multi-compartmental nature of the body.
Kt/V may disadvantage women and smaller patients in terms of the amount of dialysis received. Normal kidney function is expressed as the Glomerular filtration rate or GFR. GFR is usually normalized in people to body surface area. A man and a woman of similar body surface areas will have markedly different levels of total body water (which corresponds to V). Also, smaller people of either sex will have markedly lower levels of V, but only slightly lower levels of body surface area. For this reason, any dialysis dosing system that is based on V may tend to underdose smaller patients and women. Some investigators have proposed dosing based on surface area (S) instead of V, but clinicians usually measure the URR and then calculate Kt/V. One can "adjust" the Kt/V, to calculate a "surface-area-normalized" or "SAN"-Kt/V as well as a "SAN"-standard Kt/V. This puts a wrapper around Kt/V and normalizes it to body surface area. [5]
Importance of total weekly dialysis time and frequency
Kt/V has been criticized because quite high levels can be achieved, particularly in smaller patients, during relatively short dialysis sessions. This is especially true for small people, where "adequate" levels of Kt/V often can be achieved over 2 to 2.5 hours. One important part of dialysis adequacy has to do with adequate removal of salt and water, and also of solutes other than urea, especially larger molecular weight substances and phosphorus. A number of studies suggest that a longer amount of time on dialysis, or more frequent dialysis sessions, lead to better results. There have been various alternative methods of measuring dialysis adequacy, most of which have proposed some number based on Kt/V and number of dialysis sessions per week, e.g., the standardized Kt/V, or simply number of dialysis sessions per week squared multiplied by the hours on dialysis per session; e.g. the hemodialysis product by Scribner and Oreopoulos [6] It is not practical to give long dialysis sessions (greater than 4.5 hours) 3x/week in a dialysis center during the day. Longer sessions can be practically delivered if dialysis is done at home. Most experience has been gained with such long dialysis sessions given at night. Some centers are offering every-other-night or 3x/week nocturnal dialysis. The benefits of giving more frequent dialysis sessions is also an area of active study, and new easy-to-use machines are permitting easier use of home dialysis, where 2-3+ hr sessions can be given 4-7 days per week.
Kt/V minimums and targets for hemodialysis
One question in terms of Kt/V is, how much is enough? The answer has been based on observational studies, and the NIH-funded HEMO trial done in the United States, and also, on kinetic analysis. For a US perspective, see the [7] and for a United Kingdom perspective see [8] According to the US guidelines, for 3x/week dialysis a Kt/V (without rebound) should be 1.2 at a minimum with a target value of 1.4 (15% above the minimum values). However, there is suggestive evidence that larger amounts may need to be given to women, smaller patients, malnourished patients, and patients with clinical problems. The recommended minimum Kt/V value changes depending on how many sessions per week are given, and is reduced for patients who have a substantial degree of residual renal function.
Kt/V minimums and targets for peritoneal dialysis
For a US perspective, see [9] For the United States, the minimum weekly Kt/V target used to be 2.0. This was lowered to 1.7 in view of the results of a large randomized trial done in Mexico, the ADEMEX trial, [10] and also from reanalysis of previous observational study results from the perspective of residual kidney function.
For a United Kingdom perspective see: [11] This is still in draft form.
↑ 1.0 1.1 Gotch FA, Sargent JA. A mechanistic analysis of the National Cooperative Dialysis Study (NCDS) Kidney Int. 1985;28(3):526-34. PMID 3934452.
↑ 2.0 2.1 "Clinical practice guidelines for nutrition in chronic renal failure. K/DOQI, National Kidney Foundation". Am J Kidney Dis. 35 (6 Suppl 2): S1–140. 2000. PMID 10895784. . Available at: http://www.kidney.org/professionals/KDOQI/guidelines_updates/doqi_uptoc.html
↑ Babb AL, Popovich RP, Christopher TG, Scribner BH. The genesis of the square meter-hour hypothesis. Trans Am Soc Artif Intern Organs. 1971;17:81-91. PMID 5158139.
↑ Johnson WJ, Hagge WW, Wagoner RD, Dinapoli RP, Rosevear JW. Effects of urea loading in patients with far-advanced renal failure. Mayo Clin Proc. 1972 Jan;47(1):21-9. PMID 5008253.
↑ Daugirdas JT et al. Surface-area-normalized (SAN) adjustment to Kt/V and weekly standard Kt/V. J Am Soc Nephrol (abstract) 2006. and Appendix A. Handbook of Dialysis, 4th Edition. Daugirdas JT, Blake PB, Ing TS, editors. Lippincott Williams and Wilkins, Philadelphia, 2007.
↑ Scribner BH, Oreopoulos DG, The Hemodialysis Product (HDP): A Better Index of Dialysis Adequacy than Kt/V, Dialysis & Transplantation, 2002 Jan;31(1):13-15. Full Text, Mirror; original link no longer available.
↑ KDOQI Hemodialysis Adequacy Update 2006. [1].
↑ U.K. Renal Association Clinical Practice Guidelines, 2006 Update. [2].
↑ KDOQI Peritoneal Adequacy Update 2006. [3].
↑ Paniagua R, Amato D, Vonesh E, Correa-Rotter R, Ramos A, Moran J, Mujais S; Mexican Nephrology Collaborative Study Group. Effects of increased peritoneal clearances on mortality rates in peritoneal dialysis: ADEMEX, a prospective, randomized, controlled trial. J Am Soc Nephrol. 2002 May;13(5):1307-20. PMID 11961019
↑ U.K. Renal Association Clinical Practice Guidelines, Peritoneal Dialysis. 2006 Update. [4].
Hemodialysis Dose and Adequacy - a description of URR and Kt/V from the Kidney and Urologic Diseases Clearinghouse.
Kt/V and the adequacy of hemodialysis - UpToDate.com
Advisory on Peritoneal Dialysis - American Association of Kidney Patients
Peritoneal Dialysis Dose and Adequacy - a description of URR and Kt/V from the Kidney and Urologic Diseases Clearinghouse.
free Kt/V calculators, HD and PD - kt-v.net
Kt/V calculator - medindia.com
Kt/V - HDCN
de:Kt/V
Retrieved from "https://www.wikidoc.org/index.php?title=Kt/V&oldid=678087"
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Planet Earth/1g. Coriolis Effect: How Earth’s Spin Affects Motion Across its Surface. - Wikibooks, open books for an open world
Planet Earth/1g. Coriolis Effect: How Earth’s Spin Affects Motion Across its Surface.
1 Earth’s Inertia
2 Differences in Velocity due to Earth’s Spheroid Shape
3 Tossing a Ball on a Merry-go-round
4 The Rules of Earth’s Coriolis Force
5 A Common Misconception Regarding Flushing Toilets in Each Hemisphere
6 Why the Swirl is in the Opposite Direction to the Movement?
7 The Trajectory of Moving Objects on Earth’s Surface
Earth’s Inertia[edit | edit source]
A ball dropped on a fast moving train with zero acceleration will fall straight down.
As a spinning spheroid, Earth is constantly in motion, however one of the reasons you do not notice this spinning motion is because the Earth’s spin lacks any acceleration, and has a set speed or velocity. To offer an example that might be more familiar, imagine that Earth is a long train traveling at a constant speed down a very smooth track. The people on the train do not feel the motion. In fact, the passengers may be unaware of the motion of the train when the windows are closed and they do not have any reference to their motion from the surrounding passing landscape. If you were to drop a ball on the train it will fall straight downward from your perspective, and give the illusion that the train is not moving. This is because the train is traveling at a constant velocity, and hence has zero acceleration. If the train were to slow down, or speed up, the acceleration would change from zero, and the passengers would suddenly feel the motion of the train. If the train speeds up, and exhibits a positive acceleration, the passengers would feel a force pushing them backward. If the train slows down, and exhibits a negative acceleration, the passengers would feel a force pushing them forward. A ball dropped during these times would move in relationship to the slowing or increasing speed of the train. When the train is moving with constant velocity and zero acceleration we refer to this motion as inertia. An object with inertia has zero acceleration and a constant velocity or speed. Although the Earth’s spinning is slowing very very slightly, its acceleration is close to zero, and is in a state of inertia. As passengers on its surface we do not feel this motion because Earth’s spin is not speeding up or slowing down.
A simple animation showing two balls dropping from a height. The left ball is falling when acceleration of the box is zero (constant velocity), the right ball is falling when acceleration is changing, resulting in a curved path to its fall in relationship to the box.
Differences in Velocity due to Earth’s Spheroid Shape[edit | edit source]
Image of Earth’s shape taken from the Apollo 17 mission.
The Earth is spherical in shape, represented by a globe. Because of this shape, free moving objects that move across the Earth’s surface long distances will experience differences in their velocity due to Earth’s curvature. Objects starting near the equator and moving toward the pole will have a faster starting velocity, because they are starting at the widest part of the Earth, and slow down as they move toward the poles. Likewise, objects starting near the poles and move toward the equator will increase their velocity, as they move toward the widest part of the Earth’s spin. These actions result in accelerations that are not zero, but positive and negative.
The path and spin of large weather storms are effected by Earth’s spin and the Coriolis Effect.
As a consequence, the path of free moving objects across large distances of the spinning surface of Earth curve. This effect on the object’s path is called the Coriolis Effect. Understanding the Coriolis effect is important to understand the motion of storms, hurricanes, ocean surface currents, as well as airplane flight paths, weather balloons, rockets and even long-distance rifle shooting. Anything that moves across different latitudes of the Earth will be subjected to the Coriolis effect.
Tossing a Ball on a Merry-go-round[edit | edit source]
A ball is rolled on a moving merry-go-round from the center, from the perspective above the ball rolls straight, but from the perspective of George it has curved in relationship to the red dot (Sally’s position).
The best way to understand the Coriolis effect is to imagine a merry-go-round, which represents a single hemisphere of a spinning Earth. The merry-go-round has two people, Sally who is sitting on the edge of the spinning merry-go-round (representing a position on the Equator of the Earth), and George who is sitting at the center of the merry-go-round (representing a position on the North Pole of the Earth). George at the North Pole has a ball. He rolls the ball to Sally at the Equator. Because Sally has a faster velocity as she is sitting on the edge of the merry-go-round, by the time that the ball arrives at Sally’s position she will have moved left, and the ball rolled straight would miss her location. The ball’s path is straight from perspective above the merry-go-round, but from the perspective of George it appears that the ball’s path is moving toward the right of Sally. The key to understand this effect, is that the velocity of the ball increases as it moves from George to Sally, hence its acceleration is positive, while both George and Sally have zero acceleration. If we were to map the path of the rolled ball from the perspective of the merry-go-round, the path of the ball will curve clock-wise. Since the Earth is like a giant merry-go-round, we tend to view this change to the path of free moving objects as a Coriolis force. This force can be seen in the path of any free moving object that is moving across different latitudes. The coriolis force adheres to three rules.
The Rules of Earth’s Coriolis Force[edit | edit source]
The Coriolis force is proportional to the velocity of the object relative to the Earth; if there is no relative velocity, there is no Coriolis force.
The Coriolis force increases with increasing latitude; it is at a maximum at the North and South Poles, but with opposite signs, and is zero at the equator in respect to the mapped surface of the Earth, but does exert some upward force at the equator.
The Coriolis force always acts at right angles to the direction of motion, in the Northern Hemisphere it acts to the right of the starting observation point, and in the Southern Hemisphere to the left of the starting observation point. This results in Clockwise motion in the Northern Hemisphere and Counter-Clockwise motion in the Southern Hemisphere.
In the Northern Hemisphere the Coriolis force acts to the right (clockwise motion), while in the Southern Hemisphere the Coriolis force acts to the left (Counter-Clockwise).
These paths are difficult to empirically predict, as they appear to curve in respect to the surface of the Earth. One example where the Coriolis force comes into your daily life is when you travel by airplane crossing different latitudes. Because the Earth is moving below the airplane as it flies, the path relative to the Earth will curve, resulting in the airplane having to adjust its flight path to account for the motion of the Earth below. The Coriolis force also affects the atmosphere and ocean waters because these gasses and liquids are able to move in respect to the solid spinning Earth.
How to calculate the coriolis effect for a moving object across Earth’s surface. The red line is an object on earth traveling from point P at the velocity
{\displaystyle V_{t}}
(V_total). For the coriolis effect only the horizontal speed
{\displaystyle V_{h}}
(V_horizontal) is of importance. To get
{\displaystyle V_{h}}
{\displaystyle V_{t}}
one can say:
{\displaystyle v_{t}\cdot \sin \theta =v_{h}}
The coriolis force is then calculated by taking:
{\displaystyle coriolisforce=2mv_{t}\cdot \omega \cdot \sin \theta }
{\displaystyle \omega }
is velocity of the Earth’s spin at the Equator, and m is the mass of the object.
A Common Misconception Regarding Flushing Toilets in Each Hemisphere[edit | edit source]
A flushing toilet bowl is not effected by the Coriolis Effect because it is too small relative to Earth’s shape.
A common misconception is that because of the Coriolis effect there were differences in the direction of the swirl of draining water from sinks, toilets and basins depending on the hemisphere. This misconception arose from a famous experiment that was conducted over a hundred years ago, where water was drained from a very large wooden barrel. After the water was allowed to sit and settle for a week (so that there was no influence of any agitation in the water). A tiny plug at the base on the barrel was pulled, and the water slowly drained out of the barrel. Because of the large size of the barrel, the water moved with different velocities across the breath of the barrel. Water on the equator side of the barrel move slightly further than the water on the pole side of the barrel resulting in a path toward the right in the Northern Hemisphere, resulting in a counter-clockwise direction of the draining water.
Since this famous experiment, popular accounts of the Coriolis effect have focused on this phenomenon, despite the fact that most drains are influenced more by the shape of the basin and flow of water. Rarely do they reflect the original experimental conditions of a large basin or barrel, and water that is perfectly settled in the basin. Recently the experiment has been replicated with small child swimming pools, (see https://www.youtube.com/watch?v=mXaad0rsV38) about several feet across. Even at this small size, it was found that the water drained counter-clock-wise in the Northern Hemisphere, and clock-wise in the Southern Hemisphere. Note that the effect will be more pronounced the closer the experiment is conducted near the poles.
Why the Swirl is in the Opposite Direction to the Movement?[edit | edit source]
A drain located in the Northern Hemisphere shows the path of water flowing along the black arrows show a curvature to the right as the water flows into the low drain opening, causing the water to swirl in a counter-clockwise direction, which is opposite to the curved path of the water.
Why does the water swirl through the drain opposite to the direction of movement of the water? The motion of the water is relative to the drain plug, since the path from the equatorial side will be moving faster than at the center drain plug, the path will curve right in the Northern Hemisphere, resulting in a path that overshoots the center drain plug on the right side, from this point near the drain plug, the water will be pulled by gravity to flow through the drain toward the water’s left side resulting in a counter-clock-wise direction as it drains. In the Southern Hemisphere it will be in the opposite direction.
A storm in the North Hemisphere will curve clockwise in its path, but the storm will rotate in a counter-clockwise direction.
Satellites that track hurricanes and typhoons demonstrate that storms will curve to the right in the Northern Hemisphere and left in the Southern Hemisphere, but the clouds and winds will swirl in the opposite direction into the eye of the storm in a counter-clock-wise direction in the Northern Hemisphere and clock-wise direction in the Southern Hemisphere—similar to experiments with large barrels and child-size swimming pools. However, the storm’s overall path will be in a clock-wise direction in the Northern Hemisphere and counter-clock-wise direction in the Southern Hemisphere. This is why hurricanes hit the eastern coast and Gulf of Mexico in North America. Including the states of Florida, Louisiana, Texas, the Carolinas, Georgia and Virginia, and rarely if ever the western coast, such as California and the Baja California Peninsula of Mexico.
The Trajectory of Moving Objects on Earth’s Surface[edit | edit source]
If the trajectory of the free moving object does not cross different latitudes the path will be straight, since velocity in relationship to the Earth’s surface will remain the same, and acceleration will be zero. Coriolis effect behaves in three dimensions and the higher the altitude the more velocity will be exerted on the object. This is because the object’s position higher in the sky will result in a longer orbit around the Earth, and quicker velocity. Thus, changes in velocity can occur at any latitude if the free moving object changes altitude, and at the equator this force is vertical, leading to a peculiar net flow upward of air around the equator of the Earth, which is called the Intertropical Convergence Zone.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Planet_Earth/1g._Coriolis_Effect:_How_Earth’s_Spin_Affects_Motion_Across_its_Surface.&oldid=3930291"
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311 Tickets 2022 | Tour Dates | Concerts Schedule
Home > Artists > 311 Tickets
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311 VIP Packages 2022
311 (three hundred eleven) is the natural number following 310 and preceding 312.
311 is the 64th prime; a twin prime with 313; an irregular prime; an Eisenstein prime with no imaginary part and real part of the form
{displaystyle 3n-1}
; a Gaussian prime with no imaginary part and real part of the form
{displaystyle 4n-1}
; and a permutable prime with 113 and 131.
It can be expressed as a sum of consecutive primes in four different ways: as a sum of three consecutive primes (101 + 103 + 107), as a sum of five consecutive primes (53 + 59 + 61 + 67 + 71), as a sum of seven consecutive primes (31 + 37 + 41 + 43 + 47 + 53 + 59), and as a sum of eleven consecutive primes (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47).
311 is a strictly non-palindromic number, as it is not palindromic in any base between base 2 and base 309.
311 is the smallest positive integer d such that the imaginary quadratic field Q(√–d) has class number = 19.
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EUDML | Some properties and characterizations of -normal functions. EuDML | Some properties and characterizations of -normal functions.
Some properties and characterizations of
a
-normal functions.
Xiao, Jie. "Some properties and characterizations of -normal functions.." International Journal of Mathematics and Mathematical Sciences 20.3 (1997): 503-510. <http://eudml.org/doc/47826>.
keywords = {-normal function; -normal function},
title = {Some properties and characterizations of -normal functions.},
TI - Some properties and characterizations of -normal functions.
KW - -normal function; -normal function
a
-normal function,
a
-normal function
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A survey of local car dealers revealed that
64\%
of all cars sold last month had a Green Fang system,
28\%
had alarm systems, and
22\%
had both Green Fang and alarm systems.
What is the probability one of these cars selected at random had neither Green Fang nor an alarm system?
Make a Venn diagram or two-way table.
What is the probability that a car had Green Fang and was not protected by an alarm system?
Use the percentage of cars that have Green Fang and the percentage of cars that have both Green Fang
and an alarm to find out the percentage of cars with only Green Fang.
Are having Green Fang and an alarm system disjoint (mutually exclusive) events?
No, because
22\%
of the cars had both systems.
Use the alternative definition of independence (see the Math Notes box in Lesson 10.2.3) to determine if having Green Fang is associated with having an alarm.
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Logarithms, Studymaterial: Board Plus foundation-9 MATHS, Foundation Math - Meritnation
Let us suppose we are given 3 numbers: 2, 3 and 9.
Now, we know that 32 = 9
\sqrt{9}=3
The above two expressions are formed by combining 2 and 3, and 2 and 9 respectively to get the third number.
Is there an expression wherein we can combine 3 and 9 to get 2?
3 and 9 can be combined to get 2 as:
Here, ‘log’ is the abbreviated form of a concept called ‘Logarithms’.
The expression can be read as ‘logarithm of 9 to the base 3 is equal to 2’.
In general, if a is any positive real number (except 1), n is any rational number such that , then n is called the logarithm of b to the base a, and is written as .
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RemoveTask - Maple Help
Home : Support : Online Help : Programming : Document Tools : RemoveTask
remove a task from a Maple help database
RemoveTask(task)
string ; name of the task to remove
The RemoveTask function removes a task from a help database. Note that you can not remove tasks which are included in the standard distribution of Maple with this command.
\mathrm{with}\left(\mathrm{DocumentTools}\right):
\mathrm{RemoveTask}\left("my task"\right)
The DocumentTools[RemoveTask] command was introduced in Maple 16.
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Cell - Maple Help
Home : Support : Online Help : Programming : Document Tools : Layout : Cell
generate XML for a Cell element
Cell( c, opts )
(optional) ; content for the Cell
backgroundstyle : nonnegint:=0 ; Specifies whether the fillcolor option will be respected (1) or ignored (0).
columnspan : nonnegint:=1 ; Specifies how many columns to the right the Cell will span.
fillcolor : {list(nonnegint),symbol,string}:=[255,255,255] ; Specifies the background color of the Cell. The passed value can be either a named color or a list of three integers each between 0 and 255. A list of nonnegative integers is interpreted as RGB values in a 24bit 3-channel color space. The default value is [255,255,255] which corresponds to white.
padding : nonnegint:=NULL ; The number of pixels of padding for the Cell.
rowspan : nonnegint:=1 ; Specifies how many rows downwards the Cell will span.
visible : truefalse:=true ; Specifies whether the cell is visible or hidden
The Cell command in the Layout Constructors package returns an XML function call which represents a Cell element of a worksheet Table.
\mathrm{with}\left(\mathrm{DocumentTools}\right):
\mathrm{with}\left(\mathrm{DocumentTools}:-\mathrm{Layout}\right):
Executing the Cell command produces a function call.
C≔\mathrm{Cell}\left(\mathrm{Textfield}\left("Some text"\right)\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Table-Cell}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"columnspan"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"1"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"backgroundstyle"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"0"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"rowspan"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"1"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"fillcolor"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"\left[255,255,255\right]"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"visible"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"true"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text-field}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"alignment"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"centred"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"style"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Text"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"layout"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Normal"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Some text"}\right)\right)
The following example produces a Table, T consisting of a single column and a single row. The contents of the row is C, the Cell element created above.
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(C\right)\right):
\mathrm{xml}≔\mathrm{Worksheet}\left(T\right):
\mathrm{InsertContent}\left(\mathrm{xml}\right):
The next example illustrates using a color to fill the background of a Cell.
C≔\mathrm{Cell}\left(\mathrm{Textfield}\left("Some text"\right),\mathrm{fillcolor}="gold"\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(C\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
Using a fill color on a Cell is an effective way to show a 3D plot with a black background. The next example uses the InlinePlot element constructor for the content of the Cell.
P≔\mathrm{InlinePlot}\left(\mathrm{plot3d}\left({x}^{2}\mathrm{sin}\left(y\right),x=-1..1,y=-2\mathrm{\pi }..2\mathrm{\pi },\mathrm{axes}=\mathrm{none}\right)\right):
C≔\mathrm{Cell}\left(\mathrm{Textfield}\left(P\right),\mathrm{fillcolor}="black"\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(C\right),\mathrm{widthmode}=\mathrm{pixels},\mathrm{width}=300,\mathrm{alignment}=\mathrm{center}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
The content of the Cell can be padded by a specified number of pixels.
C≔\mathrm{Cell}\left(\mathrm{Textfield}\left("AgTy"\right),\mathrm{padding}=50\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(C\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
C≔\mathrm{Cell}\left(\mathrm{Textfield}\left("AgTy"\right),\mathrm{padding}=0\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(C\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
Vertical alignment within a Cell is a property of its parent Row, for which the default is top alignment. Note the difference in alignment between the two Cells on the left.
\mathrm{C1}≔\mathrm{Cell}\left(\mathrm{Textfield}\left("Some text"\right)\right):
\mathrm{C2}≔\mathrm{Cell}\left(\mathrm{Textfield}\left(\mathrm{style}=\mathrm{TwoDimOutput},\mathrm{Equation}\left(\frac{\mathrm{sqrt}\left(\mathrm{exp}\left(-\frac{{t}^{2}}{\mathrm{\pi }}\right)\right)}{\mathrm{\Gamma }\left(x\right)}\right)\right)\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Row}\left(\mathrm{C1},\mathrm{C2}\right),\mathrm{Row}\left(\mathrm{C1},\mathrm{C2},\mathrm{align}=\mathrm{center}\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
\frac{\sqrt{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}}}}{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
\frac{\sqrt{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}}}}{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
Horizontal alignment within a Cell is a property of the content (child element) of the Cell.
\mathrm{C1}≔\mathrm{Cell}\left(\mathrm{Textfield}\left("Some text",\mathrm{alignment}=\mathrm{right}\right)\right):
\mathrm{C2}≔\mathrm{Cell}\left(\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(\mathrm{Cell}\left(\right)\right),\mathrm{widthmode}=\mathrm{pixels},\mathrm{width}=50,\mathrm{alignment}=\mathrm{center}\right)\right):
\mathrm{C3}≔\mathrm{Cell}\left(\mathrm{Textfield}\left("Other text",\mathrm{alignment}=\mathrm{left}\right)\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Row}\left(\mathrm{C1},\mathrm{C2},\mathrm{C3}\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
A cell may span more than a single Column or Row of a Table.
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Row}\left(\mathrm{Cell}\left(\mathrm{columnspan}=2\right),\mathrm{Cell}\left(\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
Parsing of Cells is done Row by Row.
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{widthmode}=\mathrm{pixels},\mathrm{width}=90,\mathrm{alignment}=\mathrm{center},\mathrm{Row}\left(\mathrm{Cell}\left("A",\mathrm{columnspan}=2\right),\mathrm{Cell}\left("A",\mathrm{rowspan}=2\right)\right),\mathrm{Row}\left(\mathrm{Cell}\left("B",\mathrm{rowspan}=2\right),\mathrm{Cell}\left("B"\right)\right),\mathrm{Row}\left(\mathrm{Cell}\left("C",\mathrm{columnspan}=2\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{Column}\left(\right),\mathrm{widthmode}=\mathrm{pixels},\mathrm{width}=90,\mathrm{alignment}=\mathrm{center},\mathrm{Row}\left(\mathrm{Cell}\left("A",\mathrm{rowspan}=2\right),\mathrm{Cell}\left("A",\mathrm{columnspan}=2\right)\right),\mathrm{Row}\left(\mathrm{Cell}\left("B"\right),\mathrm{Cell}\left("B",\mathrm{rowspan}=2\right)\right),\mathrm{Row}\left(\mathrm{Cell}\left("C",\mathrm{columnspan}=2\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
The DocumentTools:-Layout:-Cell command was introduced in Maple 2015.
The DocumentTools:-Layout:-Cell command was updated in Maple 2016.
The visible option was introduced in Maple 2016.
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On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
1 September 2017 On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
M. Rapoport, B. Smithling, W. Zhang
In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of
p
-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a
p
-adic field. We prove our conjecture in the case of a unitary group in three variables.
M. Rapoport. B. Smithling. W. Zhang. "On the arithmetic transfer conjecture for exotic smooth formal moduli spaces." Duke Math. J. 166 (12) 2183 - 2336, 1 September 2017. https://doi.org/10.1215/00127094-2017-0003
Received: 31 March 2015; Revised: 3 November 2016; Published: 1 September 2017
Keywords: arithmetic fundamental lemma , arithmetic Gan–Gross–Prasad conjecture , Rapoport–Zink space , special cycles
M. Rapoport, B. Smithling, W. Zhang "On the arithmetic transfer conjecture for exotic smooth formal moduli spaces," Duke Mathematical Journal, Duke Math. J. 166(12), 2183-2336, (1 September 2017)
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Ideal_triangle Knowpia
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.
Three ideal triangles in the Poincaré disk model creating an ideal pentagon
Two ideal triangles in the Poincaré half-plane model
Ideal triangles have the following properties:
In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties:
Any ideal triangle has area π.[1]
Distances in an ideal triangleEdit
Dimensions related to an ideal triangle and its incircle, depicted in the Beltrami–Klein model (left) and the Poincaré disk model (right)
The inscribed circle to an ideal triangle has radius
{\displaystyle r=\ln {\sqrt {3}}={\frac {1}{2}}\ln 3=\operatorname {artanh} {\frac {1}{2}}=2\operatorname {artanh} (2-{\sqrt {3}})=}
{\displaystyle =\operatorname {arsinh} {\frac {1}{3}}{\sqrt {3}}=\operatorname {arcosh} {\frac {2}{3}}{\sqrt {3}}\approx 0.549}
The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radius r above, with equality only for the center of the inscribed circle.
The inscribed circle meets the triangle in three points of tangency, forming an equilateral contact triangle with side length
{\displaystyle d=\ln \left({\frac {{\sqrt {5}}+1}{{\sqrt {5}}-1}}\right)=2\ln \varphi \approx 0.962}
{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}
A circle with radius d around a point inside the triangle will meet or intersect at least two sides of the triangle.
The distance from any point on a side of the triangle to another side of the triangle is equal or less than
{\displaystyle a=\ln \left(1+{\sqrt {2}}\right)\approx 0.881}
, with equality only for the points of tangency described above.
a is also the altitude of the Schweikart triangle.
If the curvature is −K everywhere rather than −1, the areas above should be multiplied by 1/K and the lengths and distances should be multiplied by 1/√K.[citation needed]
Thin triangle conditionEdit
The δ-thin triangle condition used in δ-hyperbolic space
Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any hyperbolic triangle, this fact is important in the study of δ-hyperbolic space.
In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles.
In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles.
In the Beltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is circumscribed by the boundary circle. Note that in the Beltrami-Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models, is not conformal i.e. it does not preserve angles.
Real ideal triangle groupEdit
The Poincaré disk model tiled with ideal triangles
The ideal (∞ ∞ ∞) triangle group
Another ideal tiling
The real ideal triangle group is the reflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to the free product of three order-two groups (Schwartz 2001).
^ a b "What is the radius of the inscribed circle of an ideal triangle". Retrieved 9 December 2015.
Schwartz, Richard Evan (2001). "Ideal triangle groups, dented tori, and numerical analysis". Annals of Mathematics. Ser. 2. 153 (3): 533–598. arXiv:math.DG/0105264. doi:10.2307/2661362. JSTOR 2661362. MR 1836282.
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EUDML | Eigenvalue problems for the -Laplacian with indefinite weights. EuDML | Eigenvalue problems for the -Laplacian with indefinite weights.
Eigenvalue problems for the
p
-Laplacian with indefinite weights.
Cuesta, Mabel. "Eigenvalue problems for the -Laplacian with indefinite weights.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2001 (2001): Paper No. 33, 9 p., electronic only-Paper No. 33, 9 p., electronic only. <http://eudml.org/doc/121067>.
author = {Cuesta, Mabel},
keywords = {nonlinear eigenvalue roblem; -Laplacian; indefinite weight; -Laplacian},
title = {Eigenvalue problems for the -Laplacian with indefinite weights.},
AU - Cuesta, Mabel
TI - Eigenvalue problems for the -Laplacian with indefinite weights.
KW - nonlinear eigenvalue roblem; -Laplacian; indefinite weight; -Laplacian
M. Arias, J. Campos, M. Cuesta, J.-P. Gossez, An asymmetric Neumann problem with weights
Marcello Lucia, On the uniqueness and simplicity of the principal eigenvalue
M. Arias, J. Campos, M. Cuesta, J.-P. Gossez, Asymmetric elliptic problems with indefinite weights
Marco Degiovanni, Sergio Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity
nonlinear eigenvalue roblem,
p
-Laplacian, indefinite weight,
p
Articles by Cuesta
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Catmull–Clark subdivision surface - Wikipedia
Technique in 3D computer graphics
Catmull–Clark level-3 subdivision of a cube with the limit subdivision surface shown below. (Note that although it looks like the bi-cubic interpolation approaches a sphere, an actual sphere is quadric.)
Visual difference between sphere (green) and Catmull-Clark subdivision surface (magenta) from a cube
The Catmull–Clark algorithm is a technique used in 3D computer graphics to create curved surfaces by using subdivision surface modeling. It was devised by Edwin Catmull and Jim Clark in 1978 as a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology.[1]
In 2005, Edwin Catmull, together with Tony DeRose and Jos Stam, received an Academy Award for Technical Achievement for their invention and application of subdivision surfaces. DeRose wrote about "efficient, fair interpolation" and character animation. Stam described a technique for a direct evaluation of the limit surface without recursion.
