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Mean - Simple English Wikipedia, the free encyclopedia
general term for the several definitions of mean value, the sum divided by the count
(Redirected from Mean (statistics))
In mathematics and statistics, the mean is a kind of average. Besides the mean, there are other kinds of average, and there are also a few kinds of mean.
The most common mean is the arithmetic mean, which is calculated by adding all of the values together, then dividing by the number of values.
For example, if 1, 2, 2, 100, 100 is a set of numbers or scores. If we add all the numbers, the answer is 205. By dividing this number by the number of numbers (5), we find that the mean is 41. The difficulty with this particular set of numbers is that no one in this group scored anything like a 41, and it does not tell us much about what kind of scores these numbers represent.
2 Related calculations
Calculation detailsEdit
In general, to find the average of
{\displaystyle N}
numbers, the
{\displaystyle N}
numbers are added and the total is divided by
{\displaystyle N}
In symbols, if the numbers are
{\displaystyle X_{1}}
{\displaystyle X_{2}}
{\displaystyle X_{3}}
{\displaystyle X_{N}}
, the total is:
{\displaystyle X_{1}+X_{2}+X_{3}+...+X_{N}}
The total is divided by
{\displaystyle N}
to make the average:
{\displaystyle {X_{1}+X_{2}+X_{3}+...+X_{N}} \over N}
{\displaystyle X_{1}}
{\displaystyle X_{2}}
{\displaystyle X_{3}}
{\displaystyle X_{N}}
are all the numbers in a sample
{\displaystyle X}
, then this average is also called the sample mean of
{\displaystyle X}
, and represented by the symbol
{\displaystyle {\overline {X}}}
Lucy is 5 years old. Tom is 6 years old. Emily is 7 years old. To find the average age:
Add the three numbers :
{\displaystyle 5+6+7=18}
The total is 18. Divide the total 18 by three:
{\displaystyle 18/3=6}
The average of the three numbers is 6.
{\displaystyle {\frac {5+6+7}{3}}}
Therefore, the average age of Lucy, Tom and Emily is 6 years.
Related calculationsEdit
The idea behind the mean is to represent a number of measurements, or values, by one value only. But there are different ways to calculate such a representing value.
The median is the number that divides all the samples in such a way that half of the samples are below it, and the other half above. Example: 1, 10, 50, 100, 100 is a set of numbers or scores. If we look at these scores, we discover that the number 50 falls in the middle of the range of numbers, which tells us that half the numbers or scores are above this number, and half the numbers and scores are below this number. This is more information, depending on what you are trying to find out about this group of numbers, to help you find out what you want to know. It is not always possible to make the higher and lower group each exactly half of the total (for example, the equal division fails for the list 1, 2, 2).
The modus or mode is the number that occurs most often. Example: 1, 2, 2, 100, 200 is a set of numbers or scores. If we look at the numbers we discover that the number 2 recurs most often and would tell us that the number or score of 2 is the most common score or number in the group.
The arithmetic mean is just the average, the value that is the sum of all values, divided by their number. This is what is most often referred to as mean.
The geometric mean is the root of the product of all values.[2] For example, the geometric mean of 4, 6, and 9 is 6, because 4 times 6 times 9 is 216, and the cube root (because there are three values) of 216 is 6.
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. It is often used when people want a mean of rates or percentages.
The root mean square (or quadratic mean) is the square root of the arithmetic mean of the squares of the values.[2] The root mean square is at least as high as the arithmetic mean, and usually higher.[3]
If people do many different measurements, they will get many different results. Those results have a certain distribution, and they can also be centered around an average value. This average value is what mathematicians call arithmetic mean.
Mean can also stand for expected value. For a random variable
{\displaystyle X}
, this is represented by the symbol
{\displaystyle E(X)}
↑ 2.0 2.1 "Mean | mathematics". Encyclopedia Britannica. Retrieved 2020-08-21.
↑ Weisstein, Eric W. "Mean". mathworld.wolfram.com. Retrieved 2020-08-21.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Mean&oldid=8145972"
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Estimate optical flow - MATLAB estimateFlow
estimateFlow
Estimate optical flow
flow = estimateFlow(opticFlow,I)
flow = estimateFlow(opticFlow,I) estimates optical flow between two consecutive video frames.
opticFlow — Object for optical flow estimation
opticalFlowFarneback object | opticalFlowHS | object | opticalFlowLK object | opticalFlowLKDoG object
Object for optical flow estimation, specified as one of the following:
opticalFlowFarneback object
opticalFlowHS object
opticalFlowLK object
opticalFlowLKDoG object
The input opticFlow defines the optical flow estimation method and its properties used for estimating the optical flow velocity matrices.
I — Current video frame
Current video frame, specified as a 2-D grayscale image of size m-by-n. The input image is generated from the current video frame read using the VideoReader object. The video frames in RGB format must be converted to 2-D grayscale images for estimating the optical flow.
flow — Object for storing optical flow velocity matrices
opticalFlow object
Object for storing optical flow velocity matrices, returned as an opticalFlow object.
The function estimates optical flow of the input video using the method specified by the input object opticFlow. The optical flow is estimated as the motion between two consecutive video frames. The video frame T at the given instant tcurrent is referred as current frame and the video frame T-1 is referred as previous frame. The initial value of the previous frame at time tcurrent = 0 is set as a uniform image of grayscale value 0.
If you specify opticFlow as opticalFlowLKDoG object, then the estimation delays by an amount relative to the number of video frames. The amount of delay depends on the value of NumFrames defined in opticalFlowLKDoG object. The optic flow estimated for a video frame at tcurrent corresponds to the video frame at time
{t}_{flow}=\left({t}_{current}-\left(NumFrames-1\right)/2\right)
. tcurrent is the time of the current video frame.
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LMIs in Control/Applications/An LMI for Multi-Robot Systems/Consensus for Multi-Agent Systems - Wikibooks, open books for an open world
LMIs in Control/Applications/An LMI for Multi-Robot Systems/Consensus for Multi-Agent Systems
< LMIs in Control | Applications | An LMI for Multi-Robot Systems
This application gives an instance of the use of LMIs in achieving consensus for multi-agent systems. More focus is placed on the flocking algorithm, which is one of the consensus algorithms extensively used in multi-robot systems.
2 Flocking Algorithm
3 Mathematical Model:
A robot swarm is a network of multiple robots working together as a unit to achieve an objective. In order for robots in a swarm to work together, they need to come to consensus/agreement on a number of parameters. A number of algorithms called consensus algorithms exist to serve that purpose. Consensus Algorithms are algorithms that allow a set of agents to reach an agreement. Various areas such as graph theory, control theory and matrix theory overlap when it comes to designing consensus algorithms for multi-agent systems.
Flocking Algorithm[edit | edit source]
The flocking algorithm is one of the consensus algorithms used in trajectory planning for multi-agent networks. The algorithm helps in maintaining separation, so that the agents do not collide, alignment, so that the agents have the same headings, and cohesion, so that the agents do not wonder far away from each other. The flocking algorithm can be represented in a graph network form, where the nodes in the graph represent the agents or individual robots, and the edges/links represent the nature of interaction between the nodes. Consequently. the interactions between the nodes can be Below is the general equation for a flocking algorithm.
{\displaystyle {\dot {v_{i}}}-\sum _{i=N_{i}}^{n}i=(v_{i}-v_{j})+\sum _{i=N_{i}}^{n}i\Delta _{r_{i}}V_{ij}(r_{ij})}
{\displaystyle V_{ij}}
is the artificial potential functio{\displaystyle n}
is the number of robots
{\displaystyle {\dot {v_{i}}}}
is the input/acceleration control
{\displaystyle r_{i}}
is the position vector of robot
{\displaystyle i}
{\displaystyle r_{ij}=r_{i}-r_{j}}
, which is the distance between two neighboring robots
Mathematical Model:[edit | edit source]
Before we begin, we need to obtain a state space representation for our system. Based on observation, it can be deduced that the general form of a consensus algorithm bears some resemblance to the state space equation. The general equation for a consensus algorithm in terms of the Laplacian matrix is:
{\displaystyle {\dot {x}}=-Lx}
{\displaystyle L}
represents the Laplacian matrix.
Comparing the above equation to the state space equation,
{\displaystyle {\dot {x}}=Ax}
, we notice that the
{\displaystyle A}
has been substituted with a
{\displaystyle -L}
The A, B and C matrices are represented as diagonal blocks because of the decentralized nature of the control framework. Each block in the A, B and C matrices is associated with its corresponding state, input, or output vector.
{\displaystyle {\begin{aligned}\ A&={\begin{bmatrix}\ a_{11}&a_{1n}\\\ a_{n1}&a_{nn}\\\end{bmatrix}}&\ B={\begin{bmatrix}\ b_{11}&b_{1n}\\\ b_{n1}&b_{nn}\\\end{bmatrix}}&\ &C={\begin{bmatrix}\ c_{11}&c_{1n}\\\ c_{n1}&c_{nn}\\\end{bmatrix}}\end{aligned}}}
x, u and y represent the state, input and output vectors respectively.
{\displaystyle {\begin{aligned}\ x&={\begin{bmatrix}\ x_{1}&...&x_{n}\\\end{bmatrix}}^{T}&\ y&={\begin{bmatrix}\ y_{1}&...&y_{n}\\\end{bmatrix}}^{T}&\ u&={\begin{bmatrix}\ u_{1}&...&u_{n}\\\end{bmatrix}}^{T}&\end{aligned}}}
Further Discussion:[edit | edit source]
In Elham Semsar-Kazerooni and Khashayar's paper, "Optimal Consensus Seeking in a Network of Multi-Agent Systems", further work was done through a series of variable substitutions and other mathematical operations to obtain the an LMI as seen below. For more information on the LMI formulation procedure, and the details of the variables in the above LMI, please visit follow the link in the reference section to review the paper.
{\displaystyle \min trace(P)}
Subject to::
{\displaystyle {\begin{aligned}\ {\begin{bmatrix}\ \gamma &{\bar {S}}^{*}ZQ^{1/2}&{\bar {S}}^{*}W^{*}R^{1/2}\\\ Q^{1/2}Z{\bar {S}}&-I&0\\\ R^{1/2}W{\bar {S}}&0&-I\\\end{bmatrix}}\ <0,\end{aligned}}}
Seeking in Network and Multi-Agent Systems - Optimal Consensus Seeking in a Network of Multi-Agent Systems paper by Elham Semsar-Kazerooni and Khashayar.
Return to Main Page:[edit | edit source]
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Applications/An_LMI_for_Multi-Robot_Systems/Consensus_for_Multi-Agent_Systems&oldid=3789292"
Book:LMIs in Control
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The limit command has been enhanced for the case of limits of bivariate rational functions with non-isolated singularities. Many such limits that could not be determined previously are now computable. If the limit exists in such a situation, it is either
+\mathrm{∞}
-\mathrm{∞}
. Maple can also determine if the limit does not exist, and then returns
\mathrm{undefined}
In Maple 18, all the following limit calls would return unevaluated, but they can now be computed in Maple 2015.
f≔\frac{x y}{x+y}:
\mathrm{limit}\left(f,\left\{x=0,y=0\right\}\right)
\textcolor[rgb]{0,0,1}{\mathrm{undefined}}
g≔\frac{4 x y}{{\left(x-y\right)}^{2}}:
\mathrm{limit}\left(g,\left\{x=0,y=0\right\}\right)
\textcolor[rgb]{0,0,1}{\mathrm{undefined}}
h≔\frac{{x}^{4}-{x}^{2}-{y}^{2}}{{\left(x-y\right)}^{4}}:
\mathrm{limit}\left(h,\left\{x=0,y=0\right\}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
Let us plot these three functions in the neighborhood of the origin.
f
+\mathrm{∞}
on one side of the singularity
y=-x
-\mathrm{∞}
on the other side (as shown in the following plot). Therefore, the limit at the origin does not exist.
\mathrm{pf1}≔\mathrm{plot3d}\left(f, x=-0.1..0.1,y=-0.1..-x-1e-10,\mathrm{axes}=\mathrm{boxed},\mathrm{view}=-10..10\right):
\mathrm{pf2}≔\mathrm{plot3d}\left(f, x=-0.1..0.1,y=-x+1e-10..0.1,\mathrm{axes}=\mathrm{boxed},\mathrm{view}=-10..10\right):
\mathrm{plots}:-\mathrm{display}\left(\mathrm{pf1},\mathrm{pf2}\right)
Now, consider the second example.
s≔\mathrm{plots}:-\mathrm{spacecurve}\left(\left[x,-x,-1\right],x=-0.1..0.1,\mathrm{color}=\mathrm{red},\mathrm{thickness}=3\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{pg}≔\mathrm{plot3d}\left(g,x=-0.1..0.1,y=-0.1..0.1,\mathrm{axes}=\mathrm{boxed},\mathrm{view}=-10..100,\mathrm{numpoints}=40000\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{plots}:-\mathrm{display}\left(s,\mathrm{pg}\right)
g
+\mathrm{∞}
close to the singularity
y=x
. However, along the anti-diagonal
y=-x
, the limit is finite:
\mathrm{eval}\left(g,y=-x\right)
\textcolor[rgb]{0,0,1}{-1}
g
does not have a limit at the origin. In fact, any number
≥-1
can occur, namely, as the limit along the ray
y=a x
-1≤a<1
\mathrm{eval}\left(g,y=a x\right)
\frac{\textcolor[rgb]{0,0,1}{4}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{a}}{{\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{limit}\left(,x=0\right)
\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{a}}{{\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{plot}\left(,a=-3..1.1,\mathrm{view}=-2..10\right)
In the last example,
h
-\mathrm{∞}
on both sides of the singularity
y=x
\mathrm{plot3d}\left(h,x=-0.1..0.1,y=-0.1..0.1,\mathrm{axes}=\mathrm{boxed},\mathrm{view}=-100000..1\right)
However, in this case the limit along any ray
y=a x
a≠1
-\mathrm{∞}
\mathrm{eval}\left(h,y=a x\right)
\frac{\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{4}}}
\mathrm{limit}\left(,x=0\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{signum}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}{{\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{4}}}\right)\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
\mathrm{factor}\left(\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{signum}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}{{\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{4}}}\right)\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
\mathrm{simplify}\left(\right) assuming a<1;\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{simplify}\left(\right) assuming a>1
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
You can prove that the limit exists and is
-\mathrm{∞}
for any curve approaching the origin by using the theory of Lagrange multipliers. The extremal values (maxima and minima) of the function
h
r
C≔{x}^{2}+{y}^{2}-{r}^{2}:
\mathrm{with}\left(\mathrm{VectorCalculus}\right):
\mathrm{df}≔\mathrm{normal}~\left(\mathrm{Jacobian}\left(\left[h\right],\left[x,y\right]\right)\right)
\left[\begin{array}{cc}-\frac{2\left(2{x}^{3}y-{x}^{2}-xy-2{y}^{2}\right)}{{\left(x-y\right)}^{5}}& \frac{2\left(2{x}^{4}-2{x}^{2}-xy-{y}^{2}\right)}{{\left(x-y\right)}^{5}}\end{array}\right]
\mathrm{dC}≔\mathrm{Jacobian}\left(\left[C\right],\left[x,y\right]\right)
\left[\begin{array}{cc}2x& 2y\end{array}\right]
\mathrm{eq}≔\mathrm{factor}\left(\mathrm{numer}\left(\mathrm{normal}\left({\mathrm{df}}_{1,1}{\mathrm{dC}}_{1,2}-{\mathrm{df}}_{1,2}{\mathrm{dC}}_{1,1}\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{8}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\right)\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\right)
Thus, the local maximum and minimum values of
C
C=0
\mathrm{eq}=0
\mathrm{eq}=0
. However, you also need to consider the global suprema and infima, which may occur close to the singularity
y=x
In the example, the factor
{x}^{2}+{y}^{2}
\mathrm{eq}
does not admit any real paths, so there is only one critical path given by
{x}^{3}-x-y=0
y=-x+{x}^{3}
\mathrm{normal}\left(\mathrm{eval}\left(h,y=-x+{x}^{3}\right)\right)
\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}{\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{3}}}
\mathrm{limit}\left(,x=0\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
In order to certify the limit close to the singularity
y=x
as well, you cannot take the limit along the singularity. Instead, consider two curves that approach the singularity closely from the top and from the bottom, respectively:
\mathrm{c1}≔y=x+{x}^{2}
\textcolor[rgb]{0,0,1}{\mathrm{c1}}\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}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
\mathrm{c2}≔y=x-{x}^{2}
\textcolor[rgb]{0,0,1}{\mathrm{c2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
\mathrm{plot}\left(\left[x,\mathrm{rhs}\left(\mathrm{c1}\right),\mathrm{rhs}\left(\mathrm{c2}\right),-x+{x}^{3}\right],x=-0.5..0.5,\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{legend}=\left["singular path","c1","c2","critical path"\right]\right)
\mathrm{normal}\left(\mathrm{eval}\left(h,\mathrm{c1}\right)\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}}
\mathrm{limit}\left(,x=0\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
\mathrm{normal}\left(\mathrm{eval}\left(h,\mathrm{c2}\right)\right)
\frac{\textcolor[rgb]{0,0,1}{2}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}}
\mathrm{limit}\left(,x=0\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}
updates/Maple17/BivariateLimits, limit, limit/multi - multidimensional limits
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Repeated measures model class - MATLAB
RepeatedMeasuresModel class
BetweenDesign
BetweenModel
BetweenFactorNames
WithinFactorNames
Repeated measures model class
A RepeatedMeasuresModel object represents a model fitted to data with multiple measurements per subject. The object comprises data, fitted coefficients, covariance parameters, design matrix, error degrees of freedom, and between- and within-subjects factor names for a repeated measures model. You can predict model responses using the predict method and generate random data at new design points using the random method.
You can fit a repeated measures model using fitrm(t,modelspec).
Formula for model specification, specified as a character vector or string scalar of the form 'y1-yk ~ terms'. Specify the terms using Wilkinson notation. fitrm treats the variables used in model terms as categorical if they are categorical (nominal or ordinal), logical, character arrays, string arrays, or a cell array of character vectors.
BetweenDesign — Design for between-subject factors
Design for between-subject factors and values of repeated measures, stored as a table.
BetweenModel — Model for between-subjects factors
Model for between-subjects factors, stored as a character vector. This character vector is the text representation to the right of the tilde in the model specification you provide when fitting the repeated measures model using fitrm.
BetweenFactorNames — Names of variables used as between-subject factors
Names of variables used as between-subject factors in the repeated measures model, rm, stored as a cell array of character vectors.
ResponseNames — Names of variables used as response variables
Names of variables used as response variables in the repeated measures model, rm, stored as a cell array of character vectors.
WithinDesign — Values of within-subject factors
Values of the within-subject factors, stored as a table.
WithinModel — Model for within-subjects factors
Model for within-subjects factors, stored as a character vector.
You can specify WithinModel as a character vector or a string scalar using dot notation: Mdl.WithinModel = newWithinModelValue.
WithinFactorNames — Names of within-subject factors
Names of the within-subject factors, stored as a cell array of character vectors.
Coefficients — Values of estimated coefficients
Values of the estimated coefficients for fitting the repeated measures as a function of the terms in the between-subjects model, stored as a table.
fitrm' defines the coefficients for a categorical term using 'effects' coding, which means coefficients sum to 0. There is one coefficient for each level except the first. The implied coefficient for the first level is the sum of the other coefficients for the term.
You can display the coefficient values as a matrix rather than a table using coef = r.Coefficients{:,:}.
You can display marginal means for all levels using the margmean method.
Covariance — Estimated response covariances
Estimated response covariances, that is, covariance of the repeated measures, stored as a table. fitrm computes the covariances around the mean returned by the fitted repeated measures model rm.
You can display the covariance values as a matrix rather than a table using coef = r.Covariance{:,:}.
DFE — Error degrees of freedom
Error degrees of freedom, stored as a scalar value. DFE is the number of observations minus the number of estimated coefficients in the between-subjects model.
The column vector, species, consists of iris flowers of three different species: setosa, versicolor, virginica. The double matrix meas consists of four types of measurements on the flowers: the length and width of sepals and petals in centimeters, respectively.
fitrm uses the 'effects' contrasts, which means that the coefficients sum to 0. The rm.DesignMatrix has one column of 1s for the intercept, and two other columns species_setosa and species_versicolor, which are as follows:
species_setosa=\left\{\begin{array}{c}1,\phantom{\rule{1em}{0ex}}if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\\ 0,\phantom{\rule{1em}{0ex}}if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1,\phantom{\rule{1em}{0ex}}if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\phantom{\rule{1em}{0ex}}\end{array}
species_versicolor=\left\{\begin{array}{c}0,\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{1em}{0ex}}if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{1em}{0ex}}\\ 1,\phantom{\rule{1em}{0ex}}if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1,\phantom{\rule{1em}{0ex}}if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\phantom{\rule{1em}{0ex}}\end{array}.
Display the error degrees of freedom.
rm.DFE
The error degrees of freedom is the number of observations minus the number of estimated coefficients in the between-subjects model, e.g. 150 – 3 = 147.
Use these rules to specify the responses in modelspec.
Use these rules to specify terms in modelspec.
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3-way flow control in an isothermal liquid system - MATLAB - MathWorks Italia
Pressure-Compensated 3-Way Flow Control Valve (IL)
Control member travel between closed and opened orifice
3-way flow control in an isothermal liquid system
The Pressure-Compensated 3-Way Flow Control Valve (IL) block models constant-pressure flow control. When the control pressure, pA – pB, meets or exceeds the Set orifice pressure differential, the relief component of the underlying compensator valve opens to maintain the pressure in the block.
The flow control valve opening and closing is controlled by a physical signal received at port S, which determines the area of the underlying orifice block. A positive signal opens the valve. Port R vents liquid to another part of your network.
For pressure-compensated flow control without venting, see the Pressure-Compensated Flow Control Valve (IL) block.
You can choose the valve model with the Orifice parameterization setting:
Linear - area vs. control member position is an analytical formulation that assumes the valve opening area and the valve control member are related linearly.
Tabulated data - Area vs. control member position is a user-supplied data sheet that relates orifice opening area and the control member position. The block queries between data points with linear interpolation and uses nearest extrapolation for points beyond the table boundaries.
Tabulated data - Volumetric flow rate vs. control member position and pressure drop is a user-supplied data sheet that relates the control member position, orifice pressure drop, and orifice volumetric flow rate. The block queries between data points with linear interpolation and uses a nearest extrapolation for points beyond the table boundaries.
At the extremes of the orifice area and valve pressure range, you can maintain numerical robustness in your simulation by tuning the block Smoothing factor to a nonzero value less than 1. A smoothing function is applied to all calculated areas and pressures, but primarily influences the simulation at the extremes of these ranges.
\stackrel{^}{A}=\frac{\left(A-{A}_{leak}\right)}{\left({A}_{\mathrm{max}}-{A}_{leak}\right)}.
{\stackrel{^}{A}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{A}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{A}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.
{A}_{smoothed}={\stackrel{^}{A}}_{smoothed}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.
\stackrel{^}{p}=\frac{\left(p-{p}_{set}\right)}{\left({p}_{\mathrm{max}}-{p}_{set}\right)}.
{\stackrel{^}{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{p}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{p}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}},
{p}_{smoothed}={\stackrel{^}{p}}_{smoothed}\left({p}_{\mathrm{max}}-{p}_{set}\right)+{p}_{set}.
The Pressure-Compensated 3-Way Flow Control Valve (IL) is constructed from the Pressure Compensator Valve (IL) and Orifice (IL) blocks.
Three-Way Flow Control Valve Schematic
A — Valve liquid port
Liquid entry or exit port to the three-way valve.
B — Orifice liquid port
Liquid entry or exit port to the orifice.
R — Reducing valve liquid port
Liquid exit port from the reducing valve.
Orifice opening in m, set as a physical signal.
Control member travel between closed and opened orifice — Maximum control member stroke
To enable this parameter, set Orifice parameterization to Linear - area vs. control member position.
Control member position vector — Vector of opening position
[0, .002, .007] m (default) | 1-by-n vector
Control member position vector, s — Vector of control member positions
Pressure compensator valve regulation range — Area of the fully opened pressure compensator valve
Pressure compensator valve maximum area — Pressure operational region of the pressure compensating valve
Pressure-Compensated Flow Control Valve (IL) | Pressure Compensator Valve (IL) | Pressure-Reducing 3-Way Valve (IL) | Shuttle Valve (IL) | Orifice (IL)
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Pools - Sublime Docs
Pools allow a borrower to raise capital from multiple lenders. Every pool exists as an independent entity characterized by the set of parameters used to initialize it. The parameters of a pool are set by the borrower, who is also the pool creator.
The customizability of pools serves multiple use cases -
A reputed market maker wishing to raise debt on their own terms without intermediaries
DAOs issuing bonds of different seniority to meet operational expenses
Risk assessment experts wishing to raise debt to issue loans themselves
A pool is created by a user who wishes to borrow capital. Users need to be verified by one of the supported verifiers to be able to create pools - this is necessary for lenders to be able to perform due diligence before depositing assets into the pool.
The pool creator needs to set the following parameters:
Pool Size: Total borrow amount requested
Minimum Borrow Amount: Minimum amount that should be collected for the pool to go active. If this goal is not met, the pool is cancelled, and lenders can withdraw any capital they deposited in the pool
Borrow Asset: Asset being requested by the borrower
Collateral Asset: The asset borrower will put up as collateral
Collateral Ratio: The ideal collateral ratio that will be maintained by the borrower.
Interest Rate: The (simple) interest rate the borrower will be paying over the period of the loan.
Repayment Interval (
RI
): Duration between two interest instalment deadlines.
Number of Repayment Intervals (
nRI
): Number of instalments the borrower will pay over the course of the loan. The total loan duration will thus be
RI \times nRI
Pool Savings Strategy: Collateral supplied by the borrower can earn yield passively by having it deployed on of the supported savings strategy via Sublime's Savings Account. This parameter allows the pool creator to pick that strategy.
Collateral Amount: Amount (denominated in the collateral asset) that the borrower will be depositing.
Salt: Used to generate the address for the pool to be deployed for this loan request.
Verifier: Since it's possible for a user to have been verified by different verifiers, they're required to supply the verifier they wish to use (think of it as supplying one of your many possible identities) while creating this pool.
Lender Verifier: Similar to (12), pool creators can optionally specify a verifier for the lenders if they wish to restrict participation in some way.
Borrowers can have multiple pools active at the same time, allowing them to provide multiple options for the lenders to pick. Upon creation, pools enter a collection period during which lenders can start supplying liquidity. Refer Broken link to get a high-level overview of the different stages of pools.
Providing liquidity to a pool is fairly straightforward - in case a lender is satisfied with the terms offered by the borrower, they can deposit liquidity into the pool. We provide key points to make it easier for lenders to assess a borrower:
A timeline view of the borrower's repayment history: Lenders can look at the borrower's past performance on loans. They can view the amount the borrower has repaid over their lifetime, the defaults they've made, how they've responded to margin calls, etc.
Other lenders participating in the pool: Lenders can further examine the details of other lenders who've supplied liquidity in the pool. Having recognizable lenders supplying liquidity is more likely to attract other lenders who trust them.
Timeline of activity on other DeFi protocols: We query subgraphs of a few other DeFi protocols (such as Uniswap) for a user's wallet activity. This allows lenders to further gauge the borrower's borrowing capacity.
The above data points are aimed at helping lenders determine the borrower's repayment capacity and credibility. Furthermore, lenders can interact with the borrower off-chain and seek additional information.
Lenders receive ERC-20 pool tokens representing their position in a pool. Redemption of any repayments that the borrower makes is based on the number of pool tokens owned. This makes it possible for lenders to exit a pool early by selling their pool tokens to someone else, and also enables building structured products by combining positions in different pools together.
By the end of every instalment period, the borrower is expected to repay the interest that is due for the period. Repayments can be made any time, and over multiple transactions within the interval. Note that the interest accrued is the simple interest.
The principal amount is due in the very last interval.
If the borrower fails to repay an instalment by the period deadline, their loan enters a grace period, during which repayment is still possible albeit with a penalty. If the borrower fails to repay the loan during the grace period, the borrower is considered to have defaulted on their loan. Read pool liquidations to learn more about liquidations and default.
Borrowers can request a one-time extension for a given instalment. An extension pushes the instalment deadline to the next deadline. This request has to be made before the loan enters the grace period, and borrowers can only be granted one extension for a given loan. Since an extension can be used only once for a given loan, borrowers must use this option carefully.
Once an extension request is made, lenders have until the end of the interval to vote on whether the extension should be granted or not. Every lender's voting power is proportional to the amount they lent. Lenders that abstain from voting are assumed to be against the extension. Votes in favour of the extension must pass the extension threshold for the extension to be passed.
Each lender is covered by collateral in proportion to their share of the loan. If the total collateral posted by the borrower at the time the loan goes active is
C
(denominated in
collateralAsset
), then the collateral backing each lender (
c_{i}
) when the loan goes active is defined as
\begin{align} c_{i} &= \frac{l_{i}}{L} \times C \\ \text{where } ~L &= \text{total loan amount denominated in borrowAsset} \\ l_{i} &= \text{debt owed to lender } i \text{ denominated in borrowAsset} \end{align}
Once a loan becomes active,
l_i
L
change over the course of the loan due to interest accrual, repayments by the borrower, and margin calls. In case the borrower chooses to have their collateral supplied to generate yield,
C
also increases due to interest accrual.
c_{i}s
can also begin to differ from each other due to margin calls which are exercised by lenders on an individual basis. We thus define the borrower's collateral ratio against individual lenders at any given time
t
\begin{align} \text{currentCollateralRatio}_{i} &= \frac{c_{i}}{l_{i}} \end{align}
To learn more about margin calls, please refer to pool liquidation.
currentPoolCollateralRatio
\frac{\sum_{i}c_{i}}{\sum_{i}l_{i}}
thus represents the pool's overall collateral ratio at a given point in time.
Pool Liquidation
Liquidations can occur under two scenarios:
Missed instalment repayment - In case a borrower fails to repay an instalment before its period end, they enter a grace period. Repayments can still be made during the grace period, albeit with a penalty. Borrowers also have a one-time extension possibility that shifts the instalment deadline by a single instalment interval. Receiving an extension requires a vote by the lenders of the pool. Each lender's voting power is equal to the amount they lent, and total votes in favour of the extension must be greater than or equal to the threshold required for the extension to be passed. Lenders who do not vote are by default considered against the extension. In case the borrower fails to repay and doesn't win an extension, their collateral is liquidated. Capital recovered through liquidation of the collateral is distributed amongst lenders proportional to the liquidity they provided.
Margin calls - Individual lenders can exercise margin calls in case the borrower's collateral ratio falls below the pool's ideal collateral ratio set during the pool creation. Margin calls require a borrower to post extra collateral that is used to top up their collateral ratio against the lender that initiated the margin call. A margin call has to be answered within a limited time period (called the
marginCallDuration
currentCollateralRatio_{i}
poolCollateralRatio
. In case the borrower fails to recollateralize within the
marginCallDuration
period, collateral equal to
c_{i}
is liable for liquidation.
Both the scenarios for liquidation involve an element of trust - if lenders trust the borrower, they will vote in favour of the extension request, and lenders who trust the borrower might never exercise margin calls even if their collateral ratio drops significantly. At the same time, they provide ample avenues for lenders to limit their risk exposure.
Lifecycle of a pool
A pool goes through different period depending on the state of the loan:
Collection Period: Upon creation, pools enter the collection period. During the collection period, lenders can deposit liquidity into the pool. During pool creation, the borrower is expected to deposit a portion of the collateral required. Furthermore, once a lender supplies liquidity into a pool, they can only withdraw their principal at the end of the loan period (unless the pool is cancelled or terminated).
Active Period: Upon completion of the collection period, the loan enters into the active status, marking the beginning of the loan period.
Loan Withdrawal Period: The borrower can start withdrawing their loan during the loan withdrawal period after depositing the remainder collateral. This period is subsumed within the Active period (refer Fig. 1 below).
Cancelled: A pool can be cancelled by the borrower during the collection period. The funds deposited by lenders are returned and collateral deposited by the borrower is returned back to them. The borrower is charged a penalty for cancelling the pool.
Terminated: In case it is discovered that a borrower is acting maliciously (for eg, the borrower is impersonating someone else), the pool can be terminated.
Defaulted: In case the borrower ends up missing a repayment, the pool enters the default state.
Closed: A pool is closed when the borrower successfully repays the loan.
Fig. 1: Lifecycle of loan
Furthermore, the pool enters different states within the active period depending on repayments and collateral ratios:
Grace Period: In case the borrower fails to repay an instalment on time, they enter the grace period.
Default: If the borrower fails to repay their instalment during the grace period and fails to get an extension, their loan enters the default state.
Margin Call: Should the borrower's collateral ratio fall below the minimum threshold, lenders can start executing margin calls. Please note that margin calls are executed by lenders on an individual basis.
Fig. 2: Different loan states. Note that dotted lines indicate states that are lender-specific
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Symmetric_difference Knowpia
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets
{\displaystyle \{1,2,3\}}
{\displaystyle \{3,4\}}
{\displaystyle \{1,2,4\}}
{\displaystyle A\triangle B}
. The symmetric difference is the union without the intersection:
{\displaystyle ~\setminus ~}
{\displaystyle ~=~}
The symmetric difference of the sets A and B is commonly denoted by
{\displaystyle A\ominus B,}
{\displaystyle A\operatorname {\triangle } B.}
[1][2][better source needed]
The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
{\displaystyle ~(A\triangle B)\triangle C}
{\displaystyle ~\triangle ~}
{\displaystyle ~=~}
The symmetric difference is equivalent to the union of both relative complements, that is:[1]
{\displaystyle A\,\triangle \,B=\left(A\setminus B\right)\cup \left(B\setminus A\right),}
The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation:
{\displaystyle A\mathbin {\triangle } B=\{x:(x\in A)\oplus (x\in B)\}.}
The same fact can be stated as the indicator function (denoted here by
{\displaystyle \chi }
) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments:
{\displaystyle \chi _{(A\,\triangle \,B)}=\chi _{A}\oplus \chi _{B}}
or using the Iverson bracket notation
{\displaystyle [x\in A\,\triangle \,B]=[x\in A]\oplus [x\in B]}
The symmetric difference can also be expressed as the union of the two sets, minus their intersection:
{\displaystyle A\,\triangle \,B=(A\cup B)\setminus (A\cap B),}
{\displaystyle A\mathbin {\triangle } B\subseteq A\cup B}
; the equality in this non-strict inclusion occurs if and only if
{\displaystyle A}
{\displaystyle B}
are disjoint sets. Furthermore, denoting
{\displaystyle D=A\mathbin {\triangle } B}
{\displaystyle I=A\cap B}
{\displaystyle D}
{\displaystyle I}
are always disjoint, so
{\displaystyle D}
{\displaystyle I}
{\displaystyle A\cup B}
. Consequently, assuming intersection and symmetric difference as primitive operations, the union of two sets can be well defined in terms of symmetric difference by the right-hand side of the equality
{\displaystyle A\,\cup \,B=(A\,\triangle \,B)\,\triangle \,(A\cap B)}
{\displaystyle {\begin{aligned}A\,\triangle \,B&=B\,\triangle \,A,\\(A\,\triangle \,B)\,\triangle \,C&=A\,\triangle \,(B\,\triangle \,C).\end{aligned}}}
{\displaystyle {\begin{aligned}A\,\triangle \,\varnothing &=A,\\A\,\triangle \,A&=\varnothing .\end{aligned}}}
Thus, the power set of any set X becomes an abelian group under the symmetric difference operation. (More generally, any field of sets forms a group with the symmetric difference as operation.) A group in which every element is its own inverse (or, equivalently, in which every element has order 2) is sometimes called a Boolean group;[3][4] the symmetric difference provides a prototypical example of such groups. Sometimes the Boolean group is actually defined as the symmetric difference operation on a set.[5] In the case where X has only two elements, the group thus obtained is the Klein four-group.
Equivalently, a Boolean group is an elementary abelian 2-group. Consequently, the group induced by the symmetric difference is in fact a vector space over the field with 2 elements Z2. If X is finite, then the singletons form a basis of this vector space, and its dimension is therefore equal to the number of elements of X. This construction is used in graph theory, to define the cycle space of a graph.
From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed. In particular:
{\displaystyle (A\,\triangle \,B)\,\triangle \,(B\,\triangle \,C)=A\,\triangle \,C.}
This implies triangle inequality:[6] the symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C.
{\displaystyle A\cap (B\,\triangle \,C)=(A\cap B)\,\triangle \,(A\cap C),}
and this shows that the power set of X becomes a ring, with symmetric difference as addition and intersection as multiplication. This is the prototypical example of a Boolean ring.
Further properties of the symmetric difference include:
{\displaystyle A\mathbin {\triangle } B=\emptyset }
{\displaystyle A=B}
{\displaystyle A\mathbin {\triangle } B=A^{c}\mathbin {\triangle } B^{c}}
{\displaystyle A^{c}}
{\displaystyle B^{c}}
{\displaystyle A}
's complement,
{\displaystyle B}
's complement, respectively, relative to any (fixed) set that contains both.
{\displaystyle \left(\bigcup _{\alpha \in {\mathcal {I}}}A_{\alpha }\right)\triangle \left(\bigcup _{\alpha \in {\mathcal {I}}}B_{\alpha }\right)\subseteq \bigcup _{\alpha \in {\mathcal {I}}}\left(A_{\alpha }\mathbin {\triangle } B_{\alpha }\right)}
{\displaystyle {\mathcal {I}}}
is an arbitrary non-empty index set.
{\displaystyle f:S\rightarrow T}
is any function and
{\displaystyle A,B\subseteq T}
are any sets in
{\displaystyle f}
's codomain, then
{\displaystyle f^{-1}\left(A\mathbin {\triangle } B\right)=f^{-1}\left(A\right)\mathbin {\triangle } f^{-1}\left(B\right).}
{\displaystyle x\,\triangle \,y=(x\lor y)\land \lnot (x\land y)=(x\land \lnot y)\lor (y\land \lnot x)=x\oplus y.}
n-ary symmetric differenceEdit
The repeated symmetric difference is in a sense equivalent to an operation on a multiset of sets giving the set of elements which are in an odd number of sets.[clarification needed]
As above, the symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection:
{\displaystyle \triangle M=\left\{a\in \bigcup M:\left|\{A\in M:a\in A\}\right|{\text{ is odd}}\right\}.}
Evidently, this is well-defined only when each element of the union
{\textstyle \bigcup M}
is contributed by a finite number of elements of
{\displaystyle M}
{\displaystyle M=\left\{M_{1},M_{2},\ldots ,M_{n}\right\}}
is a multiset and
{\displaystyle n\geq 2}
. Then there is a formula for
{\displaystyle |\triangle M|}
, the number of elements in
{\displaystyle \triangle M}
, given solely in terms of intersections of elements of
{\displaystyle M}
{\displaystyle |\triangle M|=\sum _{l=1}^{n}(-2)^{l-1}\sum _{1\leq i_{1}<i_{2}<\ldots <i_{l}\leq n}\left|M_{i_{1}}\cap M_{i_{2}}\cap \ldots \cap M_{i_{l}}\right|.}
Symmetric difference on measure spacesEdit
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are.
First consider a finite set S and the counting measure on subsets given by their size. Now consider two subsets of S and set their distance apart as the size of their symmetric difference. This distance is in fact a metric, which makes the power set on S a metric space. If S has n elements, then the distance from the empty set to S is n, and this is the maximum distance for any pair of subsets.[7]
Using the ideas of measure theory, the separation of measurable sets can be defined to be the measure of their symmetric difference. If μ is a σ-finite measure defined on a σ-algebra Σ, the function
{\displaystyle d_{\mu }(X,Y)=\mu (X\,\triangle \,Y)}
is a pseudometric on Σ. dμ becomes a metric if Σ is considered modulo the equivalence relation X ~ Y if and only if
{\displaystyle \mu (X\,\triangle \,Y)=0}
. It is sometimes called Fréchet-Nikodym metric. The resulting metric space is separable if and only if L2(μ) is separable.
{\displaystyle \mu (X),\mu (Y)<\infty }
{\displaystyle |\mu (X)-\mu (Y)|\leq \mu (X\,\triangle \,Y)}
{\displaystyle {\begin{aligned}|\mu (X)-\mu (Y)|&=\left|\left(\mu \left(X\setminus Y\right)+\mu \left(X\cap Y\right)\right)-\left(\mu \left(X\cap Y\right)+\mu \left(Y\setminus X\right)\right)\right|\\&=\left|\mu \left(X\setminus Y\right)-\mu \left(Y\setminus X\right)\right|\\&\leq \left|\mu \left(X\setminus Y\right)\right|+\left|\mu \left(Y\setminus X\right)\right|\\&=\mu \left(X\setminus Y\right)+\mu \left(Y\setminus X\right)\\&=\mu \left(\left(X\setminus Y\right)\cup \left(Y\setminus X\right)\right)\\&=\mu \left(X\,\triangle \,Y\right)\end{aligned}}}
{\displaystyle S=\left(\Omega ,{\mathcal {A}},\mu \right)}
is a measure space and
{\displaystyle F,G\in {\mathcal {A}}}
are measurable sets, then their symmetric difference is also measurable:
{\displaystyle F\triangle G\in {\mathcal {A}}}
. One may define an equivalence relation on measurable sets by letting
{\displaystyle F}
{\displaystyle G}
be related if
{\displaystyle \mu \left(F\triangle G\right)=0}
. This relation is denoted
{\displaystyle F=G\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {D}},{\mathcal {E}}\subseteq {\mathcal {A}}}
{\displaystyle {\mathcal {D}}\subseteq {\mathcal {E}}\left[{\mathcal {A}},\mu \right]}
if to each
{\displaystyle D\in {\mathcal {D}}}
there's some
{\displaystyle E\in {\mathcal {E}}}
{\displaystyle D=E\left[{\mathcal {A}},\mu \right]}
. The relation "
{\displaystyle \subseteq \left[{\mathcal {A}},\mu \right]}
" is a partial order on the family of subsets of
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {D}}={\mathcal {E}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {D}}\subseteq {\mathcal {E}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {E}}\subseteq {\mathcal {D}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle =\left[{\mathcal {A}},\mu \right]}
" is an equivalence relationship between the subsets of
{\displaystyle {\mathcal {A}}}
The symmetric closure of
{\displaystyle {\mathcal {D}}}
is the collection of all
{\displaystyle {\mathcal {A}}}
-measurable sets that are
{\displaystyle =\left[{\mathcal {A}},\mu \right]}
{\displaystyle D\in {\mathcal {D}}}
. The symmetric closure of
{\displaystyle {\mathcal {D}}}
{\displaystyle {\mathcal {D}}}
{\displaystyle {\mathcal {D}}}
is a sub-
{\displaystyle \sigma }
{\displaystyle {\mathcal {A}}}
, so is the symmetric closure of
{\displaystyle {\mathcal {D}}}
{\displaystyle F=G\left[{\mathcal {A}},\mu \right]}
{\displaystyle \left|\mathbf {1} _{F}-\mathbf {1} _{G}\right|=0}
{\displaystyle \left[{\mathcal {A}},\mu \right]}
Hausdorff distance vs. symmetric differenceEdit
The Hausdorff distance and the (area of the) symmetric difference are both pseudo-metrics on the set of measurable geometric shapes. However, they behave quite differently. The figure at the right shows two sequences of shapes, "Red" and "Red ∪ Green". When the Hausdorff distance between them becomes smaller, the area of the symmetric difference between them becomes larger, and vice versa. By continuing these sequences in both directions, it is possible to get two sequences such that the Hausdorff distance between them converges to 0 and the symmetric distance between them diverges, or vice versa.
Separable sigma algebras
^ a b c Taylor, Courtney (March 31, 2019). "What Is Symmetric Difference in Math?". ThoughtCo. Retrieved 2020-09-05.
^ Weisstein, Eric W. "Symmetric Difference". mathworld.wolfram.com. Retrieved 2020-09-05.
^ Givant, Steven; Halmos, Paul (2009). Introduction to Boolean Algebras. Springer Science & Business Media. p. 6. ISBN 978-0-387-40293-2.
^ Humberstone, Lloyd (2011). The Connectives. MIT Press. p. 782. ISBN 978-0-262-01654-4.
^ Rotman, Joseph J. (2010). Advanced Modern Algebra. American Mathematical Soc. p. 19. ISBN 978-0-8218-4741-1.
^ Rudin, Walter (January 1, 1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill Education. p. 306. ISBN 978-0070542358.
^ Claude Flament (1963) Applications of Graph Theory to Group Structure, page 16, Prentice-Hall MR0157785
Symmetric difference of sets. In Encyclopaedia of Mathematics
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Q24 In the above figure (not to scale ) AB, BC,CF,DE and FE are chords of the - Maths - Practical Geometry - 12253161 | Meritnation.com
In the above figure (not to scale )
\overline{)\mathrm{AB}}, \overline{)\mathrm{BC}},\overline{)\mathrm{CF}},\overline{)\mathrm{DE}}
\overline{)\mathrm{FE}}
are chords of the circle. If
\angle
ABC = 100° and
\angle
FED = 110°, then
\angle
FPA =
(a) 20° (b) 30°
\mathrm{AFED} \mathrm{and} \mathrm{ABCF} \mathrm{are} \mathrm{cyclic} \mathrm{quadrilaterals} \mathrm{as} \mathrm{their} \mathrm{all} \mathrm{vertices} \mathrm{lies} \mathrm{on} \mathrm{the} \mathrm{circle}.\phantom{\rule{0ex}{0ex}}\mathrm{In} \mathrm{cyclic} \mathrm{quadrilateral} \mathrm{AFED},\phantom{\rule{0ex}{0ex}}\angle \mathrm{FED}+\angle \mathrm{FAD}=180° \left( \mathrm{Opposite} \mathrm{angles} \mathrm{of} \mathrm{cyclic} \mathrm{quadrilateral} \mathrm{are} \mathrm{supplementary} \right)\phantom{\rule{0ex}{0ex}}110°+\angle \mathrm{FAD}=180°\phantom{\rule{0ex}{0ex}}\angle \mathrm{FAD}=70°\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{FAP}=70°\phantom{\rule{0ex}{0ex}}\mathrm{In} \mathrm{cyclic} \mathrm{quadrilateral} \mathrm{ABCF},\phantom{\rule{0ex}{0ex}}\angle \mathrm{ABC}+\angle \mathrm{AFC}=180° \left( \mathrm{Opposite} \mathrm{angles} \mathrm{of} \mathrm{cyclic} \mathrm{quadrilateral} \mathrm{are} \mathrm{supplementary} \right)\phantom{\rule{0ex}{0ex}}100°+\angle \mathrm{AFC}=180°\phantom{\rule{0ex}{0ex}}\angle \mathrm{AFC}=80°\phantom{\rule{0ex}{0ex}}⇒ \angle \mathrm{AFP}=80°\phantom{\rule{0ex}{0ex}}\mathrm{Now}, \phantom{\rule{0ex}{0ex}}\mathrm{In} ∆\mathrm{AFP} , \mathrm{using} \mathrm{angle} \mathrm{sum} \mathrm{property}\phantom{\rule{0ex}{0ex}}\angle \mathrm{FAP}+\angle \mathrm{AFP}+\angle \mathrm{FPA}=180°\phantom{\rule{0ex}{0ex}}70°+80°+\angle \mathrm{FPA}=180°\phantom{\rule{0ex}{0ex}}150°+\angle \mathrm{FPA}=180°\phantom{\rule{0ex}{0ex}}\angle \mathrm{FPA}=30°\phantom{\rule{0ex}{0ex}}\mathrm{Correct} \mathrm{option} \mathrm{is} \mathrm{B}.
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Dictionary:A+B,A*B - SEG Wiki
Denotes ways (sum, product) in which AVO intercept and slope are combined to yield a single index, where amplitude is expressed as
{\displaystyle A+B\sin ^{2}\theta }
{\displaystyle \theta }
being the angle of incidence.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:A%2BB,A*B&oldid=90531"
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Radiant flux - Wikipedia
"Spectral power" redirects here. Not to be confused with Spectral power density.
Find sources: "Radiant flux" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this template message)
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), one joule per second (J/s), while that of spectral flux in frequency is the watt per hertz (W/Hz) and that of spectral flux in wavelength is the watt per metre (W/m)—commonly the watt per nanometre (W/nm).
1.1 Radiant flux
1.2 Spectral flux
2 Relationship with the Poynting vector
Radiant fluxEdit
Radiant flux, denoted Φe ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[1]
{\displaystyle \Phi _{\mathrm {e} }={\frac {\partial Q_{\mathrm {e} }}{\partial t}},}
Qe is the radiant energy emitted, reflected, transmitted or received;
Spectral fluxEdit
Spectral flux in frequency, denoted Φe,ν, is defined as[1]
{\displaystyle \Phi _{\mathrm {e} ,\nu }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \nu }},}
Spectral flux in wavelength, denoted Φe,λ, is defined as[1]
{\displaystyle \Phi _{\mathrm {e} ,\lambda }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \lambda }},}
Relationship with the Poynting vectorEdit
One can show that the radiant flux of a surface is the flux of the Poynting vector through this surface, hence the name "radiant flux":
{\displaystyle \Phi _{\mathrm {e} }=\int _{\Sigma }\mathbf {S} \cdot \mathbf {\hat {n}} \,\mathrm {d} A=\int _{\Sigma }|\mathbf {S} |\cos \alpha \,\mathrm {d} A,}
Σ is the surface;
n is a unit normal vector to that surface;
A is the area of that surface;
α is the angle between n and S.
But the time-average of the norm of the Poynting vector is used instead, because in radiometry it is the only quantity that radiation detectors are able to measure:
{\displaystyle \Phi _{\mathrm {e} }=\int _{\Sigma }\langle |\mathbf {S} |\rangle \cos \alpha \,\mathrm {d} A,}
where < • > is the time-average.
Boyd, Robert (1983). Radiometry and the Detection of Optical Radiation (Pure & Applied Optics Series). Wiley-Interscience. ISBN 978-0-471-86188-1.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Radiant_flux&oldid=1079706525"
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Globular cluster Messier 13 in Hercules
6×105[5] M☉
{\displaystyle {\begin{smallmatrix}\left[{\ce {Fe}}/{\ce {H}}\right]\end{smallmatrix}}}
One of the best-known clusters of the Northern Hemisphere
Discovery and visibilityEdit
M13 was discovered by Edmond Halley in 1714, and cataloged by Charles Messier on June 1, 1764, into his list of objects not to mistake for comets; Messier's list, including Messier 13, eventually became known as the Messier Catalog.[8]
About one third of the way from Vega to Arcturus, four bright stars in Herculēs form the Keystone asterism, the broad torso of the hero. M13 can be seen in this asterism 2⁄3 of the way north (by west) from Zeta to Eta Herculis. Although only telescopes with great light-gathering capability fully resolve the stars of the cluster, M13 may be visible to the naked eye depending on circumstances. With a low-power telescope, Messier 13 looks like a comet or fuzzy patch. The cluster is visible throughout the year from latitudes greater than 36 degrees north, with the longest visibility during Northern Hemisphere spring and summer.[9]
It is located at right ascension 16h 41.7m, declination +36° 28'. With an apparent magnitude of 5.8, it is barely visible with the naked eye on clear nights. Its diameter is about 23 arcminutes and it is readily viewable in small telescopes.[10] Nearby is NGC 6207, a 12th-magnitude edge-on galaxy that lies 28 arcminutes directly northeast. A small galaxy, IC 4617, lies halfway between NGC 6207 and M13, north-northeast of the large globular cluster's center.
In traditional binoculars, the Hercules Globular Cluster appears as a round patch of light. At least four inches of telescope aperture resolves stars in M13's outer extent as small pinpoints of light. However, only larger telescopes resolve stars further into the center of the cluster.[11]
About 145 light-years in diameter, M13 is composed of several hundred thousand stars, the brightest of which is a red giant, the variable star V11, also known as V1554 Herculis,[12] with an apparent visual magnitude of 11.95. M13 is 22,200–25,000 light-years away from Earth,[13] and the globular cluster is one of over one hundred that orbit the center of the Milky Way.[14][15]
Single stars in this globular cluster were first resolved in 1779.[13] Compared to the stars in the neighborhood of the Sun, the stars of the M13 population are more than a hundred times more densely packed.[13] They are so close together that they sometimes collide and produce new stars.[13] The newly formed, young stars, so-called "blue stragglers", are particularly interesting to astronomers.[13]
The 1974 Arecibo message, which contained encoded information about the human race, DNA, atomic numbers, Earth's position and other information, was beamed from the Arecibo Observatory radio telescope towards M13 as an experiment in contacting potential extraterrestrial civilizations in the cluster. The cluster will move through space during the transit time; opinions differ as to whether or not the cluster will be in a position to receive the message by the time that it arrives.[16][17]
The last two variables (V63 and V64) were discovered from Spain in April 2021 and March 2022 respectively.
The science fiction novellas "Sucker Bait" by Isaac Asimov and the novel Question and Answer by Poul Anderson take place on Troas, a world within M13.
In the German science fiction series Perry Rhodan, M13 is the location of Arkon, the homeworld of the race of Arkonides.
In author Dan Simmons' Hyperion Cantos the Hercules cluster is where a copy of Earth was secretly recreated after the original was destroyed.
In his novel The Sirens of Titan, Kurt Vonnegut writes "Every passing hour brings the Solar System forty-three thousand miles closer to Globular Cluster M13 in Hercules—and still there are some misfits who insist that there is no such thing as progress."