1 Recursive evaluation
2 Exact evaluation
3 Software using the algorithm
Recursive evaluation[edit]
Catmull–Clark surfaces are defined recursively, using the following refinement scheme.[1]
Start with a mesh of an arbitrary polyhedron. All the vertices in this mesh shall be called original points.
For each face, add a face point
Set each face point to be the average of all original points for the respective face
Face points (blue spheres)
For each edge, add an edge point.
Set each edge point to be the average of the two neighbouring face points (AF) and the midpoint of the edge (ME)
{\displaystyle {\frac {AF+ME}{2}}}
Edge points (magenta cubes)
For each original point (P), take the average (F) of all n (recently created) face points for faces touching P, and take the average (R) of all n edge midpoints for original edges touching P, where each edge midpoint is the average of its two endpoint vertices (not to be confused with new edge points above). (Note that from the perspective of a vertex P, the number of edges neighboring P is also the number of adjacent faces, hence n)
Move each original point to the new vertex point
{\displaystyle {\frac {F+2R+(n-3)P}{n}}}
(This is the barycenter of P, R and F with respective weights (n − 3), 2 and 1)
New vertex points (green cones)
Form edges and faces in the new mesh
Connect each new face point to the new edge points of all original edges defining the original face
New edges, 4 per face point
Connect each new vertex point to the new edge points of all original edges incident on the original vertex
3 new edges per vertex point of shifted original vertices
Define new faces as enclosed by edges
Final faces to the mesh
The new mesh will consist only of quadrilaterals, which in general will not be planar. The new mesh will generally look "smoother" (i.e. less "jagged" or "pointy") than the old mesh. Repeated subdivision results in meshes that are more and more rounded.
The arbitrary-looking barycenter formula was chosen by Catmull and Clark based on the aesthetic appearance of the resulting surfaces rather than on a mathematical derivation, although they do go to great lengths to rigorously show that the method converges to bicubic B-spline surfaces.[1]
It can be shown that the limit surface obtained by this refinement process is at least
{\displaystyle {\mathcal {C}}^{1}}
at extraordinary vertices and
{\displaystyle {\mathcal {C}}^{2}}
everywhere else (when n indicates how many derivatives are continuous, we speak of
{\displaystyle {\mathcal {C}}^{n}}
continuity). After one iteration, the number of extraordinary points on the surface remains constant.
Exact evaluation[edit]
The limit surface of Catmull–Clark subdivision surfaces can also be evaluated directly, without any recursive refinement. This can be accomplished by means of the technique of Jos Stam (1998).[3] This method reformulates the recursive refinement process into a matrix exponential problem, which can be solved directly by means of matrix diagonalization.
Software using the algorithm[edit]
CATIA (Imagine and Shape)
Creo (Freestyle)[5]
Daz Studio, 2.0
DeleD Community Edition
DeleD Designer
LightWave 3D, version 9
Power Surfacing add-in for SolidWorks
Pixar's OpenSubdiv[6][7][8][9][10]
Rhinoceros 3D - Grasshopper 3D Plugin - Weaverbird Plugin
SketchUp - Requires a Plugin.
Conway polyhedron notation - A set of related topological polyhedron and polygonal mesh operators.
^ a b c Catmull, E.; Clark, J. (1978). "Recursively generated B-spline surfaces on arbitrary topological meshes" (PDF). Computer-Aided Design. 10 (6): 350. doi:10.1016/0010-4485(78)90110-0.
^ "Catmull–Clark subdivision surface - Rosetta Code". rosettacode.org. Retrieved 2022-01-13.
^ Stam, J. (1998). "Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values" (PDF). Proceedings of the 25th annual conference on Computer graphics and interactive techniques - SIGGRAPH '98. pp. 395–404. CiteSeerX 10.1.1.20.7798. doi:10.1145/280814.280945. ISBN 978-0-89791-999-9. S2CID 2771758.
^ "Subdivision Surface Modifier". 2020-01-15.
^ Manuel Kraemer (2014). "OpenSubdiv: Interoperating GPU Compute and Drawing". In Martin Watt; Erwin Coumans; George ElKoura; et al. (eds.). Multithreading for Visual Effects. CRC Press. pp. 163–199. ISBN 978-1-4822-4356-7.
^ Meet the Experts: Pixar Animation Studios, The OpenSubdiv Project. YouTube. Archived from the original on 2021-12-11.
^ "Pixar's OpenSubdiv V2: A detailed look". 2013-09-18.
^ http://on-demand.gputechconf.com/gtc/2014/video/S4856-subdivision-surfaces-industry-standard.mp4
^ OpenSubdiv Blender demo. YouTube. Archived from the original on 2021-12-11.
Derose, T.; Kass, M.; Truong, T. (1998). "Subdivision surfaces in character animation" (PDF). Proceedings of the 25th annual conference on Computer graphics and interactive techniques - SIGGRAPH '98. pp. 85. CiteSeerX 10.1.1.679.1198. doi:10.1145/280814.280826. ISBN 978-0897919999. S2CID 1221330.
Loop, C.; Schaefer, S. (2008). "Approximating Catmull-Clark subdivision surfaces with bicubic patches" (PDF). ACM Transactions on Graphics. 27: 1–11. CiteSeerX 10.1.1.153.2047. doi:10.1145/1330511.1330519. S2CID 6068564.
Kovacs, D.; Mitchell, J.; Drone, S.; Zorin, D. (2010). "Real-Time Creased Approximate Subdivision Surfaces with Displacements" (PDF). IEEE Transactions on Visualization and Computer Graphics. 16 (5): 742–51. doi:10.1109/TVCG.2010.31. PMID 20616390. S2CID 17138394. preprint
Matthias Nießner, Charles Loop, Mark Meyer, Tony DeRose, "Feature Adaptive GPU Rendering of Catmull-Clark Subdivision Surfaces", ACM Transactions on Graphics Volume 31 Issue 1, January 2012, doi:10.1145/2077341.2077347, demo
Nießner, Matthias ; Loop, Charles ; Greiner, Günther: Efficient Evaluation of Semi-Smooth Creases in Catmull-Clark Subdivision Surfaces: Eurographics 2012 Annex: Short Papers (Eurographics 2012, Cagliary). 2012, pp 41–44.
Wade Brainerd, Tessellation in Call of Duty: Ghosts also presented as a SIGGRAPH2014 tutorial [1]
D. Doo and M. Sabin: Behavior of recursive division surfaces near extraordinary points, Computer-Aided Design, 10 (6) 356–360 (1978), (doi, pdf)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Catmull–Clark_subdivision_surface&oldid=1065433857"
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Cornelia Druțu - Wikipedia
Cornelia Druţu
Cornelia Druțu is a Romanian mathematician notable for her contributions in the area of geometric group theory.[1] She is Professor of mathematics at the University of Oxford[1] and Fellow[2] of Exeter College, Oxford.
3.1 Selected contributions
3.2 Selected publications (in the order corresponding to the results above)
3.3 Published book
Druțu was born in Iaşi, Romania. She attended the Emil Racoviță High School (now the National College Emil Racoviță[3]) in Iași. She earned a B.S. in Mathematics from the University of Iași, where besides attending the core courses she received extra curricular teaching in geometry and topology from Professor Liliana Răileanu.[2]
Druțu earned a Ph.D. in Mathematics from University of Paris-Sud, with a thesis entitled Réseaux non uniformes des groupes de Lie semi-simple de rang supérieur et invariants de quasiisométrie, written under the supervision of Pierre Pansu.[4] She then joined the University of Lille 1 as Maître de conférences (MCF). In 2004 she earned her Habilitation degree from the University of Lille 1.[5]
In 2009 she became Professor of mathematics at the Mathematical Institute, University of Oxford.[1]
She held visiting positions at the Max Planck Institute for Mathematics in Bonn, the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, the Mathematical Sciences Research Institute in Berkeley, California. She visited the Isaac Newton Institute in Cambridge as holder of a Simons Fellowship.[6] From 2013 to 2020 she chaired the European Mathematical Society/European Women in Mathematics scientific panel of women mathematicians.[7][8]
In 2009, Druțu was awarded the Whitehead Prize by the London Mathematical Society for her work in geometric group theory.[9]
In 2017, Druțu was awarded a Simons Visiting Fellowship.[6]
The quasi-isometry invariance of relative hyperbolicity; a characterization of relatively hyperbolic groups using geodesic triangles, similar to the one of hyperbolic groups.
A classification of relatively hyperbolic groups up to quasi-isometry; the fact that a group with a quasi-isometric embedding in a relatively hyperbolic metric space, with image at infinite distance from any peripheral set, must be relatively hyperbolic.
The non-distortion of horospheres in symmetric spaces of non-compact type and in Euclidean buildings, with constants depending only on the Weyl group.
The quadratic filling for certain linear solvable groups (with uniform constants for large classes of such groups).
A construction of a 2-generated recursively presented group with continuously many non-homeomorphic asymptotic cones. Under the Continuum Hypothesis, a finitely generated group may have at most continuously many non-homeomorphic asymptotic cones, hence the result is sharp.
A characterization of Kazhdan's property (T) and of the Haagerup property using affine isometric actions on median spaces.
A study of generalizations of Kazhdan's property (T) for uniformly convex Banach spaces.
A proof that random groups satisfy strengthened versions of Kazhdan's property (T) for high enough density; a proof that for random groups the conformal dimension of the boundary is connected to the maximal value of p for which the groups have fixed point properties for isometric affine actions on
{\displaystyle L^{p}}
Selected publications (in the order corresponding to the results above)[edit]
Druţu, Cornelia (2009). "Relatively hyperbolic groups: geometry and quasi-isometric invariance". Commentarii Mathematici Helvetici. 84: 503–546. arXiv:math/0605211. doi:10.4171/CMH/171. MR 2507252. S2CID 7643177. .
Behrstock, Jason; Druţu, Cornelia; Mosher, Lee (2009). "Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity". Mathematische Annalen. 344 (3): 543–595. arXiv:math/0512592. doi:10.1007/s00208-008-0317-1. MR 2501302. S2CID 640737.
Druţu, Cornelia (1997). "Nondistorsion des horosphères dans des immeubles euclidiens et dans des espaces symétriques". Geometric and Functional Analysis. 7 (4): 712–754. doi:10.1007/s000390050024. MR 1465600. S2CID 122966047.
Druţu, Cornelia (2004). "Filling in solvable groups and in lattices in semisimple groups". Topology. 43 (5): 983–1033. arXiv:math/0110107. doi:10.1016/j.top.2003.11.004. MR 2079992. S2CID 15240355.
Druţu, Cornelia; Sapir, Mark (2005). With an appendix by Denis Osin and Mark Sapir. "Tree-graded spaces and asymptotic cones of groups". Topology. 44 (5): 959–1058. arXiv:math/0405030. doi:10.1016/j.top.2005.03.003. MR 2153979. S2CID 119658274.
Chatterji, Indira; Druţu, Cornelia; Haglund, Frédéric (2010). "Kazhdan and Haagerup properties from the median viewpoint". Advances in Mathematics. 225 (2): 882–921. CiteSeerX 10.1.1.313.1428. doi:10.1016/j.aim.2010.03.012. MR 2671183.
Druțu, Cornelia; Nowak, Piotr W. (2017). "Kazhdan projections, random walks and ergodic theorems". Journal für die reine und angewandte Mathematik. 2019 (754): 49–86. arXiv:1501.03473. doi:10.1515/crelle-2017-0002. S2CID 118775530.
Druțu, Cornelia; Mackay, John (2019). "Random groups, random graphs and eigenvalues of p-Laplacians". Advances in Mathematics. 341: 188–254. doi:10.1016/j.aim.2018.10.035. MR 3872847.
Published book[edit]
Druțu, Cornelia; Kapovich, Michael (2018). Geometric Group Theory (PDF). American Mathematical Society Colloquium Publications. Vol. 63. Providence, RI: American Mathematical Society. ISBN 978-1-4704-1104-6. MR 3753580.
Ultralimit
Tree-graded space
^ a b c Cornelia Druțu. "Cornelia Druţu's Homepage".
^ a b Exeter College, Oxford. "Professor Cornelia Druțu".
^ "the National College Emil Racoviță".
^ Cornelia Druțu at the Mathematics Genealogy Project
^ Cornelia Druțu. "Habilitation Cornelia Druțu".
^ a b "Simons Visiting Fellowships".
^ "EMS/EWM Scientific Panel". Women and Mathematics. 2021-08-17. Retrieved 2022-02-28.
^ Caroline, Series (October 2013). "European Level Organisations for Women Mathematicians" (PDF). European Women in Mathematics.
^ London Mathematical Society. "Prize Winners 2009". Archived from the original on 2009-10-23. Retrieved 2010-10-31.
personal webpage, University of Oxford
"MathSciNet". Retrieved October 31, 2010.
"ArXiv.org". Retrieved October 31, 2010.
Cornelia Druţu. "Papers". Retrieved October 31, 2010.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cornelia_Druțu&oldid=1085527388"
20th-century Romanian mathematicians
21st-century Romanian mathematicians
Paris-Sud 11 University alumni
Scientists from Iași
|
Erika is tracking the depth of the water in her local creek. Her first twelve measurements are below, in inches.
16
15
13
12
17
14
11
9
11
9
10
9
Find the median, first quartile (Q1), and third quartile (Q3).
The median is the middle number of an ordered set of data.
Order the measurements in ascending order, from least to greatest. From both ends, count until you get to a middle number. Remember, if there are two middle numbers, the median is the average of the two.
Now find the median of the lower half (Q1) and upper half (Q3) of the data set. Note that if there are an odd number of data values, the median is not included in either half of the data set.
\text{Q}1=9.5
\text{median}=11.5
\text{Q}3=14.5
Create a box plot for the data. Read the Math Notes box in this lesson to review how to make a box plot.
What is the interquartile range (IQR) for Erika’s data?
Subtract Q1 from Q3 to find the IQR.
|
Multiple Choice: Jill’s car tires are spinning at a rate of
120
revolutions per minute. If her car tires’ radii are each
14
inches, how far does she travel in
5
140π\text{ in.}
8400π\text{ in.}
3360π\text{ in.}
16800π\text{ in.}
2πr = 2π(14) = 28π
So, in one revolution the tire travels
28π
\frac{28\pi\ \text{inches}}{1\ \text{revolution}}\cdot\frac{120\ \text{revolution}}{1\ \text{minute}}\cdot\frac{5\ \text{minutes}}{1\ \text{trip}}\cdot\frac{16800\pi\ \text{inches}}{1\ \text{trip}}
1680π\text{ in.}
|
Bisimulation - Wikipedia
Find sources: "Bisimulation" – news · newspapers · books · scholar · JSTOR (February 2017) (Learn how and when to remove this template message)
In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa.
Intuitively two systems are bisimilar if they, assuming we view them as playing a game according to some rules, match each other's moves. In this sense, each of the systems cannot be distinguished from the other by an observer.
2.1 Relational definition
2.2 Fixpoint definition
2.3 Ehrenfeucht–Fraïssé game definition
2.4 Coalgebraic definition
3 Variants of bisimulation
4 Bisimulation and modal logic
Given a labelled state transition system (
{\displaystyle S}
{\displaystyle \Lambda }
, →), where
{\displaystyle S}
{\displaystyle \Lambda }
is a set of labels and → is a set of labelled transitions (i.e., a subset of
{\displaystyle S\times \Lambda \times S}
), a bisimulation is a binary relation
{\displaystyle R\subseteq S\times S}
{\displaystyle R}
{\displaystyle R^{T}}
are simulations. From this follows that the symmetric closure of a bisimulation is a bisimulation, and that each symmetric simulation is a bisimulation. Thus some authors define bisimulation as a symmetric simulation.[1]
{\displaystyle R}
is a bisimulation if and only if for every pair of states
{\displaystyle (p,q)}
{\displaystyle R}
and all labels α in
{\displaystyle \Lambda }
{\displaystyle p\mathrel {\overset {\alpha }{\rightarrow }} p'}
{\displaystyle q\mathrel {\overset {\alpha }{\rightarrow }} q'}
{\displaystyle (p',q')\in R}
{\displaystyle q\mathrel {\overset {\alpha }{\rightarrow }} q'}
{\displaystyle p\mathrel {\overset {\alpha }{\rightarrow }} p'}
{\displaystyle (p',q')\in R}
Given two states
{\displaystyle p}
{\displaystyle q}
{\displaystyle S}
{\displaystyle p}
is bisimilar to
{\displaystyle q}
{\displaystyle p\,\sim \,q}
, if and only if there is a bisimulation
{\displaystyle R}
{\displaystyle (p,q)\in R}
. This means that the bisimilarity relation
{\displaystyle \,\sim \,}
is the union of all bisimulations:
{\displaystyle (p,q)\in \,\sim \,}
{\displaystyle (p,q)\in R}
for some bisimulation
{\displaystyle R}
The set of bisimulations is closed under union;[Note 1] therefore, the bisimilarity relation is itself a bisimulation. Since it is the union of all bisimulations, it is the unique largest bisimulation. Bisimulations are also closed under reflexive, symmetric, and transitive closure; therefore, the largest bisimulation must be reflexive, symmetric, and transitive. From this follows that the largest bisimulation — bisimilarity — is an equivalence relation.[2]
Note that it is not always the case that if
{\displaystyle p}
{\displaystyle q}
{\displaystyle q}
{\displaystyle p}
then they are bisimilar. For
{\displaystyle p}
{\displaystyle q}
to be bisimilar, the simulation between
{\displaystyle p}
{\displaystyle q}
must be the converse of the simulation between
{\displaystyle q}
{\displaystyle p}
. Counter-example in CCS:
{\displaystyle \,M=a.b+a\,}
{\displaystyle \,M'=a.b\ }
simulate each other but are not bisimilar.
^ Meaning the union of two bisimulations is a bisimulation.
Relational definitionEdit
Bisimulation can be defined in terms of composition of relations as follows.
Given a labelled state transition system
{\displaystyle (S,\Lambda ,\rightarrow )}
, a bisimulation relation is a binary relation
{\displaystyle R}
{\displaystyle S}
{\displaystyle R}
{\displaystyle S}
{\displaystyle S}
{\displaystyle \forall \alpha \in \Lambda }
{\displaystyle R\ ;\ {\overset {\alpha }{\rightarrow }}\quad {\subseteq }\quad {\overset {\alpha }{\rightarrow }}\ ;\ R}
{\displaystyle R^{-1}\ ;\ {\overset {\alpha }{\rightarrow }}\quad {\subseteq }\quad {\overset {\alpha }{\rightarrow }}\ ;\ R^{-1}}
From the monotonicity and continuity of relation composition, it follows immediately that the set of bisimulations is closed under unions (joins in the poset of relations), and a simple algebraic calculation shows that the relation of bisimilarity—the join of all bisimulations—is an equivalence relation. This definition, and the associated treatment of bisimilarity, can be interpreted in any involutive quantale.
Fixpoint definitionEdit
Bisimilarity can also be defined in order-theoretical fashion, in terms of fixpoint theory, more precisely as the greatest fixed point of a certain function defined below.
{\displaystyle S}
, Λ, →), define
{\displaystyle F:{\mathcal {P}}(S\times S)\to {\mathcal {P}}(S\times S)}
to be a function from binary relations over
{\displaystyle S}
to binary relations over
{\displaystyle S}
{\displaystyle R}
be any binary relation over
{\displaystyle S}
{\displaystyle F(R)}
is defined to be the set of all pairs
{\displaystyle (p,q)}
{\displaystyle S}
{\displaystyle S}
{\displaystyle \forall \alpha \in \Lambda .\,\forall p'\in S.\,p{\overset {\alpha }{\rightarrow }}p'\,\Rightarrow \,\exists q'\in S.\,q{\overset {\alpha }{\rightarrow }}q'\,{\textrm {and}}\,(p',q')\in R}
{\displaystyle \forall \alpha \in \Lambda .\,\forall q'\in S.\,q{\overset {\alpha }{\rightarrow }}q'\,\Rightarrow \,\exists p'\in S.\,p{\overset {\alpha }{\rightarrow }}p'\,{\textrm {and}}\,(p',q')\in R}
Bisimilarity is then defined to be the greatest fixed point of
{\displaystyle F}
Ehrenfeucht–Fraïssé game definitionEdit
Bisimulation can also be thought of in terms of a game between two players: attacker and defender.
"Attacker" goes first and may choose any valid transition,
{\displaystyle \alpha }
{\displaystyle (p,q)}
. I.e.:
{\displaystyle (p,q){\overset {\alpha }{\rightarrow }}(p',q)}
{\displaystyle (p,q){\overset {\alpha }{\rightarrow }}(p,q')}
The "Defender" must then attempt to match that transition,
{\displaystyle \alpha }
from either
{\displaystyle (p',q)}
{\displaystyle (p,q')}
depending on the attacker's move. I.e., they must find an
{\displaystyle \alpha }
{\displaystyle (p',q){\overset {\alpha }{\rightarrow }}(p',q')}
{\displaystyle (p,q'){\overset {\alpha }{\rightarrow }}(p',q')}
Attacker and defender continue to take alternating turns until:
The defender is unable to find any valid transitions to match the attacker's move. In this case the attacker wins.
The game reaches states
{\displaystyle (p,q)}
that are both 'dead' (i.e., there are no transitions from either state) In this case the defender wins
The game goes on forever, in which case the defender wins.
{\displaystyle (p,q)}
, which have already been visited. This is equivalent to an infinite play and counts as a win for the defender.
By the above definition the system is a bisimulation if and only if there exists a winning strategy for the defender.
Coalgebraic definitionEdit
A bisimulation for state transition systems is a special case of coalgebraic bisimulation for the type of covariant powerset functor. Note that every state transition system
{\displaystyle (S,\Lambda ,\rightarrow )}
is bijectively a function
{\displaystyle \xi _{\rightarrow }}
{\displaystyle S}
to the powerset of
{\displaystyle S}
{\displaystyle \Lambda }
{\displaystyle {\mathcal {P}}(\Lambda \times S)}
{\displaystyle p\mapsto \{(\alpha ,q)\in \Lambda \times S:p{\overset {\alpha }{\rightarrow }}q\}.}
{\displaystyle \pi _{i}\colon S\times S\to S}
{\displaystyle i}
-th projection mapping
{\displaystyle (p,q)}
{\displaystyle p}
{\displaystyle q}
respectively for
{\displaystyle i=1,2}
{\displaystyle {\mathcal {P}}(\Lambda \times \pi _{1})}
the forward image of
{\displaystyle \pi _{1}}
defined by dropping the third component
{\displaystyle P\mapsto \{(\alpha ,p)\in \Lambda \times S:\exists q.(\alpha ,p,q)\in P\}}
{\displaystyle P}
{\displaystyle \Lambda \times S\times S}
{\displaystyle {\mathcal {P}}(\Lambda \times \pi _{2})}
Using the above notations, a relation
{\displaystyle R\subseteq S\times S}
is a bisimulation on a transition system
{\displaystyle (S,\Lambda ,\rightarrow )}
if and only if there exists a transition system
{\displaystyle \gamma \colon R\to {\mathcal {P}}(\Lambda \times R)}
on the relation
{\displaystyle R}
commutes, i.e. for
{\displaystyle i=1,2}
, the equations
{\displaystyle \xi _{\rightarrow }\circ \pi _{i}={\mathcal {P}}(\Lambda \times \pi _{i})\circ \gamma }
hold where
{\displaystyle \xi _{\rightarrow }}
is the functional representation of
{\displaystyle (S,\Lambda ,\rightarrow )}
Variants of bisimulationEdit
In special contexts the notion of bisimulation is sometimes refined by adding additional requirements or constraints. An example is that of stutter bisimulation, in which one transition of one system may be matched with multiple transitions of the other, provided that the intermediate states are equivalent to the starting state ("stutters").[3]
A different variant applies if the state transition system includes a notion of silent (or internal) action, often denoted with
{\displaystyle \tau }
, i.e. actions that are not visible by external observers, then bisimulation can be relaxed to be weak bisimulation, in which if two states
{\displaystyle p}
{\displaystyle q}
are bisimilar and there is some number of internal actions leading from
{\displaystyle p}
to some state
{\displaystyle p'}
then there must exist state
{\displaystyle q'}
such that there is some number (possibly zero) of internal actions leading from
{\displaystyle q}
{\displaystyle q'}
. A relation
{\displaystyle {\mathcal {R}}}
on processes is a weak bisimulation if the following holds (with
{\displaystyle {\mathcal {S}}\in \{{\mathcal {R}},{\mathcal {R}}^{-1}\}}
{\displaystyle a,\tau }
being an observable and mute transition respectively):
{\displaystyle \forall p,q.\quad (p,q)\in {\mathcal {S}}\Rightarrow p{\stackrel {\tau }{\rightarrow }}p'\Rightarrow \exists q'.\quad q{\stackrel {\tau ^{\ast }}{\rightarrow }}q'\wedge (p',q')\in {\mathcal {S}}}
{\displaystyle \forall p,q.\quad (p,q)\in {\mathcal {S}}\Rightarrow p{\stackrel {a}{\rightarrow }}p'\Rightarrow \exists q'.\quad q{\stackrel {\tau ^{\ast }a\tau ^{\ast }}{\rightarrow }}q'\wedge (p',q')\in {\mathcal {S}}}
This is closely related to bisimulation up to a relation.
Typically, if the state transition system gives the operational semantics of a programming language, then the precise definition of bisimulation will be specific to the restrictions of the programming language. Therefore, in general, there may be more than one kind of bisimulation, (bisimilarity resp.) relationship depending on the context.
Bisimulation and modal logicEdit
Since Kripke models are a special case of (labelled) state transition systems, bisimulation is also a topic in modal logic. In fact, modal logic is the fragment of first-order logic invariant under bisimulation (van Benthem's theorem).
Checking that two finite transition systems are bisimilar can be done in polynomial time.[4] The fastest algorithms are linear time using partition refinement through a reduction to the coarsest partition problem.
^ Jančar, Petr and Srba, Jiří (2008). "Undecidability of Bisimilarity by Defender's Forcing". J. ACM. New York, NY, USA: Association for Computing Machinery. 55 (1): 26. doi:10.1145/1326554.1326559. ISSN 0004-5411. S2CID 14878621. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Milner, Robin (1989). Communication and Concurrency. USA: Prentice-Hall, Inc. ISBN 0131149849.
^ Baier, Christel; Katoen, Joost-Pieter (2008). Principles of Model Checking. MIT Press. p. 527. ISBN 978-0-262-02649-9.
^ Baier & Katoen (2008), Cor. 7.45, p. 486.
Park, David (1981). "Concurrency and Automata on Infinite Sequences". In Deussen, Peter (ed.). Theoretical Computer Science. Proceedings of the 5th GI-Conference, Karlsruhe. Lecture Notes in Computer Science. Vol. 104. Springer-Verlag. pp. 167–183. doi:10.1007/BFb0017309. ISBN 978-3-540-10576-3.
Milner, Robin (1989). Communication and Concurrency. Prentice Hall. ISBN 0-13-114984-9.
Davide Sangiorgi. (2011). Introduction to Bisimulation and Coinduction. Cambridge University Press. ISBN 9781107003637
Software toolsEdit
CADP: tools to minimize and compare finite-state systems according to various bisimulations
mCRL2: tools to minimize and compare finite-state systems according to various bisimulations
The Bisimulation Game Game
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bisimulation&oldid=1075365016"
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Home > Lessons > Lesson 8 > The Rotation Curve of the Milky Way
Deriving the Galactic Mass from the Rotation Curve [2]
Now that we have a concept of the size, stellar populations, and an overall understanding of the Milky Way as a galaxy, let us consider another property that we can determine for the Milky Way: its mass. In most instances, when we intend to calculate the mass of an astronomical object, we return to Newton's version of Kepler's third law:
\[ {P}^{2}\quad =\text{ }(4\pi {\quad }^{2}\quad x\text{ }{a}^{3})\text{ }/\text{ }G({m}_{1}\quad +\text{ }{m}_{2}) \]
The Sun is orbiting around the Galactic center, so in principle, if we can measure the Sun's distance from the Galactic Center and its orbital period, this means we can estimate the sum of the masses of the Sun and the Galaxy (at least the portion of the Galaxy that is interior to the Sun's orbit). Since we anticipate the Galaxy's mass to far exceed the Sun's mass, we can take the value that we calculate to be the Galaxy's mass. So, what is the answer? How massive is our galaxy?
The distance from the Sun to the Galactic Center can be measured using a few different techniques, but it is a difficult measurement to make. It is still the case that researchers disagree about the exact value, but it is approximately 8 kpc (that is, 8,000 parsecs). There is a related, but also difficult measurement to make, and that is the velocity of the Sun with respect to the Galactic Center. It is approximately 200 km/sec, which allows us to estimate the period of the Sun's orbit around the Galactic Center in the following way:
Assume the Sun is following a circular orbit with radius 8,000 parsecs.
Calculate the circumference of the Sun's orbit:
c\text{ }=\text{ }2\pi r\text{ }=\text{ }\left(2\pi \right)*\left(8000\text{ pc}\right)*(3.1\text{ }x\text{ }{10}^{13}\quad \text{km}/\text{pc})\text{ }=\text{ }1.6\text{ }x\text{ }{10}^{18}\text{ km}
Calculate the period of the orbit by taking the circumference and dividing by the velocity:
P\text{ }=\text{ }1.6\text{ }x\text{ }{10}^{18}\quad \text{km }/\text{ }200\text{ km/sec }=\text{ }8.0\text{ }x\text{ }{10}^{15}\quad \text{sec }\approx \text{ }250\text{ million years}
If you take the semi-major axis of the Sun's orbit to be 8 kiloparsecs and the orbital period to be 250 million years, you can determine that the Milky Way's mass interior to the Sun's orbit is approximately 1011 solar masses, or 100 billion times the mass of the Sun.
Now, let us compare and contrast motions in the Solar System of the planets and motions in the Galaxy of the stars. What we did above to calculate the period of the Sun's orbit was to use the equation:
orbital period (P) = orbit circumference (2πr) / orbital velocity (v)
We can rearrange this equation and calculate orbital velocity for any object given its period and semi-major axis. If we apply this to the planets in the Solar System, you find that as you get more distant from the Sun, the orbital velocity of the object is slower. Below is a two-dimensional plot that I created for the orbital velocities of the planets (and Pluto) as a function of their distance from the Sun. Each point is labeled with the first letter of the object's name (e.g., V = Venus). This type of plot (orbital velocity as a function of distance from the center) is referred to as a rotation curve.
Figure 8.16: Plot of the orbital velocities of the planets in the Solar System showing how they decrease faster than linearly for objects more distant from the Sun.
The behavior of the planets in the Solar System as exhibited in this plot is often referred to as Keplerian Rotation. Clearly, the Milky Way Galaxy is more complicated than the Solar System. There are at least 100 billion objects, gas clouds, and dust, and there is not one single dominant mass in the center. However, astronomers expected that as you got more distant from the center of the Galaxy, the velocities of the stars should fall off in a manner similar to the Keplerian rotation exhibited by the planets in the Solar System. However, astronomers have observed that there is a significant difference between the predicted shape of the Milky Way's rotation curve and what is actually measured. See the image below.
Figure 8.17: Rotation curve of a typical spiral galaxy: predicted (A) and observed (B). The discrepancy between the curves is attributed to dark matter.