Deliberately engineering a star in Messier 13 to go nova was part of the Cybermen's complicated plot in the 1968 Doctor Who story The Wheel in Space.
In Bill Amend's popular comic strip FoxTrot, Jason Fox mentions observing the M13 Globular Cluster.
Heart of the Hercules Globular Cluster, Hubble image
^ Paust, Nathaniel E. Q.; et al. (February 2010), "The ACS Survey of Galactic Globular Clusters. VIII. Effects of Environment on Globular Cluster Global Mass Functions", The Astronomical Journal, 139 (2): 476–491, Bibcode:2010AJ....139..476P, doi:10.1088/0004-6256/139/2/476, hdl:2152/34371.
^ a b "M 13". SIMBAD. Centre de données astronomiques de Strasbourg. Retrieved 2006-11-15.
^ Leonard, Peter J. T.; Richer, Harvey B.; Fahlman, Gregory G. (1992), "The mass and stellar content of the globular cluster M13", Astronomical Journal, 104: 2104, Bibcode:1992AJ....104.2104L, doi:10.1086/116386.
^ a b Forbes, Duncan A.; Bridges, Terry (May 2010), "Accreted versus in situ Milky Way globular clusters", Monthly Notices of the Royal Astronomical Society, 404 (3): 1203–1214, arXiv:1001.4289, Bibcode:2010MNRAS.404.1203F, doi:10.1111/j.1365-2966.2010.16373.x.
^ "Messier 13 (M13) - The Great Hercules Cluster - Universe Today". Universe Today. 2016-05-09. Retrieved 2018-04-23.
^ "M13: Great Cluster in Hercules | EarthSky.org". earthsky.org. Retrieved 2018-03-26.
^ "M 13". Messier Objects Mobile -- Charts, Maps & Photos. 2016-10-16. Retrieved 2018-04-23.
^ "How to See the Great Hercules Cluster of Stars". Space.com. Retrieved 2018-04-23.
^ Samus, N.N.; Pastukhova, E.N.; Durlevich, O.V.; Kazarovets, E.V.; Kireeva, N.N. (2020), "The 83rd Name-List of Variable Stars. Variables in Globular Clusters and Novae", Peremennye Zvezdy (Variable Stars) 40, No. 8
^ a b c d e Garner, Rob (2017-10-06). "Messier 13 (The Hercules Cluster)". NASA. Retrieved 2018-04-23.
^ "Control Telescope :: Stars & Nebulae". Retrieved 2021-11-22.
^ "Star Cluster". Retrieved 2021-11-22.
^ "It's the 25th anniversary of Earth's first attempt to phone E.T." 1999-11-12. Archived from the original on 2008-08-02. Retrieved 2018-06-28.
^ "Science 2.0". In regard to the email from. Retrieved 2015-04-15.
L199 (V63), a new variable star in M13
Variability of L261 in M13 (V64)
Rothery, David; Bauer, Amanda; Dhillon, Vik; Lawrence, Pete; Chapman, Allan; Fohring, Dora. "M13 – Hercules Globular Cluster". Deep Sky Video. Brady Haran.
16h 41m 41.44s, 36° 27′ 36.9″
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Dialectica interpretation — Wikipedia Republished // WIKI 2
In proof theory, the Dialectica interpretation[1] is a proof interpretation of intuitionistic arithmetic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal Dialectica, where Gödel's paper was published in a 1958 special issue dedicated to Paul Bernays on his 70th birthday.
2 Dialectica interpretation of intuitionistic logic
2.1 Formula translation
2.2 Proof translation (soundness)
2.3 Characterisation principles
3 Extensions of basic interpretation
3.1 Induction
3.2 Classical logic
3.3 Comprehension
4 Dialectica interpretation of linear logic
5 Variants of the Dialectica interpretation
Via the Gödel–Gentzen negative translation, the consistency of classical Peano arithmetic had already been reduced to the consistency of intuitionistic Heyting arithmetic. Gödel's motivation for developing the dialectica interpretation was to obtain a relative consistency proof for Heyting arithmetic (and hence for Peano arithmetic).
Dialectica interpretation of intuitionistic logic
The interpretation has two components: a formula translation and a proof translation. The formula translation describes how each formula
{\displaystyle A}
of Heyting arithmetic is mapped to a quantifier-free formula
{\displaystyle A_{D}(x;y)}
of the system T, where
{\displaystyle x}
{\displaystyle y}
are tuples of fresh variables (not appearing free in
{\displaystyle A}
). Intuitively,
{\displaystyle A}
{\displaystyle \exists x\forall yA_{D}(x;y)}
. The proof translation shows how a proof of
{\displaystyle A}
has enough information to witness the interpretation of
{\displaystyle A}
, i.e. the proof of
{\displaystyle A}
can be converted into a closed term
{\displaystyle t}
{\displaystyle A_{D}(t;y)}
in the system T.
The quantifier-free formula
{\displaystyle A_{D}(x;y)}
is defined inductively on the logical structure of
{\displaystyle A}
as follows, where
{\displaystyle P}
is an atomic formula:
{\displaystyle {\begin{array}{lcl}(P)_{D}&\equiv &P\\(A\wedge B)_{D}(x,v;y,w)&\equiv &A_{D}(x;y)\wedge B_{D}(v;w)\\(A\vee B)_{D}(x,v,z;y,w)&\equiv &(z=0\rightarrow A_{D}(x;y))\wedge (z\neq 0\to B_{D}(v;w))\\(A\rightarrow B)_{D}(f,g;x,w)&\equiv &A_{D}(x;fxw)\rightarrow B_{D}(gx;w)\\(\exists zA)_{D}(x,z;y)&\equiv &A_{D}(x;y)\\(\forall zA)_{D}(f;y,z)&\equiv &A_{D}(fz;y)\end{array}}}
Proof translation (soundness)
The formula interpretation is such that whenever
{\displaystyle A}
is provable in Heyting arithmetic then there exists a sequence of closed terms
{\displaystyle t}
{\displaystyle A_{D}(t;y)}
is provable in the system T. The sequence of terms
{\displaystyle t}
and the proof of
{\displaystyle A_{D}(t;y)}
are constructed from the given proof of
{\displaystyle A}
in Heyting arithmetic. The construction of
{\displaystyle t}
is quite straightforward, except for the contraction axiom
{\displaystyle A\rightarrow A\wedge A}
which requires the assumption that quantifier-free formulas are decidable.
Characterisation principles
It has also been shown that Heyting arithmetic extended with the following principles
Markov's principle
Independence of premise for universal formulas
is necessary and sufficient for characterising the formulas of HA which are interpretable by the Dialectica interpretation.[citation needed]
Extensions of basic interpretation
The basic dialectica interpretation of intuitionistic logic has been extended to various stronger systems. Intuitively, the dialectica interpretation can be applied to a stronger system, as long as the dialectica interpretation of the extra principle can be witnessed by terms in the system T (or an extension of system T).
Given Gödel's incompleteness theorem (which implies that the consistency of PA cannot be proven by finitistic means) it is reasonable to expect that system T must contain non-finitistic constructions. Indeed this is the case. The non-finitistic constructions show up in the interpretation of mathematical induction. To give a Dialectica interpretation of induction, Gödel makes use of what is nowadays called Gödel's primitive recursive functionals, which are higher-order functions with primitive recursive descriptions.
Formulas and proofs in classical arithmetic can also be given a Dialectica interpretation via an initial embedding into Heyting arithmetic followed by the Dialectica interpretation of Heyting arithmetic. Shoenfield, in his book, combines the negative translation and the Dialectica interpretation into a single interpretation of classical arithmetic.
In 1962 Spector [2] extended Gödel's Dialectica interpretation of arithmetic to full mathematical analysis, by showing how the schema of countable choice can be given a Dialectica interpretation by extending system T with bar recursion.
Dialectica interpretation of linear logic
The Dialectica interpretation has been used to build a model of Girard's refinement of intuitionistic logic known as linear logic, via the so-called Dialectica spaces.[3] Since linear logic is a refinement of intuitionistic logic, the dialectica interpretation of linear logic can also be viewed as a refinement of the dialectica interpretation of intuitionistic logic.
Although the linear interpretation in Shirahata's work [4] validates the weakening rule (it is actually an interpretation of affine logic), de Paiva's dialectica spaces interpretation does not validate weakening for arbitrary formulas.
Variants of the Dialectica interpretation
Several variants of the Dialectica interpretation have been proposed since. Most notably the Diller-Nahm variant (to avoid the contraction problem) and Kohlenbach's monotone and Ferreira-Oliva bounded interpretations (to interpret weak Kőnig's lemma). Comprehensive treatments of the interpretation can be found at [5], [6] and.[7]
^ Kurt Gödel (1958). Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica. pp. 280–287.
^ Clifford Spector (1962). Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. Recursive Function Theory: Proc. Symposia in Pure Mathematics. pp. 1–27.
^ Valeria de Paiva (1991). The Dialectica Categories (PDF). University of Cambridge, Computer Laboratory, PhD Thesis, Technical Report 213.
^ Masaru Shirahata (2006). The Dialectica interpretation of first-order classical affine logic. Theory and Applications of Categories, Vol. 17, No. 4. pp. 49–79.
^ Jeremy Avigad and Solomon Feferman (1999). Gödel's functional ("Dialectica") interpretation (PDF). in S. Buss ed., The Handbook of Proof Theory, North-Holland. pp. 337–405.
^ Ulrich Kohlenbach (2008). Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Verlag, Berlin. pp. 1–536.
^ Anne S. Troelstra (with C.A. Smoryński, J.I. Zucker, W.A.Howard) (1973). Metamathematical Investigation of intuitionistic Arithmetic and Analysis. Springer Verlag, Berlin. pp. 1–323. {{cite book}}: CS1 maint: multiple names: authors list (link)
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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SuggestVariableOrder
Suggests a variable order for decomposing a polynomial system efficiently
SuggestVariableOrder(sys)
SuggestVariableOrder(sys, vars)
SuggestVariableOrder(sys, 'decomposition'='cad')
list of set of variables
'decomposition'='cad'
The decomposition option controls the targeted type of decomposition.
If 'decomposition'='cad' is specified and vars not supplied, then the method is best appropriate for computing a cylindrical algebraic decomposition, see CylindricalAlgebraicDecompose.
The command SuggestVariableOrder(sys) computes a variable order which is expected to speed up the decomposition of the polynomial system sys when passed to one of the commands Triangularize, RealTriangularize, LazyRealTriangularize, SamplePoints, ComprehensiveTriangularize, RealComprehensiveTriangularize, CylindricalAlgebraicDecompose, RealRootClassification, PartialCylindricalAlgebraicDecomposition, GeneralConstruct.
The input argument sys is a list of constraints which can be any polynomial equation, inequation or inequality. Each constraint consisting of a polynomial (with no equality or inequality sign) is interpreted as an equation.
The output of SuggestVariableOrder(sys) is a variable list which can then be passed as argument to PolynomialRing.
If vars is given as an input argument, the following rules apply: (1) each indeterminate not appearing in sys or vars will be treated as parameters and therefore will be present in SuggestVariableOrder(sys) with a smaller rank than any indeterminate appearing in both sys and vars. (2) In addition, if vars is given as a list, then the orders among variables appearing in both vars and sys remain unchanged.
The command SuggestVariableOrder(sys) computes this variable list by means of combinatorial arguments only, say by comparing vertex degrees in a suitable graph. No algebraic computations are performed. Therefore, this variable order is determined heuristically and there is no guarantee of optimality.
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):
\mathrm{sys}≔[{v}^{4}+4xu{v}^{2}-2{y}^{2}{v}^{2}-4{x}^{2}{v}^{2}-4{y}^{2}{u}^{2}+4x{y}^{2}u+{y}^{4},4{u}^{2}-4xu-{y}^{2},4{u}^{2}-4xu-{y}^{2}]
\textcolor[rgb]{0,0,1}{\mathrm{sys}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\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}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{v}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{v}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{v}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\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}{v}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{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}{4}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{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}}]
Compute a variable order for it.
\mathrm{SuggestVariableOrder}\left(\mathrm{sys}\right)
[\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}{v}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]
Use the 'decomposition'='cad' option to confirm that this order is suitable for
\mathrm{lv}≔\mathrm{SuggestVariableOrder}\left(\mathrm{sys},\mathrm{decomposition}=\mathrm{cad}\right)
\textcolor[rgb]{0,0,1}{\mathrm{lv}}\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}{v}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]
Building a polynomial ring
R≔\mathrm{PolynomialRing}\left(\mathrm{lv}\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
Computing a cylindrical algebraic decomposition of this challenging example
\mathrm{cad}≔\mathrm{CylindricalAlgebraicDecompose}\left(\mathrm{sys},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{cad}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{c_a_d}}
The RegularChains[SuggestVariableOrder] command was introduced in Maple 16.
PartialCylindricalAlgebraicDecomposition
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Learn how to calculate the midway point for continuous probability distributions
This integral calculates the median m of a random variable. C.K.Taylor
The median of a set of data is the midway point wherein exactly half of the data values are less than or equal to the median. In a similar way, we can think about the median of a continuous probability distribution, but rather than finding the middle value in a set of data, we find the middle of the distribution in a different way.
The total area under a probability density function is 1, representing 100%, and as a result, half of this can be represented by one-half or 50 percent. One of the big ideas of mathematical statistics is that probability is represented by the area under the curve of the density function, which is calculated by an integral, and thus the median of a continuous distribution is the point on the real number line where exactly half of the area lies to the left.
This can be more succinctly stated by the following improper integral. The median of the continuous random variable X with density function f( x) is the value M such that:
0.5=\int_{m}^{-\infty}f(x)dx
0.5=∫m−∞f(x)dx
Median for Exponential Distribution
We now calculate the median for the exponential distribution Exp(A). A random variable with this distribution has density function f(x) = e-x/A/A for x any nonnegative real number. The function also contains the mathematical constant e, approximately equal to 2.71828.
Since the probability density function is zero for any negative value of x, all that we must do is integrate the following and solve for M:
0.5 = ∫0M f(x) dx
Since the integral ∫ e-x/A/A dx = -e-x/A, the result is that
0.5 = -e-M/A + 1
This means that 0.5 = e-M/A and after taking the natural logarithm of both sides of the equation, we have:
ln(1/2) = -M/A
Since 1/2 = 2-1, by properties of logarithms we write:
- ln2 = -M/A
Multiplying both sides by A gives us the result that the median M = A ln2.
Median-Mean Inequality in Statistics
One consequence of this result should be mentioned: the mean of the exponential distribution Exp(A) is A, and since ln2 is less than 1, it follows that the product Aln2 is less than A. This means that the median of the exponential distribution is less than the mean.
This makes sense if we think about the graph of the probability density function. Due to the long tail, this distribution is skewed to the right. Many times when a distribution is skewed to the right, the mean is to the right of the median.
What this means in terms of statistical analysis is that we can oftentimes predict that the mean and median do not directly correlate given the probability that data is skewed to the right, which can be expressed as the median-mean inequality proof known as Chebyshev's inequality.
As an example, consider a data set that posits that a person receives a total of 30 visitors in 10 hours, where the mean wait time for a visitor is 20 minutes, while the set of data may present that the median wait time would be somewhere between 20 and 30 minutes if over half of those visitors came in the first five hours.
Taylor, Courtney. "Exponential Distribution Medians." ThoughtCo, Aug. 26, 2020, thoughtco.com/calculate-the-median-of-exponential-distribution-3126442. Taylor, Courtney. (2020, August 26). Exponential Distribution Medians. Retrieved from https://www.thoughtco.com/calculate-the-median-of-exponential-distribution-3126442 Taylor, Courtney. "Exponential Distribution Medians." ThoughtCo. https://www.thoughtco.com/calculate-the-median-of-exponential-distribution-3126442 (accessed May 23, 2022).
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Newton (unit) - Simple English Wikipedia, the free encyclopedia
The newton (symbol: N) is the SI unit of force. It is named after Sir Isaac Newton because of his work on classical mechanics. A newton is how much force is required to make a mass of one kilogram accelerate at a rate of one metre per second squared.
{\displaystyle 1\,\mathrm {N} =1\,\mathrm {kg} \cdot \mathrm {m} /\mathrm {s} ^{2}}
1 N is the force of Earth's gravity on a mass of about 102 g.
On the Earth's surface, a mass of 1 kg pushes on its support with an average force of 9.8 N.
1 Newton is equal to the amount of force required to accelerate an object (with a mass of one kilogram) at a rate of 1 meter per second, per second.
The US Customary Unit of force is the pound (symbol: lbf). 1 pound is equal to 4.44822 newtons.
In 1946, Conférence Générale des Poids et Mesures (CGPM) set the unit of force in the MKS system of units to be the amount needed to accelerate 1 kilogram of mass at the rate of 1 metre per second each second. In 1948, the CGPM adopted the name "newton" for this force. The MKS system then became the blueprint for today's SI International System of Units. That made the newton the standard unit of force.
This SI unit is named after Isaac Newton. As with every International System of Units (SI) unit named for a person, the first letter of its symbol is upper case (N). However, when an SI unit is spelled out in English, it should always begin with a lower case letter (newton)—except in a situation where any word in that position would be capitalized, such as at the beginning of a sentence or in material using title case. Note that "degree Celsius" conforms to this rule because the "d" is lowercase.— Based on The International System of Units, section 5.2.
Newton's second law of motion states that F = m•a, where F is the force applied, m is the mass of the object receiving the force, and a is the acceleration of the object. The newton is therefore:[2]
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Newton_(unit)&oldid=7833730"
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Allantoicase - Wikipedia
Allantoicase homohexamer, Saccharomyces cerevisiae
structure of allantoicase
1sg3 / SCOPe / SUPFAM
Allantoicase is an enzyme (EC 3.5.3.4) that in humans is encoded by the ALLC gene. Allantoicase catalyzes the chemical reaction
allantoate + H2O
{\displaystyle \rightleftharpoons }
(S)-ureidoglycolate + urea
Thus, the two substrates of this enzyme are allantoate and H2O, whereas its two products are (S)-ureidoglycolate and urea.
This enzyme belongs to the family of hydrolases, those acting on carbon-nitrogen bonds other than peptide bonds, specifically in linear amidines. The systematic name of this enzyme class is allantoate amidinohydrolase. This enzyme participates in purine metabolism by facilitating the utilization of purines as secondary nitrogen sources under nitrogen-limiting conditions. While purine degradation converges to uric acid in all vertebrates, its further degradation varies from species to species. Uric acid is excreted by birds, reptiles, and some mammals that do not have a functional uricase gene, whereas other mammals produce allantoin. Amphibians and microorganisms produce ammonia and carbon dioxide using the uricolytic pathway. Allantoicase performs the second step in this pathway catalyzing the conversion of allantoate into ureidoglycolate and urea.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 1O59 and 1SG3.
The structure of allantoicase is best described as being composed of two repeats (the allantoicase repeats: AR1 and AR2), which are connected by a flexible linker. The crystal structure, resolved at 2.4A resolution, reveals that AR1 has a very similar fold to AR2, both repeats being jelly-roll motifs, composed of four-stranded and five-stranded antiparallel beta-sheets.[1] Each jelly-roll motif has two conserved surface patches that probably constitute the active site.[2]
^ Xu Q, Schwarzenbacher R, Page R, Sims E, Abdubek P, Ambing E, Biorac T, Brinen LS, Cambell J, Canaves JM, Chiu HJ, Dai X, Deacon AM, DiDonato M, Elsliger MA, Floyd R, Godzik A, Grittini C, Grzechnik SK, Hampton E, Jaroszewski L, Karlak C, Klock HE, Koesema E, Kovarik JS, Kreusch A, Kuhn P, Lesley SA, Levin I, McMullan D, McPhillips TM, Miller MD, Morse A, Moy K, Ouyang J, Quijano K, Reyes R, Rezezadeh F, Robb A, Spraggon G, Stevens RC, van den Bedem H, Velasquez J, Vincent J, von Delft F, Wang X, West B, Wolf G, Hodgson KO, Wooley J, Wilson IA (August 2004). "Crystal structure of an allantoicase (YIR029W) from Saccharomyces cerevisiae at 2.4 A resolution". Proteins. 56 (3): 619–24. doi:10.1002/prot.20164. PMID 15229895. S2CID 5688375.
^ Leulliot N, Quevillon-Cheruel S, Sorel I, Graille M, Meyer P, Liger D, Blondeau K, Janin J, van Tilbeurgh H (May 2004). "Crystal structure of yeast allantoicase reveals a repeated jelly roll motif". The Journal of Biological Chemistry. 279 (22): 23447–52. doi:10.1074/jbc.M401336200. PMID 15020593.
Florin M, Duchateau-Bosson G (1940). "Microdosage photometrique de l'allantoine en solutions pures et dans l'urine". Enzymologia. 9: 5–9.
Trijbels F, Vogels GD (May 1966). "Allantoicase and ureidoglycolase in Pseudomonas and Penicillium species". Biochimica et Biophysica Acta (BBA) - Enzymology and Biological Oxidation. 118 (2): 387–95. doi:10.1016/S0926-6593(66)80047-4. PMID 4960174.
Hildebrand F, van Griensven M, Giannoudis P, Schreiber T, Frink M, Probst C, Grotz M, Krettek C, Pape HC (2005). "Impact of hypothermia on the immunologic response after trauma and elective surgery". Surgical Technology International. 14: 41–50. PMID 16525953.
Gravenmade EJ, Vogels GD, Van der Drift C (March 1970). "Hydrolysis, racemization and absolute configuration of ureidoglycolate, a substrate of allantoicase". Biochimica et Biophysica Acta (BBA) - Enzymology. 198 (3): 569–82. doi:10.1016/0005-2744(70)90134-8. PMID 4314237.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Allantoicase&oldid=1038540910"
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Undo - Maple Help
Home : Support : Online Help : Education : Student Packages : Calculus 1 : Single Stepping : Undo
undo the last rule applied to a problem
Undo(expr)
algebraic or algebraic equation; select the problem to undo
The Undo command undoes the rule most recently applied to the problem expr. It returns the previous state of the problem. This operation can be repeated until the problem is in its initial state.
Maple commands other than the package commands Rule and Hint do not change the Calculus1 internal state of a problem. Therefore, Undo does not undo the results of applying such commands. However, you can usually recover the prior state of such a problem by calling GetProblem.
The normal input to Undo is the output from a previous call to Rule or GetProblem. To pass the output of GetProblem to Undo, use the internal option to GetProblem. However, the routine tries to match any expr to an existing problem.
This command can be used within a tutor after applying a Rule interactively by using the DiffTutor, IntTutor, or LimitTutor.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Calculus1}]\right):
\mathrm{infolevel}[\mathrm{Student}[\mathrm{Calculus1}]]≔1:
\mathrm{Rule}[\mathrm{change},u=2x]\left(\mathrm{Int}\left(\mathrm{sin}\left(3x\right),x\right)\right)
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\frac{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{u}
\mathrm{Undo}\left(\right)
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}
\mathrm{Rule}[\mathrm{change},u=3x]\left(\right)
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\frac{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)}{\textcolor[rgb]{0,0,1}{3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{u}
\mathrm{Undo}\left(\right)
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}
If you call Undo on a problem in its initial state, Maple returns an error.
\mathrm{Undo}\left(\right)
Error, (in Student:-Calculus1:-Undo) there is no previous state for this problem
|
A ship’s magnetic field camouflage method based on multi-objective genetic algorithm | JVE Journals
Dong Tian1 , Sheng-dao Liu2 , Zhi-xin Li3
1, 2, 3College of Electrical Engineering, Naval University of Engineering, Wuhan, China
In order to protect the ship’ s magnetic field information from detection, a method is proposed to camouflage both the amplitude and curve of ship’s magnetic field based on multi-objective genetic algorithm. The example shows that the method can camouflage the ship’s magnetic amplitude and curve feature effectively, the multi-objective genetic algorithm can avoid the subjective of weight distribution in the unified objective method and the alternative Pareto disaggregation can be found, which is more adaptable for application requirement.
Keywords: ship’s magnetic field, magnetic field camouflage, magnetic field amplitude, magnetic field curve, multi-objective genetic algorithm.
The ship’s magnetic field is the stable signal source of underwater martine magnetic weapons, the magnetic field features can be obtained after analysis of a large number of magnetic field data, which can be used as the accurate technologic reference (such as sensitivity) for designing and operating magnetic weapon, or to establish the date base for surveillance and identification [1-3]. To protect magnetic field form being detected, in this paper, the multi- objective genetic algorithm was used to calculate and adjust the degaussing coil’s currents to control magnetic amplitude and the features of curve into a camouflage condition. The multi-objective genetic algorithm can avoid the subjective of weight distribution in the uniform objective method. And also, the alternative Pareto disaggregation can be found, which is more adaptable for application requirement.
2. Mathematical model of ship’s magnetic field camouflage
Degaussing system decrease the ship magnetic field by the opposite magnetic field generated by degaussing coils’ current. The adjustment of degaussing coils equals that each coil is correctly distributed current to minimize the error between the magnetic field of ship and that generated by all the coil together [4]. Thus, the ship culminates in terms of degaussing. 41 observation points are chosen，which are evenly distributed within the twice of the ship length range under the keel at the standard measure depth, after degaussing system is on, the vertical magnetic field of the ship is
{H}_{z1}
{H}_{z1max}
. when degaussing system is off, the ship’s vertical magnetic field is
{H}_{z2}
{H}_{z2max}
. Because the distinction of magnetic field amplitude and magnetic field curve is closely related to the performance of magnetic weapons, the camouflage is mainly focused on the magnetic amplitude and curve. After the camouflage, the vertical magnetic field is
{H}_{z3}
{H}_{z3max}
. In order to keep the diversity of camouflage, it can preset various camouflage magnetic field amplitude
{H}_{zset}
based on the requirements, and the
{H}_{zset}
should be larger than
{H}_{z1max}
,and less than
{H}_{z2max}
. The camouflaged objectives are
1) Minimize the absolute error between
{H}_{z3max}
{H}_{zset}
minf\left(I\right)=|{H}_{z3max} -{H}_{zset} |.
2) Maximize the average error between the normalized magnetic field absolute value of
{H}_{z3}
{H}_{z1}
\mathrm{m}\mathrm{a}\mathrm{x}\alpha \left(I\right)=\frac{1}{N}\sum _{k}^{N}\left|\left|\frac{{H}_{z3k}}{{H}_{z3\mathrm{m}\mathrm{a}\mathrm{x}}}\right|-\left|\frac{{H}_{z1k}}{{H}_{z1\mathrm{m}\mathrm{a}\mathrm{x}}}\right|\right|.
The equation above can be transferred into founding the minimum:
\mathrm{m}\mathrm{i}\mathrm{n}\beta \left(I\right)=1/\left(\frac{1}{N}\sum _{k}^{N}\left|\left|\frac{{H}_{z3k}}{{H}_{z3\mathrm{m}\mathrm{a}\mathrm{x}}}\right|-\left|\frac{{H}_{z1k}}{{H}_{z1\mathrm{m}\mathrm{a}\mathrm{x}}}\right|\right|\right).
k
is the serial number of the observation points and
N=
41 is the total number of the points.