The solid line labeled B is a schematic rotation curve similar to what is measured for the Milky Way. The dashed line labeled A is the predicted rotation curve displaying Keplerian rotation. What the rotation curve B tells us is that our model of the Milky Way so far is missing something. In order for objects far from the center of the Galaxy to be moving faster than predicted, there must be significant additional mass far from the Galactic Center exerting gravitational pulls on those stars. This means that the Milky Way must include a component that is very massive and much larger than the visible disk of the Galaxy. We do not see any component in visible light or any other part of the electromagnetic spectrum, so this massive halo must be dark. Today, we refer to this as the "dark matter halo" of the Galaxy, and we will discuss dark matter more in our lesson on cosmology.
Returning to the image of the Milky Way that we studied before, the wire frame halo is actually meant to represent the extent of the dark matter halo. In the image below, compare the scale of the disk to the scale of the dark matter halo.
Figure 8.18: Schematic diagram of dark matter halo of Milky Way, captured from Partiview / Digital Universe Atlas, represented by a wire-frame sphere that completely encloses, and is much larger than, the thin disk of the Milky Way.
Source: Captured from Partiview / Digital Universe Atlas [4]
[2] http://www.astronomynotes.com/ismnotes/s7.htm
[3] http://en.wikipedia.org/wiki/Galaxy_rotation_curve
[4] http://www.haydenplanetarium.org/universe
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Data‐Driven Synthesis of Broadband Earthquake Ground Motions Using Artificial Intelligence | Bulletin of the Seismological Society of America | GeoScienceWorld
Manuel A. Florez;
Manuel A. Florez *
Seismological Laboratory, California Institute of Technology, Pasadena, California, U.S.A.
Corresponding author: mflorezt@caltech.edu
Michaelangelo Caporale;
Michaelangelo Caporale
Division of Engineering and Applied Sciences, California Institute of Technology, Pasadena, California, U.S.A.
Pakpoom Buabthong;
Zachary E. Ross;
Domniki Asimaki;
Men‐Andrin Meier
Manuel A. Florez, Michaelangelo Caporale, Pakpoom Buabthong, Zachary E. Ross, Domniki Asimaki, Men‐Andrin Meier; Data‐Driven Synthesis of Broadband Earthquake Ground Motions Using Artificial Intelligence. Bulletin of the Seismological Society of America 2022; doi: https://doi.org/10.1785/0120210264
Robust estimation of ground motions generated by scenario earthquakes is critical for many engineering applications. We leverage recent advances in generative adversarial networks (GANs) to develop a new framework for synthesizing earthquake acceleration time histories. Our approach extends the Wasserstein GAN formulation to allow for the generation of ground motions conditioned on a set of continuous physical variables. Our model is trained to approximate the intrinsic probability distribution of a massive set of strong‐motion recordings from Japan. We show that the trained generator model can synthesize realistic three‐component accelerograms conditioned on magnitude, distance, and
VS30
. Our model captures most of the relevant statistical features of the acceleration spectra and waveform envelopes. The output seismograms display clear P‐ and S‐wave arrivals with the appropriate energy content and relative onset timing. The synthesized peak ground acceleration estimates are also consistent with observations. We develop a set of metrics that allow us to assess the training process’s stability and to tune model hyperparameters. We further show that the trained generator network can interpolate to conditions in which no earthquake ground‐motion recordings exist. Our approach allows for the on‐demand synthesis of accelerograms for engineering purposes.
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EUDML | A note on -ideals in certain algebras of operators. EuDML | A note on -ideals in certain algebras of operators.
M
Cho, Chong-Man; Roh, Woo Suk
Cho, Chong-Man, and Roh, Woo Suk. "A note on -ideals in certain algebras of operators.." International Journal of Mathematics and Mathematical Sciences 23.10 (2000): 681-685. <http://eudml.org/doc/49068>.
author = {Cho, Chong-Man, Roh, Woo Suk},
keywords = {algebras of operators; -ideal; -ideal},
title = {A note on -ideals in certain algebras of operators.},
AU - Cho, Chong-Man
AU - Roh, Woo Suk
TI - A note on -ideals in certain algebras of operators.
KW - algebras of operators; -ideal; -ideal
algebras of operators,
M
-ideal,
M
-ideal
Articles by Roh
|
First, examine each shape to determine which type of shape it is. Look for characteristics such as right angles, parallel sides, and number of sides.
After you have determined the shape, recall the area formulas specific to that shape. In this case, a trapezoid and a triangle.
Try solving for the area of the parallelogram and triangle using these formulas.
459
|
EUDML | stability of conservation laws for a traffic flow model. EuDML | stability of conservation laws for a traffic flow model.
{L}^{1}
stability of conservation laws for a traffic flow model.
Li, Tong. " stability of conservation laws for a traffic flow model.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2001 (2001): Paper No. 14, 18 p., electronic only-Paper No. 14, 18 p., electronic only. <http://eudml.org/doc/121066>.
author = {Li, Tong},
keywords = { well-posedness; relaxation; continuous dependence of the solution on its initial data; equilibrium entropy solutions; zero relaxation limit; large-time behavior; equilibrium shock waves; well-posedness},
title = { stability of conservation laws for a traffic flow model.},
TI - stability of conservation laws for a traffic flow model.
KW - well-posedness; relaxation; continuous dependence of the solution on its initial data; equilibrium entropy solutions; zero relaxation limit; large-time behavior; equilibrium shock waves; well-posedness
{L}^{1}
well-posedness, relaxation, continuous dependence of the solution on its initial data, equilibrium entropy solutions, zero relaxation limit, large-time behavior, equilibrium shock waves,
{L}^{1}
|
EUDML | On belonging of trigonometric series to Orlicz space. EuDML | On belonging of trigonometric series to Orlicz space.
On belonging of trigonometric series to Orlicz space.
Tikhonov, S.
Tikhonov, S.. "On belonging of trigonometric series to Orlicz space.." JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] 5.2 (2004): Paper No. 22, 7 p., electronic only-Paper No. 22, 7 p., electronic only. <http://eudml.org/doc/124234>.
@article{Tikhonov2004,
author = {Tikhonov, S.},
keywords = {weighted Orlicz space; trigonometric series},
title = {On belonging of trigonometric series to Orlicz space.},
TI - On belonging of trigonometric series to Orlicz space.
KW - weighted Orlicz space; trigonometric series
weighted Orlicz space, trigonometric series
{L}^{p}
Articles by Tikhonov
|
EUDML | On spectra of geometric operators on open manifolds and differentiable groupoids. EuDML | On spectra of geometric operators on open manifolds and differentiable groupoids.
On spectra of geometric operators on open manifolds and differentiable groupoids.
Lauter, Robert, and Nistor, Victor. "On spectra of geometric operators on open manifolds and differentiable groupoids.." Electronic Research Announcements of the American Mathematical Society [electronic only] 7 (2001): 45-53. <http://eudml.org/doc/223135>.
@article{Lauter2001,
author = {Lauter, Robert, Nistor, Victor},
keywords = {Laplace operator; pseudodifferential operator; -algebra; groupoid; essential spectrum; -algebra},
title = {On spectra of geometric operators on open manifolds and differentiable groupoids.},
AU - Lauter, Robert
TI - On spectra of geometric operators on open manifolds and differentiable groupoids.
KW - Laplace operator; pseudodifferential operator; -algebra; groupoid; essential spectrum; -algebra
Laplace operator, pseudodifferential operator,
{C}^{*}
-algebra, groupoid, essential spectrum,
{C}^{*}
Pseudogroups and differentiable groupoids
Articles by Lauter
Articles by Nistor
|
Edge-Neighbor-Rupture Degree of Graphs
2013 Edge-Neighbor-Rupture Degree of Graphs
The edge-neighbor-rupture degree of a connected graph
G
\text{ENR}\left(G\right)=\text{max}\left\{\omega \left(G-S\right)-\left|S\right|-m\left(G-S\right):S\subseteq E\left(G\right),\omega \left(G-S\right)\ge 1\right\}
S
is any edge-cut-strategy of
G
\omega \left(G-S\right)
is the number of the components of
G-S
m\left(G-S\right)
G-S
. In this paper, the edge-neighbor-rupture degree of some graphs is obtained and the relations between edge-neighbor-rupture degree and other parameters are determined.
Ersin Aslan. "Edge-Neighbor-Rupture Degree of Graphs." J. Appl. Math. 2013 1 - 5, 2013. https://doi.org/10.1155/2013/783610
Ersin Aslan "Edge-Neighbor-Rupture Degree of Graphs," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-5, (2013)
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15 January 2012 Discrete fractional Radon transforms and quadratic forms
Duke Math. J. 161(1): 69-106 (15 January 2012). DOI: 10.1215/00127094-1507288
We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove sharp results for this class of discrete operators in all dimensions, providing necessary and sufficient conditions for them to extend to bounded operators from
{\ell }^{p}
{\ell }^{q}
. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.
Lillian B. Pierce. "Discrete fractional Radon transforms and quadratic forms." Duke Math. J. 161 (1) 69 - 106, 15 January 2012. https://doi.org/10.1215/00127094-1507288
Primary: 42B20 , 44A12
Lillian B. Pierce "Discrete fractional Radon transforms and quadratic forms," Duke Mathematical Journal, Duke Math. J. 161(1), 69-106, (15 January 2012)
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!! used as default html header if there is none in the selected theme. Animated sequences
{u}_{n}\left(x\right)
Visualize 3 6 8 10 15 20 30 50 70 100 termes from
n
x
: from to
y
Description: animated plot of a sequence (series) of functions. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, analysis, geometry, curves,functions,fourier_series,taylor_series
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Write the inequality represented by each graph.
Since the arrow is pointing to the left, that means that all those values are less than the selected point.
Therefore, any value to the left of
3
3
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. Since the arrow is pointing to the right, that means that all those values are greater than the selected point.
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Euclidean domain — Wikipedia Republished // WIKI 2
Commutative ring with a Euclidean division
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.
Euclidean domains appear in the following chain of class inclusions:
rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
Abstract Algebra | Introduction to Euclidean Domains
RNT2.3. Euclidean Domains
Abstract Algebra: Euclidean Domains, UFD begins, 11-5-18
Group Theory 91, Euclidean Domain implies Unique factorization Domain
1.1 Notes on the definition
4 Norm-Euclidean fields
Let R be an integral domain. A Euclidean function on R is a function f from R \{0} to the non-negative integers satisfying the following fundamental division-with-remainder property:
{\displaystyle f(a)=\min _{x\in R\setminus \{0\}}g(xa)}
Z, the ring of integers. Define f (n) = |n|, the absolute value of n.[3]
Z[ i ], the ring of Gaussian integers. Define f (a + bi) = a2 + b2, the norm of the Gaussian integer a + bi.
Z[ω] (where ω is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define f (a + bω) = a2 − ab + b2, the norm of the Eisenstein integer a + bω.
K [X], the ring of polynomials over a field K. For each nonzero polynomial P, define f (P) to be the degree of P.[4]
K [[X]], the ring of formal power series over the field K. For each nonzero power series P, define f (P) as the order of P, that is the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f (P) ≤ f (Q) if and only if P divides Q.
Any discrete valuation ring. Define f (x) to be the highest power of the maximal ideal M containing x. Equivalently, let g be a generator of M, and v be the unique integer such that g v is an associate of x, then define f (x) = v. The previous example K [[X]] is a special case of this.
A Dedekind domain with finitely many nonzero prime ideals P1, ... , Pn. Define
{\displaystyle f(x)=\sum _{i=1}^{n}v_{i}(x)}
Every domain that is not a principal ideal domain, such as the ring of polynomials in at least two indeterminates over a field, or the ring of univariate polynomials with integer coefficients, or the number ring Z[ √−5 ].
The ring of integers of Q( √−19 ), consisting of the numbers a + b√−19/2 where a and b are integers and both even or both odd. It is a principal ideal domain that is not Euclidean.
{\displaystyle p\in A}
{\displaystyle A^{\times }\to (A/p)^{\times }}
{\displaystyle A\to A/p}
R is a principal ideal domain (PID). In fact, if I is a nonzero ideal of R then any element a of I\{0} with minimal value (on that set) of f(a) is a generator of I.[8] As a consequence R is also a unique factorization domain and a Noetherian ring. With respect to general principal ideal domains, the existence of factorizations (i.e., that R is an atomic domain) is particularly easy to prove in Euclidean domains: choosing a Euclidean function f satisfying (EF2), x cannot have any decomposition into more than f(x) nonunit factors, so starting with x and repeatedly decomposing reducible factors is bound to produce a factorization into irreducible elements.
If Euclidean division is algorithmic, that is, if there is an algorithm for computing the quotient and the remainder, then an extended Euclidean algorithm can be defined exactly as in the case of integers.[9]
If a Euclidean domain is not a field then it has an element a with the following property: any element x not divisible by a can be written as x=ay+u for some unit u and some element y. This follows by taking a to be a non-unit with f(a) as small as possible. This strange property can be used to show that some principal ideal domains are not Euclidean domains, as not all PIDs have this property. For example, for d = −19, −43, −67, −163, the ring of integers of
{\displaystyle \mathbf {Q} ({\sqrt {d}})}
However, in many finite extensions of Q with trivial class group, the ring of integers is Euclidean (not necessarily with respect to the absolute value of the field norm; see below). Assuming the extended Riemann hypothesis, if K is a finite extension of Q and the ring of integers of K is a PID with an infinite number of units, then the ring of integers is Euclidean.[11] In particular this applies to the case of totally real quadratic number fields with trivial class group. In addition (and without assuming ERH), if the field K is a Galois extension of Q, has trivial class group and unit rank strictly greater than three, then the ring of integers is Euclidean.[12] An immediate corollary of this is that if the number field is Galois over Q, its class group is trivial and the extension has degree greater than 8 then the ring of integers is necessarily Euclidean.
Norm-Euclidean fields
Algebraic number fields K come with a canonical norm function on them: the absolute value of the field norm N that takes an algebraic element α to the product of all the conjugates of α. This norm maps the ring of integers of a number field K, say OK, to the nonnegative rational integers, so it is a candidate to be a Euclidean norm on this ring. If this norm satisfies the axioms of a Euclidean function then the number field K is called norm-Euclidean or simply Euclidean.[13][14] Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.
{\displaystyle \mathbf {Q} ({\sqrt {-5}})}
{\displaystyle \mathbf {Q} ({\sqrt {-19}})}
{\displaystyle \mathbf {Q} ({\sqrt {69}})}
Those that are norm-Euclidean, such as Gaussian integers (integers of
{\displaystyle \mathbf {Q} ({\sqrt {-1}})}
The norm-Euclidean quadratic fields have been fully classified; they are
{\displaystyle \mathbf {Q} ({\sqrt {d}})}
Valuation (algebra)
^ Rogers, Kenneth (1971), "The Axioms for Euclidean Domains", American Mathematical Monthly, 78 (10): 1127–8, doi:10.2307/2316324, JSTOR 2316324, Zbl 0227.13007
^ Fraleigh & Katz 1967, p. 377, Example 1
^ Samuel, Pierre (1 October 1971). "About Euclidean rings". Journal of Algebra. 19 (2): 282–301 (p. 285). doi:10.1016/0021-8693(71)90110-4. ISSN 0021-8693.
^ Pierre, Samuel (1964). Lectures on Unique Factorization Domains (PDF). Tata Institute of Fundamental Research. pp. 27–28.
^ "Quotient of polynomials, PID but not Euclidean domain?".
^ Fraleigh & Katz 1967, p. 377, Theorem 7.4
^ Motzkin, Theodore (1949), "The Euclidean algorithm", Bulletin of the American Mathematical Society, 55 (12): 1142–6, doi:10.1090/S0002-9904-1949-09344-8, Zbl 0035.30302
^ Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers", Proceedings of Symposia in Pure Mathematics, AMS, 24: 321–332, doi:10.1090/pspum/024/0337902, ISBN 9780821814246
^ Harper, Malcolm; Murty, M. Ram (2004), "Euclidean rings of algebraic integers" (PDF), Canadian Journal of Mathematics, 56 (1): 71–76, CiteSeerX 10.1.1.163.7917, doi:10.4153/CJM-2004-004-5
^ Hardy, G.H.; Wright, E.M.; Silverman, Joseph; Wiles, Andrew (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 978-0-19-921986-5.
^ LeVeque, William J. (2002) [1956]. Topics in Number Theory. Vol. I and II. Dover. pp. II:57, 81. ISBN 978-0-486-42539-9. Zbl 1009.11001.
Samuel, Pierre (1971). "About Euclidean rings" (PDF). Journal of Algebra. 19 (2): 282–301. doi:10.1016/0021-8693(71)90110-4.
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!! used as default html header if there is none in the selected theme. Parabolic trough calculator
Width of the reflective surface (
{L}_{e}
Distance from upper line to focal point (
{I}_{\mathrm{up}}
Thickness of the back plate (
{T}_{b}
Bearing radius (
\frac{1}{2}{D}_{o}
Description: determining the design parameters of small closed parabolic troughs interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, engineering, geometry, engineering, parabolic trough, parabola
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Atmosphere | Free Full-Text | Implications of COVID-19 Restriction Measures in Urban Air Quality of Thessaloniki, Greece: A Machine Learning Approach
Screening Approach for Short-Term PM2.5 Health Co-Benefits: A Case Study from 15 Metropolitan Cities around the World during the COVID-19 Pandemic
Spatio-Temporal Characteristics of Dry-Wet Conditions and Boundaries in Five Provinces of Northwest China from 1960 to 2020
The Impact of Large-Scale Social Restriction Phases on the Air Quality Index in Jakarta
Akritidis, D.
Papakosta, E.
Tzoumaka, P.
Kelessis, A.
Implications of COVID-19 Restriction Measures in Urban Air Quality of Thessaloniki, Greece: A Machine Learning Approach
Paraskevi Tzoumaka
Apostolos Kelessis
Department of Environment and Hydroeconomy, Directorate of Environment, Industry, Energy and Natural Resources, of the Region of Central Macedonia, 54642 Thessaloniki, Greece
Department of Environment of the Municipality of Thessaloniki, 54642 Thessaloniki, Greece
Academic Editors: Gunnar W. Schade, Nicole Mölders, Daniele Contini, Gabriele Curci, Francesca Costabile, Prashant Kumar and Chris G. Tzanis
(This article belongs to the Special Issue Coronavirus Pandemic Shutdown Effects on Urban Air Quality)
Following the rapid spread of COVID-19, a lockdown was imposed in Thessaloniki, Greece, resulting in an abrupt reduction of human activities. To unravel the impact of restrictions on the urban air quality of Thessaloniki, NO
{}_{2}
and O
{}_{3}
observations are compared against the business-as-usual (BAU) concentrations for the lockdown period. BAU conditions are modeled, applying the XGBoost (eXtreme Gradient Boosting) machine learning algorithm on air quality and meteorological surface measurements, and reanalysis data. A reduction in NO
{}_{2}
concentrations is found during the lockdown period due to the restriction policies at both AGSOFIA and EGNATIA stations of −24.9 [−26.6, −23.2]% and −18.4 [−19.6, −17.1]%, respectively. A reverse effect is revealed for O
{}_{3}
concentrations at AGSOFIA with an increase of 12.7 [10.8, 14.8]%, reflecting the reduced O
{}_{3}
titration by NO
{}_{x}
. The implications of COVID-19 lockdowns in the urban air quality of Thessaloniki are in line with the results of several recent studies for other urban areas around the world, highlighting the necessity of more sophisticated emission control strategies for urban air quality management. View Full-Text
Keywords: COVID-19; air quality; machine learning; NO2; O3; Thessaloniki; Greece COVID-19; air quality; machine learning; NO2; O3; Thessaloniki; Greece
Akritidis, D.; Zanis, P.; Georgoulias, A.K.; Papakosta, E.; Tzoumaka, P.; Kelessis, A. Implications of COVID-19 Restriction Measures in Urban Air Quality of Thessaloniki, Greece: A Machine Learning Approach. Atmosphere 2021, 12, 1500. https://doi.org/10.3390/atmos12111500
Akritidis D, Zanis P, Georgoulias AK, Papakosta E, Tzoumaka P, Kelessis A. Implications of COVID-19 Restriction Measures in Urban Air Quality of Thessaloniki, Greece: A Machine Learning Approach. Atmosphere. 2021; 12(11):1500. https://doi.org/10.3390/atmos12111500
Akritidis, Dimitris, Prodromos Zanis, Aristeidis K. Georgoulias, Eleni Papakosta, Paraskevi Tzoumaka, and Apostolos Kelessis. 2021. "Implications of COVID-19 Restriction Measures in Urban Air Quality of Thessaloniki, Greece: A Machine Learning Approach" Atmosphere 12, no. 11: 1500. https://doi.org/10.3390/atmos12111500
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Toward Seismic Metamaterials: The METAFORET Project | Seismological Research Letters | GeoScienceWorld
Philippe Roux;
ISTerre, CNRS UMR 5275, IFSTTAR, Université Grenoble Alpes, Campus Universitaire, 1381, rue de la Piscine, 38041 Grenoble Cedex 9, France, philippe.roux@univ-grenoble-alpes.fr
Dino Bindi;
Dino Bindi
Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Public Law Foundation State of Brandenburg, Telegrafenberg, D‐14473 Potsdam, Germany
Tobias Boxberger;
Tobias Boxberger
Andrea Colombi;
Fabrice Cotton;
Isabelle Douste‐Bacque;
Isabelle Douste‐Bacque
Stéphane Garambois;
Philippe Gueguen;
Gregor Hillers;
Now at Institute of Seismology, University of Helsinki, Helsinki, Finland.
Dan Hollis;
Sisprobe Inc., 24 Allée des Vulpains, 38240 Meylan, France
Thomas Lecocq;
Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium
Ildut Pondaven
Philippe Roux, Dino Bindi, Tobias Boxberger, Andrea Colombi, Fabrice Cotton, Isabelle Douste‐Bacque, Stéphane Garambois, Philippe Gueguen, Gregor Hillers, Dan Hollis, Thomas Lecocq, Ildut Pondaven; Toward Seismic Metamaterials: The METAFORET Project. Seismological Research Letters 2018;; 89 (2A): 582–593. doi: https://doi.org/10.1785/0220170196
We report on a seismic metamaterial experiment in a pine‐tree forest environment where the dense collection of trees behaves as subwavelength coupled resonators for surface seismic waves. For the METAFORET experiment, more than 1000 seismic sensors were deployed over a
120 m×120 m
area to study the properties of the ambient and induced seismic wavefield that propagates in the ground and in trees. The goal of the experiment was to establish a link between seismic‐relevant scales and microscale and mesoscale studies that pioneered the development of metamaterial physics in optics and acoustics. The first results of the METAFORET experiment show the presence of frequency band gaps for Rayleigh waves associated with compressional and flexural resonances of the trees, which confirms the strong influence that a dense collection of trees can have on the propagation of seismic waves.
Mimizan France
Vertical seismic profiling using double-beamforming processing of nonuniform anthropogenic seismic noise: The case study of Rittershoffen, Upper Rhine Graben, France
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\frac{4g}{3}
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\frac{3g}{2}
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\left(1\right) \frac{B{v}_{0}L}{3r},\frac{2}{3},\frac{B{v}_{0}L}{3r},\frac{1}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(2\right) \frac{B{v}_{0}L}{3r},\frac{1}{3},\frac{B{v}_{0}L}{3r},\frac{2}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{B{v}_{0}L}{3r},\frac{1}{3},\frac{B{v}_{0}L}{3r},\frac{1}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(4\right) \frac{B{v}_{0}L}{3r},\frac{B{v}_{0}L}{3r},\frac{B{v}_{0}L}{3r}
\omega
\left(1\right) \frac{1}{2}B\omega {l}^{2}\phantom{\rule{0ex}{0ex}}\left(2\right) B{\omega }^{2}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{3}{2}B\omega {l}^{2}\phantom{\rule{0ex}{0ex}}\left(4\right) Zero
Why choke coil can't be used for direct current??
When lightning strikes, nearby magnetic compass needles may be seen to jerk in response to the electrical discharge. No compass needle deflection results during the accumulation of electrostatic charge preceding the lightning bolt, but only when the bolt actually strikes. What does this phenomenon indicate about voltage, current, and magnetism?
the speed of em wave in a medium whose dielectric constant is 2.25 & relative permeability 4 is equal to
Solve and don't give NCERT solution ..
4. Two co-axial solenoids shown in figure. If key of primary suddenly opened then direction of instantaneous induced current in resistance 'R' which connected in secondary :
(1) L to N (2) N to L (3) Alternating (4) Zero
Shreya Kaushik asked a question
a circular loop of radius R is moved away from a current carrying wire then induced current in a circular wire will be
\frac{1}{2}B\omega {R}^{2}
\frac{3}{2}B\omega {R}^{2}
\frac{1}{4}B\omega {R}^{2}
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Ground truth comparisons. (a) Two hyperplane cuts of the multidimensional objective function
{\mathcal{L}}_{j}\left({\lambda }_{t},{\lambda }_{{t}^{\prime }}\right)
, with respect to λ= (t h , smin), with smin held constant and two values of the arguments,
{t}_{h}^{\prime }={t}_{h}+1
(red curve) and
{t}_{h}^{\prime }={t}_{h}+2
(green curve); values of the constants in
{\mathcal{L}}_{j}
were α = 0.001 and β = 1.0. (b) Various iterations of the optimization algorithm, showing the values of the area difference between ground truth and the calculated borders. The area difference is obtained by summing the difference of areas for all image slices. The plot shows that despite the initial λ, the algorithm converges towards a constant non-zero area difference. (c) Visual inspection of borders obtained with an optimal solution λ* compared with the ground truth, superposed on the corresponding germinal center image.
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Solid_immersion_lens Knowpia
A solid immersion lens (SIL) has higher magnification and higher numerical aperture than common lenses by filling the object space with a high-refractive-index solid material. SIL was originally developed for enhancing the spatial resolution of optical microscopy.[1] There are two types of SIL:
Hemispherical SIL: Theoretically capable of increasing the numerical aperture of an optical system by
{\displaystyle n}
, the index of refraction of the material of the lens.
Weierstrass SIL (superhemispherical SIL or superSIL): the height of the truncated sphere is
{\displaystyle (1+1/n)r}
, where r is the radius of the spherical surface of the lens. Theoretically capable of increasing the numerical aperture of an optical system by
{\displaystyle n^{2}}
Applications of SILEdit
Solid immersion lens microscopyEdit
All optical microscopes are diffraction-limited because of the wave nature of light. Current research focuses on techniques to go beyond this limit known as the Rayleigh criterion. The use of SIL can achieve spatial resolution better than the diffraction limit in air, for both far-field imaging [3][4] and near-field imaging.
Optical data storageEdit
Because SIL provides high spatial resolution, the spot size of laser beam through the SIL can be smaller than diffraction limit in air, and the density of the associated optical data storage can be increased.
Similar to immersion lithography, the use of SIL can increase spatial resolution of projected photolithographic patterns, creating smaller components on wafers.
Offers advantages for semiconductor wafer emission microscopy which detects faint emissions of light (Photons) from electron-hole recombination under the influence of electrical stimulation
^ S. M. Mansfield and G. S. Kino, “Solid immersion microscope”, Appl. Phys. Lett., vol. 57, no. 24, p. 2615, (1990).
^ Barnes, W., Björk, G., Gérard, J. et al. "Solid-state single photon sources: light collection strategies" Eur. Phys. J. D (2002) 18: 197. https://doi.org/10.1140/epjd/e20020024
^ R. Chen, K. Agarwal, C. Sheppard, J. Phang, and X. Chen, "A complete and computationally efficient numerical model of aplanatic solid immersion lens scanning microscope," Opt. Express 21, 14316-14330 (2013).
^ L. Hu, R. Chen, K. Agarwal, C. Sheppard, J. Phang, and X. Chen, "Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy," Opt. Express 19, 19280-19295 (2011).
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Solve system of nonlinear equations - MATLAB fsolve - MathWorks India
c
\begin{array}{c}2{x}_{1}+{x}_{2}=\mathrm{exp}\left(c{x}_{1}\right)\\ -{x}_{1}+2{x}_{2}=\mathrm{exp}\left(c{x}_{2}\right).\end{array}
c=-1
c
c
c
c
\begin{array}{c}2{x}_{1}-{x}_{2}={e}^{-{x}_{1}}\\ -{x}_{1}+2{x}_{2}={e}^{-{x}_{2}}.\end{array}
F\left(x\right) = 0
\begin{array}{c}2{x}_{1}-{x}_{2}-{e}^{-{x}_{1}}=0\\ -{x}_{1}+2{x}_{2}-{e}^{-{x}_{2}}=0.\end{array}
X
X*X*X=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]
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Information-theoretic security - Wikipedia
(Redirected from Information theoretic security)
Security of a cryptosystem which derives purely from information theory
A cryptosystem is considered to have information-theoretic security (also called unconditional security[1]) if the system is secure against adversaries with unlimited computing resources and time. In contrast, a system which depends on the computational cost of cryptanalysis to be secure (and thus can be broken by an attack with unlimited computation) is called computationally, or conditionally, secure.[2]
2 Physical layer encryption
3 Secret key agreement
An encryption protocol with information-theoretic security is impossible to break even with infinite computational power. Protocols proven to be information-theoretically secure are resistant to future developments in computing. The concept of information-theoretically secure communication was introduced in 1949 by American mathematician Claude Shannon, one of the founders of classical information theory, who used it to prove the one-time pad system was secure.[3] Information-theoretically secure cryptosystems have been used for the most sensitive governmental communications, such as diplomatic cables and high-level military communications[citation needed].
There are a variety of cryptographic tasks for which information-theoretic security is a meaningful and useful requirement. A few of these are:
Secret sharing schemes such as Shamir's are information-theoretically secure (and also perfectly secure) in that having less than the requisite number of shares of the secret provides no information about the secret.
More generally, secure multiparty computation protocols often have information-theoretic security.
Private information retrieval with multiple databases can be achieved with information-theoretic privacy for the user's query.
Reductions between cryptographic primitives or tasks can often be achieved information-theoretically. Such reductions are important from a theoretical perspective because they establish that primitive
{\displaystyle \Pi }
can be realized if primitive
{\displaystyle \Pi '}
can be realized.
Symmetric encryption can be constructed under an information-theoretic notion of security called entropic security, which assumes that the adversary knows almost nothing about the message being sent. The goal here is to hide all functions of the plaintext rather than all information about it.
Information-theoretic cryptography is quantum-safe.
Physical layer encryption[edit]
Technical limitations Algorithms which are computationally or conditionally secure (i.e., they are not information-theoretic secure) are dependent on resource limits. For example, RSA relies on the assertion that factoring large numbers is hard.
A weaker notion of security, defined by Aaron D. Wyner, established a now-flourishing area of research that is known as physical layer encryption.[4] It exploits the physical wireless channel for its security by communications, signal processing, and coding techniques. The security is provable, unbreakable, and quantifiable (in bits/second/hertz).
Wyner's initial physical layer encryption work in the 1970s posed the Alice–Bob–Eve problem in which Alice wants to send a message to Bob without Eve decoding it. If the channel from Alice to Bob is statistically better than the channel from Alice to Eve, it had been shown that secure communication is possible.[5] That is intuitive, but Wyner measured the secrecy in information theoretic terms defining secrecy capacity, which essentially is the rate at which Alice can transmit secret information to Bob. Shortly afterward, Imre Csiszár and Körner showed that secret communication was possible even if Eve had a statistically better channel to Alice than Bob did.[6] The basic idea of the information theoretic approach to securely transmit confidential messages (without using an encryption key) to a legitimate receiver is to use the inherent randomness of the physical medium (including noises and channel fluctuations due to fading) and exploit the difference between the channel to a legitimate receiver and the channel to an eavesdropper to benefit the legitimate receiver.[7] More recent theoretical results are concerned with determining the secrecy capacity and optimal power allocation in broadcast fading channels.[8][9] There are caveats, as many capacities are not computable unless the assumption is made that Alice knows the channel to Eve. If that were known, Alice could simply place a null in Eve's direction. Secrecy capacity for MIMO and multiple colluding eavesdroppers is more recent and ongoing work,[10][11] and such results still make the non-useful assumption about eavesdropper channel state information knowledge.