When it is to solve the multi-objectives optimization, it is more simple to sum up all the objectives into a uniform objective by weight distribution，while the weight is susceptible to subjectivity. The multi-objective genetic algorithm connects the Pareto domination theory in economics and the genetic algorithm, which can find out the Pareto disaggregation, so it is more pragmatic. As follows is the calculation steps of the multi-objective optimization function gamultiobj in MATLAB and improved based on NSGA-II [5].
• Make sure the constraint type;
• Generate the original population;
• Population revolution: choice, crossover, mutation, generation of sub-population, combination of original population and sub-population, calculation of rank and distance, non-dominated rank, calculate of crowding distance, trim population, calculate of average distance and spread;
• Plot the first front;
Judge the termination condition:
If satisfied, the Pareto optimum result comes out;
If not satisfied, the population revolution goes on.
The calculation steps of the coils’ currents by function gamultiobj in the process of ship magnetic field camouflage is illustrated in Fig. 1.
Fig. 1. Calculation steps of the coils’ currents in the process of camouflage
3.1. Main experiment and calculation parameters
Fig. 2 shows three kinds of ships’ degaussing coils: longitudinal coil (L coil), athwartship coil (A coil) and vertical coil (V coil) [6-8].
In this paper, the ship model is equipped with a total of 31 sets of coils. The coils’ currents range is –5 A-5 A, which the degaussing supply supplies. The magnetic field of individual coil is measured by three components magnetic sensor in the laboratory, the observation points are evenly distributed within the twice of the ship length range under the keel at the standard measure depth, then the magnetic value could be calculated when the coil current is 1 A, which is called as coil’s efficiency. The typical vertical, athwartship and vertical coil’s normalized efficiency is shown in Fig. 3, in which the normalized efficiency is the ratio of each coil efficiency to the maximal value of all the three coils efficiency.
The main parameters of gamultiobj as shown in Table 1.
Fig. 2. Schematic of degaussing coils: a) L coils; b) A coils, c) V coils, d) Assembly coils
Fig. 3. Typical normalized coils’ efficiency
Table 1. The main parameters of the algorithm
Pareto Fraction
Stall Gen Limit
During a camouflage process, preset
{H}_{zset}=1.4*{H}_{z1max}\text{,}
calculation result in this camouflage mode as shown in Fig. 4.
It is observed from Fig 4 that the multi-objective genetic algorithm can find the Pareto solutions within the specified maximum number of iterations, the distribution of
f\left(I\right)
in the solutions is wide (between 0-30 nT) but smaller magnitude, Which indicates that the amplitude of the magnetic field of the ship is close to the preset false magnetic field amplitude; the distribution of
\beta \left(I\right)
is relatively concentrated, and the change of of
\beta \left(I\right)
is no longer obvious when
f\left(I\right)
is greater than 20 nT.
Fig. 4. Pareto front distribution
Multi-objective genetic algorithm can obtain alternative solutions, which can be selected according to the actual application. For the purpose of camouflage, it has been able to effectively achieve magnetic field amplitude camouflage, when the absolute error
f\left(I\right)
between the amplitude of the ship’s magnetic field and the preset false amplitude is less than 10 nT. Therefore, the final solution of the magnetic field camouflage can be chosen according to the following steps: first of all, choose solutions when the
f\left(I\right)
less than 10 nT, and then choose the solution when
\beta \left(I\right)
is smallest, and use the coils’ currents corresponding to the
final solution to calculate the ship magnetic field in the camouflage mode. From Fig. 5, It can be seen that the amplitude of the magnetic field can be controlled well at a preset false level in the camouflage mode, and between the amplitudes of the ship magnetic field before and after the degaussing system is turned on; when camouflaged, the magnetic field curve changes obviously compared with degaussing system opened.
Fig. 5. Normalized magnetic field intensity under the keel
By controlling the coils’ currents of the onboard degaussing system, an effective camouflage of the magnetic field amplitude and the curve characteristics of the ship is realized. The multi-objective genetic algorithm can avoid the subjectivity of the weight assignment in the unified objective method, which can obtain alternative solutions, and the solutions are more in line with the actual demand.
Wang Hai-yun, Dong Da-qun Application of ship’ magnetic field model parameters to recognizing target. Ship Engineering, Vol. 5, Issue 3, 1999, p. 46-49, (in Chinese). [Search CrossRef]
Wen Wu-di, Liu Zhong-le, Li Hua Ship classifying method based on vectorial magnetic field of warship. Mine Warfare and Ship Self-Defense, Vol. 21, Issue 3, 2013, p. 42-45, (in Chinese). [Search CrossRef]
Jong D. A., Cococcioni M. Fuzzy logic generic mine model. Conference Proceedings of Undersea Defence Technology, NATO, Berlin, 2007. [Search CrossRef]
Zhu Xian-qiao, Liu Da-ming, Yang Ming-ming Degaussing coils optimal calibration method based on multi-objectives. Journal of Beijing University of Aeronautics and Astronautics, Vol. 38, Issue 11, 2012, p. 1507-1511, (in Chinese). [Search CrossRef]
Shi Feng, Wang Hui, Yu Lei, et al. MATLAB Intelligent Algorithm 30 Cases Studies. Beihang University Press, Beijing, 2011. [Search CrossRef]
Choi N., Jeung G., Yang C., Chung H., et al. Optimization of degaussing coil currents for magnetic silencing of a ship taking the ferromagnetic hull effect into account. IEEE Transactions on Applied Superconductivity, Vol. 22, Issue 3, 2012, p. 4904504. [Search CrossRef]
Choi N., Jeung G., Jung S., Yang C., Chung H., et al. Efficient methodology for optimizing degaussing coil currents in ships utilizing magnetomotive force sensitivity information. IEEE Transactions on Magnetics, Vol. 48, Issue 2, 2012, p. 419. [Search CrossRef]
Giwoo Jeung, Nak-Sun Choi, Chang-Seob Yang, et al. Indirect fault detection method for an onboard degaussing coil system exploiting underwater magnetic signals. Journal of Magnetics, Vol. 19, Issue 1, 2014, p. 72-77. [Search CrossRef]
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Edit Compensator Dynamics - MATLAB & Simulink - MathWorks España
Compensator Editor
Graphical Compensator Editing
Lead and Lag Networks
Using Control System Designer, you can manually edit compensator dynamics to achieve your design goals. In particular, you can adjust the compensator gain, and you can add the following compensator dynamics:
Real and complex poles, including integrators
Real and complex zeros, including differentiators
You can add dynamics and modify compensator parameters using the Compensator Editor or using the graphical Bode Editor, Root Locus Editor, or Nichols Editor plots.
To open the Compensator Editor dialog box, in Control System Designer, in an editor plot area, right-click and select Edit Compensator. Alternatively, in the Data Browser, in the Controllers section, right-click the compensator you want to edit and click Open Selection.
The Compensator Editor displays the transfer function for the currently selected compensator. You can select a different compensator to edit using the drop-down list. By default, the compensator transfer function displays in the time constant format. You can select a different format by changing the corresponding Control System Designer preference.
In Control System Designer, on the Control System tab, click Preferences. In the Control System Designer Preferences dialog box, on the Options tab, select a Compensator Format.
To add poles and zeros to your compensator, in the Compensator Editor, right-click in the Dynamics table and, under Add Pole/Zero, select the type of pole/zero you want to add.
The app adds a pole or zero of the selected type with default parameters.
To edit a pole or zero, in the Dynamics table, click on the pole/zero type you want to edit. Then, in the Edit Selected Dynamics section, in the text boxes, specify the pole and zero locations.
To delete poles and zeros, in the Dynamics table, click on the pole/zero type you want to delete. Then, right-click and select Delete Pole/Zero.
You can also add and adjust poles and zeros directly from Bode Editor, Root Locus Editor, or Nichols Editor plots. Use this method to roughly place poles and zeros in the correct area before fine-tuning their locations using the Compensator Editor.
To add poles and zeros directly from an editor plot, right-click the plot area and, under Add Pole/Zero, select the type of pole/zero you want to add. In the editor plot, the app displays the editable compensator poles and zeros as red X’s and O’s respectively.
In the editor plots, you can drag poles and zeros to adjust their locations. As you drag a pole or zero, the app displays the new value in the status bar, on the right side.
To delete a pole or zero, right-click the plot area and select Delete Pole/Zero. Then, in the editor plot, click the pole or zero you want to delete.
You can add the following poles and zeros to your compensator:
Real pole/zero — Specify the pole/zero location on the real axis
Complex poles/zeros — Specify complex conjugate pairs by:
Setting the real and imaginary parts directly.
Setting the natural frequency, ωn, and damping ratio, ξ.
Integrator — Add a pole at the origin to eliminate steady-state error for step inputs and DC inputs.
Differentiator — Add a zero at the origin.
You can add lead networks, lag networks, and combination lead-lag networks to your compensator.
Lead One pole and one zero on the negative real axis, with the zero having a lower natural frequency
Increase stability margins
Increase system bandwidth
Reduce rise time
Lag One pole and one zero on the negative real axis, with the pole having a lower natural frequency
Reduce high-frequency gain
Increase phase margin
Improve steady-state accuracy
Lead-Lag A combination of a lead network and a lag network Combine the effects of lead and lag networks
To add a lead-lag network, add separate lead and lag networks.
To configure a lead or lag network for your compensator, use one of the following options:
Specify the pole and zero locations. Placing the pole and zero further apart increases the amount of phase angle change.
Specify the maximum amount of phase angle change and the frequency at which this change occurs. The app automatically computes the pole and zero locations.
When graphically changing pole and zero locations for a lead or lag compensator, in the editor plot, you can drag the pole and zeros independently.
If you know that your system has disturbances at a particular frequency, you can add a notch filter to attenuate the gain of the system at that frequency. The notch filter transfer function is:
\frac{{s}^{2}+2{\xi }_{1}{\omega }_{n}s+{\omega }_{n}^{2}}{{s}^{2}+2{\xi }_{2}{\omega }_{n}s+{\omega }_{n}^{2}}
ωn is the natural frequency of the notch.
The ratio ξ2/ξ1 sets the depth of the notch.
To configure a notch filter for your compensator, in the Compensator Editor dialog box, you can specify the:
Natural Frequency — Attenuated frequency
Notch Depth and Notch Width
Damping for the complex poles and zeros of the transfer function.
When graphically editing a notch filter, in the Bode Editor, you can drag the bottom of the notch to adjust ωn and the notch depth. To adjust the width of the notch without changing ωn or the notch depth, you can drag the edges of the notch.
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Histidinol-phosphate transaminase - Wikipedia
Histidinol-phosphate transaminase homodimer, E.Coli
In enzymology, a histidinol-phosphate transaminase (EC 2.6.1.9) is an enzyme that catalyzes the chemical reaction
L-histidinol phosphate + 2-oxoglutarate
{\displaystyle \rightleftharpoons }
3-(imidazol-4-yl)-2-oxopropyl phosphate + L-glutamate
Thus, the two substrates of this enzyme are L-histidinol phosphate and 2-oxoglutarate, whereas its two products are 3-(imidazol-4-yl)-2-oxopropyl phosphate and L-glutamate.
This enzyme belongs to the family of transferases, specifically the transaminases, which transfer nitrogenous groups. The systematic name of this enzyme class is L-histidinol-phosphate:2-oxoglutarate aminotransferase. Other names in common use include imidazolylacetolphosphate transaminase, glutamic-imidazoleacetol phosphate transaminase, histidinol phosphate aminotransferase, imidazoleacetol phosphate transaminase, L-histidinol phosphate aminotransferase, histidine:imidazoleacetol phosphate transaminase, IAP transaminase, and imidazolylacetolphosphate aminotransferase. This enzyme participates in 5 metabolic pathways: histidine metabolism, tyrosine metabolism, phenylalanine metabolism, phenylalanine, tyrosine and tryptophan biosynthesis, and novobiocin biosynthesis. It employs one cofactor, pyridoxal phosphate.
As of late 2007, 11 structures have been solved for this class of enzymes, with PDB accession codes 1FG3, 1FG7, 1GEW, 1GEX, 1GEY, 1H1C, 1IJI, 1UU0, 1UU1, 1UU2, and 2F8J.
AMES BN, HORECKER BL (1956). "The biosynthesis of histidine: imidazoleacetol phosphate transaminase". J. Biol. Chem. 220 (1): 113–28. PMID 13319331.
Martin RG; Goldberger RF (1963). "Imidazolylacetolphosphate:L-glutamate aminotransferase. Purification and properties". J. Biol. Chem. 242 (6): 1168–1174. PMID 5337155.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Histidinol-phosphate_transaminase&oldid=1037530985"
|
-6x=4-2y
If you graphed this equation, what shape would the graph have? How can you tell?
If there's an
x
y
, and there are no exponents like
x^2
, what kind of graph does the equation make?
Remember that this equation could be written in
y=mx+b
Without changing the form of the equation, find the coordinates of three points that must be on the graph of this equation. Then graph the equation on graph paper.
Make a table and substitute values for
x
y
until you have enough points to graph the line.
y
. Does your answer agree with your graph? If so, how do they agree? If not, check your work to find the error.
y
-6x-4=-2y
6x+4=2y
3x+2=y
Yes, they both have the same starting value (
2
) and growth (
3
|
Ts Grewal Vol II 2018 for Class 12 Commerce Accountancy Chapter 18 - Issue Of Debentures
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Books of Vishwas Ltd.
(Debenture application money received for 2,000 debentures at Rs 25 each)
(Debenture application money transferred to 9% Debentures A/c)
(Debenture allotment money due on 2,000 Debentures at Rs 25 each)
(Debenture allotment money received)
9% Debenture First and Final Call A/c
(Debenture first and final call money due on 2,000 debentures at Rs 50 each)
(Debenture first and final call received)
(Debenture application money transferred to 9% Debenture account for 2,000 Debenture, adjusted to Debenture Allotment account for 200 Debentures and money refunded for 200 debentures)
Debenture First Call A/c
(Debenture first call money due on 2,000 9% debenture at Rs 30 each)
(Debenture first call money received)
Debenture Final Call A/c
(Debentures final call money due on 2,000 9% Debentures at Rs 30 each)
To Debenture Final Call A/c
(Debenture final call received on 2,000 9% Debenture at Rs 30 each)
ABCParticulars
Bank A/c (60,000
×
(Received application money on 60,000 Debenture)
To 10% Debentures A/c (40,000
×
(Application money transferred to Debentures A/c)
In the Books of Narain Laxmi Ltd.
Bank A/c (10,000 debentures × 135)
(Application money received on 10,000 12% debenture)
(7,500; 12% Debentures of Rs 100 each issued at a premium of Rs 35 and excess money refunded)
Books of Raj Ltd.
(Debenture allotment due on 5,000 8% Debentures at Rs 20 including premium of Rs 5)
(Debenture first and final call due on 5,000 Debentures at Rs 75 each)
Face Value of Debenture = Rs 100
Premium (Rs 100 × 10%) = Rs 10
∴ Issue Price = Rs 110
Amount Payable as:
On Application (25%)
Rs 25 including premium of Rs 10 (i.e. Rs 10 + 15)
On Allotment (85%)
Rs 85 per debenture
(Debenture application money received for 10,000 debentures at Rs 25 including premium of Rs10 each transferred to debenture account)
(Debentures allotment due on 10,000 Debentures at Rs 85 each)
Debenture Allotment
Bank A/c (7,000
×
(Received application money on 7,000 debentures)
(Transfer of application money to Debentures A/c)
Debenture Allotment A/c (7,000
×
Loss on issue of Debentures A/c (7,000
×
To 10% Debentures A/c (7,000
×
To Securities Premium Reserve A/c (7,000
×
To Premium on Redemption of Debentures A/c(7,000
×
(Allotment due on 7,000 Debentures at a premium of Rs 50 per debentures and redeemable at premium of 10%)
Vijay Laxmi Ltd. invited applications for 10,000; 12% Debentures of ₹ 100 each at a premium of ₹ 70 per debenture .The full amount was payable on application.
Applications were received for 13,500 debentures. Applications for 3,500 debentures were rejected and application money was refunded . Debentures were allotted to the remaining applications .
In the Books of Vijay Laxmi Ltd.
(10,000; 12% Debentures issued at a premium of Rs 70 and excess money refunded)
Iron Products Ltd. issued 5,000; 9% Debentures of ₹ 100 each at a premium of ₹ 40 payable as follows;
(i) ₹ 40 , including premium of ₹ 10 on applications;
(ii) ₹ 45, including premium of ₹ 15 on allotment ; and
The issue was subscribed and allotment made. Calls were made and due amount was received .
Pass Journal entries .
Bank A/c (5,000×40)
(Application money received)
To 9% Debentures A/c (5,000×30)
To Security Premium Reserve A/c (5,000×10)
(Application money adjusted)
Debenture Allotment A/c (5,000×45)
(Allotment money due)
Debenture First and Final call A/c (5,000×55)
(First call money due)
To Debenture Final and Final call A/c
X Ltd . issued 12,000; 8% Debentures of ₹ 100 each at a discount of 5% payable as 25% on application;20% on allotment and balance after three months.
Discount (Rs 100 × 5%) = Rs 5
∴ Issue Price = Rs 95
Rs 20 (25 – 5) per debenture
On First and Final Call (50%)
(Application money received for 12,000 8% Debentures at Rs 25 each)
(Debenture application money transferred to 8% Debentures account)
(Allotment money due on 12,000 8% Debentures at Rs 20 each at discount of Rs 5)
(First and final call money due on 12,000 8% Debentures at Rs 50 each)
To 8% Debentures First and Final Call A/c
Alka Ltd . issued 5,000, 10% Debentures of ₹ 1,000 each at a discount of 10% redeemable at a premium of 5% after 5 years . According to the terms of issue ₹ 500 was payable on application and the balance amount on allotment of debentures. Record necessary entries regarding issue of 10% Debentures.
×
×
Discount on issue of Debentures A/c (5,000
×
100) 5,00,000
×
×
To Premium on Redemption of Debentures A/c (5,000
×
(Allotment due on 5,000 Debentures at a discount of Rs 100 per debentures and redeemable at premium of 5%)
Amrit Ltd . was promoted by Amrit and Bhaskar with an authorised capital of ₹ 10,00,000 divide into 1,00,000 shares of ₹ 10 each.
The company decided to issue 1,000,6% Debentures of ₹ 100 each to Amrit and Bhaskar each for their services in incorporating the company.
Incorporation Cost A/c (2,000 × 100)
( Debentures issued to promoters)
A limited company bought a Building for ₹ 9,00,000 and the consideration was paid by issuing 10% Debentures of the normal (face) value of ₹ 100 each at a discount of 10%.
(Building purchased)
(Issued 10,000, 10% debentures at 10% discount)
Wye Ltd . purchased an established business for ₹ 2,00,000 payable as ₹ 65,000 by cheque and the balance by issuing 9% Debentures of ₹ 100 each at a discount of 10%.
Books of Wye Ltd.
(Business purchased)
(Amount paid to Vendor in cash)
(Issued 1,500 debentures at 10% discount)
Newton Ltd. purchased a Machinery from B for ₹ 5,76,000 to be paid by the issue of 9% Debentures of ₹ 100 each at 4% discount. Journalise the trasactions.
Books of Newton Ltd.
(Machinery purchased from B)
(Issued 6,000 debentures at 4% discount)
Reliance Ltd. purchased machinery costing ₹ 1,35,000 . It was agreed that the purchase consideration be paid by issuing 9% Debentures of ₹ 100 each . Assume debentures have been issued
(ii)at a discount of 10%.
(Machinery Purchases)
(Issued 1,350 debentures at par)
Deepak Ltd purchased furniture of ₹ 2,20,000 from M/s. Furniture Mart. 50% of the amount was paid to M/s. Furniture Mart by accepting a Bill of Exchanged and for the balance the company issued 9% Debenture of ₹ 100 each at a premium of 10% in favour of M/s. Furniture Mart.
Books of Deepak Ltd.
To Furniture Mart
(Furniture purchased from Furniture Mart)
(Bill accepted from Furniture Mart against 50% payment)
(Issued 1,000 9% Debentures of Rs 100 each at a premium of 10% to Furniture Mart)
X Ltd . took over the assets of ₹ 6,00,000 and liabilities of ₹ 80,000 of Y Ltd for an agreed purchase consideration of ₹ 6,00,000 payable 10% in cash and the balance by the issue of 12% Debentures of ₹ 100 each . Give necessary journal entries in the books of X Ltd., assuming that:
Goodwill A/c(Balancing Figure)
To Y Ltd.
(Purchase of business of Y Ltd.)
(Payment made in cash)
( Purchase consideration discharged by issue of 12% Debentures)
\text{1) Number of Debentures to issued}=\frac{5,40,000}{120}=4,500\text{ Debentures}
\text{2) Number of Debentures to issued}=\frac{5,40,000}{90}=6,000\text{ Debentures}
X Ltd. took over the assets of ₹ 6,60,000 and liabilities of ₹ 80,000 of Y Ltd . for ₹ 6,00,000. Give necessary journal entries in the books of X Ltd. assuming that:
Goodwill A/c (Balancing Figure)
(Purchase of business took over)
(Purchase consideration discharged)
( Purchase consideration discharged)
(a) 1,000 , 10% Debentures of ₹ 100 each at a discount of 10% ; and
Discount on issue of Debentures A/c (1,000×10)
To 10% Debentures A/c (1,000×100)
Books of Lotus Ltd.
To Goneby Company A/c
(Business purchased of Goneby Company)
Goneby Company A/c
(Issued 3,000 debentures at 10% premium)
Exe Ltd. purchased the assets of the book value ₹4,00,000 and took over the liabilities of ₹ 50,000 from Mohan Bros.It was agreed that the purchase consideration ,settled at ₹3,80,000 be paid by issuing debentures of ₹ 100 each.
(Asset and liabilities purchased from Mohan Bros.)
Case 1 When Debentures are issued at Par
Mohan Bros.
Case 2 When Debentures are issued at 10% discount
(Issued 4,222 Debentures of Rs 100 each at 10% discount to Mohan Bros. and fraction of debentures is paid in cash)
Case 3 When Debentures are issued at 10% premium
(Issued 3,454 Debentures of Rs 100 each at 10% premium to Mohan Bros. and fraction of debentures is paid in cash)
R Ltd. purchased the assets of S Ltd. for ₹5,00,000. It also agreed to take over the liabilities of S Ltd. amounted to ₹ 2,00,000 for a purchase consideration of ₹2,80,000 . The payment of S Ltd. was made by issue of 9% Debentures of ₹ 100 each at par.
Books of R Ltd.
To S Ltd.
(Asset purchased and liabilities took over from S Ltd.)
(Issued 2,800 9% Debentures of Rs 100 each)
Books of Romi Ltd.
To Kapil Enterprises
(Asset purchased and Creditors took over from Kapil Enterprises)
Kapil Enterprises A/c
(Issued 14,400 8% Debentures of Rs 100 each at a premium of 25% to Kapil Enterprises)
(Assets purchased and Creditors took over from Kapil Enterprises)
(Issued 20,000 8% Debentures of Rs 100 each at discount of 10% to Kapil Enterprises)
(i) To sundry persons for cash at par
—
₹ 5,00,000 nominal.
(ii) To a vendor for ₹ 5,50,000 for purchase of fixed assets
—
Best Barcode Ltd.
Loan (Secured by issue of 9% Debentures of Rs 6,00,000 as Collateral Security)
(Loan taken against issuing 9% Debentures as collateral Security)
(Issued 9% Debentures of Rs 6,00,000 as collateral security)
Posting in the Company's Balance Sheet
Loan (Secured by issue of 9% Debentures of
Rs 6,00,000 as Collateral Security)
9% Debentures (Issued as Collateral Security to Bank against loan)
Less: Debenture Suspense Account
When Debentures Issued as Collateral Security are shown separately
To Loan from Bandhan Bank Ltd.
Posting in the Company's Balance Sheet (When Debentures Issued as Collateral Security are shown separately)
Loan from Bandhan Bank (Secured by issue of Debentures of Rs 4,00,000)
Alternative Method: When debentures Issued as Collateral Security are not shown separately
(Loan taken from Bandhan Bank secured by issuing Debentures as collateral security)
(When Debentures Issued as Collateral Security are not shown separately)
|
Free object - Wikipedia
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices.
The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms.
3 Free universal algebras
4 Free functor
5 List of free objects
Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function u : E1 → E2 between vector spaces is entirely determined by its values on a basis of the vector space E1. The following definition translates this to any category.
A concrete category is a category that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with faithful functor F : C → Set. Let X be an object in Set (that is, X is a set, here called a basis), let A be an object in C, and let i : X → F(A) be an injective map between the sets X and F(A) (called the canonical insertion). Then A is said to be the free object on X (with respect to i) if and only if it satisfies the following universal property:
for any object B in C and any map between sets f : X → F(B), there exists a unique morphism g : A → B in C such that f = F(g) ∘ i. That is, the following diagram commutes:
{\displaystyle {\begin{array}{c}X{\xrightarrow {\quad i\quad }}F(A)\\{}_{f}\searrow \quad \swarrow {}_{F(g)}\\F(B)\quad \\\end{array}}}
In this way the free functor that builds the free object A from the set X becomes left adjoint to the forgetful functor.
The creation of free objects proceeds in two steps. For algebras that conform to the associative law, the first step is to consider the collection of all possible words formed from an alphabet. Then one imposes a set of equivalence relations upon the words, where the relations are the defining relations of the algebraic object at hand. The free object then consists of the set of equivalence classes.
Consider, for example, the construction of the free group in two generators. One starts with an alphabet consisting of the five letters
{\displaystyle \{e,a,b,a^{-1},b^{-1}\}}
. In the first step, there is not yet any assigned meaning to the "letters"
{\displaystyle a^{-1}}
{\displaystyle b^{-1}}
; these will be given later, in the second step. Thus, one could equally well start with the alphabet in five letters that is
{\displaystyle S=\{a,b,c,d,e\}}
. In this example, the set of all words or strings
{\displaystyle W(S)}
will include strings such as aebecede and abdc, and so on, of arbitrary finite length, with the letters arranged in every possible order.
In the next step, one imposes a set of equivalence relations. The equivalence relations for a group are that of multiplication by the identity,
{\displaystyle ge=eg=g}
, and the multiplication of inverses:
{\displaystyle gg^{-1}=g^{-1}g=e}
. Applying these relations to the strings above, one obtains
{\displaystyle aebecede=aba^{-1}b^{-1},}
where it was understood that
{\displaystyle c}
is a stand-in for
{\displaystyle a^{-1}}
, an{\displaystyle d}
{\displaystyle b^{-1}}
, whil{\displaystyle e}
is the identity element. Similarly, one has
{\displaystyle abdc=abb^{-1}a^{-1}=e.}
Denoting the equivalence relation or congruence by
{\displaystyle \sim }
, the free object is then the collection of equivalence classes of words. Thus, in this example, the free group in two generators is the quotient
{\displaystyle F_{2}=W(S)/\sim .}
This is often written as
{\displaystyle F_{2}=W(S)/E}
{\displaystyle W(S)=\{a_{1}a_{2}\ldots a_{n}\,\vert \;a_{k}\in S\,;\,n\in \mathbb {N} \}}
is the set of all words, and
{\displaystyle E=\{a_{1}a_{2}\ldots a_{n}\,\vert \;e=a_{1}a_{2}\ldots a_{n}\,;\,a_{k}\in S\,;\,n\in \mathbb {N} \}}
is the equivalence class of the identity, after the relations defining a group are imposed.