Still other work is less theoretical by attempting to compare implementable schemes. One physical layer encryption scheme is to broadcast artificial noise in all directions except that of Bob's channel, which basically jams Eve. One paper by Negi and Goel details its implementation, and Khisti and Wornell computed the secrecy capacity when only statistics about Eve's channel are known.[12][13]
Parallel to that work in the information theory community is work in the antenna community, which has been termed near-field direct antenna modulation or directional modulation.[14] It has been shown that by using a parasitic array, the transmitted modulation in different directions could be controlled independently.[15] Secrecy could be realized by making the modulations in undesired directions difficult to decode. Directional modulation data transmission was experimentally demonstrated using a phased array.[16] Others have demonstrated directional modulation with switched arrays and phase-conjugating lenses.[17][18][19]
That type of directional modulation is really a subset of Negi and Goel's additive artificial noise encryption scheme. Another scheme using pattern-reconfigurable transmit antennas for Alice called reconfigurable multiplicative noise (RMN) complements additive artificial noise.[20] The two work well together in channel simulations in which nothing is assumed known to Alice or Bob about the eavesdroppers.
Secret key agreement[edit]
The different works mentioned in the previous part employ, in one way or another, the randomness present in the wireless channel to transmit information-theoretically secure messages. Conversely, we could analyze how much secrecy one can extract from the randomness itself in the form of a secret key. That is the goal of secret key agreement.
In this line of work, started by Maurer[21] and Ahlswede and Csiszár,[22] the basic system model removes any restriction on the communication schemes and assumes that the legitimate users can communicate over a two-way, public, noiseless, and authenticated channel at no cost. This model has been subsequently extended to account for multiple users[23] and a noisy channel[24] among others.
Leftover hash lemma (privacy amplification)
^ Diffie, Whitfield; Hellman, Martin E. (November 1976). "New Directions in Cryptography" (PDF). IEEE Transactions on Information Theory. IT-22 (6): 646. Retrieved 8 December 2021.
^ Maurer, Ueli (August 1999). "Information-Theoretic Cryptography". Advances in Cryptology — CRYPTO '99, Lecture Notes in Computer Science. 1666: 47–64.
^ Shannon, Claude E. (October 1949). "Communication Theory of Secrecy Systems" (PDF). Bell System Technical Journal. 28 (4): 656–715. doi:10.1002/j.1538-7305.1949.tb00928.x. hdl:10338.dmlcz/119717. Retrieved 2011-12-21.
^ Koyluoglu (16 July 2010). "Information Theoretic Security". Retrieved 11 August 2010.
^ Wyner, A. D. (October 1975). "The Wire-Tap Channel" (PDF). Bell System Technical Journal. 54 (8): 1355–1387. doi:10.1002/j.1538-7305.1975.tb02040.x. S2CID 21512925. Archived from the original (PDF) on 2014-02-04. Retrieved 2013-04-11.
^ Csiszár, I.; Körner, J. (May 1978). "Broadcast Channels with Confidential Messages". IEEE Transactions on Information Theory. IT-24 (3): 339–348. doi:10.1109/TIT.1978.1055892.
^ Liang, Y.; Vincent Poor, H.; Shamai, S. (2008). "Information Theoretic Security". Foundations and Trends in Communications and Information Theory. 5 (4–5): 355–580. doi:10.1561/0100000036.
^ Liang, Yingbin; Poor, Vincent; Shamai (Shitz), Shlomo (June 2008). "Secure Communication Over Fading Channels". IEEE Transactions on Information Theory. 54 (6): 2470–2492. arXiv:cs/0701024. doi:10.1109/tit.2008.921678. S2CID 7249068.
^ Gopala, P.; Lai, L.; El Gamal, H. (October 2008). "On the Secrecy Capacity of Fading Channels". IEEE Transactions on Information Theory. 54 (10): 4687–4698. arXiv:cs/0610103. doi:10.1109/tit.2008.928990. S2CID 3264079.
^ Khisti, Ashish; Wornell, Gregory (November 2010). "Secure Transmission with Multiple Antennas II: The MIMOME Wiretap Channel". IEEE Transactions on Information Theory. 56 (11): 5515–5532. arXiv:1006.5879. Bibcode:2010arXiv1006.5879K. doi:10.1109/tit.2010.2068852. S2CID 1428.
^ Oggier, F.; Hassibi, B. (August 2011). "The Secrecy Capacity of the MIMO Wiretap Channel". IEEE Transactions on Information Theory. 57 (8): 4961–4972. arXiv:0710.1920. doi:10.1109/tit.2011.2158487. S2CID 1586.
^ Negi, R.; Goel, S. (2008). "Guaranteeing secrecy using artificial noise". IEEE Transactions on Wireless Communications. 7 (6): 2180–2189. doi:10.1109/twc.2008.060848. S2CID 5430424.
^ Khisti, Ashish; Wornell, Gregory (Jul 2010). "Secure transmission with multiple antennas I: The MISOME wiretap channel". IEEE Transactions on Information Theory. 56 (7): 3088–3104. CiteSeerX 10.1.1.419.1480. doi:10.1109/tit.2010.2048445. S2CID 47043747.
^ Daly, M.P.; Bernhard, J.T. (Sep 2009). "Directional modulation technique for phased arrays". IEEE Transactions on Antennas and Propagation. 57 (9): 2633–2640. Bibcode:2009ITAP...57.2633D. doi:10.1109/tap.2009.2027047. S2CID 27139656.
^ Babakhani, A.; Rutledge, D.B.; Hajimiri, A. (Dec 2008). "Transmitter architectures based on near-field direct antenna modulation" (PDF). IEEE Journal of Solid-State Circuits. IEEE. 76 (12): 2674–2692. Bibcode:2008IJSSC..43.2674B. doi:10.1109/JSSC.2008.2004864. S2CID 14595636.
^ Daly, M.P.; Daly, E.L.; Bernhard, J.T. (May 2010). "Demonstration of directional modulation using a phased array". IEEE Transactions on Antennas and Propagation. 58 (5): 1545–1550. Bibcode:2010ITAP...58.1545D. doi:10.1109/tap.2010.2044357. S2CID 40708998.
^ Hong, T.; Song, M.-Z.; Liu, Y. (2011). "RF directional modulation technique using a switched antenna array for physical layer secure communication applications". Progress in Electromagnetics Research. 116: 363–379. doi:10.2528/PIER11031605.
^ Shi, H.; Tennant, A. (April 2011). Direction dependent antenna modulation using a two element array. Proceedings 5th European Conference on Antennas and Propagation(EUCAP). pp. 812–815.
^ Malyuskin, O.; Fusco, V. (2012). "Spatial data encryption using phase conjugating lenses". IEEE Transactions on Antennas and Propagation. 60 (6): 2913–2920. Bibcode:2012ITAP...60.2913M. doi:10.1109/tap.2012.2194661. S2CID 38743535.
^ Daly, Michael (2012). Physical layer encryption using fixed and reconfigurable antennas (Ph.D.). University of Illinois at Urbana-Champaign.
^ Maurer, U. M. (May 1993). "Secret key agreement by public discussion from common information". IEEE Transactions on Information Theory. 39 (3): 733–742. doi:10.1109/18.256484.
^ Ahlswede, R.; Csiszár, I. (July 1993). "Common randomness in information theory and cryptography. I. Secret sharing". IEEE Transactions on Information Theory. 39 (4): 1121–1132. doi:10.1109/18.243431.
^ Narayan, Prakash; Tyagi, Himanshu (2016). "Multiterminal Secrecy by Public Discussion". Foundations and Trends in Communications and Information Theory. 13 (2–3): 129–275. doi:10.1561/0100000072.
^ Bassi, G.; Piantanida, P.; Shamai, S. (2019). "The Secret Key Capacity of a Class of Noisy Channels with Correlated Sources". Entropy. 21 (8): 732. Bibcode:2019Entrp..21..732B. doi:10.3390/e21080732.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Information-theoretic_security&oldid=1076775934"
Information-theoretically secure algorithms
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Spectre and Meltdown Testing on a E3-1270 v3 – Open Fluids
In Benchmarks, CentOS 7Tags ANSYS, CFD-ACE+, MATLAB, meltdown, spectrePublish Date 18/01/2018 374 Views
Following on from the previous post concerning OpenFOAM benchmarks against spectre and meltdown fixes, the next set come from a Dell T1700 (E3-1270 v3, 16GB RAM) but this time we’ve looked at more applications, specifically:
CFD-ACE+ 2015
The methodology for the tests was to run the benchmarks three or four times to get an average reading (with standard deviation) for three cases:
Base system as it was;
Fully patched system but with RedHat tuneables off, and;
Fully patched system with RedHat tuneables on.
For refernce the base system was running Dell’s A19 BIOS and RedHat’s kernel 3.10.0-514.6.1.el7.x86_64. Whereas the final system was running A24 and 3.10.0-693.11.6.el7.x86_64, respectively. The results are all relative to the base case with the “Relative Difference” column computed as:
\frac{{t}_{new}}{{t}_{base}}
with the standard deviation column being a normalise sample standard deviation.
Interestingly, the results indicated that patching a very old BIOS may increase the performance of your code. For example, the MATLAB benchmarks ran faster across the board after applying both the BIOS patches and the OS patches.
Alternatively, both CFD-ACE+ and ANSYS, in general, saw a speed increase with the BIOS patches, as would be expected with the more efficient microcode updates in place, but then decreased after applying RedHat’s suggested mitigation tunables.
Again, the conclusion is that for these workloads there is relatively little hit for applying the BIOS and OS patches, especially if you are already running an older BIOS without modern microcode updates.
First up were the MATLAB 2017B results, as shown in the table below.
Base BIOS Only BIOS + Kernel
Ref STD Dev Rel Diff STD Dev Rel Diff STD Dev
Matrix 1 2.058% 0.965 0.823% 0.903 0.028%
FFT 1 0.577% 0.831 0.193% 0.827 2.061%
Wavelet 1 0.071% 0.997 0.061% 0.975 0.028%
Followed with the CFD-ACE+ results, as listed below.
VOF 1 0.841% 0.984 0.031% 1.023 0.111%
Multi4 1 0.219% 0.994 0.233% 1.009 0.253%
Finally the ANSYS 18.2 results, as listed below.
Couple 1 2.000% 1.007 1.147% 0.993 1.162%
V18sp-1 1 2.244% 0.948 1.624% 0.967 2.030%
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(Redirected from Bonus experience weekend)
26 April 2022 20 May 2022 (12:00) 30 May 2022 (12:00) The daily rewards have changed to the following (not including the doubling mentioned in the login message):
One hour of double XP can be traded with Nic the trader and Vic the trader for Credits to use at Nic's Store and Vic's Store.
250 credits for members
50 credits for free-to-play players.
Players are limited to one trade per day.
Other mechanics are as during the previous event.
7 February 2022 18 February 2022 (12:00) 28 February 2022 (12:00) | Mechanics from the previous event are repeated.
18 October 2021 5 November 2021 (12:00) 15 November 2021 (12:00) Mechanics from the previous event are repeated.
27 July 2021 6 August 2021 (12:00) 16 August 2021 (12:00)
Players get 10 days to use 48 hours of double XP time, down from 21 days during the previous Double XP weekend.
23 April 2021 3 May 2021 (12:00) 24 May 2021 (12:00)
Players still get 48 hours of double XP time, but the event now lasts for 21 days instead of 10.
5 February 2021 19 February 2021 (12:00) 1 March 2021 (12:00) [4] Mechanics from the previous event are repeated.
21 October 2020 6 November 2020 (12:00) 16 November 2020 (12:00) Archaeology now also enjoys the bonuses.[5]
17 July 2020 7 August 2020 (12:00) 17 August (12:00) [6] Most of the mechanics of the event are otherwise the same as previous event except for a new pause button with a one hour cooldown, and the daily rewards have changed to the following (not including the doubling mentioned in the login message):
{\displaystyle \left({\frac {x-10}{7.5}}\right)^{2}+1.1}
{\displaystyle x}
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Global regularity for the energy-critical NLS on $ {\mathbb{S}}^{3}$
{𝕊}^{3}
We establish global existence for the energy-critical nonlinear Schrödinger equation on
{𝕊}^{3}
. This follows similar lines to the work on
{𝕋}^{3}
but requires new extinction results for linear solutions and bounds on the interaction of a Euclidean profile and a linear wave of much higher frequency that are adapted to the new geometry.
author = {Pausader, Benoit and Tzvetkov, Nikolay and Wang, Xuecheng},
title = {Global regularity for the energy-critical {NLS} on $ {\mathbb{S}}^{3}$},
AU - Pausader, Benoit
AU - Wang, Xuecheng
TI - Global regularity for the energy-critical NLS on $ {\mathbb{S}}^{3}$
Pausader, Benoit; Tzvetkov, Nikolay; Wang, Xuecheng. Global regularity for the energy-critical NLS on $ {\mathbb{S}}^{3}$. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 315-338. doi : 10.1016/j.anihpc.2013.03.006. http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.006/
[1] T. Alazard, R. Carles, Loss of regularity for super-critical nonlinear Schrödinger equations, Math. Ann. 343 (2009), 397-420 | MR 2461259 | Zbl 1161.35047
[2] N. Anantharaman, F. Macia, The dynamics of the Schrödinger flow from the point of view of semiclassical measures, Spectral Geometry, Proc. Sympos. Pure Math. vol. 84, Amer. Math. Soc., Providence, RI (2012), 93-116 | MR 2985311 | Zbl 1317.53080
[3] R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. Math. France 136 no. 1 (2008), 27-65 | EuDML 272498 | Numdam | MR 2415335 | Zbl 1157.35100
[4] R. Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. Partial Differential Equations 33 no. 10–12 (2008), 1862-1889 | MR 2475322 | Zbl 1157.35101
[5] V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations 32 no. 10 (2007), 1643-1677 | MR 2372482 | Zbl 1143.35091
[6] V. Banica, R. Carles, T. Duyckaerts, On scattering for NLS: from Euclidean to hyperbolic space, Discrete Contin. Dyn. Syst. 24 no. 4 (2009), 1113-1127 | MR 2505694 | Zbl 1168.35316
[7] H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 no. 1 (1999), 131-175 | MR 1705001 | Zbl 0919.35089
[8] V. Banica, L. Ignat, Dispersion for the Schrödinger equation on networks, J. Math. Phys. 52 (2011), 083703 | MR 2858075 | Zbl 1272.81079
[9] M.D. Blair, H.F. Smith, C.D. Sogge, On Strichartz estimates for Schrödinger operators in compact manifolds with boundary, Proc. Amer. Math. Soc. 136 (2008), 247-256 | MR 2350410 | Zbl 1169.35012
[10] M.D. Blair, H.F. Smith, C.D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann. 354 (2012), 1397-1430, http://dx.doi.org/10.1007/s00208-011-0772-y | MR 2993000 | Zbl 1262.58015
[11] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107-156 | EuDML 58115 | MR 1209299 | Zbl 0787.35097
[12] J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal. 3 (1993), 157-178 | EuDML 58117 | MR 1209300 | Zbl 0787.35096
[13] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145-171 | MR 1626257 | Zbl 0958.35126
[14] J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, arXiv:1107.1129 | MR 3038558 | Zbl 1271.42039
[15] N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), 569-605 | MR 2058384 | Zbl 1067.58027
[16] N. Burq, P. Gérard, N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4) 38 no. 2 (2005), 255-301 | EuDML 82659 | Numdam | MR 2144988 | Zbl 1116.35109
[17] N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), 187-223 | MR 2142336 | Zbl 1092.35099
[18] N. Burq, P. Gérard, N. Tzvetkov, Global solutions for the nonlinear Schrödinger equation on three-dimensional compact manifolds, Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud. vol. 163, Princeton Univ. Press, Princeton, NJ (2007), 111-129 | MR 2333209 | Zbl 1180.35475
[19] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI (2003) | MR 2002047 | Zbl 1055.35003
[20] M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrodinger and wave equations, Ann. Inst. H. Poincaré (2013), arXiv:math/0311048
[21] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in
{ℝ}^{3}
, Ann. of Math. 167 (2008), 767-865 | MR 2415387 | Zbl 1178.35345
[22] B. Dodson, Global well-posedness and scattering for the defocusing, energy-critical, nonlinear Schrödinger equation in the exterior of a convex obstacle when
d=4
[23] P. Gérard, V. Pierfelice, Nonlinear Schrödinger equation on four-dimensional compact manifolds, Bull. Soc. Math. France 138 (2010), 119-151 | EuDML 272391 | Numdam | MR 2638892 | Zbl 1183.35251
[24] M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations 25 (2000), 1827-1844 | MR 1778782 | Zbl 0970.35134
[25] M. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 3 (2009), 917-941, http://dx.doi.org/10.1016/j.anihpc.2010.01.006 | EuDML 78874 | Numdam | Zbl 1169.35372
[26] Z. Hani, Global well-posedness of the cubic nonlinear Schrödinger equation on compact manifolds without boundary, Anal. PDE 5 no. 2 (2012), 339-363 | MR 2942981
[27] Z. Hani, B. Pausader, On scattering for the quintic defocusing nonlinear Schrödinger equation on
ℝ×{𝕋}^{2}
, Comm. Pure Appl. Math. (2013) | MR 3245101
[28] S. Herr, The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds, Amer. J. Math. (2013) | MR 3117307 | Zbl 1278.58006
[29] S. Herr, D. Tataru, N. Tzvetkov, Global well-posedness of the energy critical nonlinear Schrödinger equation with small initial data in
{H}^{1}\left({T}^{3}\right)
, Duke Math. J. 159 no. 2 (2011), 329-349 | Zbl 1230.35130
[30] S. Herr, D. Tataru, N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Reine Angew. Math. (2013), http://dx.doi.org/10.1515/crelle-2012-0013 | MR 3200335 | Zbl 1293.35299
[31] A.D. Ionescu, B. Pausader, G. Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE 5 no. 4 (2012), 705-746 | MR 3006640 | Zbl 1264.35215
[32] A.D. Ionescu, B. Pausader, Global wellposedness of the energy-critical defocusing NLS on
ℝ×{𝕋}^{3}
, Comm. Math. Phys. 312 no. 3 (2012), 781-831 | MR 2925134 | Zbl 1253.35159
[33] A.D. Ionescu, B. Pausader, The energy-critical defocusing NLS on
{𝕋}^{3}
, Duke Math. J. 161 no. 8 (2012), 1581-1612 | MR 2931275 | Zbl 1245.35119
[34] O. Ivanovici, F. Planchon, On the energy critical Schrödinger equation in 3D non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 5 (2010), 1153-1177 | Numdam | MR 2683754 | Zbl 1200.35066
[35] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), 353-392 | MR 1855973 | Zbl 1038.35119
[36] R. Killip, M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE 5 no. 4 (2012), 855-885, http://dx.doi.org/10.2140/apde.2012.5.855 | MR 3006644 | Zbl 1264.35219
[37] R. Killip, M. Visan, X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, arXiv:1208.4904 | MR 3553392 | Zbl 06707101
[38] C.E. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), 645-675 | MR 2257393 | Zbl 1115.35125
[39] H. Koch, D. Tataru, Dispersive estimates for principally normal pseudo-differential operators, Comm. Pure Appl. Math. 58 no. 2 (2005), 217-284 | MR 2094851 | Zbl 1078.35143
[40] D. Li, H. Smith, X. Zhang, Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data, arXiv:1109.4194 | MR 2923187 | Zbl 1307.35281
[41] F. Merle, L. Vega, Compactness at blow-up time for
{L}^{2}
solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not. 1998 no. 8 (1998), 399-425 | MR 1628235 | Zbl 0913.35126
[42] F. Planchon, On the cubic NLS on 3D compact domains, J. Inst. Math. Jussieu 2 (2013), 1-18 | MR 3134013 | Zbl 1290.35249
[43] C.D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 no. 1 (1986), 43-65 | MR 835795 | Zbl 0636.42018
[44] C.D. Sogge, Concerning the
{L}^{p}
norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-138 | MR 930395 | Zbl 0641.46011
[45] C.D. Sogge, Kakeya–Nikodym averages and
{L}^{p}
-norms of eigenfunctions, Tohoku Math. J. (2) 63 no. 4 (2011), 519-538 | MR 2872954 | Zbl 1234.35156
[46] H. Takaoka, N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on
ℝ×𝕋
, J. Funct. Anal. 182 no. 2 (2001), 427-442 | MR 1828800 | Zbl 0976.35085
[47] N. Tzvetkov, N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces, Comm. Partial Differential Equations 37 (2012), 125-135 | MR 2864809 | Zbl 1247.35004
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Physics problem related to laws of motion and gravity
Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. It is seen that the center of mass, in accordance with the law of conservation of momentum, remains in place.
1.1 Restricted three-body problem
2.2 Special-case solutions
4 Other problems involving three bodies
5 n-body problem
The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions
{\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})}
of three gravitationally interacting bodies with masses
{\displaystyle m_{i}}
{\displaystyle {\begin{aligned}{\ddot {\mathbf {r} }}_{\mathbf {1} }&=-Gm_{2}{\frac {\mathbf {r_{1}} -\mathbf {r_{2}} }{|\mathbf {r_{1}} -\mathbf {r_{2}} |^{3}}}-Gm_{3}{\frac {\mathbf {r_{1}} -\mathbf {r_{3}} }{|\mathbf {r_{1}} -\mathbf {r_{3}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {2} }&=-Gm_{3}{\frac {\mathbf {r_{2}} -\mathbf {r_{3}} }{|\mathbf {r_{2}} -\mathbf {r_{3}} |^{3}}}-Gm_{1}{\frac {\mathbf {r_{2}} -\mathbf {r_{1}} }{|\mathbf {r_{2}} -\mathbf {r_{1}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {3} }&=-Gm_{1}{\frac {\mathbf {r_{3}} -\mathbf {r_{1}} }{|\mathbf {r_{3}} -\mathbf {r_{1}} |^{3}}}-Gm_{2}{\frac {\mathbf {r_{3}} -\mathbf {r_{2}} }{|\mathbf {r_{3}} -\mathbf {r_{2}} |^{3}}}.\end{aligned}}}
{\displaystyle G}
is the gravitational constant.[3][4] This is a set of nine second-order differential equations. The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions
{\displaystyle \mathbf {r_{i}} }
{\displaystyle \mathbf {p_{i}} }
{\displaystyle {\frac {d\mathbf {r_{i}} }{dt}}={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p_{i}} }},\qquad {\frac {d\mathbf {p_{i}} }{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {r_{i}} }},}
{\displaystyle {\mathcal {H}}}
is the Hamiltonian:
{\displaystyle {\mathcal {H}}=-{\frac {Gm_{1}m_{2}}{|\mathbf {r_{1}} -\mathbf {r_{2}} |}}-{\frac {Gm_{2}m_{3}}{|\mathbf {r_{3}} -\mathbf {r_{2}} |}}-{\frac {Gm_{3}m_{1}}{|\mathbf {r_{3}} -\mathbf {r_{1}} |}}+{\frac {\mathbf {p_{1}} ^{2}}{2m_{1}}}+{\frac {\mathbf {p_{2}} ^{2}}{2m_{2}}}+{\frac {\mathbf {p_{3}} ^{2}}{2m_{3}}}.}
{\displaystyle {\mathcal {H}}}
is simply the total energy of the system, gravitational plus kinetic.
Restricted three-body problem[edit]
The circular restricted three-body problem is a valid approximation of elliptical orbits found in the Solar System, and this can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation (Coriolis effects are dynamic and not shown). The Lagrange points can then be seen as the five places where the gradient on the resultant surface is zero (shown as blue lines), indicating that the forces are in balance there.
Mathematically, the problem is stated as follows. Let
{\displaystyle m_{1,2}}
be the masses of the two massive bodies, with (planar) coordinates
{\displaystyle (x_{1},y_{1})}
{\displaystyle (x_{2},y_{2})}
{\displaystyle (x,y)}
be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to
{\displaystyle 1}
. Then, the motion of the planetoid is given by
{\displaystyle {\begin{aligned}{\frac {d^{2}x}{dt^{2}}}=-m_{1}{\frac {x-x_{1}}{r_{1}^{3}}}-m_{2}{\frac {x-x_{2}}{r_{2}^{3}}},\\{\frac {d^{2}y}{dt^{2}}}=-m_{1}{\frac {y-y_{1}}{r_{1}^{3}}}-m_{2}{\frac {y-y_{2}}{r_{2}^{3}}},\end{aligned}}}
{\displaystyle r_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}}
. In this form the equations of motion carry an explicit time dependence through the coordinates
{\displaystyle x_{i}(t),y_{i}(t)}
. However, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.
General solution[edit]
While a system of 3 bodies interacting gravitationally is chaotic, a system of 3 bodies interacting elastically isn't.
Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the three-body problem.
Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).
Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by
{\displaystyle \sigma ={\frac {e^{\frac {\pi s}{2\beta }}-1}{e^{\frac {\pi s}{2\beta }}+1}}.}
Special-case solutions[edit]
An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259.[11]
Numerical approaches[edit]
Other problems involving three bodies[edit]
^ a b c Barrow-Green, June (2008), "The Three-Body Problem", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), The Princeton Companion to Mathematics, Princeton University Press, pp. 726–728
^ "Historical Notes: Three-Body Problem". Retrieved 19 July 2017.
^ a b Barrow-Green, June (1997). Poincaré and the Three Body Problem. American Mathematical Society. pp. 8–12. Bibcode:1997ptbp.book.....B. ISBN 978-0-8218-0367-7.
^ "The Three-Body Problem" (PDF).
^ a b Cartwright, Jon (8 March 2013). "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem". Science Now. Retrieved 2013-04-04.
^ Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203.
^ Beloriszky, D. (1930). "Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps". Bulletin Astronomique. Série 2. 6: 417–434. Bibcode:1930BuAst...6..417B.
^ Burrau (1913). "Numerische Berechnung eines Spezialfalles des Dreikörperproblems". Astronomische Nachrichten. 195 (6): 113–118. Bibcode:1913AN....195..113B. doi:10.1002/asna.19131950602.
^ Victor Szebehely; C. Frederick Peters (1967). "Complete Solution of a General Problem of Three Bodies". Astronomical Journal. 72: 876. Bibcode:1967AJ.....72..876S. doi:10.1086/110355.
^ a b Šuvakov, M.; Dmitrašinović, V. "Three-body Gallery". Retrieved 12 August 2015.
^ Here the gravitational constant G has been set to 1, and the initial conditions are r1(0) = −r3(0) = (−0.97000436, 0.24308753); r2(0) = (0,0); v1(0) = v3(0) = (0.4662036850, 0.4323657300); v2(0) = (−0.93240737, −0.86473146). The values are obtained from Chenciner & Montgomery (2000).
^ Moore, Cristopher (1993), "Braids in classical dynamics" (PDF), Physical Review Letters, 70 (24): 3675–3679, Bibcode:1993PhRvL..70.3675M, doi:10.1103/PhysRevLett.70.3675, PMID 10053934
^ Chenciner, Alain; Montgomery, Richard (2000). "A remarkable periodic solution of the three-body problem in the case of equal masses". Annals of Mathematics. Second Series. 152 (3): 881–902. arXiv:math/0011268. Bibcode:2000math.....11268C. doi:10.2307/2661357. JSTOR 2661357. S2CID 10024592.
^ Montgomery, Richard (2001), "A new solution to the three-body problem" (PDF), Notices of the American Mathematical Society, 48: 471–481
^ Heggie, Douglas C. (2000), "A new outcome of binary–binary scattering", Monthly Notices of the Royal Astronomical Society, 318 (4): L61–L63, arXiv:astro-ph/9604016, Bibcode:2000MNRAS.318L..61H, doi:10.1046/j.1365-8711.2000.04027.x
^ Hudomal, Ana (October 2015). "New periodic solutions to the three-body problem and gravitational waves" (PDF). Master of Science Thesis at the Faculty of Physics, Belgrade University. Retrieved 5 February 2019.
^ Li, Xiaoming; Liao, Shijun (December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits". Science China Physics, Mechanics & Astronomy. 60 (12): 129511. arXiv:1705.00527. Bibcode:2017SCPMA..60l9511L. doi:10.1007/s11433-017-9078-5. ISSN 1674-7348. S2CID 84838204.
^ Li, Xiaoming; Jing, Yipeng; Liao, Shijun (13 September 2017). "The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum". arXiv:1709.04775. doi:10.1093/pasj/psy057. {{cite journal}}: Cite journal requires |journal= (help)
^ Li, Xiaoming; Liao, Shijun (2019). "Collisionless periodic orbits in the free-fall three-body problem". New Astronomy. 70: 22–26. arXiv:1805.07980. Bibcode:2019NewA...70...22L. doi:10.1016/j.newast.2019.01.003. S2CID 89615142.
^ Technion (6 October 2021). "A Centuries-Old Physics Mystery? Solved". SciTechDaily. SciTech. Retrieved 12 October 2021.
^ Ginat, Yonadav Barry; Perets, Hagai B. (23 July 2021). "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters". Physical Review. arXiv:2011.00010. doi:10.1103/PhysRevX.11.031020. Retrieved 12 October 2021.
^ The 1747 memoirs of both parties can be read in the volume of Histoires (including Mémoires) of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French):
The peculiar dating is explained by a note printed on page 390 of the "Memoirs" section: "Even though the preceding memoirs, of Messrs. Clairaut and d'Alembert, were only read during the course of 1747, it was judged appropriate to publish them in the volume for this year" (i.e. the volume otherwise dedicated to the proceedings of 1745, but published in 1749).
^ Jean le Rond d'Alembert, in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol. 2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp. 329–312, at sec. VI, p. 245.
^ "Coplanar Motion of Two Planets, One Having a Zero Mass". Annals of Mathematics, Vol. III, pp. 65–73, 1887.
^ Breen, Philip G.; Foley, Christopher N.; Boekholt, Tjarda; Portegies Zwart, Simon (2019). "Newton vs the machine: Solving the chaotic three-body problem using deep neural networks". arXiv:1910.07291. doi:10.1093/mnras/staa713. S2CID 204734498. {{cite journal}}: Cite journal requires |journal= (help)
^ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. p. 311. ISBN 978-0-13-111892-8. OCLC 40251748.
^ a b Crandall, R.; Whitnell, R.; Bettega, R. (1984). "Exactly soluble two-electron atomic model". American Journal of Physics. 52 (5): 438–442. Bibcode:1984AmJPh..52..438C. doi:10.1119/1.13650.
^ Calogero, F. (1969). "Solution of a Three-Body Problem in One Dimension". Journal of Mathematical Physics. 10 (12): 2191–2196. Bibcode:1969JMP....10.2191C. doi:10.1063/1.1664820.
^ Musielak, Z. E.; Quarles, B. (2014). "The three-body problem". Reports on Progress in Physics. 77 (6): 065901. arXiv:1508.02312. Bibcode:2014RPPh...77f5901M. doi:10.1088/0034-4885/77/6/065901. ISSN 0034-4885. PMID 24913140. S2CID 38140668.
^ Florin Diacu. "The Solution of the n-body Problem", The Mathematical Intelligencer, 1996.
^ Qin, Amy (November 10, 2014). "In a Topsy-Turvy World, China Warms to Sci-Fi". The New York Times. Archived from the original on December 9, 2019. Retrieved February 5, 2020.
Aarseth, S. J. (2003). Gravitational n-Body Simulations. New York: Cambridge University Press. ISBN 978-0-521-43272-6.
Bagla, J. S. (2005). "Cosmological N-body simulation: Techniques, scope and status". Current Science. 88: 1088–1100. arXiv:astro-ph/0411043. Bibcode:2005CSci...88.1088B.
Chambers, J. E.; Wetherill, G. W. (1998). "Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions". Icarus. 136 (2): 304–327. Bibcode:1998Icar..136..304C. CiteSeerX 10.1.1.64.7797. doi:10.1006/icar.1998.6007.
Efstathiou, G.; Davis, M.; White, S. D. M.; Frenk, C. S. (1985). "Numerical techniques for large cosmological N-body simulations". Astrophysical Journal. 57: 241–260. Bibcode:1985ApJS...57..241E. doi:10.1086/191003.
Hulkower, Neal D. (1978). "The Zero Energy Three Body Problem". Indiana University Mathematics Journal. 27 (3): 409–447. Bibcode:1978IUMJ...27..409H. doi:10.1512/iumj.1978.27.27030.
Hulkower, Neal D. (1980). "Central Configurations and Hyperbolic-Elliptic Motion in the Three-Body Problem". Celestial Mechanics. 21 (1): 37–41. Bibcode:1980CeMec..21...37H. doi:10.1007/BF01230244. S2CID 123404551.
Li, Xiaoming; Liao, Shijun (2014). "On the stability of the three classes of Newtonian three-body planar periodic orbits". Science China Physics, Mechanics & Astronomy. 57 (11): 2121–2126. arXiv:1312.6796. Bibcode:2014SCPMA..57.2121L. doi:10.1007/s11433-014-5563-5. S2CID 73682020.
Moore, Cristopher (1993). "Braids in Classical Dynamics" (PDF). Physical Review Letters. 70 (24): 3675–3679. Bibcode:1993PhRvL..70.3675M. doi:10.1103/PhysRevLett.70.3675. PMID 10053934.
Poincaré, H. (1967). New Methods of Celestial Mechanics (3 vol. English translated ed.). American Institute of Physics. ISBN 978-1-56396-117-5.
Šuvakov, Milovan; Dmitrašinović, V. (2013). "Three Classes of Newtonian Three-Body Planar Periodic Orbits". Physical Review Letters. 110 (10): 114301. arXiv:1303.0181. Bibcode:2013PhRvL.110k4301S. doi:10.1103/PhysRevLett.110.114301. PMID 25166541. S2CID 118554305.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Three-body_problem&oldid=1085972782"
Equations of astronomy
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EUDML | Nonlinear harmonic measures on trees. EuDML | Nonlinear harmonic measures on trees.
Nonlinear harmonic measures on trees.
Kaufman, Robert, Llorente, José G., and Wu, Jang-Mei. "Nonlinear harmonic measures on trees.." Annales Academiae Scientiarum Fennicae. Mathematica 28.2 (2003): 279-302. <http://eudml.org/doc/123345>.
@article{Kaufman2003,
author = {Kaufman, Robert, Llorente, José G., Wu, Jang-Mei},
keywords = {-regular forward branching tree; averaging operator; strong maximum principle; -harmonic; harmonic measure; -superharmonic; monotone elliptic operator; -regular forward branching tree; -harmonic; -superharmonic},
title = {Nonlinear harmonic measures on trees.},
TI - Nonlinear harmonic measures on trees.
KW - -regular forward branching tree; averaging operator; strong maximum principle; -harmonic; harmonic measure; -superharmonic; monotone elliptic operator; -regular forward branching tree; -harmonic; -superharmonic
José G. Llorente, Juan J. Manfredi, Jang-Mei Wu,
p
-harmonic measure is not additive on null sets
\kappa
-regular forward branching tree, averaging operator, strong maximum principle,
F
-harmonic, harmonic measure,
F
-superharmonic, monotone elliptic operator,
\kappa
-regular forward branching tree,
F
-harmonic,
F
-superharmonic
Articles by Kaufman
Articles by Llorente
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EUDML | Local heights on Abelian varieties and rigid analytic uniformization. EuDML | Local heights on Abelian varieties and rigid analytic uniformization.
Local heights on Abelian varieties and rigid analytic uniformization.
Werner, Annette. "Local heights on Abelian varieties and rigid analytic uniformization.." Documenta Mathematica 3 (1998): 301-319. <http://eudml.org/doc/223632>.
keywords = {Mazur-Tate pairing; -adic local height pairings; abelian variety; split semistable reduction; Raynaud extension; semistable ordinary reduction; -adic local height pairings},
title = {Local heights on Abelian varieties and rigid analytic uniformization.},
AU - Werner, Annette
TI - Local heights on Abelian varieties and rigid analytic uniformization.
KW - Mazur-Tate pairing; -adic local height pairings; abelian variety; split semistable reduction; Raynaud extension; semistable ordinary reduction; -adic local height pairings
Mazur-Tate pairing,
p
-adic local height pairings, abelian variety, split semistable reduction, Raynaud extension, semistable ordinary reduction,
p
-adic local height pairings
>1
Articles by Werner
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Quaternion frame rotation - MATLAB rotateframe - MathWorks Switzerland
Rotate Frame Using Quaternion Vector
Rereference Group of Points using Quaternion
Quaternion frame rotation
rotationResult = rotateframe(quat,cartesianPoints)
rotationResult = rotateframe(quat,cartesianPoints) rotates the frame of reference for the Cartesian points using the quaternion, quat. The elements of the quaternion are normalized before use in the rotation.
Define a point in three dimensions. The coordinates of a point are always specified in the order x, y, and z. For convenient visualization, define the point on the x-y plane.
Create a quaternion vector specifying two separate rotations, one to rotate the frame 45 degrees and another to rotate the point -90 degrees about the z-axis. Use rotateframe to perform the rotations.
0,0,-pi/2],'euler','XYZ','frame');
rereferencedPoint = rotateframe(quat,[x,y,z])
rereferencedPoint = 2×3
Plot the rereferenced points.
plot(rereferencedPoint(1,1),rereferencedPoint(1,2),'bo')
plot(rereferencedPoint(2,1),rereferencedPoint(2,2),'go')
Define two points in three-dimensional space. Define a quaternion to rereference the points by first rotating the reference frame about the z-axis 30 degrees and then about the new y-axis 45 degrees.
Use rotateframe to reference both points using the quaternion rotation operator. Display the result.
rP = rotateframe(quat,[a;b])
Quaternion that defines rotation, specified as a scalar quaternion or vector of quaternions.
rotationResult — Re-referenced Cartesian points
Cartesian points defined in reference to rotated reference frame, returned as a vector or matrix the same size as cartesianPoints.
The data type of the re-referenced Cartesian points is the same as the underlying data type of quat.
Quaternion frame rotation re-references a point specified in R3 by rotating the original frame of reference according to a specified quaternion:
{L}_{q}\left(u\right)={q}^{*}uq
For convenience, the rotateframe function takes a point in R3 and returns a point in R3. Given a function call with some arbitrary quaternion, q = a + bi + cj + dk, and arbitrary coordinate, [x,y,z],
point = [x,y,z];
rereferencedPoint = rotateframe(q,point)
the rotateframe function performs the following operations:
{u}_{q}=0+xi+yj+zk
{q}_{n}=\frac{q}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}+{d}^{2}}}
{v}_{q}={q}^{*}{u}_{q}q
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Maximum Likelihood Estimation of regARIMA Models - MATLAB & Simulink - MathWorks í•œêµ
For regression models with ARIMA time series errors in Econometrics Toolbox™, εt = σzt, where:
εt is the innovation corresponding to observation t.
σ is the constant variance of the innovations. You can set its value using the Variance property of a regARIMA model.
zt is the innovation distribution. You can set the distribution using the Distribution property of a regARIMA model. Specify either a standard Gaussian (the default) or standardized Student’s t with ν > 2 or NaN degrees of freedom.
If εt has a Student’s t distribution, then
{z}_{t}={T}_{\mathrm{ν}}\sqrt{\frac{\mathrm{ν}â2}{\mathrm{ν}}},
where Tν is a Student’s t random variable with ν > 2 degrees of freedom. Subsequently, zt is t-distributed with mean 0 and variance 1, but has the same kurtosis as Tν. Therefore, εt is t-distributed with mean 0, variance σ, and has the same kurtosis as Tν.
estimate builds and optimizes the likelihood objective function based on εt by:
Estimating c and β using MLR
Inferring the unconditional disturbances from the estimated regression model,
{\stackrel{^}{u}}_{t}={y}_{t}â\stackrel{^}{c}â{X}_{t}\stackrel{^}{\mathrm{β}}
Estimating the ARIMA error model,
{\stackrel{^}{u}}_{t}={\mathrm{Î}}^{â1}\left(L\right)\mathrm{Î}\left(L\right){\mathrm{ε}}_{t},
where H(L) is the compound autoregressive polynomial and N(L) is the compound moving average polynomial
Inferring the innovations from the ARIMA error model,
{\stackrel{^}{\mathrm{ε}}}_{t}={\stackrel{^}{\mathrm{Î}}}^{â1}\left(L\right)\stackrel{^}{\mathrm{Î}}\left(L\right){\stackrel{^}{u}}_{t}
f\left({\mathrm{ε}}_{1},...,{\mathrm{ε}}_{T}\text{|}{H}_{Tâ1}\right)=\underset{t=1}{\overset{T}{â}}f\left({\mathrm{ε}}_{t}{\text{|H}}_{tâ1}\right),
logL=â\frac{T}{2}\mathrm{log}\left(2\mathrm{Ï}\right)â\frac{T}{2}\mathrm{log}{\mathrm{Ï}}^{2}â\frac{1}{2{\mathrm{Ï}}^{2}}\underset{t=1}{\overset{T}{â}}{\mathrm{ε}}_{t}^{2}.
If zt is a standardized Student’s t, then the loglikelihood objective function is
logL=T\mathrm{log}\left[\frac{\mathrm{Î}\left(\frac{\mathrm{ν}+1}{2}\right)}{\mathrm{Î}\left(\frac{\mathrm{ν}}{2}\right)\sqrt{\mathrm{Ï}\left(\mathrm{ν}â2\right)}}\right]â\frac{T}{2}{\mathrm{Ï}}^{2}â\frac{\mathrm{ν}+1}{2}\underset{t=1}{\overset{T}{â}}log\left[1+\frac{{\mathrm{ε}}_{t}^{2}}{{\mathrm{Ï}}^{2}\left(\mathrm{ν}â2\right)}\right].
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!! used as default html header if there is none in the selected theme. Coincidence Param
t
F\left(t\right),G\left(t\right)
Description: find the best approximation of a parametric curve. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, geometry, functions, curves, parametric_curves
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Estimation of actual maximum kVA demand - Electrical Installation Guide
HomeGeneral rules of electrical installation designPower loading of an installationEstimation of actual maximum kVA demand
1 Factor of maximum utilization (ku)
2 Diversity factor - Coincidence factor (ks)
3 Diversity factor for an apartment block
4 Rated Diversity Factor for distribution switchboards
5 Diversity factor according to circuit function
All individual loads are not necessarily operating at full rated nominal power nor necessarily at the same time. Factors ku and ks allow the determination of the maximum power and apparent-power demands actually required to dimension the installation.
This factor must be applied to each individual load, with particular attention to electric motors, which are very rarely operated at full load.
In an industrial installation this factor may be estimated on an average at 0.75 for motors.
For Electric Vehicle the utilization factor will be systematically estimated to 1, as it takes a long time to load completely the batteries (several hours) and a dedicated circuit feeding the charging station or wall box will be required by standards.
Diversity factor - Coincidence factor (ks)
It is a matter of common experience that the simultaneous operation of all installed loads of a given installation never occurs in practice, i.e. there is always some degree of diversity and this fact is taken into account for estimating purposes by the use of a factor (ks).
This factor is defined in IEC60050 - International Electrotechnical Vocabulary, as follows:
Coincidence factor = The ratio, expressed as a numerical value or as a percentage, of the simultaneous maximum demand of a group of electrical appliances or consumers within a specified period, to the sum of their individual maximum demands within the same period. As per this definition, the value is always ≤ 1 and can be expressed as a percentage
Diversity factor = The reciprocal of the coincidence factor. It means it will always be ≥ 1.
Note: In practice, the most commonly used term is the diversity factor, but it is used in replacement of the coincidence factor, thus will be always ≤ 1. The term "simultaneity factor" is another alternative that is sometimes used.
The factor ks is applied to each group of loads (e.g. being supplied from a distribution or sub-distribution board).
The following tables are coming from local standards or guides, not from international standards. They should only be used as examples of determination of such factors. See also the specific case of Electric Vehicle charging application.
Some typical values for this case are given in Figure A11, and are applicable to domestic consumers without electrical heating, and supplied at 230/400 V (3-phase 4-wires). In the case of consumers using electrical heat-storage units for space heating, a factor of 0.8 is recommended, regardless of the number of consumers.
Fig. A11 – Example of diversity factors for an apartment block as defined in French standard NFC14-100, and applicable for apartments without electrical heating
Number of downstream consumers
Diversity factor (ks)
Example (see Fig. A12):
5 storeys apartment building with 25 consumers, each having 6 kVA of installed load.
From Figure A12, it is possible to determine the magnitude of currents in different sections of the common main feeder supplying all floors. For vertical rising mains fed at ground level, the cross-sectional area of the conductors can evidently be progressively reduced from the lower floors towards the upper floors.
These changes of conductor size are conventionally spaced by at least 3-floor intervals.
In the example, the current entering the rising main at ground level is:
{\displaystyle {\frac {150\times 0.46\times 10^{3}}{400{\sqrt {3}}}}=100A}
the current entering the third floor is:
{\displaystyle {\frac {\left(36+24\right)\times 0.63\times 10^{3}}{400{\sqrt {3}}}}=55A}
Fig. A12 – Application of the diversity factor (ks) to an apartment block of 5 storeys
The standards IEC61439-1 and 2 define in a similar way the Rated Diversity Factor for distribution switchboards (in this case, always ≤ 1)
IEC61439-2 also states that, in the absence of an agreement between the assembly manufacturer (panel builder) and user concerning the actual load currents (diversity factors), the assumed loading of the outgoing circuits of the assembly or group of outgoing circuits may be based on the values in Fig. A13.
If the circuits are mainly for lighting loads, it is prudent to adopt ks values close to unity.
Fig. A13 – Rated diversity factor for distribution boards (cf IEC61439-2 table 101)
Assumed loading factor
Distribution - 6 to 9 circuits 0.7
ks factors which may be used for circuits supplying commonly-occurring loads, are shown in Figure A14. It is provided in French practical guide UTE C 15-105.
Fig. A14 – Diversity factor according to circuit function (see UTE C 15-105 table AC)
Socket-outlets 0.1 to 0.2[a]
Lifts and catering hoist[b] For the most powerful motor 1
For all motors 0.60
^ In certain cases, notably in industrial installations, this factor can be higher.
^ The current to take into consideration is equal to the nominal current of the motor, increased by a third of its starting current.
Retrieved from "http://www.electrical-installation.org/enw/index.php?title=Estimation_of_actual_maximum_kVA_demand&oldid=27701"
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Price Elasticity of Supply - Course Hero
Microeconomics/Elasticity/Price Elasticity of Supply
Learn all about price elasticity of supply in just a few minutes! Professor Jadrian Wooten of Penn State University explains how price elasticity of supply information impacts a firm's supply decisions.
Much as consumers respond to changes in the price of a good by changing their demand for it, producers respond to changes in the price of a good by changing the amount of it supplied. The law of supply, which states that all other things being equal, an increase in the price of a good will result in an increase in the quantity of the good supplied, means the quantity supplied has a positive or direct relationship with price. The price elasticity of supply is a measure of how responsive the quantity supplied of a good is to changes in its price. Price elasticity of supply (ES) is calculated by dividing the percentage change in quantity supplied (QS) by the percentage change in price (P).
\text{E}_{\text{S}}=\frac{\%\Delta \text{Q}_{\text{S}}}{\%\Delta \text{P}}
Due to the positive relationship between price and quantity supplied, the price elasticity of supply will usually be greater than or equal to zero.
For example, if the price of fingernail clippers suddenly dropped by 50%, the number of fingernail clippers produced (the supply) would also drop. If the quantity supplied dropped by 30%, the price elasticity of supply would be
-30\%/-50\%=+0.6
Classifying the Price Elasticity of Supply
Like demand, supply can be classified as as being elastic or inelastic. Elastic supply occurs when the quantity of a good or service that producers supply is relatively sensitive to changes in price; the percentage change in price is not as large as the percentage change in quantity supplied. This means that if the price of a good or service decreases by 10%, sellers reduce their quantity supplied by more than 10%. In this case, sellers are relatively sensitive to changes in the price of their product because the price greatly impacts the change in quantity.
Inelastic supply arises when the supply of a good or service does not change according to the price of that good or service. When it is calculated, the price elasticity of supply is less than 1. In this case, a 10% increase in the price of a good causes sellers to increase the quantity supplied by less than 10%. Thus, the sellers are relatively insensitive to price changes, because price changes do not greatly impact the quantity.
For example, suppose there is a 10% increase in the price of goat cheese. Farms then decide to increase their production by 5%. This means there is a price elasticity of supply (which is inelastic):
\frac{+5\%}{+10\%}=+0.5
Extreme Cases of the Price Elasticity of Supply
When any change in price causes an infinite change in supply, there is a perfectly elastic supply. In this case, sellers are so sensitive to a drop in the price of their product that they reduce the quantity supplied to zero. The supply curve is a horizontal line. This is a hypothetical extreme that does not exist in reality, because it is difficult to make an infinite amount of a product, and situations often change.
A perfectly inelastic supply does not change in response to changes in price; that is, the price elasticity of supply is equal to zero. Here, any change in price of a good or service leads to no change in the quantity supplied; the supply curve is vertical. For example, there is only one of each painting by Leonardo da Vinci; each piece is all that can be supplied to the market, regardless of price.
When the price elasticity of supply is infinite, supply is a perfectly elastic. Sellers are so sensitive to a drop in the price that they reduce the quantity supplied to zero. When the price elasticity of supply is equal to zero, supply is perfectly inelastic. Regardless of a change in price, there is no change in quantity supplied.
<Cross-Price Elasticity of Demand>Suggested Reading
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EUDML | Homotopy type of symplectomorphism groups of . EuDML | Homotopy type of symplectomorphism groups of .
Homotopy type of symplectomorphism groups of
{S}^{2}×{S}^{2}
Anjos, Silvia. "Homotopy type of symplectomorphism groups of .." Geometry & Topology 6 (2002): 195-218. <http://eudml.org/doc/122986>.
@article{Anjos2002,
author = {Anjos, Silvia},
keywords = {symplectomorphism group; Pontryagin ring; homotopy equivalence},
title = {Homotopy type of symplectomorphism groups of .},
AU - Anjos, Silvia
TI - Homotopy type of symplectomorphism groups of .
KW - symplectomorphism group; Pontryagin ring; homotopy equivalence
Jarek Kędra, Fundamental group of
Symp\left(M,\omega \right)
with no circle action
symplectomorphism group, Pontryagin ring, homotopy equivalence
Homology and cohomology of
H
Articles by Anjos
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EUDML | Eigenvalue distribution of integral operators defined by Orlicz kernels. EuDML | Eigenvalue distribution of integral operators defined by Orlicz kernels.
Eigenvalue distribution of integral operators defined by Orlicz kernels.
Elshobaky, E.; Faragallah, M.
Elshobaky, E., and Faragallah, M.. "Eigenvalue distribution of integral operators defined by Orlicz kernels.." Southwest Journal of Pure and Applied Mathematics [electronic only] 2 (1998): 1-6. <http://eudml.org/doc/228619>.
@article{Elshobaky1998,
author = {Elshobaky, E., Faragallah, M.},
keywords = {eigenvalue distribution; interpolation of vector-valued - and Besov-sequence spaces; Orlicz-type sequence spaces; generalized Schatten classes; interpolation of vector-valued - and Besov-sequence spaces},
title = {Eigenvalue distribution of integral operators defined by Orlicz kernels.},
AU - Elshobaky, E.
AU - Faragallah, M.
TI - Eigenvalue distribution of integral operators defined by Orlicz kernels.
KW - eigenvalue distribution; interpolation of vector-valued - and Besov-sequence spaces; Orlicz-type sequence spaces; generalized Schatten classes; interpolation of vector-valued - and Besov-sequence spaces
eigenvalue distribution, interpolation of vector-valued
{\ell }_{p}
- and Besov-sequence spaces, Orlicz-type sequence spaces, generalized Schatten classes, interpolation of vector-valued
{\ell }_{p}
- and Besov-sequence spaces
Banach sequence spaces
p
Articles by Elshobaky
Articles by Faragallah
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How to Calculate Bond Equivalent Yield: 11 Steps (with Pictures)
How to Calculate Bond Equivalent Yield
2 Calculating Bond Equivalent Yield
3 Using Bond Equivalent Yield
Bond equivalent yield (or BEY) is a tool for determining the annual yield on a discount bond or note. For bonds that do not have an annual yield clearly stated, investors can convert the stated yield into an annual yield by using the bond equivalent yield calculation. BEY is useful in comparing different bonds for the purpose of analysis and investing, as it allows the analyst to make useful comparisons between bonds with annual payments and those with more frequent payments (such as quarterly or monthly).[1] X Research source
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Decide to use bond equivalent yield. The BEY calculation is used to compare fixed-income securities (bonds and notes) with other fixed-income securities in a relative way, regardless of how frequently they make payments. To calculate BEY, you will need the price of the bond, the par value (face value), and the number of days to maturity.
For example, the BEY would be useful if you wanted to compare returns on two bonds, perhaps a six month bond and a twelve month bond with otherwise similar terms. Obviously, the overall return on the six month bond would usually be lower than the other bond. However, the BYE calculation allows you to calculate the annual yield for both bonds and compare them on a relative basis.
You can also consider using an annual percentage rate (APR) calculation. The APR is commonly used in evaluating consumer debt. This valuation can also be useful for investors and others who want to calculate bond yield. The annual percentage rate is simply the interest rate expressed in annual terms and can be complementary to a BEY calculation.
The APR calculation can be used if the yield to maturity is known.[2] X Research source
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Note the par value, or face value, of the bond. This is the amount that will be paid to the bond holder at maturity. Bonds often have a par value of $1,000 or, more rarely, $100. Bonds sold at a discount, like those being calculated here, are sold at a lower price than the par value. For example, a $1,000 bond might sell for $975.
The par value is clearly stated in the bond offering.
Par value is also used to calculate coupon payments (interest payouts) in coupon-paying bonds.[3] X Research source
Determine the price of the bond. The price of the bond is the price that the bondholder pays to acquire the bond in the market. Again, for discount bonds this number is lower than the par value of the bond. If you already own the bond, you should use your purchase price, which should be in your records. If not, you should use the current stated market price.[4] X Research source
Calculate the days to maturity. Start by finding the maturity date, which should be listed in the bond offering. This date represents the day when the par value of the bond is paid out to the bondholder. To find days to maturity, calculate the actual number of days from today until that date.[5] X Research source
Calculating Bond Equivalent Yield
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Learn the bond equivalent yield equation. The equation for BEY is essentially two calculations that are multiplied together. One side gives the return on investment and the other converts that return into an annualized yield. The equation is as follows:
{\displaystyle BEY={\frac {FV-P}{P}}*{\frac {365}{d}}}
In the equation, the variables stand for the following:
BEY is the bond equivalent yield.
FV is the face value (also called the par value).
P is the purchase price of the bond.
d is the days to maturity.[6] X Research source
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Input your variables. Place your values for the bond you are calculating BEY for into the equation. Double check it to make sure everything is in the right place; it's easy to mix up par value and purchase price when inputting these figures.
For example, imagine you are calculating BEY for a bond with a $1,000 face value, $980 purchase price, and 90 days until maturity. Your completed equation would look like this:
{\displaystyle BEY={\frac {\$1,000-\$980}{\$980}}*{\frac {365}{90}}}
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Solve the right side of the equation. Subtract the top part of the right side first. Then, divide the result by the bottom value to solve the right side of the equation.
The example equation would look like this after the first calculation:
{\displaystyle BEY={\frac {\$20}{\$980}}*{\frac {365}{90}}}
And then after the second:
{\displaystyle BEY=0.0204*{\frac {365}{90}}}
Note that this result, 0.0204, is a rounded figure. If you do not round this number, your final result may be different.
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Divide the right side of the equation. Simply divide 365 by the days to maturity to solve the right side of the equation.
The example equation solves like this:
{\displaystyle BEY=0.0204*4.056}
This result, 4.056 is also rounded.
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Multiply the two sides together. Solve for BEY by multiplying the results of your previous calculations together. In the example, this would give
{\displaystyle BEY=0.827}
This result has also been rounded.
The BEY is usually expressed as a percentage. To convert it, multiply your answer by 100. So, the example would be 0.827*100, or 8.27%.[7] X Research source
Using Bond Equivalent Yield
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Compare various bonds using the bond equivalent yield. With the BEY in hand, you can compare semiannual, quarterly, monthly or other bonds to annual government bonds or other stable debt instruments.[8] X Research source Comparing your bond to other bonds is also simple with BEY because newspapers quote yields in BEY terms.[9] X Research source
Learn about the use of BEY with shorter-term debt instruments. For example, a money-market product generally has a much shorter maturity time than the average bond. Finance pros use more complicated BEY calculations to compare a long-term bond in its last several months before maturity or to compare a money-market product to something else like a long-term government bond.
You can also calculate bond yields quickly using an online calculator. Try searching for them using a search engine.[10] X Research source
↑ http://www.investopedia.com/terms/b/bey.asp
↑ http://finance.zacks.com/compute-bond-equivalent-yield-effective-annual-rate-6069.html
↑ http://www.financeformulas.net/Bond_Equivalent_Yield.html
↑ http://www.investinganswers.com/financial-dictionary/bonds/bond-equivalent-yield-bey-729
↑ http://www.miniwebtool.com/bond-equivalent-yield-calculator/
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Solve Sudoku Puzzles Via Integer Programming: Solver-Based - MATLAB & Simulink - MathWorks Switzerland
Write the Rules for Sudoku
Call the Sudoku Solver
This example shows how to solve a Sudoku puzzle using binary integer programming. For the problem-based approach, see Solve Sudoku Puzzles Via Integer Programming: Problem-Based.
This approach is particularly simple because you do not give a solution algorithm. Just express the rules of Sudoku, express the clues as constraints on the solution, and then intlinprog produces the solution.
The key idea is to transform a puzzle from a square 9-by-9 grid to a cubic 9-by-9-by-9 array of binary values (0 or 1). Think of the cubic array as being 9 square grids stacked on top of each other. The top grid, a square layer of the array, has a 1 wherever the solution or clue has a 1. The second layer has a 1 wherever the solution or clue has a 2. The ninth layer has a 1 wherever the solution or clue has a 9.
The objective function is not needed here, and might as well be 0. The problem is really just to find a feasible solution, meaning one that satisfies all the constraints. However, for tie breaking in the internals of the integer programming solver, giving increased solution speed, use a nonconstant objective function.
x
x
x\left(i,j,1\right),...,x\left(i,j,9\right)
i
j
\sum _{k=1}^{9}x\left(i,j,k\right)=1.
i
i
k
\sum _{j=1}^{9}x\left(i,j,k\right)=1.
j
j
k
\sum _{i=1}^{9}x\left(i,j,k\right)=1.
1\le i\le 3
1\le j\le 3
1\le k\le 9
\sum _{i=1}^{3}\sum _{j=1}^{3}x\left(i,j,k\right)=1.
i
j
\sum _{i=1}^{3}\sum _{j=1}^{3}x\left(i+U,j+V,k\right)=1,
U,V\phantom{\rule{0.5em}{0ex}}ϵ\phantom{\rule{0.5em}{0ex}}\left\{0,3,6\right\}.
\left(i,j\right)
m
1\le m\le 9
x\left(i,j,m\right)=1
\sum _{k=1}^{9}x\left(i,j,k\right)=1
x\left(i,j,k\right)=0
k\ne m
Although the Sudoku rules are conveniently expressed in terms of a 9-by-9-by-9 solution array x, linear constraints are given in terms of a vector solution matrix x(:). Therefore, when you write a Sudoku program, you have to use constraint matrices derived from 9-by-9-by-9 initial arrays.
Here is one approach to set up Sudoku rules, and also include the clues as constraints. The sudokuEngine file comes with your software.
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EUDML | Some analogies between number theory and dynamical systems on foliated spaces. EuDML | Some analogies between number theory and dynamical systems on foliated spaces.
Some analogies between number theory and dynamical systems on foliated spaces.
Deninger, Christopher. "Some analogies between number theory and dynamical systems on foliated spaces.." Documenta Mathematica (1998): 23-46. <http://eudml.org/doc/229449>.
keywords = {-functions of motives; leafwise cohomology; dynamical systems; foliations; -functions; zeta functions; -functions of motives; -functions},
title = {Some analogies between number theory and dynamical systems on foliated spaces.},
TI - Some analogies between number theory and dynamical systems on foliated spaces.
KW - -functions of motives; leafwise cohomology; dynamical systems; foliations; -functions; zeta functions; -functions of motives; -functions
Christopher Deninger, Wilhelm Singhof, A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas
Jeffrey C. Lagarias, Li coefficients for automorphic
L
L
-functions of motives, leafwise cohomology, dynamical systems, foliations,
L
-functions, zeta functions,
L
-functions of motives,
L
Abelian varieties of dimension
>1
L
𝐙
𝐑
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-Step Derivations on -Groupoids: The Case
N. O. Alshehri, Hee Sik Kim, J. Neggers, " -Step Derivations on -Groupoids: The Case ", The Scientific World Journal, vol. 2014, Article ID 726470, 6 pages, 2014. https://doi.org/10.1155/2014/726470
N. O. Alshehri,1 Hee Sik Kim ,2 and J. Neggers3
1Department of Mathematics, King Abdulaziz University, Faculty of Science for Girls, Jeddah, Saudi Arabia
2Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea
3Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA
Academic Editor: Xiao-Long Xin
We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping is a self-derivation at each step. Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet on , consisting of a two-step derivation and its square , for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.
The notion of derivations arising in analytic theory is extremely helpful in exploring the structures and properties of algebraic systems. Several authors [1, 2] studied derivations in rings and near rings. Jun and Xin [3] applied the notion of derivation in ring and near ring theory to -algebras. In [4], the concept of derivation for lattices was introduced and some of its properties are investigated. For more details, the reader is referred to [3, 5–7].
Iséki and Tanaka introduced two classes of abstract algebras: -algebras and -algebras [8, 9]. Neggers and Kim introduced the notion of -algebras which is another useful generalization of -algebras and then investigated several relations between -algebras and -algebras as well as several other relations between -algebras and oriented digraphs [10]. Kim and Neggers [11] introduced the notion of and obtained a semigroup structure. Bell and Kappe [1] studied rings in which derivations satisfy certain algebraic conditions. Alshehri [12] applied the notion of derivations in incline algebras.
The present authors [13] introduced the notion of ranked bigroupoids and discussed -self-(co)derivations. In addition, they defined rankomorphisms and -scalars for ranked bigroupoids and obtained some properties of these as well. Recently, Jun et al. [14] obtained further results on derivations of ranked bigroupoids, and Jun et al. [15] introduced the notion of generalized coderivations in ranked bigroupoids and showed that new generalized coderivations of ranked bigroupoids are obtained by combining a generalized self-coderivation with a rankomorphism.