A simpler example are the free monoids. The free monoid on a set X, is the monoid of all finite strings using X as alphabet, with operation concatenation of strings. The identity is the empty string. In essence, the free monoid is simply the set of all words, with no equivalence relations imposed. This example is developed further in the article on the Kleene star.
In the general case, the algebraic relations need not be associative, in which case the starting point is not the set of all words, but rather, strings punctuated with parentheses, which are used to indicate the non-associative groupings of letters. Such a string may equivalently be represented by a binary tree or a free magma; the leaves of the tree are the letters from the alphabet.
The algebraic relations may then be general arities or finitary relations on the leaves of the tree. Rather than starting with the collection of all possible parenthesized strings, it can be more convenient to start with the Herbrand universe. Properly describing or enumerating the contents of a free object can be easy or difficult, depending on the particular algebraic object in question. For example, the free group in two generators is easily described. By contrast, little or nothing is known about the structure of free Heyting algebras in more than one generator.[1] The problem of determining if two different strings belong to the same equivalence class is known as the word problem.
As the examples suggest, free objects look like constructions from syntax; one may reverse that to some extent by saying that major uses of syntax can be explained and characterised as free objects, in a way that makes apparently heavy 'punctuation' explicable (and more memorable).[clarification needed]
Free universal algebrasEdit
Main article: Term algebra
{\displaystyle S}
be any set, and let
{\displaystyle \mathbf {A} }
be an algebraic structure of type
{\displaystyle \rho }
{\displaystyle S}
. Let the underlying set of this algebraic structure
{\displaystyle \mathbf {A} }
, sometimes called its universe, be
{\displaystyle A}
{\displaystyle \psi :S\to A}
be a function. We say that
{\displaystyle (A,\psi )}
(or informally just
{\displaystyle \mathbf {A} }
) is a free algebra (of type
{\displaystyle \rho }
) on the set
{\displaystyle S}
of free generators if, for every algebra
{\displaystyle \mathbf {B} }
{\displaystyle \rho }
and every function
{\displaystyle \tau :S\to B}
{\displaystyle B}
is a universe of
{\displaystyle \mathbf {B} }
, there exists a unique homomorphism
{\displaystyle \sigma :A\to B}
{\displaystyle \sigma \circ \psi =\tau .}
Free functorEdit
The most general setting for a free object is in category theory, where one defines a functor, the free functor, that is the left adjoint to the forgetful functor.
Consider a category C of algebraic structures; the objects can be thought of as sets plus operations, obeying some laws. This category has a functor,
{\displaystyle U:\mathbf {C} \to \mathbf {Set} }
, the forgetful functor, which maps objects and functions in C to Set, the category of sets. The forgetful functor is very simple: it just ignores all of the operations.
The free functor F, when it exists, is the left adjoint to U. That is,
{\displaystyle F:\mathbf {Set} \to \mathbf {C} }
takes sets X in Set to their corresponding free objects F(X) in the category C. The set X can be thought of as the set of "generators" of the free object F(X).
For the free functor to be a left adjoint, one must also have a Set-morphism
{\displaystyle \eta :X\to U(F(X))\,\!}
. More explicitly, F is, up to isomorphisms in C, characterized by the following universal property:
Whenever A is an algebra in C, and g : X → U(A) is a function (a morphism in the category of sets), then there is a unique C-morphism h : F(X) → A such that U(h) ∘ η = g.
Concretely, this sends a set into the free object on that set; it is the "inclusion of a basis". Abusing notation,
{\displaystyle X\to F(X)}
(this abuses notation because X is a set, while F(X) is an algebra; correctly, it is
{\displaystyle X\to U(F(X))}
The natural transformation
{\displaystyle \eta :\operatorname {id} _{\mathbf {Set} }\to UF}
is called the unit; together with the counit
{\displaystyle \varepsilon :FU\to \operatorname {id} _{\mathbf {C} }}
, one may construct a T-algebra, and so a monad.
The cofree functor is the right adjoint to the forgetful functor.
There are general existence theorems that apply; the most basic of them guarantees that
Whenever C is a variety, then for every set X there is a free object F(X) in C.
Here, a variety is a synonym for a finitary algebraic category, thus implying that the set of relations are finitary, and algebraic because it is monadic over Set.
Other types of forgetfulness also give rise to objects quite like free objects, in that they are left adjoint to a forgetful functor, not necessarily to sets.
For example, the tensor algebra construction on a vector space is the left adjoint to the functor on associative algebras that ignores the algebra structure. It is therefore often also called a free algebra. Likewise the symmetric algebra and exterior algebra are free symmetric and anti-symmetric algebras on a vector space.
List of free objectsEdit
See also: Category:Free algebraic structures
Specific kinds of free objects include:
free associative algebra
free commutative algebra
free partially commutative group
free Kleene algebra
free distributive lattice
free Heyting algebra
free modular lattice
free module, and in particular, vector space
free commutative monoid
free semiring
free commutative semiring
^ Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, ISBN 0-521-23893-5. (A treatment of the one-generator free Heyting algebra is given in chapter 1, section 4.11)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Free_object&oldid=1076554937"
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Use your calculator as shown above to evaluate each of the radical expressions. Then repeat each problem using a fractional exponent instead of a root to check.
\sqrt [ 3 ] { 4 }
( \sqrt [ 10 ] { 10 } ) ^ { 4 }
\sqrt [ 10 ] { 10,000 }
Refer to the Math Notes in this lesson.
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It’s bad enough when Myriah has to do the dishes. But Myriah really hates doing the dishes several days in a row! Myriah and her mom have agreed to roll two dice. When the sum of the dice is
6
or less, Myriah has to do the dishes. If the sum is
7
or more, one of her parents does the dishes. Myriah wants to know how many times in the next two months she will end up doing the dishes
3
or more days in a row. 11-41 HW eTool (CPM).
Run a simulation of rolling the two dice and record the sum of each roll to simulate the
60
2
months).
How often can Myriah expect to have to do the dishes
3
or more days in a row during two months?
Answers will vary, but should be
≈ 7
8
Click on the dice in the eTool below to simulate problem 11-41.
Click on the link at right for the full eTool version: CCA2 11-41 HW eTool.
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Experimental and Numerical Cross-Over Jet Impingement in an Airfoil Trailing-Edge Cooling Channel | J. Turbomach. | ASME Digital Collection
A. Nongsaeng
Taslim, M. E., and Nongsaeng, A. (April 20, 2011). "Experimental and Numerical Cross-Over Jet Impingement in an Airfoil Trailing-Edge Cooling Channel." ASME. J. Turbomach. October 2011; 133(4): 041009. https://doi.org/10.1115/1.4002984
Trailing edge cooling cavities in modern gas turbine airfoils play an important role in maintaining the trailing-edge temperature at levels consistent with airfoil design life. In this study, local and average heat transfer coefficients were measured in a test section, simulating the trailing-edge cooling cavity of a turbine airfoil using the steady-state liquid crystal technique. The test rig was made up of two adjacent channels, each with a trapezoidal cross-sectional area. The first channel, simulating the cooling cavity adjacent to the trailing-edge cavity, supplied the cooling air to the trailing-edge channel through a row of racetrack-shaped slots on the partition wall between the two channels. Eleven crossover jets issued from these slots entered the trailing-edge channel and exited from a second row of race-track shaped slots on the opposite wall in staggered or inline arrangement. Two jet angles were examined. The baseline tests were for zero angle between the jet axis and the trailing-edge channel centerline. The jets were then tilted toward one wall (pressure or suction side) of the trailing-edge channel by 5 deg. Results of the two set of tests for a range of local jet Reynolds number from 10,000 to 35,000 were compared. The numerical models contained the entire trailing-edge and supply channels with all slots to simulate exactly the tested geometries. They were meshed with all-hexa structured mesh of high near-wall concentration. A pressure-correction based, multiblock, multigrid, unstructured/adaptive commercial software was used in this investigation. Standard high Reynolds number
k−ε
turbulence model in conjunction with the generalized wall function for most parts was used for turbulence closure. Boundary conditions identical to those of the experiments were applied and several turbulence model results were compared. The numerical analyses also provided the share of each cross-over and each exit hole from the total flow for different geometries. The major conclusions of this study were (a) except for the first and last cross-flow jets which had different flow structures, other jets produced the same heat transfer results on their target surfaces, (b) jets tilted at an angle of 5 deg produced higher heat transfer coefficients on the target surface. The tilted jets also produced the same level of heat transfer coefficients on the wall opposite the target wall, and (c) the numerical predictions of impingement heat transfer coefficients were in good agreement with the measured values for most cases; thus, computational fluid dynamics could be considered a viable tool in airfoil cooling circuit designs.
aerodynamics, cooling, gas turbines, jets, numerical analysis, turbulence
Airfoils, Computational fluid dynamics, Cooling, Flow (Dynamics), Heat transfer, Heat transfer coefficients, Jets, Reynolds number, Turbulence, Pressure, Numerical analysis, Temperature, Liquid crystals, Gas turbines, Cavities
Impingement Heat Transfer From Rib Roughened Surface Within Arrays of Circular Jet: The Effect of the Relative Position of the Jet Hole to the Ribs
An Experimental Evaluation of Advanced Leading-Edge Impingement Cooling Concepts
Experimental and Numerical Study of Impingement on an Airfoil Leading-Edge With and Without Showerhead and Gill Film Holes
Bethka
An Experimental and Numerical Investigation of Impingement Heat Transfer in Airfoils Leading-Edge Cooling Channel
International Symposium on Heat Transfer in Gas Turbine Systems
, Aug. 9–14, Antalya, Turkey.
Heat Transfer and Flow Friction Characteristics of Very Rough Transverse Ribbed Surfaces With and Without Pin Fins
Pressure Drop and Heat Transfer Coefficient Distributions in Serpentine Passages With and Without Turbulence Promoters
The Eighth International Heat Transfer Conference
Turbulent Heat Transfer and Friction in Pin Fin Channels With Lateral Flow Ejection
An Experimental and Numerical Investigation of Jet Impingement on Ribs in an Airfoil Trailing-Edge Cooling Channel
, Honolulu, HI, Paper No. ISROMAC12-2008-20238.
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A\mathrm{sin}\left(x\right)
A
x
t
n
p
p
p
The trace=n option specifies that a number of previous frames of the animation be kept visible. When
n
n+1
n=5
When
is a list of integers, then the frames in those positions are the frames that remain visible. Each integer in
n=0
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=5,\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=[30,35,40,45,50],\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot3d},[A\left({x}^{2}+{y}^{2}\right),x=-3..3,y=-3..3],A=-2..2,\mathrm{style}=\mathrm{patchcontour}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+{y}^{2}={r}^{2},x=-3..3,y=-3..3],r=1..3,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+Axy-{y}^{2}=1,x=-2..2,y=-3..3],A=-2..2,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{sin}\left(t\right),\mathrm{sin}\left(t\right)\mathrm{exp}\left(-\frac{t}{5}\right)],t=0..x],x=0..6\mathrm{\pi },\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t=0..A]],A=0..2\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\frac{1-{t}^{2}}{1+{t}^{2}},\frac{2t}{1+{t}^{2}},t=-10..A]],A=-10..10,\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50,\mathrm{view}=[-1..1,-1..1]\right)
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{numpoints}=100,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),\left(2+\mathrm{sin}\left(A\right)\right)t],t=0..20,\mathrm{opts}],A=0..2\mathrm{\pi }\right)
B≔\mathrm{plot3d}\left(1-{x}^{2}-{y}^{2},x=-1..1,y=-1..1,\mathrm{style}=\mathrm{patchcontour}\right):
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[t,t,1-2{t}^{2}],t=-1..A,\mathrm{opts}],A=-1..1,\mathrm{frames}=11,\mathrm{background}=B\right)
\mathrm{animate}\left(\mathrm{ball},[0,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=100\right)
\mathrm{sinewave}≔\mathrm{plot}\left(\mathrm{sin}\left(x\right),x=0..4\mathrm{\pi }\right):
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{background}=\mathrm{sinewave},\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{trace}=10,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(F,[\mathrm{\theta }],\mathrm{\theta }=0..2\mathrm{\pi },\mathrm{background}=\mathrm{plot}\left([\mathrm{cos}\left(t\right)-2,\mathrm{sin}\left(t\right),t=0..2\mathrm{\pi }]\right),\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{none}\right)
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Initial pore pressures under the Lusi mud volcano, Indonesia | Interpretation | GeoScienceWorld
, Australian School of Petroleum, Adelaide, South Australia,
. E-mail: mark.tingay@adelaide.edu.au.
Mark Tingay; Initial pore pressures under the Lusi mud volcano, Indonesia. Interpretation 2014;; 3 (1): SE33–SE49. doi: https://doi.org/10.1190/INT-2014-0092.1
The Lusi mud volcano of East Java, Indonesia, remains one of the most unusual geologic disasters of modern times. Since its sudden birth in 2006, Lusi has erupted continuously, expelling more than 90 million cubic meters of mud that has displaced approximately 40,000 people. This study undertakes the first detailed analysis of the pore pressures immediately prior to the Lusi mud volcano eruption by compiling data from the adjacent (150 m away) Banjar Panji-1 wellbore and undertaking pore pressure prediction from carefully compiled petrophysical data. Wellbore fluid influxes indicate that sequences under Lusi are overpressured from only 350 m depth and follow an approximately lithostat-parallel pore pressure increase through Pleistocene clastic sequences (to 1870 m depth) with pore pressure gradients up to
17.2 MPa/km
. Most unusually, fluid influxes, a major kick, connection gases, elevated background gases, and offset well data confirm that high-magnitude overpressures also exist in the Plio-Pleistocene volcanic sequences (1870 to approximately 2833 m depth) and Miocene (Tuban Formation) carbonates, with pore pressure gradients of
17.2–18.4 MPa/km
. The varying geology under the Lusi mud volcano poses a number of challenges for determining overpressure origin and undertaking pore pressure prediction. Overpressures in the fine-grained and rapidly deposited Pleistocene clastics have a petrophysical signature typical of disequilibrium compaction and can be reliably predicted from sonic, resistivity, and drilling exponent data. However, it is difficult to establish the overpressure origin in the low-porosity volcanic sequences and Miocene carbonates. Similarly, the volcanics do not have any clear porosity anomaly, and thus pore pressures in these sequences are greatly underestimated by standard prediction methods. The analysis of preeruption pore pressures underneath the Lusi mud volcano is important for understanding the mechanics, triggering, and longevity of the eruption, as well as providing a valuable example of the unknowns and challenges associated with overpressures in nonclastic rocks.
Tuban Formation
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Though many students think the following statements are true, each of them is actually FALSE. Confirm this by substituting simple numbers for the variables and doing the arithmetic. Note: Do not use
0
1
; they are too special.
REMEMBER: EACH OF THESE STATEMENTS IS FALSE.
Try substituting numbers such as 2, 3, 4, or 5.
\left(x + y\right)^{2} = x^{2} + y^{2}
\sqrt{p^2+q^2}=p+q
3w^{−2}=\frac{1}{3w^2}
(a^{-1} + b^{-1})^{−1} = a + b
3 · 2^{x} = 6 ^{x}
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SBONDS FAQ - Scarab.Finance
"What are SBOND (Bonds)?"
Bonds are unique tokens that can be utilized to help stabilize SCARAB price around peg (1 SCARAB = 1 FTM) by reducing circulating supply of SCARAB if the TWAP (time-weighted-average-price) goes below peg (1 SCARAB = 1 FTM).
"When can I buy SBOND (Bonds)?"
SBOND can be purchased only on contraction periods, when TWAP of SCARAB is below 1.
Every new epoch on contraction periods, SBONDs are issued in the amount of 3% of current SCARAB circulating supply, with a max debt amount of 35%. This means that if bonds reach 35% of circulating supply of SCARAB, no more bonds will be issued.
Note: SBOND TWAP (time-weighted average price) is based on SCARAB price TWAP from the previous epoch as it ends. This mean that SCARAB TWAP is real-time and SBOND TWAP is not.
"Where can I buy SBOND (Bonds)?"
You can buy SBONDs if any are available, through skarab.finance, anyone can buy as many SBONDs as they want as long as they have enough SCARAB to pay for them.
There is a limit amount (3% of SCARAB current circulating supply) of available SBONDs per epoch while on contraction periods, and are sold as first come first serve.
"Why should I buy SBOND (Bonds)?"
First and most important reason is Bonds help maintain the peg, but will not be the only measure use to keep the protocol on track. We also have a DAO fund which will step in and buy SCARAB to get it back to peg.
SBONDs don't have a expiration date, so you can view them as a investment on the protocol, because longterm you get benefits from holding bonds.
Incentives for holding SBOND
The idea is to reward SBOND buyers for helping the protocol, while also protecting the protocol from being manipulated from big players.
So after you buy SBOND using SCARAB, you get 2 possible ways to get your SCARAB back:
Sell back your SBOND for SCARAB while peg is between 1 - 1.1 (1 SCARAB = 1 FTM) with no redemption bonus. This to prevent instant dump after peg is recovered
Sell back your SBOND for SCARAB while peg is above 1.1 (1 SCARAB = 1FTM) with a bonus redemption rate
The longer you hold, the more both the protocol and you benefit from SBOND.
When SCARAB = 0.8, burn 1 SCARAB to get 1 SBOND (SBOND price = 0.8)
When SCARAB = 1.15, redeem 1 SBOND to get 1.105 SCARAB (SBOND price = 1.27)
If I buy SCARAB at 0.8, and hold it until 1.15 and then sell, I'm getting +0.35$ per SCARAB
But, if I buy SCARAB at 0.8, burn it for SBOND, and redeem it at 1.15, I'm getting 1.105 SCARAB * 1.15 (SCARAB current price) = 1,271 (+0.47$) per SBOND redeemed.
We are going to adjust our use cases, to have different behaviors on contraction and expansion periods to benefit SCARAB and SBOND holders when needed.
"When can I swap SBOND for a bonus?"
SBOND TWAP (time-weighted average price) is based on SCARAB price TWAP from the previous epoch as it ends. This mean that SCARAB TWAP is real-time and SBOND TWAP is not. In other words, you can redeem SBOND for a bonus when the previous epoch's TWAP > 1.1.
"When can I swap $SCARAB for $SBOND?"
$SBOND will only become available in the Desert following epochs in which the Time Weighted Average Price (TWAP) of $SCARAB is under peg. This means that $SCARAB's price will have had to have been under 1 $FTM per 1 $SCARAB for the majority of the previous epoch in order to trigger the Desert to "open".
The Desert will always open at the very beginning of a new epoch, and remain open for the entire epoch — the Desert can not and will never open mid-epoch — and during epochs in which the Desert is open, $SCARAB will not be printed in the Temple.
"What is the formula to calculate the redemption bonus for $SBOND?"
To encourage redemption of $SBOND for $SCARAB when $SCARAB's TWAP > 1.1, and in order to incentivize users to redeem at a higher price, $SBOND redemption will be more profitable with a higher $SCARAB TWAP value. The $SBOND to $SCARAB ratio will be 1:R, where R can be calculated in the formula as shown below:
R=1+[(SCARABtwapprice)-1)*coeff)9)]
coeff = 0.7
To further illustrate why the longer you hold $SBOND the more profitable it is, let's take an initial $1000 investment into consideration. In this example, say this $1000 is used to buy $SCARAB when $SCARAB TWAP is 0.95 and then swapped for $SBOND. If these $SBOND are redeemed when: -$SCARAB TWAP is 1.5, your investment would now be worth $1421. -$SCARAB TWAP is 2, your investment would now be worth $1789. -$SCARAB TWAP is 3, your investment would now be worth $2526. -$SCARAB TWAP is 5, your investment would now be worth $4000.
"I expected $SBOND to be issued in the desert, but there is none. Why?"
"When can I swap $SBOND back to $SCARAB?"
You can swap it back again when the following two criteria are met:
1: $SCARAB TWAP is above peg and
2. There is enough in the treasury to cover it the redemption.
"Is $SBOND right for me?"
Like anything else in crypto, obtaining $SBOND is not risk-free. Just like in the real world, you are purchasing debt from the protocol with the expectation that you will be redeemed at a premium in the future. To date, this has occurred after all contractions, but past performance does not guarantee the same future outcomes. $SBOND is ideal for those with a medium to long-term time preference, as it incentivizes hodling in exchange for potentially extremely lucrative rewards. If you are looking for a quick flip or have short-term time preference, $SBOND may not be the right investment option for you.
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Remarks on the Pressure Regularity Criterion of the Micropolar Fluid Equations in Multiplier Spaces
Fengjun Guo, "Remarks on the Pressure Regularity Criterion of the Micropolar Fluid Equations in Multiplier Spaces", Abstract and Applied Analysis, vol. 2012, Article ID 618084, 10 pages, 2012. https://doi.org/10.1155/2012/618084
Fengjun Guo 1
Academic Editor: Beong In Yun
This study is devoted to investigating the regularity criterion of weak solutions of the micropolar fluid equations in . The weak solution of micropolar fluid equations is proved to be smooth on when the pressure satisfies the following growth condition in the multiplier spaces , . The previous results on Lorentz spaces and Morrey spaces are obviously improved.
Consider the Cauchy problem of the three-dimensional (3D) micropolar fluid equations with unit viscosities associated with the initial condition: where , and are the unknown velocity vector field and the microrotation vector field. is the unknown scalar pressure field. and represent the prescribed initial data for the velocity and microrotation fields.
Micropolar fluid equations introduced by Eringen [1] are a special model of the non-Newtonian fluids (see [2–6]) which is coupled with the viscous incompressible Navier-Stokes model, microrotational effects, and microrotational inertia. When the microrotation effects are neglected or , the micropolar fluid equations (1.1) reduce to the incompressible Navier-Stokes flows (see, e.g., [7, 8]): That is to say, Navier-Stokes equations are viewed as a subclass of the micropolar fluid equations.
Mathematically, there is a large literature on the existence, uniqueness and large time behaviors of solutions of micropolar fluid equations (see [9–15] and references therein); however, the global regularity of the weak solution in the three-dimensional case is still a big open problem. Therefore it is interesting and important to consider the regularity criterion of the weak solutions under some assumptions of certain growth conditions on the velocity or on the pressure.
On one hand, as for the velocity regularity criteria, by means of the Littlewood-Paley decomposition methods, Dong and Chen [16] proved the regularity of weak solutions under the velocity condition: with Moreover, the result is further improved by Dong and Zhang [17] in the margin case:
On the other hand, as for the pressure regularity criteria, Yuan [18] investigated the regularity criterion of weak solutions of the micropolar fluid equations in Lebesgue spaces and Lorentz spaces: where is the Lorents space (see the definitions in the next section).
Recently, Dong et al. [19] improved the pressure regularity of the micropolar fluid equations in Morrey spaces: where Furthermore, Jia et al. [20] refined the regularity from Morrey spaces to Besov spaces: with One may also refer to some interesting results on the regularity criteria of Newtonian and non-Newtonian fluid equations (see [21–27] and references therein).
The aim of the present study is to investigate the pressure regularity criterion of the three-dimensional micropolar fluid equations in the multiplier spaces which are larger than the Lebesgue spaces, Lorentz spaces, and Morrey spaces.
Throughout this paper, we use to denote the constants which may change from line to line. with denote the usual Lebesgue space and Sobolev space. denote the fractional Sobolev space with
Consider a measurable function and define for the Lebesgue measure of the set . The Lorentz space is defined by if and only if
We defined , the homogeneous Morrey space associated with norm
We now recall the definition and some properties of the multiplier space .
Definition 2.1 (see Lemarié-Rieusset [28]). For , the space is defined as the space of such that
According the above definition of the multiplier space, it is not difficult to verify the homogeneity properties. For all
When , it is clear that (see Lemarié-Rieusset [28]) where denotes the homogenous space of bounded mean oscillations associated with the norm
In particular, the following imbedding (see Lemarié-Rieusset [28]) holds true.
In order to state our main results, we recall the definition of the weak solution of micropolar flows (see, e.g., Łukaszewicz [9]).
Definition 2.2. Let , , and . is termed as a weak solution to the 3D micropolar flows (1.1) and (1.2) on , if satisfies the following properties:(i);(ii)equations (1.1) and (1.2) are valid in the sense of distributions.
Our main results are now read as follows.
Theorem 2.3. Suppose , , and in the sense of distributions. Assume that is a weak solution of the 3D micropolar fluid flows (1.1) and (1.2) on . If the pressure satisfies the logarithmically growth condition: then the weak solution is regular on .
Thanks to it is easy to deduce the following pressure regularity criterion of the three-dimensional micropolar equations (1.1) and (1.2).
Corollary 2.4. On the substitution of the pressure condition (2.10) by the following conditions: the conclusion of Theorem 2.3 holds true.
Remark 2.5. According to the embedding relation (2.9), our results obviously largely improve the previous results (1.7) and (1.8). Moreover, it seems incomparable with the Besov space (1.10).
Remark 2.6. Furthermore, since we have no additional growth condition on the microrotation vector field , Theorem 2.3 is also valid for the pressure regularity problem of the three-dimensional Navier-Stokes equations (see, e.g., Zhou [29, 30]).
In order to prove our main results, we first recall the following local existence theorem of the three-dimensional micropolar fluid equations (1.1) and (1.2).
Lemma 3.1 (see Dong et al. [19]). Assume and with in the sense of distributions. Then there exist a constant and a unique strong solution of the 3D micropolar fluid equations (1.1) and (1.2) such that
By means of the local existence result, (1.1) and (1.2) with admit a unique -strong solution on a maximal time interval. For the notation simplicity, we may suppose that the maximal time interval is . Thus, to prove Theorem 2.3, it remains to show that This will lead to a contradiction to the estimates to be derived below. We now begin to follow these arguments.
Taking the inner product of the second equation of (1.1) with and the third equation of (1.1) with , respectively, and integrating by parts, it follows that where we have used the following identities due to the divergence free property of the velocity field :
Furthermore, applying Young inequality, Hölder inequality, and integration by parts, we have
Combining the above inequalities, it follows that
In order to estimate the last term of the right-hand side of (3.6), taking the divergence operator to the first equation of (1.1) produces the expression of the pressure: Employing Calderón-Zygmund inequality and the divergence free condition of the velocity derives the estimate of the pressure:
Therefore, we estimate the pressure term as Now we estimate the integral on the right-hand side of (3.9). By the Hölder inequality and the Young inequality we have where we have used the following interpolation inequality:
Hence, combining the above inequalities, we derive
Furthermore, we have the second term of the right-hand side of (3.13) rewritten as Inserting (3.14) into (3.13) and applying the Gronwall inequality, one shows that which implies
Hence we complete the proof of Theorem 2.3.