In this paper, we extend the theory of derivations on a ranked bigroupoid to that of a type of derivation on ranked trigroupoids, that is, two-step derivations on ranked trigroupoids considered as a couple of ranked bigroupoids and with such a two-step derivation on if it is a self-derivation on both and . The role of the operation in this definition is the more interesting one since it acts as the minor operation in and the major operation in . From the results obtained below it is clear that it is indeed possible to obtain meaningful insights, especially via the notion of a couplet on a ranked trigroupoid consisting of a pair of mappings , satisfying a natural condition (6) stated below which arises in a rather natural way from the context and is seen to be of interest in this study and presumably of any followup as well.
An -algebra [10] is a nonempty set with a constant and a binary operation “ ” satisfying the following axioms:(A) ,(B) ,(C) and imply for all .
A -algebra is a -algebra satisfying the following additional axioms:(D) ,(E) for all .
Given a nonempty set , we let denote the collection of all groupoids , where is a map and where is written in the usual product form. Given elements and of , define a product “ ” on these groupoids as follows: where for any . Using that notion, Kim and Neggers proved the following theorem.
Theorem 1 (see [11]). is a semigroup; that is, the operation “ ” as defined in general is associative. Furthermore, the left-zero-semigroup is the identity for this operation.
A ranked bigroupoid is an algebraic system where is a nonempty set and “ ” and “ ” are binary operations defined on . We may consider the first binary operation as the major operation, and the second binary operation as the minor operation.
Example 2 (see [16]). A -algebra is defined as an algebraic system where is a group and where . Hence every -algebra is a ranked bigroupoid.
Example 3 (see [13]). We construct a ranked bigroupoid from any -algebra. In fact, given a -algebra , if we define a binary operation “ ” on by for any , then is a ranked bigroupoid.
We introduce the notion of “ranked bigroupoids” to distinguish two bigroupoids and . Even though in the sense of bigroupoids, the same is not true in the sense of ranked bigroupoids. This is analogous to the situation for sets, where but in general.
Given an element , has a natural associated ranked bigroupoid ; that is, the major operation and the minor operation coincide.
Given a ranked bigroupoid , a map is called an -self-derivation if for all , In the same setting, a map is called an -self-coderivation if for all ,
Note that if is a commutative groupoid, then -self-derivations are -self-coderivations. A map is called an abelian- -self-derivation if it is both an -self-derivation and an -self-coderivation.
3. Two-Step Derivations and Couplets on Trigroupoids
An algebraic system is said to be a ranked trigroupoid if algebraic systems and are ranked bigroupoids. A two-step derivation on a ranked trigroupoid is a mapping such that is both an -self-derivation and an -self-derivation.
Obviously, if one considers ranked -groupoids , then one may consider -step derivations for which one has as an -self-derivation for .
In this paper we will mostly be interested in the case of two-step-derivations on ranked trigroupoids and some related pairs of maps which we call couplets. For ranked -groupoids where , we obtain triplets, quadruplets, and so forth, as the appropriate generalizations.
Let be a two-step derivation on a ranked trigroupoid . Then, for any , we have
If we let , then it follows that
We call a couplet on a ranked trigroupoid if it satisfies condition (6), and if contains a constant 0.
Example 4. Let be the set of all real numbers and let be the ranked trigroupoid where and are usual multiplication, addition, and subtraction, respectively. If we let be a couplet on the ranked trigroupoid , then
If we let in (7), then
Thus for all , whence for all . If we let in (7), then . It follows that, for any , which implies that . Hence, by (7), we have ; that is,
If we let and in (9), then we have . Hence for all .
In Example 4, if is a two-step derivation on , then and for any . It follows that , which proves that for all .
Example 5. Let and let be a ranked trigroupoid where , + are usual multiplication and addition, respectively. Define a map satisfying for all . For , , , we define a map by where and . Then is a -self-derivation. In fact, if , for some , then Assume that is a couplet on the ranked trigroupoid for some self- -derivation . Then, Since , we obtain
Proposition 6. There is no two-step derivation on the ranked trigroupoid .
Proof. Assume that is a two-step derivation on . Then, for any ,
If we let in (15), then If we let in (16), then . On the other hand, if we let and in (15), then , proving that is a contradiction. Hence there is no two-step derivation on the ranked trigroupoid .
Proof. Assume that is a two-step derivation on . Then, for any , If we let in (19), then and hence for all . If we let in (18), then , for any , is a contradiction.
Proposition 8. Let be a couplet on a ranked trigroupoid . If and , then for all . In particular, if , then is a -self-derivation.
Proof. If is a couplet on a ranked trigroupoid , then, for any , we have Since , if we let in (20), then . It follows that Hence for any . If we change into for any , then we obtain for all . In particular, if , then and hence ; that is, is a -self-derivation.
4. Frame Algebras and -Algebras
A groupoid is said to be a frame algebra if it satisfies the axioms , and(F) ,
Example 9. Every -algebra is a frame algebra.
Every lattice implication algebra (see [17, 18]) is a frame algebra.
The collection of frame algebras includes the collection of -algebras and it is a variety. In a frame algebra the element 0 is unique. Indeed, if and are both zeros, then , yields .
Proposition 10. The collection of all frame algebras forms a subsemigroup of the semigroup .
Proof. Given frame algebras , if we let , then for all . It follows that , , and , proving that is a frame algebra. This proves the proposition.
Given groupoids , we define
Proposition 11. Let and be frame algebras. If , then is a frame algebra.
Proof. If , then for all . It follows that implies that . Moreover, shows that , and shows that , proving that is a frame algebra.
A ranked trigroupoid is called an fr( )-algebra if(G) are frame algebras,(H) .
Theorem 12. Let be an fr -algebra. If is a two-step derivation on , then (i) ,(ii) for all .
Proof. (i) If is a two-step derivation on , then for any , we have It follows that ; that is, , for all . If we let , then by applying (A) we obtain .
(ii) Given , we have and .
Given a two-step derivation on a trigroupoid , we denote its kernel by .
Proposition 13. Let be an fr -algebra. If is a two-step derivation on , then (i) , ,(ii) implies that , ,(iii) ,for all .
Proof. (i) If we let in (24) and (25), respectively, then an , proving that , for any .
(ii) If , then and for any , proving that , .
(iii) If , then by Theorem 12, which shows that .
Proposition 14. Let be an fr -algebra. If is a two-step derivation on , then for all .
Proof. If , then and hence for all , which proves that .
Note that may not be computable unless the behavior of is specified, since for some .
Proposition 15. Let be an fr -algebra. If is a couplet of , then .
Proof. If is a couplet of and if , then for any . It follows from (6) that , proving that .
Let be a poset with minimal element 0. Define a binary operation “ ” on by Then is a -algebra, called a standard -algebra inherited from the poset .
Proposition 16. Let be a standard -algebra. Let be an fr -algebra and let be a couplet of . If for some , then and .
Proof. Let . We claim that . Suppose that . Since is a fr -algebra, by applying (6), we obtain that is a contradiction. Since is a standard -algebra, we obtain . We claim that . If , then is a contradiction. It follows that , proving the proposition.
5. Classification of Linear Ranked Trigroupoids
Let be the real field and let be a ranked trigroupoid, where and are linear groupoids; that is, , , and for any , where (fixed).
Thus, if is a two-step derivation on , then for any , we have and also in a similar manner we obtain
If for all , then (29) and (30) become for any . If we let , in (31), then we obtain , . Hence, (31) reduces to for all . If we let , and let , in (32) and (33), respectively, then we obtain From this information, we obtain the following propositions.
Proposition 17. Let be the real field and let be a ranked trigroupoid, where are linear groupoids; that is, , , and for any , where (fixed). Let be a two-step derivation such that for all . (1)If and , then , , ;(2)If and , then , .
Proof. (i) If , then it follows from (34) that we obtain , . If , then we have and .
(ii) If and , then and is arbitrary, and hence we obtain the result.
Proposition 18. Let be the real field and let be a ranked trigroupoid, where are linear groupoids; that is, , , and for any , where (fixed). Let be a two-step derivation such that for all :(i)if and , then , , ;(ii)if , , then , , ;(iii)if , , , then , , ;(iv)if , , , then , , ;(v)if , , , then , , .
Proof. The proof is similar to Proposition 17 and we omit it.
In Propositions 17 and 18, we observe that there are 6 different types of linear ranked trigroupoids in the special case of for all , and most of them are classified by the properties of in .
The notion of two-step derivations is a generalization of derivations. This leads to the study of trigroupoids, and we explore some relations with several algebras, for example, -algebras, frame algebras, and so forth. The classification of linear ranked trigroupoids then explains a number of concrete algebraic structures with derivations.
The authors are grateful to the referee for valuable suggestions and help.
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Copyright © 2014 N. O. Alshehri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Elliptic filter - Wikipedia
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not).[citation needed] Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.
As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter.
The gain of a lowpass elliptic filter as a function of angular frequency ω is given by:
{\displaystyle G_{n}(\omega )={1 \over {\sqrt {1+\epsilon ^{2}R_{n}^{2}(\xi ,\omega /\omega _{0})}}}}
where Rn is the nth-order elliptic rational function (sometimes known as a Chebyshev rational function) and
{\displaystyle \omega _{0}}
is the cutoff frequency
{\displaystyle \epsilon }
is the ripple factor
{\displaystyle \xi }
is the selectivity factor
The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple.
2 Poles and zeroes
3 Minimum Q-factor elliptic filters
4 Comparison with other linear filters
The frequency response of a fourth-order elliptic low-pass filter with ε = 0.5 and ξ = 1.05. Also shown are the minimum gain in the passband and the maximum gain in the stopband, and the transition region between normalized frequency 1 and ξ
A closeup of the transition region of the above plot.
In the passband, the elliptic rational function varies between zero and unity. The gain of the passband therefore will vary between 1 and
{\displaystyle 1/{\sqrt {1+\epsilon ^{2}}}}
In the stopband, the elliptic rational function varies between infinity and the discrimination factor
{\displaystyle L_{n}}
which is defined as:
{\displaystyle L_{n}=R_{n}(\xi ,\xi )\,}
The gain of the stopband therefore will vary between 0 and
{\displaystyle 1/{\sqrt {1+\epsilon ^{2}L_{n}^{2}}}}
{\displaystyle \xi \rightarrow \infty }
the elliptic rational function becomes a Chebyshev polynomial, and therefore the filter becomes a Chebyshev type I filter, with ripple factor ε
Since the Butterworth filter is a limiting form of the Chebyshev filter, it follows that in the limit of
{\displaystyle \xi \rightarrow \infty }
{\displaystyle \omega _{0}\rightarrow 0}
{\displaystyle \epsilon \rightarrow 0}
{\displaystyle \epsilon \,R_{n}(\xi ,1/\omega _{0})=1}
the filter becomes a Butterworth filter
{\displaystyle \xi \rightarrow \infty }
{\displaystyle \epsilon \rightarrow 0}
{\displaystyle \omega _{0}\rightarrow 0}
{\displaystyle \xi \omega _{0}=1}
{\displaystyle \epsilon L_{n}=\alpha }
, the filter becomes a Chebyshev type II filter with gain
{\displaystyle G(\omega )={\frac {1}{\sqrt {1+{\frac {1}{\alpha ^{2}T_{n}^{2}(1/\omega )}}}}}}
Poles and zeroes[edit]
Log of the absolute value of the gain of an 8th order elliptic filter in complex frequency space (s = σ + jω) with ε = 0.5, ξ = 1.05 and ω0 = 1. The white spots are poles and the black spots are zeroes. There are a total of 16 poles and 8 double zeroes. What appears to be a single pole and zero near the transition region is actually four poles and two double zeroes as shown in the expanded view below. In this image, black corresponds to a gain of 0.0001 or less and white corresponds to a gain of 10 or more.
An expanded view in the transition region of the above image, resolving the four poles and two double zeroes.
The zeroes of the gain of an elliptic filter will coincide with the poles of the elliptic rational function, which are derived in the article on elliptic rational functions.
The poles of the gain of an elliptic filter may be derived in a manner very similar to the derivation of the poles of the gain of a type I Chebyshev filter. For simplicity, assume that the cutoff frequency is equal to unity. The poles
{\displaystyle (\omega _{pm})}
of the gain of the elliptic filter will be the zeroes of the denominator of the gain. Using the complex frequency
{\displaystyle s=\sigma +j\omega }
{\displaystyle 1+\epsilon ^{2}R_{n}^{2}(-js,\xi )=0\,}
{\displaystyle -js=\mathrm {cd} (w,1/\xi )}
where cd() is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:
{\displaystyle 1+\epsilon ^{2}\mathrm {cd} ^{2}\left({\frac {nwK_{n}}{K}},{\frac {1}{L_{n}}}\right)=0\,}
{\displaystyle K=K(1/\xi )}
{\displaystyle K_{n}=K(1/L_{n})}
. Solving for w
{\displaystyle w={\frac {K}{nK_{n}}}\mathrm {cd} ^{-1}\left({\frac {\pm j}{\epsilon }},{\frac {1}{L_{n}}}\right)+{\frac {mK}{n}}}
where the multiple values of the inverse cd() function are made explicit using the integer index m.
The poles of the elliptic gain function are then:
{\displaystyle s_{pm}=i\,\mathrm {cd} (w,1/\xi )\,}
As is the case for the Chebyshev polynomials, this may be expressed in explicitly complex form (Lutovac & et al. 2001, § 12.8) harv error: no target: CITEREFLutovacet_al.2001 (help)
{\displaystyle s_{pm}={\frac {a+jb}{c}}}
{\displaystyle a=-\zeta _{n}{\sqrt {1-\zeta _{n}^{2}}}{\sqrt {1-x_{m}^{2}}}{\sqrt {1-x_{m}^{2}/\xi ^{2}}}}
{\displaystyle b=x_{m}{\sqrt {1-\zeta _{n}^{2}(1-1/\xi ^{2})}}}
{\displaystyle c=1-\zeta _{n}^{2}+x_{i}^{2}\zeta _{n}^{2}/\xi ^{2}}
{\displaystyle \zeta _{n}}
{\displaystyle n,\,\epsilon }
{\displaystyle \xi }
{\displaystyle x_{m}}
are the zeroes of the elliptic rational function.
{\displaystyle \zeta _{n}}
is expressible for all n in terms of Jacobi elliptic functions, or algebraically for some orders, especially orders 1,2, and 3. For orders 1 and 2 we have
{\displaystyle \zeta _{1}={\frac {1}{\sqrt {1+\epsilon ^{2}}}}}
{\displaystyle \zeta _{2}={\frac {2}{(1+t){\sqrt {1+\epsilon ^{2}}}+{\sqrt {(1-t)^{2}+\epsilon ^{2}(1+t)^{2}}}}}}
{\displaystyle t={\sqrt {1-1/\xi ^{2}}}}
The algebraic expression for
{\displaystyle \zeta _{3}}
is rather involved (See Lutovac & et al. (2001, § 12.8.1) harvtxt error: no target: CITEREFLutovacet_al.2001 (help)).
The nesting property of the elliptic rational functions can be used to build up higher order expressions for
{\displaystyle \zeta _{n}}
{\displaystyle \zeta _{m\cdot n}(\xi ,\epsilon )=\zeta _{m}\left(\xi ,{\sqrt {{\frac {1}{\zeta _{n}^{2}(L_{m},\epsilon )}}-1}}\right)}
{\displaystyle L_{m}=R_{m}(\xi ,\xi )}
Minimum Q-factor elliptic filters[edit]
The normalized Q-factors of the poles of an 8-th order elliptic filter with ξ = 1.1 as a function of ripple factor ε. Each curve represents four poles, since complex conjugate pole pairs and positive-negative pole pairs have the same Q-factor. (The blue and cyan curves nearly coincide). The Q-factor of all poles are simultaneously minimized at εQmin = 1 / √Ln = 0.02323...
See Lutovac & et al. (2001, § 12.11, 13.14) harvtxt error: no target: CITEREFLutovacet_al.2001 (help).
Elliptic filters are generally specified by requiring a particular value for the passband ripple, stopband ripple and the sharpness of the cutoff. This will generally specify a minimum value of the filter order which must be used. Another design consideration is the sensitivity of the gain function to the values of the electronic components used to build the filter. This sensitivity is inversely proportional to the quality factor (Q-factor) of the poles of the transfer function of the filter. The Q-factor of a pole is defined as:
{\displaystyle Q=-{\frac {|s_{pm}|}{2\mathrm {Re} (s_{pm})}}=-{\frac {1}{2\cos(\arg(s_{pm}))}}}
and is a measure of the influence of the pole on the gain function. For an elliptic filter, it happens that, for a given order, there exists a relationship between the ripple factor and selectivity factor which simultaneously minimizes the Q-factor of all poles in the transfer function:
{\displaystyle \epsilon _{Qmin}={\frac {1}{\sqrt {L_{n}(\xi )}}}}
This results in a filter which is maximally insensitive to component variations, but the ability to independently specify the passband and stopband ripples will be lost. For such filters, as the order increases, the ripple in both bands will decrease and the rate of cutoff will increase. If one decides to use a minimum-Q elliptic filter in order to achieve a particular minimum ripple in the filter bands along with a particular rate of cutoff, the order needed will generally be greater than the order one would otherwise need without the minimum-Q restriction. An image of the absolute value of the gain will look very much like the image in the previous section, except that the poles are arranged in a circle rather than an ellipse. They will not be evenly spaced and there will be zeroes on the ω axis, unlike the Butterworth filter, whose poles are arranged in an evenly spaced circle with no zeroes.
Comparison with other linear filters[edit]
Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients:
Daniels, Richard W. (1974). Approximation Methods for Electronic Filter Design. New York: McGraw-Hill. ISBN 0-07-015308-6.
Lutovac, Miroslav D.; Tosic, Dejan V.; Evans, Brian L. (2001). Filter Design for Signal Processing using MATLAB and Mathematica. New Jersey, USA: Prentice Hall. ISBN 0-201-36130-2.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Elliptic_filter&oldid=1068932392"
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Shannon graphs, subshifts and lambda-graph systems
October, 2002 Shannon graphs, subshifts and lambda-graph systems
Wolfgang KRIEGER, Kengo MATSUMOTO
The relationship between presentations of subshifts by Shannon graphs and by
\lambda
-graph systems is studied. A class of presentations of subshifts by 2-graph systems is characterized. A notion of synchronization is introduced. A class of presentations by
\lambda
-graph systems, that are specifically associated to subshifts that fall under this notion, is characterized.
Wolfgang KRIEGER. Kengo MATSUMOTO. "Shannon graphs, subshifts and lambda-graph systems." J. Math. Soc. Japan 54 (4) 877 - 899, October, 2002. https://doi.org/10.2969/jmsj/1191591995
Keywords: $\lambda$-graph systems , Shannon graphs , subshifts
Wolfgang KRIEGER, Kengo MATSUMOTO "Shannon graphs, subshifts and lambda-graph systems," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 54(4), 877-899, (October, 2002)
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63
drinks at Joe’s Java and Juice Hut,
42
contain coffee and
21
contain dairy products. If Alexei randomly chooses a drink, what is the probability of getting a drink with both coffee and a dairy product? What is the probability of getting neither coffee nor a dairy product? Assume that choosing a coffee drink is independent of choosing a dairy-product drink. See if you can answer these questions without making a two-way table first.
If events A and B are independent, the
\text{P}(\text{A and B})=\text{P}(\text{A})·P(\text{B})
\text{P}(\text{coffee})=\frac{42}{63}
\text{P}(\text{no coffee})
\text{P}(\text{dairy})=\frac{42}{63}
\text{P}(\text{no dairy})
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\begin{aligned} &\text{EBIT}\ =\ \text{Revenue}\ -\ \text{COGS}\ -\ \text{Operating Expenses}\\ &\text{Or}\\ &\text{EBIT}\ =\ \text{Net Income}\ +\ \text{Interest}\ +\ \text{Taxes}\\ &\textbf{where:}\\ &\text{COGS}\ =\ \text{Cost of goods sold} \end{aligned}
EBIT = Revenue − COGS − Operating ExpensesOrEBIT = Net Income + Interest + Taxeswhere:
\begin{aligned} &\text{Revenue: } \$10,000,000\\ &\text{Cost Of Goods Sold: } \$3,000,000\\ &\text{Gross Profit: } \$7,000,000 \end{aligned}
Revenue: $10,000,000Cost Of Goods Sold: $3,000,000
SG\&A: \$2,000,000
SG&A:$2,000,000
\begin{aligned} \text{EBIT: } &\$5,000,000\\ &\text{or }(\$10,000,000\ -\ \$3,000,000\ -\ \$2,000,000) \end{aligned}
EBIT: $5,000,000
\begin{aligned} &\text{EBIT}\ =\ \text{NS}\ -\ \text{COGS}\ -\ \text{SG\&A}\ +\ \text{NOI}\ +\ \text{II}\\ &\begin{aligned} \text{EBIT}\ &=\ \$65,299\ -\ \$32,909\ -\ \$18,949\ +\ \$325\\ &\quad+\ \$182\ =\ \$13,948 \end{aligned}\\ &\textbf{where:}\\ &\text{NS}\ =\ \text{Net sales}\\ &\text{SG\&A}\ =\ \text{Selling, general, and administrative expenses}\\ &\text{NOI}\ =\ \text{Non-operating income}\\ &\text{II}\ =\ \text{Interest income} \end{aligned}
EBIT = NS − COGS − SG&A + NOI + IIEBIT = $65,299 − $32,909 − $18,949 + $325+ $182 = $13,948where:NS = Net salesSG&A = Selling, general, and administrative expensesNOI = Non-operating incomeII = Interest income
\begin{aligned} &\text{EBIT}\ =\ \text{NE}\ -\ \text{NEDO}\ +\ \text{IT}\ +\ \text{IE}\\ &\begin{aligned} \text{Therefore, EBIT}\ &=\ \$10,604\ -\ \$577\ +\ \$3,342\\ &\quad +\ \$579\ =\ \$13,948\end{aligned}\\ &\textbf{where:}\\ &\text{NE}\ =\ \text{Net earnings}\\ &\text{NEDO}\ =\ \text{Net earnings from discontinued operations}\\ &\text{IT}\ =\ \text{Income taxes}\\ &\text{IE}\ =\ \text{Interest expense} \end{aligned}
EBIT = NE − NEDO + IT + IETherefore, EBIT = $10,604 − $577 + $3,342+ $579 = $13,948where:NE = Net earningsNEDO = Net earnings from discontinued operationsIT = Income taxesIE = Interest expense
U.S. Securities and Exchange Commission. "The Procter & Gamble Company Schedule 10-K 2016."
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Algebra/Parabola - Wikibooks, open books for an open world
Algebra/Parabola
← Complex Numbers Parabola Circle →
A parabola is the set of points that is the same distance away from a single point called the focus and a line called the directrix. The distance to the line is taken as the perpendicular distance.
One important point on the parabola itself is called the vertex, which is the point which has the smallest distance between both the focus and the directrix. Parabolas are symmetric, and their lines of symmetry pass through the vertex. Also due to this symmetry, only the vertex and one other point are necessary to completely define a parabola.
To derive an equation for a parabola, let's suppose (for now) that the directrix is horizontal and therefore has the equation:
{\displaystyle y=a}
Also suppose that the focus is given by (x,y) = (b,c). The vertex is then, by definition and inspection, located at:
{\displaystyle (h,k)=\left(b,{\frac {|a+c|}{2}}\right)}
Now let's examine the distance between some point in the x-y plane (x*,y*) and focus and directrix. The distance between the point and the focus is given by the distance formula:
{\displaystyle D_{1}={\sqrt {(x^{*}-b)^{2}+(y^{*}-c)^{2}}}}
Since the directrix is horizontal, the distance between the point and the line is simply
{\displaystyle D_{2}=|{y^{*}-a}|}
By the definition of a parabola, both of these distances must be equal, and therefore:
{\displaystyle {\sqrt {(x^{*}-b)^{2}+(y^{*}-c)^{2}}}=|y^{*}-a|}
This is the equation of a parabola, but let's make it a little nicer to work with. First square both sides, and notice the absolute values are no longer needed since the square of a number is always positive:
{\displaystyle (x^{*}-b)^{2}+(y^{*}-c)^{2}=(y^{*}-a)^{2}}
FOILing and then simplifying:
{\displaystyle (x^{*}-b)^{2}+y^{*2}-2y^{*}c+c^{2}=y^{*2}-2y^{*}a+a^{2}}
{\displaystyle (x^{*}-b)^{2}-2y^{*}c+c^{2}=-2y^{*}a+a^{2}}
{\displaystyle (x^{*}-b)^{2}+2y^{*}(a-c)+c^{2}-a^{2}=0}
. By differences of two squares we arrive at:
{\displaystyle (x^{*}-b)^{2}+2y^{*}(a-c)+(c-a)(c+a)=0}
. Factoring:
{\displaystyle (x^{*}-b)^{2}+(c-a)((c+a)-2y^{*})=0}
Now notice that the quantity
{\displaystyle a+c}
is twice the y-coordinate of the vertex, and that b is the x-coordinate of the vertex. Therefore:
{\displaystyle (x^{*}-h)^{2}+(c-a)(2k-2y^{*})=0}
Factoring a 2 out of the second term,
{\displaystyle (x^{*}-h)^{2}+2(c-a)(k-y^{*})=0}
or, letting D be the distance between the distance between the vertex and the focus (which is
{\displaystyle {\frac {c-a}{2}}}
) we arrive at the most useful form of a parabolic equation:
{\displaystyle (x^{*}-h)^{2}=4D(y^{*}-k)}
where D is the distance from the vertex to the focus, and (h,k) is the vertex
When written in this form, the latus rectum has a length of 4D. In addition, the coordinates of the vertex itself are (x,y)=(h,k). Using this information, and the symmetry of the parabola, it is straightforward to graph it.
But, the formula that is written in the Algebra Textbook is usually
{\displaystyle x^{2}=4py}
if the parabola is vertical and
{\displaystyle y^{2}=4px}
if the parabola is horizontal. And, the focus is (0,p) and the directrix is -p if the parabola is vertical as the focus is (p,0) and the directrix is -p if the parabola is horizontal.
Alternate formsEdit
The standard form of a parabola is:
{\displaystyle y=Ax^{2}+Bx+C}
where A, B, and C are constants. From this form we can deduce that the y-intercept of the parabola is C. It can be shown (and will be shown in a later section) that the x-coordinate of the vertex is
{\displaystyle {\frac {-B}{2A}}}
Retrieved from "https://en.wikibooks.org/w/index.php?title=Algebra/Parabola&oldid=3247733"
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Large Eddy Simulations of Flow and Heat Transfer in Rotating Ribbed Duct Flows | J. Heat Transfer | ASME Digital Collection
Mayank Tyagi, Research Associate,
Mayank Tyagi, Research Associate
Mechanical Engineering Department Louisiana State University Baton Rouge, LA 70803
Sumanta Acharya, Professor
Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 3, 2004; revision received, December 8, 2004. Review conducted by: P. M. Ligrani.
Tyagi, M., and Acharya, S. (May 25, 2005). "Large Eddy Simulations of Flow and Heat Transfer in Rotating Ribbed Duct Flows ." ASME. J. Heat Transfer. May 2005; 127(5): 486–498. https://doi.org/10.1115/1.1861924
Large eddy simulations are performed in a periodic domain of a rotating square duct with normal rib turbulators. Both the Coriolis force as well as the centrifugal buoyancy forces are included in this study. A direct approach is presented for the unsteady calculation of the nondimensional temperature field in the periodic domain. The calculations are performed at a Reynolds number (Re) of 12,500, a rotation number (Ro) of 0.12, and an inlet coolant-to-wall density ratio
Δρ/ρ
of 0.13. The predicted time and space-averaged Nusselt numbers are shown to compare satisfactorily with the published experimental data. Time sequences of the vorticity components and the temperature fields are presented to understand the flow physics and the unsteady heat transfer behavior. Large scale coherent structures are seen to play an important role in the mixing and heat transfer. The temperature field appears to contain a low frequency mode that extends beyond a single inter-rib geometric module, and indicates the necessity of using at least two inter-rib modules for streamwise periodicity to be satisfied. Proper orthogonal decomposition (POD) of the flowfield indicates a low dimensionality of this system with almost 99% of turbulent energy in the first 80 POD modes.
flow simulation, heat transfer, pipe flow, turbulence, vortices, flow instability
Ducts, Flow (Dynamics), Heat transfer, Temperature, Large eddy simulation, Coolants, Turbulence, Vortices, Vorticity, Heat flux, Physics, Rotation
Morris, W. D., 1981, Heat Transfer and Fluid Flow in Rotating Coolant Channels, Research Studies Press, Hertfordshire, UK.
Rahmat-Abadi
Convective Heat Transfer in Rotating Ribbed Tubes
Flow and Heat Transfer in Dimpled Two-pass Channel, in Heat Transfer in Gas Turbine Systems
Zhou, F., Lagrone J., and Acharya, S., 2004, “Internal Cooling in 4:1 AR Passages at High Rotation Numbers,” Paper no. GT2004-53501, IGTI Turbo Expo, Vienna.
Recent Developments in Turbine Blade Internal Cooling, in Heat Transfer in Gas Turbine Systems
Bredberg, J., 1997, Turbine Blade Internal Cooling, Chalmers University, Goteborg.
Computational Fluid Dynamics Applied to Internal Gas-Turbine Blade Cooling: A Review
Calculation of Fully Developed Turbulent Flow in Rectangular Ducts with Two Opposite Roughened Walls
Computation of Flow and Heat Transfer Through Rotating Ribbed Passages
Parniex
Experimental and Numerical Study of Developed Flow and Heat Transfer in Coolant Channels with 45 Degree Ribs
Prediction of Pressure Loss and Heat Transfer in Internal Cooling Passages
Pallares, J., Grau, F. X., and Davidson, L., 2001, “A Model for Estimating Three-Dimensional Boundary Layers in Rotating Duct Flow at High Rotation Rates,” 2nd International Symposium on Turbulence and Shear Flow Phenomena, Vol. 1, KTH-Stockholm, pp. 359–364.
Roclawski, H., Jacob, J. D., Yang, T., and McDonough, J. M., 2001, “Experimental and Computational Investigation of Flow in Gas Turbine Blade Cooling Passages,” AIAA paper 2925.
Numerical Simulation of Channel Flow with a Rib-Roughened Wall
Saha, A., and Acharya, S., 2004, “Unsteady Computations for Flow and Heat Transfer in 1:1, 4:1, 1:4 AR Ribbed Coolant Passages with Rotation,” ASME paper No. GT2004-53986.
Convective Heat Transfer in Periodic Wavy Passages
A Dynamic Subgrid-Scale Model for Compressible Turbulence and Scalar Transport
On the Formulation of the Dynamic Mixed Subgrid Scale Model
Large Eddy Simulations of Turbulent Flown in Complex and Moving Rigid Geometries With the Immersed Boundary Method
Tyagi, M., 2003, “Large Eddy Simulation of Turbulent Flows in Complex Geometries,” Ph.D. thesis, Mechanical Engineering Department, Louisiana State University.
Wray, A. A., and Hunt, J. C. R., 1989, “Algorithms for Classification of Turbulent Structures, Topological Fluid Mechanics,” Proceedings of the IUTAM symposium, H. K. Moffat and A. Tsinober, eds., Cambridge University Press, Cambridge, UK, pp. 95–104.
Characterization of Vortex Tubes and Sheets
On Coherent-Vortex Identification in Turbulence
Holmes, P., Lumley, J. L., and Berkooz, G., 1996, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press,
Turbulence and the Dynamics of Coherent Structures, Part I–III
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Low voltage tariff and metering - Electrical Installation Guide
HomeConnection to the LV utility distribution networkLow voltage tariff and metering
1 Reduction of losses
2 Reduction of peak power demand
4 Principle of kVA maximum demand metering
No attempt will be made in this guide to discuss particular tariffs, since there appears to be as many different tariff structures around the world as there are utilities.
Some tariffs are very complicated in detail but certain elements are basic to all of them and are aimed at encouraging consumers to manage their power consumption in a way which reduces the cost of generation, transmission and distribution.
The two predominant ways in which the cost of supplying power to consumers can be reduced, are:
Reduction of power losses in the generation, transmission and distribution of electrical energy. In principle the lowest losses in a power system are attained when all parts of the system operate at unity power factor
Reduction of the peak power demand, while increasing the demand at low-load periods, thereby exploiting the generating plant more fully, and minimizing plant redundancy
Although the ideal condition noted in the first possibility mentioned above cannot be realized in practice, many tariff structures are based partly on kVA demand, as well as on kWh consumed. Since, for a given kW loading, the minimum value of kVA occurs at unity power factor, the consumer can minimize billing costs by taking steps to improve the power factor of the load (as discussed in Chapter Power Factor Correction). The kVA demand generally used for tariff purposes is the maximum average kVA demand occurring during each billing period, and is based on average kVA demands, over fixed periods (generally 10, 30 or 60 minute periods) and selecting the highest of these values. The principle is described below in “principle of kVA maximum-demand metering”.
Reduction of peak power demand
The second aim, i.e. that of reducing peak power demands, while increasing demand at low-load periods, has resulted in tariffs which offer substantial reduction in the cost of energy at:
Certain hours during the 24-hour day
Certain periods of the year
The simplest example is that of a residential consumer with a storage-type water heater (or storage-type space heater, etc.). The meter has two digital registers, one of which operates during the day and the other (switched over by a timing device) operates during the night. A contactor, operated by the same timing device, closes the circuit of the water heater, the consumption of which is then indicated on the register to which the cheaper rate applies. The heater can be switched on and off at any time during the day if required, but will then be metered at the normal rate. Large industrial consumers may have 3 or 4 rates which apply at different periods during a 24-hour interval, and a similar number for different periods of the year. In such schemes the ratio of cost per kWh during a period of peak demand for the year, and that for the lowest-load period of the year, may be as much as 10: 1.
It will be appreciated that high-quality instruments and devices are necessary to implement this kind of metering, when using classical electro-mechanical equipment. Recent developments in electronic metering and micro-processors, together with remote ripple-control[1] from an utility control centre (to change peak-period timing throughout the year, etc.) are now operational, and facilitate considerably the application of the principles discussed.
In most countries, some tariffs, as noted above, are partly based on kVA demand, in addition to the kWh consumption, during the billing periods (often 3-monthly intervals). The maximum demand registered by the meter to be described, is, in fact, a maximum (i.e. the highest) average kVA demand registered for succeeding periods during the billing interval.
Figure C11 shows a typical kVA demand curve over a period of two hours divided into succeeding periods of 10 minutes. The meter measures the average value of kVA during each of these 10 minute periods.
Fig. C11 – Maximum average value of kVA over an interval of 2 hours
Principle of kVA maximum demand metering
A kVAh meter is similar in all essentials to a kWh meter but the current and voltage phase relationship has been modified so that it effectively measures kVAh (kilo-volt-ampere-hours). Furthermore, instead of having a set of decade counter dials, as in the case of a conventional kWh meter, this instrument has a rotating pointer. When the pointer turns it is measuring kVAh and pushing a red indicator before it. At the end of 10 minutes the pointer will have moved part way round the dial (it is designed so that it can never complete one revolution in 10 minutes) and is then electrically reset to the zero position, to start another 10 minute period. The red indicator remains at the position reached by the measuring pointer, and that position, corresponds to the number of kVAh (kilo-volt-ampere-hours) taken by the load in 10 minutes. Instead of the dial being marked in kVAh at that point however it can be marked in units of average kVA. The following figures will clarify the matter.
Supposing the point at which the red indicator reached corresponds to 5 kVAh. It is known that a varying amount of kVA of apparent power has been flowing for 10 minutes, i.e. 1/6 hour.
If now, the 5 kVAh is divided by the number of hours, then the average kVA for the period is obtained.
In this case the average kVA for the period will be:
{\displaystyle 5\times {\frac {1}{\frac {1}{6}}}={5\times 6}={30\ {\mbox{kVA}}}}
Every point around the dial will be similarly marked i.e. the figure for average kVA will be 6 times greater than the kVAh value at any given point. Similar reasoning can be applied to any other reset-time interval.
At the end of the billing period, the red indicator will be at the maximum of all the average values occurring in the billing period.
The red indicator will be reset to zero at the beginning of each billing period. Electro-mechanical meters of the kind described are rapidly being replaced by electronic instruments. The basic measuring principles on which these electronic meters depend however, are the same as those described above.
^ Ripple control is a system of signalling in which a voice frequency current (commonly at 175 Hz) is injected into the LV mains at appropriate substations. The signal is injected as coded impulses, and relays which are tuned to the signal frequency and which recognize the particular code will operate to initiate a required function. In this way, up to 960 discrete control signals are available.
Retrieved from "http://www.electrical-installation.org/enw/index.php?title=Low_voltage_tariff_and_metering&oldid=26446"
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!! used as default html header if there is none in the selected theme. Graphic derivative
f\left(x\right)
f\prime (x)
f\prime (-x)
-f\prime (x)
-f\prime (-x)
Description: recognize the graph of the derivative of a function. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, analysis, functions, graphing, derivative
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Relaxation of Proton Conductivity and Stress in Proton Exchange Membranes Under Strain | J. Eng. Mater. Technol. | ASME Digital Collection
Macromolecular Science and Engineering, Virginia Polytechnic Institute and State University
e-mail: danl@vt.edu
Michael A. Hickner,
Department of Chemical and Biological Systems,
e-mail: mahickn@sandia.gov
Engineering Science and Mechanics, Virginia Polytechnic Institute and
e-mail: scase@exchange.vt.edu
e-mail: jlesko@exchange.vt.edu
Liu, D., Hickner, M. A., Case, S. W., and Lesko, J. J. (June 6, 2006). "Relaxation of Proton Conductivity and Stress in Proton Exchange Membranes Under Strain." ASME. J. Eng. Mater. Technol. October 2006; 128(4): 503–508. https://doi.org/10.1115/1.2345441
The stress relaxation and proton conductivity of Nafion 117 membrane (N117-H) and sulfonated poly(arylene ether sulfone) copolymer membrane with 35% sulfonation (BPSH35) in acid forms were investigated under uniaxial loading conditions. The results showed that when the membranes were stretched, their proton conductivities in the direction of the strain initially increased compared to the unstretched films. The absolute increases in proton conductivities were larger at higher temperatures. It was also observed that proton conductivities relaxed exponentially with time at
30°C
. In addition, the stress relaxation of N117-H and BPSH35 films under both atmospheric and an immersed (in deionized water) condition was measured. The stresses were found to relax more rapidly than the proton conductivity at the same strains. An explanation for the above phenomena is developed based on speculated changes in the channel connectivity and length of proton conduction pathway in the hydrophilic channels, accompanied by the rotation, reorientation, and disentanglements of the polymer chains in the hydrophobic domains.
stress relaxation, ionic conductivity, polymer blends
Proton conductivity, Protons, Relaxation (Physics), Stress, Temperature, Water, Electrical conductivity, Thermal conductivity, Proton exchange membranes, Heat conduction, Membranes
Ionic Conductivity of an Extruded Nafion 1100 EW Series of Membranes
Determination of Water Diffusion Coefficients in Perfluorosulfonate Ionomeric Membranens
Polymeric Proton Conducting Membranes For Medium Temperature Fuel Cells (110-160°C)
Effect of Acidification Treatment and Morphological Stability of Sulfonated Poly(arylene ether sulfone) Copolymer Proton-Exchange Membranes for Fuel-Cell Use above 100°C
A Mathematical Model of the Solid Polymer Electrolyte Fuel Cell
Three-Dimensional Computational Analysis of Transport Phenomena in a PEM Fuel Cell
P. C. v. d.
Orientation of Drawn Nafion at Molecular and Mesoscopic Scales
Relaxation of Drawn Nafion Films Studied With Birefingence Experiments
Tensile Behavior of Nafion and Sulfonated Poly(arylene ether sulfone) Copolymers and its Morphological Correlations
State of Water in Disulfonated Poly(arylene ether sulfone) Copolymers and a Perfluorosulfonic Acid Copolymer (Nafion) and Its Effect on Physical and Electrochemical Properties
Ion Transport and Clustering in Nafion Perfluorinated Membranes
Ethilton
Pauraj
Protonic Conductivity and Photoconductivity Studies on H3PW12O40×21H2O Single Crystals
Sulfonated Polyether Ether Ketone Membranes Cured With Different Methods For Direct Methanol Fuel Cells
Proc. ICCN 2002 International Conference on Computational Nanoscience
Transport in Polymer-Electrolyte Membranes, I: Physical Model
Thermal Behavior of Nafion Membranes
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Collateral System - dVest Labs
In dSynth, minting a Synth is technically opening a Collateralized Dept Position (CDP) working as a loan. If a user wishes to mint a synth they have to come up with a deposit in Ether to create the underlying collateralization.
All synths have to be minted. In order to create value and establish an underlying value for a minted asset, the user has to provide a certain amount of ETH as collateral in order to take the loan.
When opening a loan, dSynth aims for a 120% collateral ratio. On the loan tab, a user can see exactly how much ETH they have to deposit in order to get to a certain C-Ratio.
If you create a synth you can specify your own collateral ratio between 110% and 1000%.
After the loan is taken out, the specified amount is then minted to the user's wallet and the ETH is transferred into the smart contract where the loan collateral is being held.
If a user wants to get their collateral back, they have the option to close the loan. Therefore, they will have to hold the number of synths minted by the loan in their wallet. The synths are then being burnt and the collateral gets unlocked and sent back to the borrower.
It is also possible to withdraw parts of the collateral. This will affect the C-Ratio and a user cant withdraw ETH that will make them fall under the desired C-Ratio.
If an assets price drops to or below 0 then it will not be possible to open any more loans.
Also, if the asset is an inverse asset and the price moves to 10% or below 10% of the deployment price the asset will not allow opening any loans until the price moves back above the threshold again as a safety mechanism.
The collateral acts as the underlying value of each loan and is deposited as ETH. In dSynth the system is set to keep a fixed C-Ratio overall loans. If the ratio of a user loan drops below the C-Ratio the loan is suspect to liquidation.
It is the borrowers' responsibility to keep the collateralization level above. If the value of the collateral drops or the value of the asset increases the user has the option to either deposit more ETH as collateral into the loan or also to repay parts of the loan (burn some of their synths) to fix their C-Ratio and don't get into the risk of liquidation.
If the collateralization ratio drops below 120%, a loan is at risk of liquidation. Any address can then liquidate or partially liquidate the loan.
If an address wants to liquidate a loan at risk, they have to cover parts of their debt in the form of the synth.
The amount to be liquidated is calculated as follows:
V = Value of ETH Collateral in USD
D = Value of the Synth in USD
t = Liquidation Ratio
S = Amount of USD debt to liquidate
S = (t* D -V) / ( t- (1+P))
On top of the liquidation amount, the liquidator will also get awarded a liquidation penalty of 10%. This amount is taken directly from the borrowers collateral.
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Introduction to arithmetic progression — lesson. Mathematics State Board, Class 10.
Let us see a moment to look around our surrounding. We might have observed specific patterns in nature, such as the patterns in the sunflowers, seashell covers and a honeycomb's holes. (Click to know more)
We can observe similar patterns in mathematics as well. For instance, say the Fibonacci series, the Arithmetic Progression, and the Geometric Progression.
In this chapter, we deep dive and explore the Arithmetic Progression with examples.
Let us see a scenario before we go into Arithmetic Progression.
Nancy is a teacher, got a job with a starting monthly pay of \(₹\)25000 and an annual raise of \(₹\)3000 per year. During the first, second, and third years, she will be paid \(₹\)25000, \(₹\)28000, and \(₹\)31000, respectively.
Now we calculate the difference of the salaries for the successive years, we get:
\(₹(\)28000 \(–\) 25000 \(–\) \() =\) \(₹\)3000;
\(₹(\)31000 \(–\) 28000\() =\) \(₹\)3000.
Thus the difference between the successive numbers (salaries) is always \(₹\)3000.
From the above scenario, we can understand that the number sequence undergoes a certain sequence or a pattern by the constant common difference. And this kind of sequence is called arithmetic progression.
The sequence of numbers in which each term is differ by the common difference from the previous term through out the sequence is known as arithmetic progression.
Let \(a\) and \(d\) be real numbers. Then the numbers of the form
a,a+d,a+2d,a+3d,a+4d,a+5d
,... is said to form Arithmetic progression. And it is denoted by \(A\).\(P\).
The number ‘\(a\)’ is called the first term and ‘\(d\)’ is called the common difference.
Arithmetic Progression in real-life:
Let us see few scenarios where we can observe the Arithmetic progression in real -life.
1. In a theatre, the arrangement of seats form an arithmetic progression. For instance, the first row might have \(12\) seats; the second row has \(14\) seats, and the third row contains \(16\).
2. Most of the banks provide a fixed interest amount for saving account. For example, if we keep \(₹\)\(1000\) in a saving account for one year, we would receive a specific percentage amount as interest at the end of the year. So, this saving amount plus interest amount added will become our new saving amount that will interest the proceeding years.
3. When you ride a taxi, you will be charged an initial rate and then a per mile or kilometres. And for every kilometre, you will be charged a certain amount extra plus the initial rate. This shows that how the arithmetic sequence works when determining the cost.
4. The lowest temperatures (in degrees Celsius) recorded in a city for a week in December, given in ascending order are:
\(– 4.1, – 3.8, – 3.5, – 3.2, – 2.9, – 2.6, – 2.3\)
Now we learned about the fundamentals of Arithmetic Progression \((A.P)\). In the upcoming lessons, we will learn the terms of Arithmetic Progression \((A.P).\) and practice the same concept.
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!! used as default html header if there is none in the selected theme. Prog modular arithmetics
{\displaystyle \frac{\mathbb{Z}}{\mathrm{NZZ}}}
N
Description: programming exercises on modular arithmetics. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algorithmics, arithmetics, modular calculation, modular arithmetics
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x=f\left(t\right),y=g\left(t\right)
{t}_{0}
t
f\prime ({t}_{0})=g\prime ({t}_{0})=0
Description: parametrize a parametric curve so that it has a cusp. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, geometry, analysis, curves, parametric_curves, cusp, singularity
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EUDML | An Axiomatization of the Algebra of Transformations Over a Set. EuDML | An Axiomatization of the Algebra of Transformations Over a Set.
An Axiomatization of the Algebra of Transformations Over a Set.
DAVIS, A.S.. "An Axiomatization of the Algebra of Transformations Over a Set.." Mathematische Annalen 164 (1966): 372-378. <http://eudml.org/doc/161413>.
author = {DAVIS, A.S.},
title = {An Axiomatization of the Algebra of Transformations Over a Set.},
AU - DAVIS, A.S.
TI - An Axiomatization of the Algebra of Transformations Over a Set.
A
Articles by A.S. DAVIS
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Parabolic SAR - Wikipedia
A parabola below the price is generally bullish, while a parabola above is generally bearish. A parabola below the price may be used as support, whereas a parabola above the price may represent resistance.[2]
{\displaystyle {SAR}_{n+1}={SAR}_{n}+\alpha (EP-{SAR}_{n})}
The α value represents the acceleration factor. Usually, this is set initially to a value of 0.02, but can be chosen by the trader. This factor is increased by 0.02 each time a new EP is recorded, which means that every time a new EP is observed, it will make the acceleration factor go up. The rate will then quicken to a point where the SAR converges towards the price. To prevent it from getting too large, a maximum value for the acceleration factor is normally set to 0.20. The traders can set these numbers depending on their trading style and the instruments being traded. Generally, it is preferable in stocks trading to set the acceleration factor to 0.01, so that it is not too sensitive to local decreases. For commodity or currency trading, the preferred value is 0.02.
If the next period's SAR value is inside (or beyond) the current period or the previous period's price range, the SAR must be set to the closest price bound. For example, if in an upward trend, the new SAR value is calculated and if it results to be more than today's or yesterday's lowest price, it must be set equal to that lower boundary.
If the next period's SAR value is inside (or beyond) the next period's price range, a new trend direction is then signaled. The SAR must then switch sides.
Upon a trend switch, the first SAR value for this new trend is set to the last EP recorded on the prior trend, EP is then reset accordingly to this period's maximum, and the acceleration factor is reset to its initial value of 0.02.
Statistical ResultsEdit
The parabolic SAR showed results at a 95% confidence level in a study of 17 years of data.[3]
^ J. Welles Wilder, Jr. (June 1978). New Concepts in Technical Trading Systems. Greensboro, NC: Trend Research. ISBN 978-0-89459-027-6.
^ "Parabolic SAR (PSAR)". Day Trading Encyclopedia. Investors Underground. Retrieved 11 November 2016.
^ COMPARISON OF THREE TECHNICAL TRADING METHODS VS. BUY-AND-HOLD FOR THE S&P 500 MARKET | Timothy C. Pistole | Graduate Student of Finance, University of Houston – Victoria
Retrieved from "https://en.wikipedia.org/w/index.php?title=Parabolic_SAR&oldid=1003218285"
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{\displaystyle O(n^{2})}
comparisons,
{\displaystyle O(n)}
{\displaystyle O(n^{2})}
{\displaystyle O(1)}
{\displaystyle O(n^{2})}
{\displaystyle O(n)}
{\displaystyle O(1)}
In computer science, selection sort is an in-place comparison sorting algorithm. It has an O(n2) time complexity, which makes it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity and has performance advantages over more complicated algorithms in certain situations, particularly where auxiliary memory is limited.
4 Comparison to other sorting algorithms
/* a[0] to a[aLength-1] is the array to sort */
int aLength; // initialise to a's length
/* (could do i < aLength-1 because single element is also min element) */
for (i = 0; i < aLength-1; i++)
/* find the min element in the unsorted a[i .. aLength-1] */
int jMin = i;
/* test against elements after i to find the smallest */
for (j = i+1; j < aLength; j++)
if (a[j] < a[jMin])
if (jMin != i)
swap(a[i], a[jMin]);
Selection sort is not difficult to analyze compared to other sorting algorithms, since none of the loops depend on the data in the array. Selecting the minimum requires scanning
{\displaystyle n}
elements (taking
{\displaystyle n-1}
comparisons) and then swapping it into the first position. Finding the next lowest element requires scanning the remaining
{\displaystyle n-1}
elements and so on. Therefore, the total number of comparisons is
{\displaystyle (n-1)+(n-2)+...+1=\sum _{i=1}^{n-1}i}
{\displaystyle \sum _{i=1}^{n-1}i={\frac {(n-1)+1}{2}}(n-1)={\frac {1}{2}}n(n-1)={\frac {1}{2}}(n^{2}-n)}
which is of complexity
{\displaystyle O(n^{2})}
in terms of number of comparisons. Each of these scans requires one swap for
{\displaystyle n-1}
elements (the final element is already in place).
Among quadratic sorting algorithms (sorting algorithms with a simple average-case of Θ(n2)), selection sort almost always outperforms bubble sort and gnome sort. Insertion sort is very similar in that after the kth iteration, the first
{\displaystyle k}
elements in the array are in sorted order. Insertion sort's advantage is that it only scans as many elements as it needs in order to place the
{\displaystyle k+1}
st element, while selection sort must scan all remaining elements to find the
{\displaystyle k+1}
st element.
While selection sort is preferable to insertion sort in terms of number of writes (
{\displaystyle n-1}
swaps versus up to
{\displaystyle n(n-1)/2}
swaps, with each swap being two writes), this is roughly twice the theoretical minimum achieved by cycle sort, which performs at most n writes. This can be important if writes are significantly more expensive than reads, such as with EEPROM or Flash memory, where every write lessens the lifespan of the memory.
Finally, selection sort is greatly outperformed on larger arrays by
{\displaystyle \Theta (n\log n)}
divide-and-conquer algorithms such as mergesort. However, insertion sort or selection sort are both typically faster for small arrays (i.e. fewer than 10–20 elements). A useful optimization in practice for the recursive algorithms is to switch to insertion sort or selection sort for "small enough" sublists.
Heapsort greatly improves the basic algorithm by using an implicit heap data structure to speed up finding and removing the lowest datum. If implemented correctly, the heap will allow finding the next lowest element in
{\displaystyle \Theta (\log n)}
time instead of
{\displaystyle \Theta (n)}
for the inner loop in normal selection sort, reducing the total running time to
{\displaystyle \Theta (n\log n)}
Selection sort can be implemented as a stable sort if, rather than swapping in step 2, the minimum value is inserted into the first position and the intervening values shifted up. However, this modification either requires a data structure that supports efficient insertions or deletions, such as a linked list, or it leads to performing
{\displaystyle \Theta (n^{2})}
writes.
bingo(array A)
{ This procedure sorts in ascending order by
repeatedly moving maximal items to the end. }
last := length(A) - 1;
{ The first iteration is written to look very similar to the subsequent ones,
but without swaps. }
nextMax := A[last];
for i := last - 1 downto 0 do
if A[i] > nextMax then
nextMax := A[i];
while (last > 0) and (A[last] = nextMax) do
while last > 0 do begin
prevMax := nextMax;
if A[i] <> prevMax then
^ This article incorporates public domain material from the NIST document: Black, Paul E. "Bingo sort". Dictionary of Algorithms and Data Structures.
The Wikibook Algorithm implementation has a page on the topic of: Selection sort
Animated Sorting Algorithms: Selection Sort at the Wayback Machine (archived 7 March 2015) – graphical demonstration
Retrieved from "https://en.wikipedia.org/w/index.php?title=Selection_sort&oldid=1089213189"
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EUDML | Continuous spatial semigroups of completely positive maps of . EuDML | Continuous spatial semigroups of completely positive maps of .
Continuous spatial semigroups of completely positive maps of
𝔅\left(H\right)
Powers, Robert T.. "Continuous spatial semigroups of completely positive maps of .." The New York Journal of Mathematics [electronic only] 9 (2003): 165-269. <http://eudml.org/doc/124296>.
@article{Powers2003,
author = {Powers, Robert T.},
keywords = {completely positive maps; *-endomorphisms; -semigroups; -semigroups},
title = {Continuous spatial semigroups of completely positive maps of .},
AU - Powers, Robert T.
TI - Continuous spatial semigroups of completely positive maps of .
KW - completely positive maps; *-endomorphisms; -semigroups; -semigroups
completely positive maps, *-endomorphisms,
{E}_{0}
{E}_{0}
{C}^{*}
{W}^{*}
Derivations, dissipations and positive semigroups in
{C}^{*}
|
Open formula — Wikipedia Republished // WIKI 2
An open formula is a formula that contains at least one free variable.[citation needed]
An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like true or false. An open formula can be transformed into a closed formula by applying quantifiers or specifying of the domain of discourse of individuals for each free variable denoted x, y, z....or x1, x2, x3.... This transformation is called capture of the free variables to make them bound variables, bound to a domain of individual constants.
For example, when reasoning about natural numbers, the formula "x+2 > y" is open, since it contains the free variables x and y. In contrast, the formula "∃y ∀x: x+2 > y" is closed, and has truth value true.
An example of closed formula with truth value false involves the sequence of Fermat numbers
{\displaystyle F_{n}=2^{2^{n}}+1,}
Microsoft Excel 2013/2016 pt 1 (Enter/Delete/Edit cells, Save, Open, Formula)
prize bond 15000 first open figers best formula 01,04,2017 by ustaad
prize bond guess bond 200 city rawalpindi first open formula tak and juft open figers
Higher-order logic
Quantifier (logic)
Wolfgang Rautenberg (2008), Einführung in die Mathematische Logik (in German) (3. ed.), Wiesbaden: Vieweg+Teubner, ISBN 978-3-8348-0578-2
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Home | Programs | Public Engagement | PhysicsQuest | Past PhysicsQuest Projects | Circular Motion
We have already looked at motion of objects falling in one direction when you drop it, or in two directions as part of a pendulum. We also looked at what happens when objects fall while also moving in the horizontal direction. In this last activity, we are combining all that knowledge to see what happens when objects move in circular motion: there are constant changes in the horizontal and vertical direction at all times during the movement, making the object follow a circular path. Examples of circular motion are carousels or merry-go-rounds in parks, a car going around a roundabout, the moon orbiting around the Earth or the Earth revolving around the Sun.
In this paper Katherine Johnson was working on finding the right moment to turn off a satellite’s engine so that it would land on a specific position on Earth. Satellites in space orbit the Earth, just like the Moon orbits around the Earth. For this particular equation, Johnson worked with elementary celestial mechanics, a branch of astronomy that deals with objects in outer space. The equation that she used is one where a body (such as one like our satellite) acted upon by a radial force of attraction proportional to the inverse square of the distance are first reviewed.
\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{r}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{=}\frac{\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{p}}{\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{1}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{+}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{ }\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{e}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{ }\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{c}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{o}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{s}\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{ }\textcolor[rgb]{0.137254901960784,0.12156862745098,0.125490196078431}{\theta }}
This complex equation is just the start in finding the distance of the satellite for Earth’s center. Though you may think that an ellipse is a different shape from a circle, circles are actually a special type of elliptical shape. An object that moves in a circular motion will have the same average velocity and altitude of an object in an elliptical orbit.
Mass: A measure of the amount of stuff (or matter) an object has. Not to be confused with weight or volume. This is a measure of how much actual stuff there is, not how big it is or how hard something is pulling on it.
Weight: Mass (amount of stuff) times how hard the planet is pulling on it (gravity). This means that your weight on the moon will be 1/6 of that on Earth (gravity on the moon is 0.166 times of that on Earth). However, your mass will still be the same.
Centripetal force: The generic name of any force that pulls objects into a circular path.
Drag force: A generic name for any force that resists the movement of the objects. For example, the air resistance that we saw when objects fell in activity one, or the resistance you feel while swimming in a pool when the water slows you down.
Tangential velocity: The instantaneous velocity in a straight line of an object moving in a circular motion. It is said to be instantaneous because, in a circular motion, the direction is constantly changing, which is why there is an acceleration and a force.
Acceleration: How fast the velocity is changing. When something accelerates, it changes how fast it is going or the direction in which it is moving. For a positive change in acceleration means that the object is moving faster, a car going from 30mph to 40 mph. A negative change means the object is moving slower, the car is going from 40mph to 30 mph. Finally, a change in the direction of the object’s velocity without changing speed, such as if a car is moving North and turns East still moving, then the car accelerated because the direction of the car’s velocity changed. Remember that velocity is a vector with direction and magnitude, therefore changes in any (or both) of those factors will produce an acceleration.
What happens when the centripetal force is removed?
Before the activity students should know
The definitions of both force, velocity, and acceleration
Gravity from the Earth makes things fall and is always pulling straight down towards the floor
Force is mass times acceleration and acceleration means change in speed or direction of velocity of the object.
What variables and forces affect the circular motion and what variables have no effect.
The force causing circular motion is directed toward the center of the circle. The speed of the object does not change, but the direction of movement is constantly changing.
Understand the direction of motion for an object traveling in a circle and direction of motion as soon as it leaves the circle.
The Science Behind Circular Motion
In the case of the falling objects from previous activities, we learned that the forces acting on the objects were the gravity pulling the objects down and the drag force pushing the objects in the opposite direction to the fall. In the case of the pendulum, there was a force caused by the pulling of the string and the gravity force pulling the objects down. In the case of circular motion, there is a force that it is pulling objects inward, towards the center of the circle. We call it centripetal force. The existence of a force tells us that there is an acceleration: Even if the object is moving at a constant speed, like the Moon around the Earth, the acceleration comes in the form of the change in the direction of the velocity, without changing the speed. Whatever the object, if it moves in a circle, there is a force acting upon it to cause it to deviate from its straight-line path, accelerate inwards and move along a circular path. In the case of the Moon orbiting around the Earth, the force of gravity from the Earth pulls the Moon towards the Earth as it is also moving forward, causing the circular motion. The word centripetal only means center seeking, so a centripetal force is any force that pulls the object into a circular path. With a swinging yo-yo, it is the string holding the yo-yo and pulling it towards the pivot point. In the case of orbiting planets or moons, it is the force of gravity pulling the orbiting object towards the object that they are orbiting, that means the force of gravity of the Earth pulls the Moon inwards.
balls of different sizes and materials (use the different balls from previous experiments to see if there is a difference)
plates with a rim
You can substitute the materials with any small balls or even make your own from playdough. For the plate, you can use cardboard to make a rim, the idea is to create a wall that will guide the movement of the ball and that you can also cut a hole from it to see what happens when the ball escapes through the hole.
In a classroom, get students to work in groups. Give them some time to think about examples of circular motion and have a class discussion about what forces are involved in the motion of the ball. As a class, decide how you are going to test if there is a force pulling the ball to move in circular motion — science happens in community!
To understand what happens when you remove the centripetal force, students first need to develop an understanding of what makes objects move in circular motion. For example, a marble can be made to go in a circle when you give the marble a push while it is constrained by a circular object, such as a plate. The marble will follow a circular path, because the surface of the marble is pushing on the surface of the plate.
Take a plate with a rim and put one of the balls on the plate. Give the ball a little push and observe the trajectory of the movement.
Think about what forces are causing the movement — you can prompt students to think about the contact of the ball with the plate and how it differs from the force of the air in contact with the piece of falling paper from activity one. Think about what the direction of movement is, the direction of the velocity, and how that connects with the direction of the force.
Use a second plate and make a cut out on the rim of the plate. Repeat the steps from before: give the ball a little push and observe its movement.
Is the ball still moving in circular motion? What happens to the ball when it gets to the cut out part? Ask students to draw a prediction of which direction the ball will go as it goes out through the cut out.
Does the ball always go out through the cut out? Does it change depending on how fast or slow the ball is moving? When the ball goes out, does it always take the same path or does that change depending on how fast it is moving?
Experiment 2: Spinning the Wiffleball
For experiment two, we are asking students to design an experiment to test what direction a wiffleball will fly if you let go of the string when you are spinning it. As with experiment one, we want students to reflect on what the forces are that are acting on the wiffleball when they are spinning it and what will happen if they let the string go. They should also consider whether the ball will behave differently if they spin it vertically or horizontally. Note that if they give the ball any extra push before letting the string go, the movement of the wiffleball will be affected. Students should consider the difference between just letting the ball t go versus giving it an extra push. The idea of the two experiments is to understand movement of objects in circular motion and how, in circular motion, the force is actually caused by a constant change in the direction of the velocity.
Again we ask students to consider what questions they are trying to answer, look at what materials they have, and design an experiment using those materials to answer their research question. We ask the students to use the scientific method as a guideline to design the experiment.
Make sure you have plenty of space around you to swing the wiffle ball.
A camera to record the experiment (optional)
You can substitute the materials with any small object you can attach a string to or even make your own from playdough.
Take the string and attach it to the wiffleball. Make sure the string is tight so that the wiffleball will not fly off.
Spin the string with the wiffleball attached. Try spinning it in the horizontal and vertical directions. Ask the students to predict where the wiffleball will fly off if they let the string go and if the direction will change depending on the position of the hand/wiffleball when they let the string go.