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B. Q. Dong and Y. Li, “Large time behavior to the system of incompressible non-Newtonian fluids in
{ℝ}^{2}
C. Zhao and Y. Li, “
{H}^{2}
-compact attractor for a non-Newtonian system in two-dimensional unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 7, pp. 1091–1103, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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Y. Zhou, “On regularity criteria in terms of pressure for the Navier-Stokes equations in
{ℝ}^{3}
Y. Zhou, “On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in
{ℝ}^{n}
,” Zeitschrift für Angewandte Mathematik und Physik, vol. 57, no. 3, pp. 384–392, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright © 2012 Fengjun Guo. 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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Operating Earnings Definition
Understanding Operating Earnings
Example of Operating Earnings
Operating earnings is a corporate finance and accounting term that isolates the profits realized from a business's core operations. Specifically, it refers to the amount of profit realized from revenues after you subtract those expenses that are directly associated with running the business, such as the cost of goods sold (COGS), general and administration (G&A) expenses, selling and marketing, research and development, depreciation, and other operating costs.
Operating earnings are an important measure of corporate profitability. Because the metric excludes non-operating expenses, such as interest payments and taxes, it enables an assessment of how well the company's chief lines of business are doing.
Operating earnings is a measure of the amount of profit realized from a business's core operations.
Operating earnings is a useful figure since it doesn't include taxes and other one-off items that might skew net income in a specific accounting period.
A commonly used variant of operating earnings is the operating margin, a percentage figure that represents operating earnings divided by total revenue.
Operating earnings lie at the heart of both internal and external analysis of how a company is making money, as well as how much money it's making. The individual components of operating costs can be measured relative to total operating costs or total revenues to assist management in running a company.
Operating earnings are usually found within a company's financial statements —specifically, towards the end of the income statement. Though it gets close to the nitty-gritty, operating earnings aren't quite the famed "bottom line" that truly signals how well—or how poorly–a firm is faring. That status belongs to a company's net income, "net" indicating what remains after deducting taxes, debt repayments, interest charges, and all the other non-operating debits a business has encountered.
Operating earnings is a term that can be used interchangeably with operating income, operating profit, and earnings before interest and taxes (EBIT).
Operating Earnings vs. Operating Margin
Many variants of metrics stemming from operating earnings can also be used to compare a given company's profitability with those of its industry peers. One of the most important of these metrics is the operating margin, which is closely tracked by management and investors from one quarter to the next for an indication of the trend in profitability.
Expressed as a percentage, operating margin is calculated by dividing operating earnings by total revenues. Or, as a formula:
\begin{aligned} \text{Operating Margin}=\frac{\text{Operating Earnings}}{\text{Revenue}} \end{aligned}
Operating Margin=RevenueOperating Earnings
Management uses this measure of earnings to gauge the profitability of various business decisions over time. External lenders and investors also pay close attention to a company's operating margin because it shows the proportion of revenues that are left over to cover non-operating costs, such as paying interest on debt obligations.
Highly variable operating margins are a prime indicator of business risk. By the same token, looking at a company's past operating margins and trends over time is a good way to gauge whether a big increase in earnings is likely to last.
Assume Gadget Co. had $10 million in revenues in a given quarter, $5 million in operating expenses, $1 million in interest expense, and $2 million in taxes. Gadget Co.'s operating earnings would be $5 million ($10 million in revenue – $5 million in operating expenses). Its operating margin is 50% ($5 million in operating earnings/$10 million in revenue).
Net income would then be derived by subtracting interest expenses and taxes and then netting out any one-time or unusual gains and losses from the operating earnings. Gadget Co.'s net income is, therefore, $2 million.
Sometimes a company presents a non-GAAP "adjusted" operating earnings figure to account for one-off costs that management believes are not part of recurring operating expenses.
Non-GAAP earnings are an alternative accounting method that varies from the Generally Accepted Accounting Principles (GAAP) that U.S. firms are required to use on financial statements.
Many companies report non-GAAP earnings in addition to their earnings based on GAAP.
A prime example is expenses stemming from restructuring (a type of corporate action taken that involves significantly modifying the debt, operations, or organization of a company as a way of limiting financial harm and improving the business.) Management may add back these costs to present higher operating earnings on an adjusted basis. However, critics could point out that restructuring costs should not be classified as one-offs if they occur with some regularity.
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Axial Compressive Loading Capacity of Pressurized Energy Pipeline With Corrosion Defects | IPC | ASME Digital Collection
Dinaer Bolati,
Dinaer Bolati
National Engineering Laboratory for Pipeline Safety
Hang An,
Jinxu Jiang
Liu, B, Tan, X, Bolati, D, An, H, & Jiang, J. "Axial Compressive Loading Capacity of Pressurized Energy Pipeline With Corrosion Defects." Proceedings of the 2020 13th International Pipeline Conference. Volume 1: Pipeline and Facilities Integrity. Virtual, Online. September 28–30, 2020. V001T03A027. ASME. https://doi.org/10.1115/IPC2020-9616
Corrosion defects are dreadfully damaging to the stability of pipelines. Using the finite element (FE) simulation method, a model of API 5L X65 steel pipeline is established in this work to study its buckling behavior subjected to axial compressive loading. The local buckling state of the pipe at the ultimate axial compressive capacity was captured. Compared with the global compressive strain capacity (CSCglobal), the local compressive strain capacity (CSClocal) is more conservative. Extensive parametric analysis, including approximately 115 FE cases, was conducted to study the influence of the corrosion defect sizes and internal pressure on the corroded pipe’s compressive loading capacity (CLC) and CSC. Results show that the enlarged size of the corrosion defect decreases both the CLC and the CSC of the pipeline, but the CLC almost keeps unchanged as the length of corrosion defects increases. The CLC decreases with the increase of the length of corrosion defects when the length is less than 1.5
Dt
and greater than 0.7
Dt
. The CSC drops significantly until the length of the corrosion defect reached 1.8
Dt
. The deeper the corrosion defect, the smaller the CLC and the CSC. An increase in the width of corrosion defects tends to correspond to a decrease in the CLC and the CSC. With the increase of internal pressure, the CSC of the pipe gets greater while the CLC gets smaller. Based on the 115 FE results, a machine learning model based on support vector regression theory was developed to predict the pipe’s CSC. The regression coefficient between SVR predicted value and FEM actual value is 98.87%, which proves that the SVR model can predict the CSC with high accuracy and efficiency.
X65 pipeline, corrosion defect, buckling, compressive loading capacity, compressive strain capacity, supported vector regression
Buckling, Corrosion, Pipelines
Pipeline Walking Interaction With Buckle Formations Along Routes With Significant Seabed Features
Asymptotic Analysis for Buckling of Undersea Corroded Pipelines
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Create Stream Particle Animations - MATLAB & Simulink
Projectile Path Over Time
What Particle Animations Can Show
1. Specify Starting Points of the Data Range
2. Create Stream Lines to Indicate Particle Paths
4. Calculate the Stream Particle Vertices
This example shows how to display the path of a projectile as a function of time using a three-dimensional quiver plot.
Show the path of the following projectile using constants for velocity and acceleration, vz and a. Calculate z as the height as time varies from 0 to 1.
z\left(t\right)={v}_{z}t+\frac{a{t}^{2}}{2}
vz = 10; % velocity constant
a = -32; % acceleration constant
z = vz*t + 1/2*a*t.^2;
Calculate the position in the x-direction and y-direction.
x = vx*t;
y = vy*t;
Compute the components of the velocity vectors and display the vectors using a 3-D quiver plot. Change the viewpoint of the axes to [70,18].
u = gradient(x);
v = gradient(y);
w = gradient(z);
quiver3(x,y,z,u,v,w,scale)
A stream particle animation is useful for visualizing the flow direction and speed of a vector field. The “particles” (represented by any of the line markers) trace the flow along a particular stream line. The speed of each particle in the animation is proportional to the magnitude of the vector field at any given point along the stream line.
This example determines the region of the volume to plot by specifying the appropriate starting points. In this case, the stream plots begin at x = 100 and y spans 20 to 50 in the z = 5 plane, which is not the full volume bounds.
[sx sy sz] = meshgrid(100,20:2:50,5);
This example uses stream lines (stream3, streamline) to trace the path of the animated particles, which adds a visual context for the animation.
sl = streamline(verts);
While all the stream lines start in the z = 5 plane, the values of some spiral down to lower values. The following settings provide a clear view of the animation:
The viewpoint (view) selected shows both the plane containing most stream lines and the spiral.
Selecting a data aspect ratio (daspect) of [2 2 0.125] provides greater resolution in the z-direction to make the stream particles more easily visible in the spiral.
Set the axes limits to match the data limits (axis) and draw the axis box (box).
daspect([2 2 0.125])
Determine the vertices along the stream line where a particle will be drawn. The interpstreamspeed function returns this data based on the stream line vertices and the speed of the vector data. This example scales the velocities by 0.05 to increase the number of interpolated vertices.
Set the axes SortMethod property to childorder so the animation runs faster.
The streamparticles function sets the following properties:
Animate to 10 to run the animation 10 times.
ParticleAlignment to on to start all particle traces together.
MarkerEdgeColor to none to draw only the face of the circular marker. Animations usually run faster when marker edges are not drawn.
MarkerFaceColor to red.
Marker to o, which draws a circular marker. You can use other line markers as well.
iverts = interpstreamspeed(x,y,z,u,v,w,verts,0.01);
set(gca,'SortMethod','childorder');
streamparticles(iverts,15,...
'Animate',10,...
'ParticleAlignment','on',...
'Marker','o');
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Lévy C curve - Wikipedia
"C curve" redirects here. For the weighting curve, see C-weighting.
In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Paul Lévy, who was the first to describe its self-similarity properties as well as to provide a geometrical construction showing it as a representative curve in the same class as the Koch curve. It is a special case of a period-doubling curve, a de Rham curve.
1 L-system construction
2 IFS construction
3 Sample Implementation of Levy C Curve
L-system construction[edit]
First eight stages in the construction of a Lévy C curve
Lévy C curve (from a L-system, after the first 12 stages)
If using a Lindenmayer system then the construction of the C curve starts with a straight line. An isosceles triangle with angles of 45°, 90° and 45° is built using this line as its hypotenuse. The original line is then replaced by the other two sides of this triangle.
At the second stage, the two new lines each form the base for another right-angled isosceles triangle, and are replaced by the other two sides of their respective triangle. So, after two stages, the curve takes the appearance of three sides of a rectangle with the same length as the original line, but only half as wide.
At each subsequent stage, each straight line segment in the curve is replaced by the other two sides of a right-angled isosceles triangle built on it. After n stages the curve consists of 2n line segments, each of which is smaller than the original line by a factor of 2n/2.
This L-system can be described as follows:
Variables: F
Constants: + −
Start: F
Rules: F → +F−−F+
where "F" means "draw forward", "+" means "turn clockwise 45°", and "−" means "turn anticlockwise 45°".
The fractal curve that is the limit of this "infinite" process is the Lévy C curve. It takes its name from its resemblance to a highly ornamented version of the letter "C". The curve resembles the finer details of the Pythagoras tree.
The Hausdorff dimension of the C curve equals 2 (it contains open sets), whereas the boundary has dimension about 1.9340 [1].
The standard C curve is built using 45° isosceles triangles. Variations of the C curve can be constructed by using isosceles triangles with angles other than 45°. As long as the angle is less than 60°, the new lines introduced at each stage are each shorter than the lines that they replace, so the construction process tends towards a limit curve. Angles less than 45° produce a fractal that is less tightly "curled".
IFS construction[edit]
Lévy C curve (from IFS, infinite levels)
If using an iterated function system (IFS, or the chaos game IFS-method actually), then the construction of the C curve is a bit easier. It will need a set of two "rules" which are: Two points in a plane (the translators), each associated with a scale factor of 1/√2. The first rule is a rotation of 45° and the second −45°. This set will iterate a point [x, y] from randomly choosing any of the two rules and use the parameters associated with the rule to scale/rotate and translate the point using a 2D-transform function.
Put into formulae:
{\displaystyle f_{1}(z)={\frac {(1-i)z}{2}}}
{\displaystyle f_{2}(z)=1+{\frac {(1+i)(z-1)}{2}}}
from the initial set of points
{\displaystyle S_{0}=\{0,1\}}
Sample Implementation of Levy C Curve[edit]
// Java Sample Implementation of Levy C Curve
public class C_curve extends JPanel {
public float x, y, len, alpha_angle;
public int iteration_n;
c_curve(x, y, len, alpha_angle, iteration_n, g2d);
public void c_curve(double x, double y, double len, double alpha_angle, int iteration_n, Graphics2D g) {
double fx = x;
double fy = y;
double length = len;
double alpha = alpha_angle;
int it_n = iteration_n;
if (it_n > 0) {
length = (length / Math.sqrt(2));
c_curve(fx, fy, length, (alpha + 45), (it_n - 1), g); // Recursive Call
fx = (fx + (length * Math.cos(Math.toRadians(alpha + 45))));
fy = (fy + (length * Math.sin(Math.toRadians(alpha + 45))));
c_curve(fx, fy, length, (alpha - 45), (it_n - 1), g); // Recursive Call
Color[] A = {Color.RED, Color.ORANGE, Color.BLUE, Color.DARK_GRAY};
g.setColor(A[ThreadLocalRandom.current().nextInt(0, A.length)]); //For Choosing Different Color Values
g.drawLine((int) fx, (int) fy, (int) (fx + (length * Math.cos(Math.toRadians(alpha)))), (int) (fy + (length * Math.sin(Math.toRadians(alpha)))));
C_curve points = new C_curve();
points.x = 200; // Stating x value
points.y = 100; // Stating y value
points.len = 150; // Stating length value
points.alpha_angle = 90; // Stating angle value
points.iteration_n = 15; // Stating iteration value
Wikimedia Commons has media related to Lévy C curve.
Paul Lévy, Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole (1938), reprinted in Classics on Fractals Gerald A. Edgar ed. (1993) Addison-Wesley Publishing ISBN 0-201-58701-7.
E. Cesaro, Fonctions continues sans dérivée, Archiv der Math. und Phys. 10 (1906) pp 57–63.
G. Faber, Über stetige Funktionen II, Math Annalen, 69 (1910) pp 372–443.
S. Bailey, T. Kim, R. S. Strichartz, Inside the Lévy dragon, American Mathematical Monthly 109(8) (2002) pp 689–703
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lévy_C_curve&oldid=1027602161"
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Harold sorted his jellybeans into two jars. He likes the purple ones best and the black ones next best, so these are both in one jar. His next favorites are yellow, orange, and white, and these are in another jar. He gave all the rest to his little sister. Harold allows himself to eat only one jellybean from each jar per day. He wears a blindfold when he selects his jellybeans so he cannot choose his favorites first.
Make an area model that represents the probabilities. What is the probability that Harold gets one black jellybean and one orange jellybean if the first jar has
60\%
black and
40\%
purple jellybeans, and the second jar has
30\%
50\%
orange, and
20\%
white jellybeans? .
Use the eTool below to Make an area model.
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Programmatic Fitting - MATLAB & Simulink - MathWorks Italia
MATLAB Functions for Polynomial Models
Linear Model with Nonpolynomial Terms
Fit a Polynomial to the Data
Plot and Calculate Confidence Bounds
Two MATLAB® functions can model your data with a polynomial.
Polynomial Fit Functions
polyfit(x,y,n) finds the coefficients of a polynomial p(x) of degree n that fits the y data by minimizing the sum of the squares of the deviations of the data from the model (least-squares fit).
polyval(p,x) returns the value of a polynomial of degree n that was determined by polyfit, evaluated at x.
If you are trying to model a physical situation, it is always important to consider whether a model of a specific order is meaningful in your situation.
This example shows how to fit data with a linear model containing nonpolynomial terms.
When a polynomial function does not produce a satisfactory model of your data, you can try using a linear model with nonpolynomial terms. For example, consider the following function that is linear in the parameters
{a}_{0}
{a}_{1}
{a}_{2}
, but nonlinear in the
t
y={a}_{0}+{a}_{1}{e}^{-t}+{a}_{2}t{e}^{-t}.
You can compute the unknown coefficients
{a}_{0}
{a}_{1}
{a}_{2}
by constructing and solving a set of simultaneous equations and solving for the parameters. The following syntax accomplishes this by forming a design matrix, where each column represents a variable used to predict the response (a term in the model) and each row corresponds to one observation of those variables.
Enter t and y as column vectors.
Form the design matrix.
Calculate model coefficients.
Therefore, the model of the data is given by
y=1.3983-0.8860{e}^{-t}+0.3085t{e}^{-t}.
Now evaluate the model at regularly spaced points and plot the model with the original data.
This example shows how to use multiple regression to model data that is a function of more than one predictor variable.
When y is a function of more than one predictor variable, the matrix equations that express the relationships among the variables must be expanded to accommodate the additional data. This is called multiple regression.
Measure a quantity
{x}_{1}
{x}_{2}
. Store these values in vectors x1, x2, and y, respectively.
A model of this data is of the form
y={a}_{0}+{a}_{1}{x}_{1}+{a}_{2}{x}_{2}.
Multiple regression solves for unknown coefficients
{a}_{0}
{a}_{1}
{a}_{2}
by minimizing the sum of the squares of the deviations of the data from the model (least-squares fit).
Construct and solve the set of simultaneous equations by forming a design matrix, X.
Solve for the parameters by using the backslash operator.
The least-squares fit model of the data is
y=0.1018+0.4844{x}_{1}-0.2847{x}_{2}.
To validate the model, find the maximum of the absolute value of the deviation of the data from the model.
This value is much smaller than any of the data values, indicating that this model accurately follows the data.
This example shows how to use MATLAB functions to:
Load sample census data from census.mat, which contains U.S. population data from the years 1790 to 1990.
This adds the following two variables to the MATLAB workspace.
cdate is a column vector containing the years 1790 to 1990 in increments of 10.
pop is a column vector with the U.S. population numbers corresponding to each year in cdate.
The plot shows a strong pattern, which indicates a high correlation between the variables.
In this portion of the example, you determine the statistical correlation between the variables cdate and pop to justify modeling the data. For more information about correlation coefficients, see Linear Correlation.
Calculate the correlation-coefficient matrix.
The diagonal matrix elements represent the perfect correlation of each variable with itself and are equal to 1. The off-diagonal elements are very close to 1, indicating that there is a strong statistical correlation between the variables cdate and pop.
This portion of the example applies the polyfit and polyval MATLAB functions to model the data.
Calculate fit parameters.
Evaluate the fit.
Plot the data and the fit.
The plot shows that the quadratic-polynomial fit provides a good approximation to the data.
Calculate the residuals for this fit.
Notice that the plot of the residuals exhibits a pattern, which indicates that a second-degree polynomial might not be appropriate for modeling this data.
Confidence bounds are confidence intervals for a predicted response. The width of the interval indicates the degree of certainty of the fit.
This portion of the example applies polyfit and polyval to the census sample data to produce confidence bounds for a second-order polynomial model.
The following code uses an interval of
±2\Delta
, which corresponds to a 95% confidence interval for large samples.
Evaluate the fit and the prediction error estimate (delta).
Plot the data, the fit, and the confidence bounds.
The 95% interval indicates that you have a 95% chance that a new observation will fall within the bounds.
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draw a rectangle ABCD, bc=5 2cm and angle dbc = 30 degree - Maths - Practical Geometry - 9999591 | Meritnation.com
draw a rectangle ABCD, bc=5.2cm and angle dbc = 30 degree
\angle
DBC = 30
°
\angle
°
as we know angle between same base and diagonals are same .
We follow these steps :
Step 1 : Draw a line BC = 5.2 cm
Step 2 : Take any radius ( Less than half of BC ) and center " B " draw a semicircle that intersect our line BC at " P " . Now with same radius and center " P " draw an arc that intersect our semicircle at " Q " . With same radius and center " Q " draw an arc that intersect our semicircle at " R " . Now with same radius and center " Q " and " R " draw an arcs these arcs intersect at " S " .
Step 3 : Join BS , we get
\angle
CBS = 90
°
Step 4 : Take any radius ( Less than half of BC ) and center " C " draw a semicircle that intersect our line BC at " E " . Now with same radius and center " E " draw an arc that intersect our semicircle at " F " . With same radius and center " F " draw an arc that intersect our semicircle at " G " . Now with same radius and center " F " and " G " draw an arcs these arcs intersect at " H " .
Step 5 : Join BJ , we get
\angle
ABH = 90
°
Step 6 : With same radius ( Used in step 2 ) and center " P " and " Q " draw two arcs and these arcs intersect at " I " . Join BI and extend it so that line intersect line CH at " D " .
Step 7 : With same radius ( Used in step 4 ) and center " E " and " F " draw two arcs and these arcs intersect at " J " . Join CJ and extend it so that line intersect line BS at " A " .
Step 8 : Join Ad and we get our required rectangle ABCD , As :
Hope this information will clear your doubts about Practical Geometry.
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A Facile and Generic Strategy to Synthesize Large-Scale Carbon Nanotubes
Yong Hu, Ting Mei, Libo Wang, Haisheng Qian, "A Facile and Generic Strategy to Synthesize Large-Scale Carbon Nanotubes", Journal of Nanomaterials, vol. 2010, Article ID 415940, 5 pages, 2010. https://doi.org/10.1155/2010/415940
Yong Hu,1,2 Ting Mei,2 Libo Wang,2 and Haisheng Qian1
1Zhejiang Key Laboratory for Reactive Chemistry on Solid Surfaces and Institute of Physical Chemistry, Zhejiang Normal University, Jinhua 321004, China
An easy method to prepare carbon nanotubes (CNTs) has been demonstrated using a two-step refluxing and calcination process. First, a readily available inorganic salt, N i ( N O 3 ) 2 ⋅ 6 H 2 O , used as the catalyst precursor was dissolved in the high-boiling-point organic solvents (alcohols or polyhydric alcohol) by refluxing at 1 9 0 ∘ C for 3 hours. After refluxing, NiO nanoparticles obtained in the solution act as the catalyst, and the organic refluxing solvents are used as the carbon source for the growth of CNTs. Second, CNTs are prepared by calcining the refluxed solution at 8 0 0 ∘ C in an N 2 atmosphere for 3 hours. Results show that CNT growth possibly originates from carbon rings, with the nanotube walls growing perpendicular to these rings and forming a closed tube at the end.
Since their emergence, carbon nanotubes (CNTs) [1] have demonstrated their versatility in a variety of applications such as supportedcatalysts for hydrogenation reactions [2], fuel cells [3], field emission devices [4], and nanoelectronic devices [5], due to their unique mechanical, chemical, and electrochemical properties [6, 7]. Consequently, demand for the consumption of CNT materials is huge, making desirable their fast, efficient, high yield and cost effective fabrication.
Previously, a number of techniques have been developed for fabricating CNTs, including electric arc-discharge [8], laser ablation [9], chemical vapor deposition (CVD) [10, 11], pyrolysis [12], plasma enhanced CVD (PECVD) [13], laser assisted CVD (LACVD) [14], two-stage fluid bed reactor [15], aerosol method [16], solvothermal method [17, 18], and high pressure CO disproportionation process (HiPCO) [19]. Recently, Tang et al. reported a novel catalytic combustion method of synthesizing CNTs in situ in high yield from polypropylene as a carbon source in the presence of organic-modified clay and a supported Ni catalyst [20]. Most of the methods originate from the idea of obtaining adequate active atomic carbon species or clusters from carbon sources and assembling them into CNTs on catalysts. Although great efforts have been made in the development of synthesis methods, their numerous steps and the adoption of expensive or nonrenewable materials still make the methods complicated. Thus we look for a simple, low-cost, and effective method for the synthesis of CNTs.
Herein, we report a simple and low-cost strategy of CNT synthesis that adopts a refluxing process with readily available inorganic salts Ni(NO3)2·6H2O as the catalyst precursor, and high-boiling-point (BP) di(ethylene glycol) (DEG), glycerol and ethylene glycol (EG), as solvents and carbon source. After refluxing, NiO nanoparticles obtained in the solution act as the catalyst for the growth of CNTs. The final CNT products are successfully prepared by calcining the refluxed solution at 800°C in an N2 atmosphere for 3 hours. The whole process is kept simple and easy to control because there are not too many parameters required for monitoring.
2.1. Synthesis of NiO Nanoparticles and CNTs
Chemical reagents purchased from Sigma-Aldrich were used in the experiments with no further purification. For a typical synthesis procedure, 0.0025 mol (0.73 g) of the metal salt Ni(NO3)2·6H2O was introduced and dissolved in 50 mL of DEG, glycerol, and EG solvent, in three round bottom flasks. The solutions were heated in an oil bath to form a clear solution under reflux with vigorous stirring. When the reflux temperature was increased to 190°C, the clear solution then turned turbid. The reaction was maintained with stirring for aging for at least 3 hours at this fixed temperature. The flasks were then cooled down to room temperature after the heat source was removed. Subsequently, the three refluxed solutions were injected into quartz crucibles with lid. The final CNT products were obtained by calcination at 800°C in an N2 atmosphere for 3 hours with a heating rate of 1°C min−1. After calcination, the black precipitates were collected and washed with absolute ethanol, dilute HCl aqueous solution, and deionized water in sequence. Finally, the obtained samples were dried in vacuum at 60°C for 6 hours.
Morphological and structural examinations of the as-prepared products were performed using field-emission scanning electron microscopy (FE-SEM, JEOL JSM-6340F); transmission electron microscopy (TEM) and high resolution TEM (HRTEM) were conducted at 200 keV with a JEM-2100F field emission machine, after dispersing the sample in ethanol and depositing several drops of the suspension on holey-carbon films supported by copper grids. Energy-dispersive X-ray spectroscopy (EDS) was performed on a JEM-2100F TEM. The Raman spectrum was recorded at ambient temperature on a Witec Alpha 300 Raman spectrometer with an argon-ion laser at an excitation wavelength of 488 nm.
The FE-SEM images of CNTs synthesized by calcining DEG solution are shown in Figure 1(a) (lowmagnification) and Figure 1(b) (highmagnification). Figures 1(c) and 1(d) show the images of the products prepared by calcining glycerol and EG solution. These images demonstrate that the approach presented in this paper offers a large-scale production yield for CNTs. These CNT samples present a uniform tube diameter ( ∼ 15 nm) and a tube length ranging from hundreds of nanometers to several microns.
FE-SEM images of CNTs prepared by calcining DEG solution: (a) low magnification and (b) high magnification; FE-SEM images of CNTs prepared by calcining (c) glycerol solution and (d) EG solution.