Get your camera ready for recording
Do several tests of letting the string go when you are spinning the string vertically and horizontally. While still maintaining circular motion, you’ll want to spin the wiffleball as slowly as possible so that your group can actually determine the proper release point. Let the string go at different positions, for example when spinning up or down, or when it is farthest from you when spinning horizontally. Does the wiffleball move in different directions? Does it matter if you are spinning clockwise or counter clockwise?
Test out your guess with the wiffle ball. Do the experiment several times to make sure you know what actually happens.
Students should reflect on objects going in a circular motion, and determine what direction is the force, and what direction is the velocity.
NGSS performance standard HS-PS2-1, where Newton’s second law of motion is applied to the motion of an object in a circle, MP8, and SP3.
https://colab.research.google.com/drive/198i8UxglkqkBV2Ed8l0JPXlvSgwSyIGg?usp=sharing
Katherine Johnson related Activity: Math for Circular Motion
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Topics in approximate inference
Update: another major update of this monograph document is in. Some small updates are still in schedule to finish before the end of 2020.
Current version (Dec 2020) (pdf)
Next major update will be on applications of approximate inference in machine learning.
Read the following explainers before you decide to delve in this subject 🤔
What is approximate inference? more
To me, approximate inference finds the best approximation to an integration problem confronting computational constraints.
Imagine you have a distribution
p(x)
and you want to compute the integral
\int p(x) F(x) dx
F(x)
of interest. We call the computation of this integral as inference. Examples include Bayesian inference where now
p(x)
is some posterior distribution and
F(x)
is the likelihood function of
x
on unseen data. Or if
p(x)
is unnormalised, taking
F(x) = 1
would return the integral as the normalising constant (or partition function) of
p
. Unfortunately for many complicated models we are fancy on now (say neural networks) this integral is intractable, and here intractability means you can't compute the exact value of the integral due to computational constraints (say running time, memory usage, precision, etc). So instead we use approximate inference to approximate that integral.
Why I should care about it? more
Now you might ask: why I should care about this approximate integral in the first place? Couldn't we just forget about integrals and do point estimations instead? Or couldn't we just use models that can do exact inference (e.g. stacking invertible transformations, auto-regressive models, sum-product networks)? Well for the first question it's more like a debate between Frequentist and Bayesian which I don't think I should say too much about it here. But I shall emphasize that integration is almost everywhere in probability and statistics, say calculating expectations and computing marginal distributions. For the second question, those models are usually more computationally demanding (both in time and memory), which is exactly the point for not using them and do approximate inference instead if you don't have that much computational resource. Approximate inference only makes sense when you do have computational constraints, and the way you do approximate inference also depends on how much precision you want and computational resource you have.
What this list is about? more
There are mainly two ways to do approximate inference: directly approximating the integral you want, or, finding an accurate approximation
q
to the target distribution
p
and using it for integration later. The first approach is mainly dominated by sampling methods (and maybe adding student models to distill it) and quadrature. The second one, sometimes referred as the indirect approach, will be the focus of this list.
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Of the students who choose to live on campus at Coastal College,
10
% are seniors. The most desirable dorm rooms are in the newly constructed OceanView dorm, and
60
% of the seniors live there, while
20
% of the rest of the students live there.
Represent these probabilities in a two-way table.
Not Ocean View
(0.60)(0.10)=\mathbf{0.06}
0.10
(0.20)\mathbf{(0.90)}=\mathbf{0.18}
1.00
Compare the percentage of seniors who live in OceanView to the percentage of total students who live in OceanView.
25\%
Use the alternative definition of independence (see the Math Notes box in Lesson 10.2.3) to determine if being a senior is associated with living in the Ocean View dorm.
\text{A}
\text{B}
\text{P}(\text{A and B})=\text{P}(\text{A})·P(\text{B})
The converse of this statement is also true:
\text{P}(\text{A and B})=\text{P}(\text{A})·P(\text{B})
\text{A}
\text{B}
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Root mean square value of input or sequence of inputs - Simulink - MathWorks Switzerland
Find the RMS value over
Root mean square value of input or sequence of inputs
The RMS block computes the root mean square (RMS) value of each row or column of the input, or along vectors of a specified dimension of the input. It can also compute the RMS value of the entire input. You can specify the dimension using the Find the RMS value over parameter. The RMS block can also track the RMS value in a sequence of inputs over a period of time. To track the RMS value in a sequence of inputs, select the Running RMS parameter.
The Running mode in the RMS block will be removed in a future release. To compute the running RMS in Simulink®, use the Moving RMS block instead.
This port is unnamed until you select the Running RMS parameter and set the Reset port parameter to any option other than None.
Specify the reset event that causes the block to reset the running RMS. The sample time of the Rst input must be a positive integer multiple of the input sample time.
To enable this port, select the Running RMS parameter and set the Reset port parameter to any option other than None.
Port_1 — RMS value along the specified dimension
When you do not select the Running RMS parameter, the block computes the RMS value in each row or column of the input, or along vectors of a specified dimension of the input. It can also compute the RMS value of the entire input at each individual sample time. Each element in the output array y is the RMS value of the corresponding column, row, or entire input. The output array y depends on the setting of the Find the RMS value over parameter. Consider a three-dimensional input signal of size M-by-N-by-P. When you set Find the RMS value over to:
Entire input — The output at each sample time is a scalar that contains the RMS value of the M-by-N-by-P input matrix.
Each row — The output at each sample time consists of an M-by-1-by-P array, where each element contains the RMS value of each vector over the second dimension of the input. For an M-by-N matrix input, the output at each sample time is an M-by-1 column vector.
Each column — The output at each sample time consists of a 1-by-N-by-P array, where each element contains the RMS value of each vector over the first dimension of the input. For an M-by-N matrix input, the output at each sample time is a 1-by-N row vector.
Specified dimension — The output at each sample time depends on the value of the Dimension parameter. If you set the Dimension to 1, the output is the same as when you select Each column. If you set the Dimension to 2, the output is the same as when you select Each row. If you set the Dimension to 3, the output at each sample time is an M-by-N matrix containing the RMS value of each vector over the third dimension of the input.
When you select Running RMS, the block tracks the RMS value of each channel in a time sequence of inputs. In this mode, you must also specify a value for the Input processing parameter.
Elements as channels (sample based) — The block treats each element of the input as a separate channel. For a three-dimensional input signal of size M-by-N-by-P, the block outputs an M-by-N-by-P array. Each element yijk of the output contains the RMS value of the element uijk for all inputs since the last reset.
When a reset event occurs, the running RMS yijk in the current frame is reset to the element uijk.
Columns as channels (frame based) — The block treats each column of the input as a separate channel. This option does not support input signals with more than two dimensions. For a two-dimensional input signal of size M-by-N, the block outputs an M-by-N matrix. Each element yij of the output contains the RMS value of the elements in the jth column of all inputs since the last reset, up to and including the element uij of the current input.
When a reset event occurs, the running RMS for each channel becomes the RMS value of all the samples in the current input frame, up to and including the current input sample.
Running RMS — Option to select running RMS
When you select the Running RMS parameter, the block tracks the RMS value of each channel in a time sequence of inputs.
Find the RMS value over — Dimension over which the block computes the RMS value
Each column — The block outputs the RMS value over each column.
Each row — The block outputs the RMS value over each row.
Entire input — The block outputs the RMS value over the entire input.
Specified dimension — The block outputs the RMS value over the dimension specified in the Dimension parameter.
To enable this parameter, clear the Running RMS parameter.
Specify the dimension (one-based value) of the input signal over which the RMS value is computed. The value of this parameter must be greater than 0 and less than the number of dimensions in the input signal.
To enable this parameter, set Find the RMS value over to Specified dimension.
When your inputs are of variable size, and you select the Running RMS parameter, then:
To enable this parameter, select the Running RMS parameter.
The block resets the running RMS whenever a reset event is detected at the optional Rst port. The reset sample time must be a positive integer multiple of the input sample time.
When a reset event occurs while the Input processing parameter is set to Elements as channels (sample based), the running RMS for each channel is initialized to the value in the corresponding channel of the current input. Similarly, when the Input processing parameter is set to Columns as channels (frame based), the running RMS for each channel becomes the RMS value of all the samples in the current input frame, up to and including the current input sample.
Use the RMS block to compute the RMS of a noisy square wave signal.
The RMS value of a discrete-time signal is the square root of the arithmetic mean of the squares of the signal sample values.
For an M-by-N input matrix u, the RMS value of the jth column of the input is given by:
{y}_{j}=\sqrt{\frac{\sum _{i=1}^{M}{|{u}_{ij}|}^{2}}{M}}\text{ }1\le j\le N
When you clear the Running RMS parameter in the block and specify a dimension, the block produces results identical to the MATLAB® rms function, when it is called as y = rms(u,D).
y is the RMS value.
The RMS value along the entire input is identical to calling the rms function as y = rms(u(:)).
When inputs are complex, the block computes the RMS value of the magnitude of the complex input.
Moving RMS | Mean
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Return_on_equity Knowpia
The return on equity (ROE) is a measure of the profitability of a business in relation to the equity. Because shareholder's equity can be calculated by taking all assets and subtracting all liabilities, ROE can also be thought of as a return on assets minus liabilities. ROE measures how many dollars of profit are generated for each dollar of shareholder's equity. ROE is a metric of how well the company utilizes its equity to generate profits.
{\displaystyle ROE={\frac {\text{Net Income}}{\text{ Equity}}}}
ROE is equal to a fiscal year net income (after preferred stock dividends, before common stock dividends), divided by total equity (excluding preferred shares), expressed as a percentage.
ROE is especially used for comparing the performance of companies in the same industry. As with return on capital, a ROE is a measure of management's ability to generate income from the equity available to it. ROEs of 15–20% are generally considered good.[2] ROE is also a factor in stock valuation, in association with other financial ratios. While higher ROE ought intuitively to imply higher stock prices, in reality, predicting the stock value of a company based on its ROE is dependent on too many other factors to be of use by itself.[3]
Sustainable growthEdit
The sustainable growth model shows that when firms pay dividends, earnings growth lowers. If the dividend payout is 20%, the growth expected will be only 80% of the ROE rate.
The growth rate will be lower if earnings are used to buy back shares. If the shares are bought at a multiple of book value (a factor of x times book value), the incremental earnings returns will be reduced by that same factor (ROE/x).
ROE is calculated from the company perspective, on the company as a whole. Since much financial manipulation is accomplished with new share issues and buyback, the investor may have a different recalculated value 'per share' (earnings per share/book value per share).
The DuPont formulaEdit
The DuPont formula, also known as the strategic profit model, is a common way to decompose ROE into three important components. Essentially, ROE will equal the net profit margin multiplied by asset turnover multiplied by accounting leverage which is total assets divided by the total assets minus total liabilities. Splitting return on equity into three parts makes it easier to understand changes in ROE over time. For example, if the net margin increases, every sale brings in more money, resulting in a higher overall ROE. Similarly, if the asset turnover increases, the firm generates more sales for every unit of assets owned, again resulting in a higher overall ROE. Finally, increasing accounting leverage means that the firm uses more debt financing relative to equity financing. Interest payments to creditors are tax-deductible, but dividend payments to shareholders are not. Thus, a higher proportion of debt in the firm's capital structure leads to higher ROE.[2] Financial leverage benefits diminish as the risk of defaulting on interest payments increases. If the firm takes on too much debt, the cost of debt rises as creditors demand a higher risk premium, and ROE decreases.[4] Increased debt will make a positive contribution to a firm's ROE only if the matching return on assets (ROA) of that debt exceeds the interest rate on the debt.[5]
{\displaystyle \mathrm {ROE} ={\frac {\mbox{Net income}}{\mbox{Sales}}}\times {\frac {\mbox{Sales}}{\mbox{Total Assets}}}\times {\frac {\mbox{Total Assets}}{\mbox{Shareholder Equity}}}}
^ http://www.investopedia.com/terms/r/returnonequity.asp Investopedia Return on Equity
^ a b "Profitability Indicator Ratios: Return On Equity", Richard Loth Investopedia
^ Rotblut, Charles; Investing, Intelligent (January 18, 2013). "Beware: Weak Link Between Return On Equity And High Stock Price Returns". Forbes. Retrieved November 4, 2018.
^ Woolridge, J. Randall and Gray, Gary; Applied Principles of Finance (2006)
^ Bodie, Kane, Markus, "Investments"
Annual Ratio Definitions
Return On Equity Screener- figures from financial statements
Online Return On Equity Calculator
Return On Equity Explained
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Ask Answer - Mechanical Properties Of Solids - Expert Answered Questions for School Students
Is rigidity and elasticity the same thing from the chapter mechanical properties of solids???
Q.40. In which of the following electrical devices electromagnetic motor is used?
(B) Heater
derive poisons ratio
what is hooke's law and what is compressibility
15. \mathrm{A} \mathrm{uniform} \mathrm{chain} \mathrm{of} \mathrm{lenght} \mathrm{L} \mathrm{and} \mathrm{mass} \mathrm{M} \mathrm{is} \mathrm{hanging} \mathrm{from} \mathrm{ceiling} \mathrm{as} \mathrm{shown} \mathrm{in} \mathrm{figure}. \mathrm{Tension} \mathrm{in} \mathrm{the} \mathrm{chain} \mathrm{at} \mathrm{a} \mathrm{distance} \mathrm{x}=\frac{\mathrm{L}}{4} \mathrm{from} \mathrm{the} \mathrm{lower} \mathrm{end} \mathrm{is} :\phantom{\rule{0ex}{0ex}}1. \frac{\mathrm{M}\left(\mathrm{l}-\mathrm{x}\right)\mathrm{g}}{\mathrm{l}} 2. \mathrm{Mg} 3. \mathrm{Zero} 4. \frac{\mathrm{mxg}}{\mathrm{l}}
Two exactly similar wires of steel and copper are stretched by equal forces. If the total elongation is 1 cm, find how much each wire is elongated. Given Y for steel is 20 x 10^10 N/m^2 and copper is 12 x 10^10 N/m^2
Calculate the percentage increase in length of a wire of diameter 2.2 mm stretched by a load of 100kg. Young’s modulus of wire is 12.5 x 10^10 N/m^2
This is the question of hc verma pg 300 exercise q4. As the stress is same in both wires Y and strain are inversely related. So the answer given is wrong
Shedha Bilal
Please answer
Please answer the following -
16) A body of mass 3.14 kg is suspended form one end of a wire of length 10 m. The radius if cross - section of the wire is changing uniformly from
5×{10}^{-4} m
at the top (i.e. point of suspension) to
9.8×{10}^{-4} m
at the bottom. Young's modulus of elasticity is
2×{10}^{11} N/{m}^{2}.
The change of length of the wire is.
\left(a\right) 4×{10}^{-3 } m\phantom{\rule{0ex}{0ex}}\left(b\right) 3×{10}^{-3} m\phantom{\rule{0ex}{0ex}}\left(c\right) {10}^{-3} m\phantom{\rule{0ex}{0ex}}\left(d\right) 2×{10}^{-3} m
Q.10. If the work done in stretching a wire by 1 mm is 2 J, then work necessary for stretching another wire of same material but with double radius of cross-section and half of the length by 1 mm is
Show that value of poisson's ratio lies -1 and o.5
Ishika Sahay
graph of stress and strain
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Home : Support : Online Help : Programming : Audio Processing : Record
record audio data from a microphone to an audio data object
Record(duration, component, opts)
duration of time to record
(optional) string; name of the Microphone component to use.
options, described below
The sample rate with which to record. When set to auto, the sample rate associated with the Microphone component is used. If no units are specified, values are presumed to be in hertz (Hz).
The Record command records audio data of a specified duration from a Microphone component and returns an Audio object containing the recorded data.
If the component parameter is not specified, the default component name Microphone0 will be used.
In order to record, open the Components Palette and insert a microphone component.
\mathrm{with}\left(\mathrm{AudioTools}\right):
Record from the microphone for 1.5 seconds.
\mathrm{audio1}≔\mathrm{Record}\left(1.5,"Microphone0"\right)
Record from the microphone for 2500 milliseconds (2.5 s) at CD quality (44.1 kHz).
\mathrm{audio2}≔\mathrm{Record}\left(2500\mathrm{Unit}\left(\mathrm{ms}\right),"Microphone0",\mathrm{rate}=44.1\mathrm{Unit}\left(\mathrm{kHz}\right)\right)
The AudioTools[Record] command was introduced in Maple 2015.
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!! used as default html header if there is none in the selected theme. Base converter
\sqrt{2}
Description: converts a number between different numbering systems, arbitrary precision. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, number, numeration, conversion
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Lidar Point Cloud Semantic Segmentation Using SqueezeSegV2 Deep Learning Network - MATLAB & Simulink - MathWorks France
The training procedure for this example is for organized point clouds. For an example showing how to convert unorganized to organized point clouds, see Unorganized to Organized Conversion of Point Clouds Using Spherical Projection.
\mathit{r}=\sqrt{{\mathit{x}}^{2\text{\hspace{0.17em}}}+{\mathit{y}}^{2}+{\mathit{z}}^{2}}
Create a pixel label datastore using pixelLabelDatastore to store pixel-wise labels from the pixel label images. The object maps each pixel label to a class name. In this example, the vegetation, ground, road, road markings, sidewalk, cars, trucks, other vehicles, pedestrian, road barrier, signs, and buildings are the objects of interest; all other pixels are the background. Specify these classes and assign a unique label ID to each class.
Create a standard SqueezeSegV2 [1] network by using the squeezesegv2Layers function. In the SqueezeSegV2 network, the encoder subnetwork consists of FireModules interspersed with max-pooling layers. This arrangement successively decreases the resolution of the input image. In addition, the SqueezeSegV2 network uses the focal loss function to mitigate the effect of the imbalanced class distribution on network accuracy. For more details on how to use the focal loss function in semantic segmentation, see focalLossLayer.
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In mathematics, the opposite or additive inverse of a number
{\displaystyle k}
is a number
{\displaystyle n}
which, when added to
{\displaystyle k}
, results in 0. The opposite of
{\displaystyle a}
{\displaystyle -a}
.[1][2] For example, −7 is the opposite of 7, because
{\displaystyle 7+-7=0}
The opposite of a positive number is the negative version of the number. The opposite of a negative number is the positive version of the number. The opposite of 0 is 0.
Two opposite numbers have the same absolute value.
↑ Weisstein, Eric W. "Additive Inverse". mathworld.wolfram.com. Retrieved 2020-08-27.
↑ "Additive Inverse". www.learnalberta.ca. Retrieved 2020-08-27.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Opposite_number&oldid=7881256"
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Elements of Frequency-domain Analysis
Frequency-domain analysis is a commonly used technique in signal processing. For example, in heart rate variation (HRV) analysis, high frequency (HF) and low frequency (LF) components are usually extracted from RR intervals based on power spectral analysis. As an industrial engineering, I am not familiar with this area, but I need this tool in my research. Although a lot of open-source software/libraries are available, I think it still necessary to understand what is going on. Therefore, I spent some time in frequency-domain analysis and make a little summary here.
Fourier transform is the basis of frequency-domain analysis. Fourier transform maps a signal from time domain to frequency domain. There are four kinds of Fourier transforms:
X(k)=\frac{1}{T} \int_0^T x(t) e^{-j2\pi k t/T} dt
X(f)=\sum_{n=-\infty}^{\infty} x(n) e^{-j2\pi fn}
Continuous-time Fourier transform:
X(f)=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt
X(k)=\sum_{n=0}^{N-1} x(n) e^{-j2\pi kn/N}
Each kind of Fourier transform has its inverse Fourier transform, which reconstruct the original signal based on the frequency components.
Take Fourier series as an example. It can be shown that any periodic function
x(t)
can be represented as linear combination of
\cos(2\pi m t/T)
\sin(2\pi n t/T)
T
is the period of
x(t)
x(t+T)=x(t)
t
m,n
x(t)=\sum_{m=0}^{\infty} a_m \cos(2\pi mt/T) + \sum_{n=1}^{\infty} b_n \sin(2\pi nt/T)
a_m
b_n
are real numbers. According to Euler's formula
e^{jx}=\cos x + j \sin x
j=\sqrt{-1}
x(t)=\sum_{k=-\infty}^{\infty} X(k) e^{j2\pi k t/T}
x(t)
is represented by a linear combination of
e^{j2\pi k t/T}
X(k)
are the coefficients (complex numbers),
k
is integer. Recall the Euler's formula,
e^{j2\pi k t/T}
T/k
or frequency
k/T
x(t)
has been discomposited into combination of components in different frequency! It can be shown that the coefficient
X(k)
X(k)=\frac{1}{T} \int_0^T x(t) e^{-j2\pi k t/T} dt
This equation is Fourier series. Therefore, Fourier series calculates the coefficients for each frequency component. And the previous equation is inverse Fourier series, which reconstruct the original time-domain signal based on the coefficients of frequency components.
Other kinds of Fourier transforms are similar. And the characteristics of each Fourier transform are listed in the following table.
Fourier series continuous periodic discrete nonperiodic
Discrete-time Fourier transform discrete nonperiodic continuous periodic
Continuous-time Fourier transform continuous nonperiodic continuous nonperiodic
Discrete Fourier transform discrete periodic discrete periodic
For example, Fourier series takes a continuous periodic function
x(t)
as input, and produces discrete non-periodic coefficients
X(k)
as output. The table shows something interesting. Discrete in one domain always related to periodic in the other domain, while continuous in one domain always related to nonperiodic in the other domain.
Since digital signals are stored in discrete values in computer, discrete-time Fourier transform (DTFT) and discrete Fourier transform (DFT) are more commonly used. DTFT takes discrete nonperiodic signal as input, and produces its frequency components as continuous periodic series. DFT takes discrete periodic signal as input, and produces its frequency components as a discrete periodic signal. If some conditions are satisfied, e.g.
N=2^n
, fast Fourier transform (FFT) algorithms can be employed to calculate DFT efficiently.
Fourier transform can be considered from the point of view function basis and kernel method, which is the concern of my another article.
There are other transforms such as z-transform and Laplace transform. They have very similar form with Fourier transform. For example, in z-transform
X(z)=\sum_{n=-\infty}^{\infty} x(n) z^{-n}
z
is a complex number. Let
z=e^{jw}
we get DTFT.
3. Energy Spectrum and Power Spectrum
Energy spectrum is defined as the energy on a specific frequency
S_x (f) = |X(f)|^2
If we sum up or integrate the energy spectrum over
(-\infty, \infty)
for Fourier series and continuous Fourier series, and over
[-1, 1]
for DFT and DTFT, we can get the total energy.
However, sometimes a signal does not have Fourier transform (continuous Fourier transform or DTFT), i.e.
|X(f)|
go to infinity, we define power spectrum as the
S_x (f)
divided by the integral interval of Fourier transform.
R_x (f)=\lim_{T \rightarrow \infty} \frac{1}{2T} |\mathcal{F} (x)|^2
\mathcal{F} (x)
applies Fourier transform to
x(t)
[-T, T]
f
R_x (f)
can be calculated as the Fourier transform of
r_x(\tau)
R_x (f)=\mathcal{F} (r_x(\tau))
r_x(\tau)=\lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^T x(t+\tau) x(t) dt
4. Stationary Stochastic Process
Until now we are talking about deterministic signals, i.e.
x(t)
is a specific function of
t
. Sometimes it is necessary to consider stochastic process, i.e.
x_t
is a random variable. Here I use subscript to distinguish stochastic process from deterministic signal.
x_1, x_2, \cdots, x_N
, if the correlation between any two variables depends only on the difference in time
Cor(x_{n+k}, x_n)=Cor(x_{1+k},x_1)=r(k)
n
, and the expected value is certain
E(x_n)=\mu
The stochastic process is weakly stationary or wide-sense stationary (WSS).
For WSS process, we cannot apply Fourier transform to the original series directly since they are random numbers. But we can apply Fourier transform to its autocorrelation function
r(k)
. Recall Section 3,
r_x(k)=\lim_{T \rightarrow \infty} \frac{1}{N} \sum_{0}^{N-1} x(t+k) x(t) dt
is the autocorrelation for WSS process with zero mean and variance 1 (for other WSS process, it will be proportional). The resulting value
R_x (f)=\mathcal{F} (r_k)
is the power spectrum.
5. Power Spectrum Estimation
Power spectrum is calculated as the Fourier transform of the autocorrelation function (ACF). However, in practice, the series is usually not infinite, and we rarely know the true ACF. Thus we need to estimate power spectrum. There are two categories of methods for power spectrum estimation: non-parametric method and parametric method.
In non-parametric method, ACF is estimated directly by data. Periodogram is a non-parametric estimation, where
\hat{r}(k)=\frac{1}{N} \sum_{n=0}^{N-1-k} x(n)x(n+k)
N
is the length of the signal. It is equivalent to the ACF of an infinite series
y(n)
y(n)=x(n)
n=0,1,\cdots,N-1
y(n)=0
for other
n
In parametric method, a model, e.g. autoregressive moving average (ARMA) model, is fitted at first. Then the ACF can be calculated.
The constraint in power spectrum analysis is that the process should be stationary. However, in a lot of situations, this assumption is not satisfied. Furthermore, power spectrum analysis need a long series of data to obtain good estimation. Therefore, maybe state-space model is a better choice in a number of situations.
About ARMA model:
Cryer, J. D., & Chan, K. S. (2008). Time series analysis: with applications in R. Springer.
About Fourier transform:
About power spectrum:
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A Variational Boundary Integral Method for the Analysis of Three-Dimensional Cracks of Arbitrary Geometry in Anisotropic Elastic Solids | J. Appl. Mech. | ASME Digital Collection
Department of Mechanical Engineering, University of California, Riverside, CA 92521
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Oct. 22, 1999; final revision, Nov. 26, 1999. Associate Technical Editor: M. Ortiz. Discussion on the paper should be addressed to the Technical Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Xu, G. (November 26, 1999). "A Variational Boundary Integral Method for the Analysis of Three-Dimensional Cracks of Arbitrary Geometry in Anisotropic Elastic Solids ." ASME. J. Appl. Mech. June 2000; 67(2): 403–408. https://doi.org/10.1115/1.1305276
A variational boundary integral method is developed for the analysis of three-dimensional cracks of arbitrary geometry in general anisotropic elastic solids. The crack is modeled as a continuous distribution of dislocation loops. The elastic energy of the solid is obtained from the known expression of the interaction energy of a pair of dislocation loops. The crack-opening displacements, which are related to the geometry of loops and their Burgers vectors, are then determined by minimizing the corresponding potential energy of the solid. In contrast to previous methods, this approach results in the symmetric system of equations with milder singularities of the type
1/R,
which facilitate their numerical treatment. By employing six-noded triangular elements and displacing midside nodes to quarter-point positions, the opening profile near the front is endowed with the accurate asymptotic behavior. This enables the direct computation of stress intensity factors from the opening displacements. The performance of the method is assessed by the analysis of an elliptical crack in the transversely isotropic solid. It also illustrates that the conventional average schemes of elastic constants furnish quite inaccurate results when the material is significantly anisotropic. [S0021-8936(00)02702-1]
crack-edge stress field analysis, dislocation loops, integral equations, elastic constants
Anisotropy, Fracture (Materials), Solids, Dislocations, Geometry, Stress, Elastic constants, Integral equations
Critical Configurations for Dislocation From Crack Tips
A First Order Perturbation Analysis on Crack-Trapping by Arrays of Obstacles
A Three-Dimensional Analysis of Crack Trapping and Bridging by Tough Particles
Clifton, R. J., 1989, “Three-Dimensional Fracture-Propagation Models,” J. L. Gidley, ed., Hydraulic Fracturing, SPE Monograph.
An Integral Equation Method for Solving the Problem of a Plane Crack of Arbitrary Shape
Three-Dimensional Crack Analysis
Cruse, T. A., 1988, Boundary Element Analysis in Computational Fracture Mechanics, Kluwer, Dordrecht, The Netherlands.
Three-Dimensional Curved Crack in an Elastic Body
Growth and Stability of Interacting Surface Flaws of Arbitrary Shape
A Variational Boundary Integral Method for the Analysis of 3-D Cracks of Arbitrary Geometry Modeled as Continuous Distributions of Dislocation Loops
Dislocations in Anisotropic Media: The Interaction Energy
Billy, B. A., and Eshelby, J. D., 1968, Fracture: An Advanced Treatise, Vol. 1, H. Liebowitz, ed., Academic Press, San Diego, CA.
The Fracture Mechanics of Slit-Like Cracks in Anisotropic Elastic Media
Kassir, M. K., and Sih, G. C., 1975, Mechanics of Fracture: Three-Dimensional Crack Problems, Noordhoff, Groningen.
Hirth, J. P., and Lothe, J., 1982, Theory of Dislocations, 2nd Ed., John Wiley and Sons, New York.
Weakening of an Elastic Solid by a Rectangular Array of Cracks
Effect of Orthotropy on Singular Stresses for a Finite Crack
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!! used as default html header if there is none in the selected theme. Graphic ODE phase graph
\{\begin{array}{ll}x\prime =& Ax+By\\ y\prime =& Cx+Dy\end{array}
A,B,C,D
Description: recognize the phase graph of a given ODE. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, analysis, functions, ode,differential_equation
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H. Y. Huang, A. Singh, C. Y. Mou, S. Johnston, A. F. Kemper, J. van den Brink, P. J. Chen, T. K. Lee, J. Okamoto, Y. Y. Chu, J. H. Li, S. Komiya, A. C. Komarek, A. Fujimori, C. T. Chen, D. J. Huang
Quantum phase transitions play an important role in shaping the phase diagram of high-temperature cuprate superconductors. These cuprates possess intertwined orders which interact strongly with superconductivity. However, the evidence for the quantum critical point associated with the charge order in the superconducting phase remains elusive. Here, we reveal the short-range charge orders and the spectral signature of the quantum fluctuations in
{\mathrm{La}}_{2-x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}
(LSCO) near the optimal doping using high-resolution resonant inelastic x-ray scattering. On performing calculations through a diagrammatic framework, we discover that the charge correlations significantly soften several branches of phonons. These results elucidate the role of charge order in the LSCO compound, providing evidence for quantum critical scaling and discommensurations associated with charge order.
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RefiningPartition - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ConstructibleSetTools Subpackage : RefiningPartition
compute the coprime factorization of a list of constructible sets
RefiningPartition(lcs, R)
list of constructible sets
The command RefiningPartition(lcs, R) returns a list of pairwise disjoint constructible sets out_lcs such that every constructible set of lcs can be written as a disjoint union of several constructible sets in out_lcs.
This is represented by a matrix showing the constructible set and its associated indices where the constructible set comes from.
This function can also be seen as a set theoretical instance of the co-prime factorization problem.
This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RefiningPartition(..) only after executing the command with(RegularChains[ConstructibleSetTools]). However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][RefiningPartition](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):
R≔\mathrm{PolynomialRing}\left([x,y,u,v]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
F≔[vxy+u{x}^{2}+x,u{y}^{2}+{x}^{2}]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}]
\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(F,[u],R\right)
\textcolor[rgb]{0,0,1}{\mathrm{cs1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(F,[v],R\right)
\textcolor[rgb]{0,0,1}{\mathrm{cs2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{cs3}≔\mathrm{Projection}\left(F,2,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{cs3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{RefiningPartition}\left([\mathrm{cs1},\mathrm{cs2}],R\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{1}]\\ \textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\\ \textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{2}]\end{array}]
\mathrm{RefiningPartition}\left([\mathrm{cs1},\mathrm{cs2},\mathrm{cs3}],R\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{3}]\\ \textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\\ \textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\\ \textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}& [\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\end{array}]
Chen, C.; Golubitsky, O.; Lemaire, F.; Moreno Maza, M.; and Pan, W. "Comprehensive Triangular Decomposition." Proc. CASC 2007, LNCS, Vol. 4770, pp. 73-101. Springer, 2007.
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