The TEM images (Figure 2) reveal that the CNTs synthesized by this method are multiwall CNTs (MWCNTs) with strong graphitic structure. Figures 2(a), 2(b), and 2(c) show the characteristics of the MWCNTs with open tips as produced in all solutions. The nanotube sizes observed in these images are in good agreement with those observed using FE-SEM. A further investigation on the tubular structure of the HRTEM image shown in Figure 3(a) clearly reveals an open tip and a closed bottom, which validates that the growth of the nanotubes originated from carbon rings resulting in the formation of closed bottoms. Figures 3(b) and 3(c) show the closed bottom of nanotube and a section of nanotube wall, which are clearly graphitized. The interlayer spacing in the multiwalls is about 0.34 nm, corresponding to the lattice parameter of graphite carbon in the (002) plane [1]. EDS performed on a ring tip (Figure 3(d)) indicates that the tip only contains carbon; no other element was found. Figures 3(e) and 3(f) show the open tip of a carbon ring at low magnification and at high magnification.
TEM images of CNTs prepared by calcining (a) DEG solution; (b) glycerol solution; (c) EG solution.
HRTEM images of CNTs: (a) nanotubes prepared by calcining DEG solution; (b) the closed bottom of a nanotube; (c) a section of nanotube wall; (d) the EDS pattern of a carbon ring; (e) the open tip of a carbon ring at low magnification and (f) at high magnification.
In the refluxing process, NiO nanoparticles used as catalyst could be obtained via the pyrolysis of common metal salts (Ni(NO3)2·6H2O) in a high BP organic solvent which acts as a carbon source in the next step. In the calcination process, nucleations of multilayer carbon rings were being formed under the catalyst effect of the NiO nanoparticles, and nanotubes were assembled along the ring axial growth. Meanwhile, because they were observed products at 700°C (Figure 4), we believe that the great amount of ring-like products further favored the growth of nanotubes from the multi-layer carbon rings. In addition, for MWCNTs, it is quite likely that the presence of the outer wall stabilizes the inner wall, keeping it open for continuous growth [21]. A possible CNT growth mechanism was depicted in detail in Figure 5.
TEM image of CNTs prepared by calcination at 700°C.
Schematic of CNT growth: (a) the formation of the original multi-layer carbon rings; (b) the growth of multi-layer nanotube walls grow perpendicular to the carbon rings; (c) the formed nanotube with an open tip and a closed bottom.
The representative Raman spectrum (Figure 6) of the product calcined in refluxing DEG solution at 800°C shows the typical features of MWCNTs [1]. The spectrum exhibits two peaks at 1318 and 1570 cm-1, indicating the graphite structure of the nanotubes. According to the analysis of Kasuya et al. [22], the complex structure as characterized by the 1540–1600 cm-1 region can be understood by zone-folding of the graphite phonon dispersion relation. The IR spectrum demonstrates the peak frequencies of the graphite (G) mode at 1570 cm-1 and contains disorder modes at 1318 cm-1 (D).
Raman spectrum of CNTs.
We have successfully synthesized CNTs of uniform diameter on a large scale through a novel refluxing and calcination solution process, in which reflux solvents were used as the carbon source and NiO nanoparticles obtained after refluxing were used as the catalyst. The products prepared in this approach were no toxic and corrosive reagents. This approach allows adoption of other metal salts and high BP organic solvents for further exploration. Due to merits such as its simplicity, low cost, high purity, good controllability, and high yield, we believe that this method can be exploited at the scale of industrial production.
This material is based on research supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Singapore National Research Foundation under CRP Award no. NRF-G-CRP 2007-01.
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Copyright © 2010 Yong Hu 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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Stimpmeter - Wikipedia
3 Sloped greens
It was designed in 1935 by golfer Edward S. Stimpson, Sr. (1904–1985).[1][2][3] The Massachusetts state amateur champion and former Harvard golf team captain, Stimpson was a spectator at the 1935 U.S. Open at Oakmont near Pittsburgh, where the winning score was 299 (+11). After witnessing a putt by a top professional (Gene Sarazen, a two-time champion) roll off a green, Stimpson was convinced the greens were unreasonably fast, but wondered how he could prove it. He developed a device, made of wood, now known as the Stimpmeter, which is an angled track that releases a ball at a known velocity so that the distance it rolls on a green's surface can be measured.[4]
In 1976, it was redesigned from aluminum by Frank Thomas of the United States Golf Association (USGA). It was first used by the USGA during the 1976 U.S. Open at Atlanta and made available to golf course superintendents in 1978. The 1976 version is painted green.
In January 2013, the USGA announced a third generation device based on work by Steven Quintavalla, a senior research engineer at the USGA labs.[5] A second hole in this version enables the option of a shorter run-out.[5] This version is painted blue, and is manufactured to a higher engineering tolerance to improve accuracy and precision.[5]
The 1976 device is an extruded aluminum bar, 36 inches (91 cm) long and 1.75 inches (4.4 cm) wide, with a 145° V-shaped groove extending along its entire length, supporting the ball at two points, 0.50 in (1.27 cm) apart. It is tapered at one end by removing metal from its underside to reduce the bounce of the ball as it rolls onto the green. It has a notch at a right angle to the length of the bar 30 inches (76 cm) from the lower tapered end where the ball is placed. The notch may be a hole completely through the bar or just a depression in it. The ball is pulled out of the notch by gravity when the device is slowly raised to an angle of about 20°, rolling onto the green at a repeatable velocity of 6.00 ft/s (1.83 m/s).[6] The distance travelled by the ball in feet is the 'speed' of the putting green. Six distances, three in each of two opposite directions, should be averaged on a flat section of the putting green. The three balls in each direction must be within 8 inches (20 cm) of each other for USGA validation of the test.[7]
Sloped greens[edit]
One problem is finding a near level surface as required in the USGA handbook. Many greens cannot be correctly measured as there may not be an area where the measured distance or green speed in opposing directions is less than a foot, particularly when they are very fast and thus require a very long level surface. A formula, based on the work of Isaac Newton, as derived and extensively tested by A. Douglas Brede, solves that problem. The formula is:
{\displaystyle {\frac {2\times S\uparrow \times \ S\downarrow }{S\uparrow +\ S\downarrow }}}
(where S↑ is speed up the slope and S↓ is speed down the slope on the same path). This eliminates the effect of the slope and provides a true green speed even on severely sloped greens.[8]
Slow 8 feet (2.4 m)
Medium 10 feet (3.0 m)
Fast 12 feet (3.7 m)
Slow 10 feet (3.0 m)
The greens at Oakmont Country Club (where the device was conceived) are some of the fastest in the world, with readings of 15 feet (4.6 m).[9]
^ a b c Frank Thomas (October 2001). "Equipment Extra: Eddie Stimpson's slant on putting". Golf Digest.
^ "Edward S. Stimpson". New York Times. UPI. March 28, 1985. Retrieved June 15, 2016.
^ Duca, Rob (June 6, 1998). "How fast is that green? Thanks to Ed Stimpson, we now know". Cape Cod Times. Hyannis, Massachusetts. Retrieved June 15, 2016.
^ Dvorchak, Robert (June 13, 2007). "Reading the greens". Pittsburgh Post-Gazette. p. E-6.
^ a b c John Paul Newport (January 26, 2013). "Ta-Da! Stimpmeter Makeover". The Wall Street Journal. p. A16.
^ Holmes, Brian W. (October 1986). "Dialogue concerning the Stimpmeter". The Physics Teacher. 24 (7): 401–404. Bibcode:1986PhTea..24..401H. doi:10.1119/1.2342065.
^ USGA Stimpmeter Instruction Booklet
^ A. Douglas Brede (November 1990). "Measuring green speed on sloped putting greens" (PDF).
^ "Oakmont: Rock & roll (& roll & roll & roll) nightmare". Pittsburgh Post-Gazette. 2007-06-10. Retrieved 2007-06-10.
A Better Stimpmeter And Calculator. CSG, Computer Support Group, Inc. and CSGNetwork.Com
How to build your own Stimpmeter
The Stimpmeter by the Rambling Man (with a picture)
"Up with the Stimpmeter by Stanley J. Zontek" (PDF). (566 KB)
"The Stimpmeter – A management tool by Patrick M. O'Brien" (PDF). (457 KB)
"Green speed physics by Authur P. Weber" (PDF). (901 KB)
"Utilizing the Stimpmeter for its intended use by Michael Morris" (PDF). (256 KB)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Stimpmeter&oldid=1086986459"
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Unit vector - Simple English Wikipedia, the free encyclopedia
A unit vector is any vector that is one unit in length. Unit vectors are often notated the same way as normal vectors, but with a mark over the letter (e.g.
{\displaystyle \mathbf {\hat {v}} }
being the unit vector of
{\displaystyle \mathbf {v} }
.)[1][2]
To make a vector into a unit vector, one just needs to divide it by its length:
{\displaystyle {\hat {\mathbf {v} }}=\mathbf {v} /\lVert \mathbf {v} \rVert }
.[3] The resulting unit vector will be in the same direction as the original vector.[4]
In component form[change | change source]
Three common unit vectors used in component form are
{\displaystyle \mathbf {\hat {i}} }
{\displaystyle \mathbf {\hat {j}} }
{\displaystyle \mathbf {\hat {k}} }
, referring to the three-dimensional unit vectors for the x-, y- and z-axes, respectively. They are commonly just notated as i, j and k.
They can be written as follows:
{\displaystyle \mathbf {\hat {i}} ={\begin{bmatrix}1&0&0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0&1&0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0&0&1\end{bmatrix}}}
For the unit vector corresponding to the
{\displaystyle i}
-th basis vector of a vector space, the symbol
{\displaystyle e_{i}}
{\displaystyle {\hat {e}}_{i}}
) may be used.[4]
↑ "Unit Vector". www.mathsisfun.com. Retrieved 2020-08-19.
↑ Weisstein, Eric W. "Unit Vector". mathworld.wolfram.com. Retrieved 2020-08-19.
↑ 4.0 4.1 "Unit Vectors | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-19.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Unit_vector&oldid=7074787"
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Intermembrane space - cellbio
The intermembrane space is a space between the two membranes of a mitochondrion: the outer mitochondrial membrane and inner mitochondrial membrane.
Type of organisms whose cells contain the intermembrane space Same as the organisms whose cells contain mitochondria: eukaryotic cells only, including plant cells, animal cells, and the cells of protists and fungi
Type of cells within the organisms that contain the intermembrane space Same as the cells that contain mitochondria: all cells except red blood cells in mammals (other vertebrates do have mitochondria in their red blood cells).
Number of intermembrane spaces per cell Same as the number of mitochondria: 1 to 1000s, depending on the energy needs of the cell
{\displaystyle ~200}
angstrom or
{\displaystyle ~20nm}
thickness (very approximate), accounting for less than 5% of the diameter (less than 10% even if you consider that it's on both sides).
Location within the mitochondrion It is right inside of the boundary of the mitochondrion (the boundary is the outer mitochondrial membrane).
What's on both sides of it Inside: inner mitochondrial membrane, outside: outer mitochondrial membrane
Structural components The intracristal space is the part of the intermembrane space between the folds (cristae) of the inner mitochondrial membrane. The peripheral space is the part of the intermembrane space farther out of the inner mitochondrial membrane.
pH About 7.0 to 7.4. Although still a little alkaline, it is less so than the mitochondrial matrix and less so than the rest of the cell, due to the pumping out of protons from the mitochondrial matrix as part of the electron transport chain.
Retrieved from "https://cellbio.subwiki.org/w/index.php?title=Intermembrane_space&oldid=229"
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Schwarz_lantern Knowpia
In mathematics, the Schwarz lantern is a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra.[1] It consists of a family of polyhedral approximations to a right circular cylinder that converge pointwise to the cylinder but whose areas do not converge to the area of the cylinder. It is also known as the Chinese lantern, because of its resemblance to a cylindrical paper lantern, or as Schwarz's boot. The "Schwarz lantern" and "Schwarz's boot" names are from mathematician Hermann Schwarz.
Schwarz boot on display in the German Museum of Technology Berlin.
The sum of the angles at each vertex is equal to two flat angles (
{\displaystyle 2\pi }
radians). This has as a consequence that the Schwarz lantern can be folded out of a flat piece of paper. The crease pattern for this folded surface, a tessellation of the paper by isosceles triangles, has also been called the Yoshimura pattern,[2] after the work of Y. Yoshimura on the Yoshimura buckling pattern of cylindrical surfaces under axial compression, which can be similar in shape to the Schwarz lantern.[3]
Animation of Schwarz lantern convergence (or lack thereof) for various relations between its two parameters
The discrete polyhedral approximation considered by Schwarz can be described by two parameters,
{\displaystyle m}
{\displaystyle n}
. The cylinder is sliced by parallel planes into
{\displaystyle 2n}
circles. Each of these circles contains
{\displaystyle 2m}
vertices of the Schwarz lantern, placed with equal spacing around the circle at (for unit circles) a circumferential distance of
{\displaystyle \pi /m}
from each other. Importantly, the vertices are placed so they shift in phase by
{\displaystyle \pi /2m}
with each slice.[4][5]
From these vertices, the Schwarz lantern is defined as a polyhedral surface formed from isosceles triangles. Each triangle has as its base two consecutive vertices along one of the circular slices, and as its apex a vertex from an adjacent cycle. These triangles meet edge-to-edge to form a polyhedral manifold, topologically equivalent to the cylinder that is being approximated.
As Schwarz showed, it is not sufficient to simply increase
{\displaystyle m}
{\displaystyle n}
if we wish for the surface area of the polyhedron to converge to the surface area of the curved surface. Depending on the relation of
{\displaystyle m}
{\displaystyle n}
the area of the lantern can converge to the area of the cylinder, to a limit arbitrarily larger than the area of the cylinder, to infinity or in other words to diverge. Thus, the Schwarz lantern demonstrates that simply connecting inscribed vertices is not enough to ensure surface area convergence.[4][5]
In the work of Archimedes it already appears that the length of a circle can be approximated by the length of regular polyhedra inscribed or circumscribed in the circle.[6][7] In general, for smooth or rectifiable curves their length can be defined as the supremum of the lengths of polygonal curves inscribed in them. The Schwarz lantern shows that surface area cannot be defined as the supremum of inscribed polyhedral surfaces.[8]
Schwarz devised his construction in the late 19th century as a counterexample to the erroneous definition in J. A. Serret's book Cours de calcul differentiel et integral,[9] which incorrectly states that:
Soit une portion de surface courbe terminee par un contour
{\displaystyle C}
; nous nommerons aire de cette surface la limite
{\displaystyle S}
vers laquelle tend l'aire d'une surface polyedrale inscrite formee de faces triangulaires et terminee par un contour polygonal
{\displaystyle \Gamma }
ayant pour limite le contour
{\displaystyle C}
Il faut demontrer que la limite
{\displaystyle S}
existe et qu'elle est independante de la loi suivant laquelle decroissent les faces de la surface polyedrale inscrite'.
Let a portion of curved surface be bounded by a contour
{\displaystyle C}
; we will define the area of this surface to be the limit
{\displaystyle S}
tended towards by the area of an inscribed polyhedral surface formed from triangular faces and bounded by a polygonal contour
{\displaystyle \Gamma }
whose limit is the contour
{\displaystyle C}
It must be shown that the limit
{\displaystyle S}
exists and that it is independent of the law according to which the faces of the inscribed polyhedral surface shrink.
Independently of Schwarz, Giuseppe Peano found the same counterexample. At the time, Peano was a student of Angelo Genocchi, who already knew about the difficulty on defining surface area from communication with Schwarz. Genocchi informed Charles Hermite, who had been using Serret's erroneous definition in his course. Hermite asked Schwarz for details, revised his course, and published the example in the second edition of his lecture notes (1883). The original note from Schwarz was not published until the second edition of his collected works in 1890.[10]
Limits of the areaEdit
A straight circular cylinder of radius
{\displaystyle r}
{\displaystyle h}
can be parametrized in Cartesian coordinates using the equations
{\displaystyle x=r\cos(u)}
{\displaystyle y=r\sin(u)}
{\displaystyle z=v}
{\displaystyle 0\leq u\leq 2\pi }
{\displaystyle 0\leq v\leq h}
. The Schwarz lantern is a polyhedron with
{\displaystyle 4mn}
triangular faces inscribed in the cylinder.
The vertices of the polyhedron correspond in the parametrization to the points
{\displaystyle u={\frac {2\mu \pi }{m}}}
{\displaystyle v={\frac {\nu h}{n}}}
and the points
{\displaystyle u={\frac {(2\mu +1)\pi }{m}}}
{\displaystyle v={\frac {(2\nu +1)h}{2n}}}
{\displaystyle \mu =0,1,2,\ldots ,m-1}
{\displaystyle \nu =0,1,2,\ldots ,n-1}
. All the faces are isosceles triangles congruent to each other. The base and the height of each of these triangles have lengths
{\displaystyle 2r\sin \left({\frac {\pi }{m}}\right){\text{ and }}{\sqrt {r^{2}\left[1-\cos \left({\frac {\pi }{m}}\right)\right]^{2}+\left({\frac {h}{2n}}\right)^{2}}}}
respectively. This gives a total surface area for the Schwarz lantern
{\displaystyle S(m,n)=4mnr\sin \left({\frac {\pi }{m}}\right){\sqrt {4r^{2}\sin ^{4}\left({\frac {\pi }{2m}}\right)+\left({\frac {h}{2n}}\right)^{2}}}}
Simplifying sines when
{\displaystyle m\to \infty }
{\displaystyle S(m,n)\simeq 4\pi nr{\sqrt {\left({\frac {\pi ^{2}r}{2m^{2}}}\right)^{2}+\left({\frac {h}{2n}}\right)^{2}}}=2\pi r{\sqrt {\left(\pi ^{2}r{\frac {n}{m^{2}}}\right)^{2}+h^{2}}}}
From this formula it follows that:
{\displaystyle n=am}
{\displaystyle a}
{\displaystyle S(m,am)\to 2\pi rh}
{\displaystyle m\to \infty }
. This limit is the surface area of the cylinder in which the Schwarz lantern is inscribed.
{\displaystyle n=am^{2}}
{\displaystyle a}
{\displaystyle S(m,am^{2})\to 2\pi r{\sqrt {\pi ^{4}r^{2}a^{2}+h^{2}}}}
{\displaystyle m\to \infty }
. This limit depends on the value of
{\displaystyle a}nd can be made equal to any number not smaller than the area of the cylinder
{\displaystyle 2r\pi h}
{\displaystyle n=am^{3}}
{\displaystyle S(m,am^{3})\to \infty }
{\displaystyle m\to \infty }
Runge's phenomenon, another example of failure of convergence
^ Zames, Frieda (September 1977). "Surface area and the cylinder area paradox". The Two-Year College Mathematics Journal. 8 (4): 207–211. doi:10.2307/3026930. JSTOR 3026930.
^ Miura, Koryo; Tachi, Tomohiro (2010). "Synthesis of rigid-foldable cylindrical polyhedra" (PDF). Symmetry: Art and Science, 8th Congress and Exhibition of ISIS. Gmünd.
^ Yoshimura, Yoshimaru (July 1955). On the mechanism of buckling of a circular cylindrical shell under axial compression. Technical Memorandum 1390. National Advisory Committee for Aeronautics.
^ a b Dubrovsky, Vladimir (March–April 1991). "In search of a definition of surface area" (PDF). Quantum. 1 (4): 6-9 and 64.
^ a b Berger, Marcel (1987). Geometry I. Universitext. Springer-Verlag, Berlin. pp. 263–264. doi:10.1007/978-3-540-93815-6. ISBN 978-3-540-11658-5. MR 2724360.
^ Traub, Gilbert (1984). The Development of the Mathematical Analysis of Curve Length from Archimedes to Lebesgue (Doctoral dissertation). New York University. p. 470. MR 2633321.
^ Brodie, Scott E. (1980). "Archimedes' axioms for arc-length and area". Mathematics Magazine. 53 (1): 36–39. doi:10.1080/0025570X.1980.11976824. JSTOR 2690029. MR 0560018.
^ Makarov, Boris; Podkorytov, Anatolii (2013). "Section 8.2.4". Real analysis: measures, integrals and applications. Universitext. Springer-Verlag, Berlin. pp. 415–416. doi:10.1007/978-1-4471-5122-7. ISBN 978-1-4471-5121-0. MR 3089088.
^ J. A. Serret, Cours de calcul differentiel et integral, Vol. II, page 296 of the first edition or page 298 of the second edition
^ Schwarz, H. A. (1890). "Sur une définition erronée de l'aire d'une surface courbe". Gesammelte Mathematische Abhandlungen von H. A. Schwarz (in French). Verlag von Julius Springer. pp. 309–311.
Bogomolny, Alexander. "The Schwarz Lantern Explained". Cut-the-knot.
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Correspondence to: #E-mail: ocs@kumoh.ac.kr, TEL: +82-54-478-7323
Elastic foundation stiffness, Fatigue life, Finite element analysis, Fractography, Main shaft, Mechanical press
탄성지반강성, 피로수명, 유한요소해석, 파면해석, 주축, 기계식 프레스
There are many high-capacity presses with compressive loads of more than 1 MN in the industrial manufacturing fields of powder metallurgy, metal forming, and metal shaping [1-4]. Mechanical presses are still preferred over hydraulic or electromagnetic presses, owing to their cost-effectiveness, and thus are widely used in industrial fields such as automotive component manufacturing [4,5]. A crankshaft or a cam and shaft is utilized to convert the rotational motion of a motor into the reciprocating motion of a ram [6,7]. In the case of an ultrahigh-capacity press, a cam and follower is advantageous in terms of manufacturing cost and ease of maintenance. A main shaft holding cams is vulnerable to failure, owing to a large number of stress concentration sources caused by carrying adjoining components as well as the simultaneous application of both flexural loads transmitted through the cams and torsional loads from the motor. Henceforth, main shafts should be designed to have an infinite life, but failures still occur in high-capacity mechanical presses.
Several previous studies used metallurgical analysis, such as fractography and material characterization, as well as finite element simulations to elucidate the causes of failure of various shafts for several machines, such as a linear motor compressor and a positive displacement motor [8-10]. Metallurgical analyses reveal traces of actual failure that are not controversial if performed correctly [11,12]. On the other hand, finite element analyses of shafts tend to produce different results depending on the modeling methods such as mesh size, especially with respect to the shaft and bearing support modeling. Therefore, it is necessary to improve the accuracy of analysis through comparison with real cases. Seifi and Abbasi [13] modified their initial finite element model by updating the friction coefficient at the contact surface of the interference shaft and bush joints through conducting suitable experiments. Guddad and Venkataram [14] conducted a tooth contact analysis of a gear pair, considering shaft and bearing flexibilities for the varying center distance between a gear and pinion and backlash to evaluate the stress in gears. Yang, et al. [15] suggested a dynamic model of a rotor-bearing-casing system with a nonlinear rolling element bearing. They considered nonlinear factors such as Hertzian contact, time-varying characteristics, clearance, and slippage in the bearing model. Recently, Engel and Al-Maeeni [16] proposed an integrated reverse engineering approach using nondestructive inspections, analytical calculations, and finite element analysis for recovering the shaft of a rotary draw bending machine.
A 3D model is shown in Figs. 1 to illustrate the power transmission mechanism 1(a) and the major components to be analyzed 1(b). The main shaft suffered significant stress concentrations owing to the three sets of keyways to secure the adjoining components of eccentric cams, gears, and an ejection cam. The main shaft including a set of two flanges holding four metal bushes was the structure to be analyzed in this study.
The necessary metallurgical and tensile test samples were taken from the region C, as indicated by the dashed rectangle in Fig. 2(b). The chemical compositions based on weight percentage are given in Table 1. The compositions of C, Cr, and Mo were slightly lower than the lower specification designated by ASTM A29M [18], but they met the local standard. The tensile tests were conducted thrice based on ASTM E8M [19]. The averaged 0.2% proof stress was 645, tensile strength was 823 MPa, elongation was 23.6%, and Rockwell hardness was 24.8. The chemical and mechanical properties satisfied all the necessary requirements. The microstructures were observed using samples collected at three locations, as shown in Fig. 3(a). The martensitic microstructure was observed near the surface Figs. 3(b) and the Bainitic microstructure was observed in the interior 3(c) and 3(d). No problematic defects or microstructures were found with respect to the metallurgy.
To study the effects of elastic foundation stiffness (EFS), the main shaft was simplified to a cylinder of diameter 200 and length 1,060 mm. In Fig. 2(a), the section of length 175 mm at the left end was not loaded at all, which was thus removed from the total length of 1,235 to obtain the shortened shaft of length 1,060 mm. ANSYS Workbench (Ver. 2021 R1) was used to conduct the finite element analysis. The cylinder was meshed with hexahedral elements, and the number of nodes and elements were 23,091 and 5,244, respectively. The boundary and load conditions are represented in Fig. 6(a). The four metal bushes (or thrust bearings) were modeled by elastic foundation (or elastic support) and the maximum load of 2 MN was applied, with 50% of the bearing load applied to each end of the shaft. Additionally, in order to prevent the rigid body motion of the cylinder, the displacement of the central point of the cylinder was fixed using the remote displacement. The magnitude and location of the maximum principal stress varied with the EFS values, as shown in Fig. 6(b). The maximum principal stress (σ1) was selected as a key parameter to predict failure, because the fracture surface exhibited brittle fracture, as shown in Fig. 5. As the value of EFS increases, the maximum stress point tended to move from the interior to the exterior of the shaft. In the actual main shaft analysis, the optimum EFS value should be found and used to simulate the actual failure locations.
Based on the loading cycle diagram and previous analyses, the main shaft was modeled as shown in Fig. 10(a) because it suffered the maximum bending load at the position shown in the figure. The maximum principal stresses of the three main portions (Left, Middle, Right) denoted in Figs. 10(b) were calculated by varying the EFS from 10 to 100 N/mm3, as shown in 11. The left and right stresses were not significantly affected by the EFS value, but the stress at the middle changed significantly. The EFS value was chosen as the optimal value of 60 N/mm3 to simulate the actual failure behavior and was held constant for all the analyses. The maximum principal stress values denoted in Fig. 10(b) were calculated with the ESF value of 60 N/mm3. The location of the maximum principal stress in the middle portion depends on the size of the mesh used, but when the mesh size was 1.7 mm or less, the maximum principal stress occurred near A, the root of the notch groove, not point in Fig. 5(b). Since the fatigue life was in the high-cycle fatigue regime, the conventional component stress-life (S–N) approach, or Basquin’s equation in Eq. (1), can predict fatigue life by adopting the universal slope of -0.085. The fatigue strength coefficient (
{S}_{f}^{\text{'}}
) was required to predict fatigue life; thus, it was calculated by Eq. (1).
\begin{array}{c}{S}_{a}={S}_{f}^{\text{'}}{\left({N}_{f}\right)}^{b}\\ 428 \text{MPa}={S}_{f}^{\text{'}}{\left(8,900,000\right)}^{-0.085}\\ {S}_{f}^{\text{'}}=1,668 \text{MPa}\end{array}
\begin{array}{c}{S}_{a}={S}_{f}^{\text{'}}{\left({N}_{f}\right)}^{b}\\ 392 \text{MPa}={1,668\left({N}_{f1}\right)}^{-0.085}\\ {N}_{f1}=25,057,684 \text{cycles}\end{array}
\begin{array}{c}{S}_{a}={S}_{f}^{\text{'}}{\left({N}_{f}\right)}^{b}\\ 352 \text{MPa}={1,668\left({N}_{f2}\right)}^{-0.085}\\ {N}_{f2}=88,892,133 \text{cycles}\end{array}
A systematic and efficient durability estimation method based on the failure analysis of a broken main shaft and a meaningful life prediction using finite element analysis were proposed. The broken main shaft showed a brittle fracture surface and was found to be damaged by low flexural stress. The main shaft plays a key role in a mechanical press, and it was found that the finite element analysis should be performed by considering displacement control, rather than force control. Thrust bearings (or bushes) used to support the main shaft could be modeled by elastic foundations, and the EFS value of 60 N/mm3 was obtained by comparing the finite element analysis results with actual failure behavior, especially with respect to failure locations and failure sequence. Two new designs were suggested to improve the durability of the main shaft. As in the first proposed model, the fatigue life of the main shaft can be effectively improved by relieving the stress concentration by increasing the fillet radius. Fatigue life can be significantly increased if the shaft assembly is redesigned so that the source of stress concentration is in the neutral plane or in the compression region when the maximum bending load is applied. In the future, it is necessary to verify the validity of the proposed method using miniaturized models.
{S}_{f}^{\text{'}}
: Fatigue Strength Coefficient
Akhtar, S., Saad, M., Misbah, M. R., Sati, M. C., (2018), Recent advancements in powder metallurgy: A review, Materials Today: Proceedings, 5(9), 18649-18655. [https://doi.org/10.1016/j.matpr.2018.06.210]
Cristofolini, I., Molinari, A., Zago, M., Amirabdollahian, S., Coube, O., Dougan, M., Larsson, M., Schneider, M., Valler, P., Voglhuber, J., (2019), Design for powder metallurgy: predicting anisotropic dimensional change on sintering of real parts, International Journal of Precision Engineering and Manufacturing, 20(4), 619-630. [https://doi.org/10.1007/s12541-019-00030-2]
Durand, C., Bigot, R., Baudouin, C., (2018), Contribution to characterization of metal forming machines: application to screw presses, Procedia Manufacturing, 15, 1024-1032. [https://doi.org/10.1016/j.promfg.2018.07.391]
Osakada, K., Mori, K., Altan, T., Groche, P., (2011), Mechanical servo press technology for metal forming, CIRP Annals, 60(2), 651-672. [https://doi.org/10.1016/j.cirp.2011.05.007]
Cheng, J., Zhou, Z., Feng, Y., Liu, Z., Zhang, Y., (2018), Thermo-Mechanical coupling analysis of the actuating mechanism in a high speed press, International Journal of Precision Engineering and Manufacturing, 19(5), 643-653. [https://doi.org/10.1007/s12541-018-0078-z]
Halicioglu, R., Dulger, L. C., Bozdana, A. T., (2016), Structural design and analysis of a servo crank press, Engineering Science and Technology, an International Journal, 19(4), 2060-2072. [https://doi.org/10.1016/j.jestch.2016.08.008]
Angelopoulos, V., (2015), A model-based design approach to redesign a crankshaft for powder metal manufacturing, M.Sc. Thesis, KTH Royal Institute of Technology.
Lanzutti, A., Andreatta, F., Raffaelli, A., Magnan, M., Zuliani, L., Fantoni, M., Fedrizzi, L., (2017), Failure analysis of a continuous press component in MDF production plant, Engineering Failure Analysis, 82, 493-500. [https://doi.org/10.1016/j.engfailanal.2017.03.016]
Khot, M., Gawali, B., (2015), Finite element analysis and optimization of flexure bearing for linear motor compressor, Physics Procedia, 67, 379-385. [https://doi.org/10.1016/j.phpro.2015.06.044]
Liu, Y., Lian, Z., Xia, C., Qian, L., Liu, S., (2019), Fracture failure analysis and research on drive shaft of positive displacement motor, Engineering Failure Analysis, 106, 104145. [https://doi.org/10.1016/j.engfailanal.2019.08.011]
George, F. V., (2004), Metallography and microstructures, ASM Handbook, 9.
ASM International, (1987), Fractograhy, ASM Handbook, 12.
Seifi, R., Abbasi, K., (2015), Friction coefficient estimation in shaft/bush interference using finite element model updating, Engineering Failure Analysis, 57, 310-322. [https://doi.org/10.1016/j.engfailanal.2015.08.006]
Guddad, R., Venkataram, N., (2017), Effect of shaft and bearing flexibilities on gear tooth contact analysis, Materials Today: Proceedings, 4(10), 10823-10829. [https://doi.org/10.1016/j.matpr.2017.08.034]
Yang, Y., Yang, W., Jiang, D., (2018), Simulation and experimental analysis of rolling element bearing fault in rotor-bearing-casing system, Engineering Failure Analysis, 92, 205-221. [https://doi.org/10.1016/j.engfailanal.2018.04.053]
Engel, B., Al-Maeeni, S. S. H., (2019), An integrated reverse engineering and failure analysis approach for recovery of mechanical shafts, Procedia CIRP, 81, 1083-1088. [https://doi.org/10.1016/j.procir.2019.03.257]
Imaoka, S., (2012), Elastic foundation stiffness, ANSYS. https://dokumen.tips/documents/efs-ansys-tutorial.html
ASTM A29M-20, (2020), Standard specification for general requirements for steel bars, carbon and alloy, hot-wrought.
ASTM E8M-21, (2021), Standard test methods for tension testing of metallic materials.
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Linear Regression is the process of fitting a line that best describes a set of data points.
Let's say you are trying to predict the Grade $g$ of students, based on how many hours $h$ they spend playing CSGO, and their IQ scores $i$. So you collected the data for a couple of students as follows:
Hours on CSGO (h)
IQ (i)
You then lay out this data as a system of equations such as: $$f(h,i) = h.\theta_1 + i.\theta_2=g$$ where $\theta_1$ and $\theta_2$ are what you are trying to learn to have a predictive model. So based on our data, now we have: $$2 \theta_1 + 85 \theta_2=80$$ and $$ 4 \theta_1 + 100 \theta_2=90$$ We can then easily calculate $\theta_1=-2.5$ and $\theta_2=1$.
So now we can plot $f(h,i)=-2.5h+i$
def grade(h, i):
return -2.5 * h + i
h = np.array([2, 4]) # hours on CSGO from 0 to 10
i = np.array([85, 100]) # IQ from 70 to 130
grades = grade(h, i)
ax.plot(h, i, grades)
ax.scatter([2, 4],[85,100], [80, 90], s=100, c='red') # plotting our sample points
ax.set_xlabel("Hours on CSGO (h)", fontsize=14)
ax.set_ylabel("IQ (i)", fontsize=14)
ax.set_zlabel("Grade (g)", fontsize=14)
plt.title(r"$f(h,i)$", fontsize=24)
What we did so far can be represented with matrix operations. We refer to features or predictors as capital $X$, because there are more than one dimensions usually (for example hours on CSGO is one dimension, and IQ is another). We refer to the target variable (in this case the grades of the students) as small $y$ because target variable is usually one dimension (in our example it is grade). So, in matrix format, that would be: $$X\theta=y$$ THIS EQUATION IS THE NUTSHELL OF SUPERVISED MACHINE LEARNING
Let's expand this matrix-format equation and generalize it.
Do we need to draw a line? using:
Intercept and slope.
We don't typically have just two points as our data have ton of points and not all of them are on the same line. We are just trying to approximate a line that captures the trend of the data.
Intercept: what y is when x is 0
Slope: how much does y change when x changes
Image(filename="slope-equation.png",width = 300, height = 100)
As we said earlier, we don't just have one predictor (small $x$), we have many predictors (aka features). In the previous example, we had two variables $x_1$ (hours spent on CSGO) and $x_2$ (the studet's IQ). But we can have more, many many more variables. In other words, $y$ is the linaer combination of all predictors $x_i$ $$y\approx f(x_1, x_2, x_3, ..., x_k) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + ... + \beta_n x_k$$ Where $\beta_0$ is the intercept, and the remaining $\beta$s are the $k$ coefficients of our linear regression model, one for each of the $k$ predictors (aka features).
When we have hundreds of thousands of points, there does not exist a line that can pass through them all. This is where we use line-fitting.
We start by setting the $\theta$ values randomly.
We use the current value of $\theta$ to get the predictions.
We calculate the error by taking the mean of all the squared differenes between the predictions and labels (also called mean squared error MSE) $$MSE=\frac{1}{n}\sum^n_{i=1}{(y_i-\hat{y_i})^2}$$ where $n$ is the number of data points, $y_i$ is one label, and $\hat{y_i}$ is the prediction for that label.
We use the error calculated to update $\theta$ and repeat from 2 to 3 until $\theta$ stops changing.
Data: Boston housing prices dataset
We will use Boston house prices data set. A typical dataset for regression models.
X, y= load_boston(return_X_y=True) # we want both features matrix X, and labels vector y
X.shape # the dataset has 506 houses with 13 features (or predictors) for a house price in boston
To use any predictive model in sklearn, we need exactly three steps:
Initialize the model by just calling its name.
Fitting (or training) the model to learn the parameters (In case of Linear Regression these parameters are the intercept and the $\beta$ coefficients.
Use the model for predictons!
# we pass in the features as well as the labels we want to map to (remember the CGSO and IQ = GPA example?)
# we can now use the model for predictions! We will just give the same predictors
predictions = lr.predict(X)
Well there are 13 features, meaning the data has 13 dimensions, so we can't visualize them as we did with the CSGO+IQ=GPA example.
But let's see the coefficients of the model, and the intercept too!
# here are the coefficients
Let us check the Linear Regression intercept.
# the intercept
The coefficients simultaneously reflect the importance of each feature in predicting the target (which is the house price in this case), but ONLY IF the features are all on the same scale. Say you can only spend 3 to 10 hours on CSGO daily, but IQ values of a student can range from 80 to 110 for example. Predicting the GPA as a linear combination of these two predictors has to give a relatively bigger coefficient to CSGO than IQ, for example, 0.5 for CSGO daily hours of 4 and 0.01 for IQ of 100 will give a nice GPA of 2.1. That's why we sometimes need to scale the features to have all of them range from 0 to 1. Stay tuned!
Linear Regression Loss Function
There are different ways of evaluating the errors. For example, if you predicted that a student's GPA is 3.0, but the student actual GPA is 1.0, the difference between the actual and predicted GPAs is $1.0 - 3.0 = -2.0$. However, there can't be a negative distance, can it be? So what can we do?
Well, you can either take the absolute difference, which is just $2.0$. Alternatively, you can take the squared difference , which is $2.0^2 = 4.0$. If you can't decide which one to use, you can add them together, it is not the end of the world, so it will be $1.0+4.0 = 5.0$. Well, each of these distance calculation techniques (aka distance metrics) result in a differently behaving linear regression model. To escape the ambiguity about the distance between the actual and the predicted value, we use the term residual, which refers to the error, regardless of how it is calculated. So let's put all residual calculation techinques in a table for you, with their formal names and formulas.
Absolute Lasso L1 |$d$|
Squared Ridge L2 $d^2$
Both Elastic Net EN |$d$| + $d^2$
The function we want to normalize when we are fitting a linear regression model is called the loss function, which is the sum of all the squared residuals on the training data, formally called Residual Sum of Squares (RSS): $$RSS = \sum_{i=1}^n{\bigg(y_i-\beta_0-\sum_{j=1}^k{\beta_jx_{ij}}\bigg)^2}$$ Notice the similarity between this equation and the MSE equation defined above. MSE is used to evaluate the performance of the model at the end, and it doesn't not depend on how $\hat{y_i}$ (i.e. the predicted value) is calculated. Whereas, RSS, uses the SS (Sum of Squares) to calculate the residual of all data points in training time.
What: Regularization is used to constraint (or regularize) the estimated coefficients towards 0. This protects the model from learning exceissively that can easily result overfit the training data. Even though we are aiming to fit a line, having a combination of many features can be quite complex, it is not exactly a line, it is the k-dimensional version of a line (e.g. k is 13 for our model on the Boston dataset)! Just to approximate the meaning on a visualizable number of dimensions...
Image(filename="regularization.png")
Regularization is used to prevent overfitting
too much regularization can result in underfitting.
We introduce this regularization to our loss function, the RSS, by simply adding all the (absolute, squared, or both) coefficients together. Yes, absolute, squared, or both, this is where we use Lasso, Ridge, or ElasticNet regressions respectively :)
So our new loss function(s) would be:
Lasso=RSS+λk∑j=1|βj| Ridge=RSS+λk∑j=1β2j ElasticNet=RSS+λk∑j=1(|βj|+β2j)
This \lambda
is a constant we use to assign the strength of our regularization. You see if \lambda=0
, we end up with good ol' linear regression with just RSS in the loss function. And if \lambda=\inf
the regularization term would dwarf RSS, which in turn, because we are trying to minimize the loss function, all coefficients are going to be zero, to counter attack this huge \lambda
., resuling in underfitting.
But hold on! We said if the features are not on the same scale, also coefficients are not going to be on the same scale, would that confuse the regularization. Yes it would :( So we need to normalize all the data to be on the same scale. The formula used to do this is for each feature $j$ for a data point $x_i$ from a total of $n$ data points:
$$\tilde{x_{ij}} = \frac{x_{ij}}{\sqrt{\frac{1}{2}\sum_{i=1}^{n}{(x_{ij}-\bar{x_j})^2}}}$$
Where $\bar{x_j}$ is the mean value for that feature over all data points.
If we can't visualize the data, how are we going to evaluate whether or not the model has overfitted or underfitted?
If it overfitted, that means it would get a very low residual error on the training set, but it might fail miserably on new data. So we split the data into training and testing splits.
Image(filename="model_complexity_error_training_test.jpg")
# we set aside 20% of the data for testing, and use the remaining 80% for training
Now we can see the performance of the model with different regularization strengths, and analyze the difference between each type of regularization.
from sklearn.linear_model import ElasticNet, Lasso, Ridge
from sklearn.metrics import mean_squared_error # we will use MSE for evaluation
def plot_errors(lambdas, train_errors, test_errors, title):
plt.plot(lambdas, train_errors, label="train")
plt.plot(lambdas, test_errors, label="test")
plt.xlabel("$\\lambda$", fontsize=14)
plt.ylabel("MSE", fontsize=14)
def evaluate_model(Model, lambdas):
training_errors = [] # we will store the error on the training set, for using each different lambda
testing_errors = [] # and the error on the testing set
# in sklearn, they refer to lambda as alpha, the name is different in different literature
# Model will be either Lasso, Ridge or ElasticNet
model = Model(alpha=l, max_iter=1000) # we allow max number of iterations until the model converges
training_predictions = model.predict(X_train)
training_mse = mean_squared_error(y_train, training_predictions)
training_errors.append(training_mse)
testing_predictions = model.predict(X_test)
testing_mse = mean_squared_error(y_test, testing_predictions)
testing_errors.append(testing_mse)
return training_errors, testing_errors
Lasso L1 Regularization
$$\text{Lasso} = RSS + \lambda \sum_{j=1}^k {|\beta_j|}$$
# let's generate different values for lambda from 0 (no-regularization) and (10 too much regularization)
lambdas = np.arange(0, 10, step=0.1)
lasso_train, lasso_test = evaluate_model(Lasso, lambdas)
plot_errors(lambdas, lasso_train, lasso_test, "Lasso")
sklearn is already warning us about using 0, the model is to complex it could not even converge to a solution! Just our of curiosty, what about negative $\lambda$? a sort of counter-regularization.
We notice increasing $\lambda$ adds too much regularization that the model starts adding error on both training and testing sets, which means it is underfitting. Using a very low $\lambda$ (e.g. 0.1) seems to gain the least testing error.
lambdas = np.arange(-10, 0.2, step=0.1)
Wow, the error jumped to 4000! Lasso increases the error monotonously with negative $\lambda$ values.
Ridge L2 Regularization
$$\text{Ridge} = RSS + \lambda \sum_{j=1}^k {\beta_j^2}$$
ridge_train, ridge_test = evaluate_model(Ridge, lambdas)
plot_errors(lambdas, ridge_train, ridge_test, "Ridge")
Ridge is noticeably smoother than Lasso, that goes to the fact that the square value introduces a larger error to minimize than just the absolute value, for example ($|-10| = 10$) but ($(-10)^2 = 100$).
Wow, the error jumped to 1400 then came back to erros similarly small with the positive $\lambda$s.
$$\text{ElasticNet} = RSS + \lambda \sum_{j=1}^k {(|\beta_j| + \beta_j^2)}$$
elastic_train, elastic_test = evaluate_model(ElasticNet, lambdas)
plot_errors(lambdas, elastic_train, elastic_test, "Elastic Net")
ElasticNet performance if remarkably comparable with Lasso.
Negative values of $\lambda$ break Elastic Net, so let's not do that.
Regularization Techniques Comparison
Lasso: will eliminate many features, and reduce overfitting in your linear model.
Ridge: will reduce the impact of features that are not important in predicting your y values.
Elastic Net: combines feature elimination from Lasso and feature coefficient reduction from the Ridge model to improve your model’s predictions.
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Some reflections on Cambridge Math Admission Test (STEP) | Matheart
I attempted the Oxford Math Admission Test (MAT) in Nov 4th, 2020 and got 82 (did not get interview invite unfortunately), now I am preparing for the Cambridge math admission test (STEP) just for extra challenge.
In this blog I would give some reflections on STEP and share my personal experience, the MAT blog would be posted later. Hope this blog would help you :)
STEP takes place in June every year, aiming at shortlisting the most talented candidates during the university application process. So unlike other university entrance exams like Alevel, HKDSE etc., STEP problems are much more difficult and challenging.
There are three papers in the examination, namely STEP1, STEP2 and STEP3, ranked according to difficulty (means 1 is the easiest and 3 is the most difficult). In recent years STEP1 is cancelled and two other exams still run as normal.
As I attempt STEP2 this year, I would mainly talk about STEP2 in this blog.
The syllabus of STEP2 can be found in this website: STEP 2021 Specifications. In general the pure math section of the exam would cover Calculus, Matrices, Polynomials, Number theory and many other topics.
I finished my last DSE exam - Math Extended Part 2 (Calculus and Algebra) in May 17th, and STEP2 would be in June 14th, so there was not much time left for preparing the exam.
Since I studied under the HKDSE curriculum, some content in STEP’s curriculum is not covered. Therefore, I bought Alevel Further Math’s textbook and started self-study. I learnt topics like relationship between roots in polynomial, linear transformation, and first-order differential equation on my own.
After that, I began practising real STEP problems by year, after several days I’ve done five past papers (2016 ~ 2020). Then I started to do the exercises provided in the book Advanced Problems in mathamatics. Most of the problems there are selected from STEP1 questions so they are not very difficult, but by doing that you could be familar with various topics, and could employ relevant knowledge to solve the problems in that topic during the actual exam.
In the last several days, I began doing STEP problems by topic, I classified topics that I has the most confidence, and selected relevant problems on STEP database for practising purposes.
In one STEP problem, there are always many subproblems, starting from trivial ones to challenging ones, and guiding you through the whole question by giving hints. So the problems would not be very difficult if you can notice them, it is highly possible that you could use the results of previous subproblems to help solve the current one. Therefore, when you are attempting the question, make sure that you check the previous subproblems and see if they could give some hints and guidence.
P \Rightarrow Q
, then Q is necessary condition of P, P is sufficient condition of Q. If
P \Leftrightarrow Q
, then P is necessary and sufficient condition of Q.
The calculus questions in STEP mainly test basic calculus manipulation skills such as integration by substitution, by parts etc. And sometimes would include reduction formula as well.
For by substitution questions, the first subproblem would always tell you what to substitute, for example:
u = 1/x
, show that for
b > 0
\int_{1/b}^b{\frac{x \ln x}{(a^2+x^2)(a^2x^2+1)}}\, dx = 0
( STEP2 2014 Question 4 )
And in the last subproblem would ask you to choose suitable substitution to prove or evaluate something.
By using a substituion of the form
u = k/x
, for suitable k, show that:
\int_{0}^{\infty}{\frac{1}{(a^2+x^2)^2}}\, dx = \frac{\pi}{4a^3}
In such problem, you would have two hints: the first is the hint from the previous subproblems, and the second is the upper and lower limit of the integral, they give you idea about what to substitute.
Example 2 ( STEP2 1998 Q7 ):
Prove that, if
I = \int_{0}^{1}{\frac{(1+x^2)^k}{(1+x)^{k+1}}}\, dx
I = \int_{0}^{\frac{1}{4}\pi}{\frac{d\theta}{[\sqrt{2} \cos \theta \cos (\frac{1}{4}\pi - \theta)]^{k+1}}} \, dx
Show further that:
I = 2\int_{0}^{\frac{1}{8}\pi}{\frac{d\theta}{[\sqrt2 \cos \theta \cos (\frac{1}{4}\pi - \theta)]^{k+1}}} \, dx = 2\int_{0}^{\sqrt{2} - 1}{\frac{(1+x^2)^k}{(1+x)^{k+1}}}\, dx
For integration by parts problem, a lot of calculations are required so make sure that you do not make careless mistakes, STEP2 1994 Question 2 is one of the very challenging questions. Lower and upper limit of the integral is still very useful in the problem, 2003 STEP2 Question 7 is a case in point:
n > 0
\int_{e^{1/n}}^{\infty}{\frac{\ln x}{x^{n+1}}} \, dx = \frac{2}{n^2 e}
From the problem we could discover that the lower limit of the integral is
e^{1/n}
, but the result contains
\frac{1}{e}
, that means the result of the integral should contain
x^{-n}
, it could only be obtained by integrating
x^{-(n+1)}
, therefore we could do the following procedure:
\int_{e^{1/n}}^{\infty}{\frac{\ln x}{x^{n+1}}} \, dx = \int_{e^{1/n}}^{\infty}{\ln x \frac{d(-\frac{1}{n}x^{-n})}{dx}} \, dx = ... = \frac{2}{n^2 e}
Trapezium rule is in syllabus too, STEP2 1987 Question 7 is worth attempting, which tests approximation of the integral.
Sometimes the problem needs to evaluate integral which upper or lower limit contains infinity, for example:
\int_{0}^{\infty} x^n e^{-x}\, dx = [-x^n e^{-x}]^{\infty}_{0} + n\int_{0}^{\infty}{x^{n-1} e^{-x}} \, dx
-x^n e^{-x}
, it equals to
\frac{-x^n}{e^x}
e^x
x^n
x \to \infty
, it approaches zero.
Linear transformation is an important concept in STEP, including rotation, shearing, reflection etc.
STEP2 1998 Question 5, STEP2 1993 Question 10, STEP2 1989 Question 7 are good practice materials.
Besides, STEP problems may test matrices in an algebaric way, for example:
If M is a
2\times 2
matrix, prove that:
\mathrm{Tr}(M^2) = \mathrm{Tr}(M)^2 - 2\mathrm{Det}(M)
Link: https://matheart.github.io/2021/06/11/Some-reflections-on-Cambridge-Math-Admission-Tests/
3. My Preparation
4. Problem Characteristics
5. Reflections by topic
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Divide quaternion by another quaternion - Simulink - MathWorks España
The Quaternion Division block divides a given quaternion by another. Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention. The output is the resulting quaternion from the division or vector of resulting quaternions from division. For the quaternion forms used, see Algorithms.
q — Dividend quaternion
Dividend quaternions in the form of [q0, p0, ..., q1, p1, ... , q2, p2, ... , q3, p3, ...], specified as a quaternion or vector of quaternions.
r — Divisor quaternion
Divisor quaternions in the form of [s0, r0, ..., s1, r1, ... , s2, r2, ... , s3, r3, ...], specified as a quaternion or vector of quaternions.
q/r — Output quaternion
Output quaternion or vector of resulting quaternions from division.
q={q}_{0}+i{q}_{1}+j{q}_{2}+k{q}_{3}
r={r}_{0}+i{r}_{1}+j{r}_{2}+k{r}_{3}.
t=\frac{q}{r}={t}_{0}+i{t}_{1}+j{t}_{2}+k{t}_{3},
\begin{array}{l}{t}_{0}=\frac{\left({r}_{0}{q}_{0}+{r}_{1}{q}_{1}+{r}_{2}{q}_{2}+{r}_{3}{q}_{3}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\\ {t}_{1}=\frac{\left({r}_{0}{q}_{1}-{r}_{1}{q}_{0}-{r}_{2}{q}_{3}+{r}_{3}{q}_{2}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\\ {t}_{2}=\frac{\left({r}_{0}{q}_{2}+{r}_{1}{q}_{3}-{r}_{2}{q}_{0}-{r}_{3}{q}_{1}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\\ {t}_{3}=\frac{\left({r}_{0}{q}_{3}-{r}_{1}{q}_{2}+{r}_{2}{q}_{1}-{r}_{3}{q}_{0}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\end{array}
Quaternion Conjugate | Quaternion Inverse | Quaternion Modulus | Quaternion Multiplication | Quaternion Norm | Quaternion Normalize | Quaternion Rotation
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This Release (#1) was created on Apr 17, 2018 at 7:30 PM ( 4 years ago )
The latest Release (#2) was created on Apr 17, 2018 at 7:31 PM ( 4 years ago ).
\frac{dI_{\mathrm{DR}}^{\mu}}{d \Omega_{\mathrm{f}}} = \frac{\alpha^2}{16 \pi} \frac{F^2_{\mu}}{\rho \nu_\mathbf{F} \omega_\mathbf{q}} \left(\frac{\nu_\mathrm{\mathbf{F}}}{c} \frac{E_L}{\omega_\mathbf{q}}\right)^2 \frac{n_{\mathrm{i}}|U_\mu(\textbf{q})|^2}{\nu^2_\mathrm{\mathbf{F}}}\mathrm{ln}(\frac{\omega_{\mathbf{q}}}{\gamma})
I_{\mathrm{G}} \propto E^4_{\mathrm{L}}
I_D=\alpha^2 \lambda_K \frac{\nu^2_F}{c^2} \frac{E_{\mathrm{L}}}{\omega^2_{\mathbf{q}}} \frac{\nu_FL_e}{A} \mathrm{ln} (\frac{\omega_{\mathbf{q}}}{2\gamma})
\gamma = \gamma^{\mathrm{d}} + \gamma^{\mathrm{ep}} \begin{cases} \gamma^{\mathrm{ep}}>\gamma^{\mathrm{d}} & \to I_{\mathrm{D}} \propto n_{\mathrm{i}}\\ \gamma^{\mathrm{ep}}<\gamma^{\mathrm{d}} & \to \frac{dI_{\mathrm{D}}}{dn_{\mathrm{i}}} = 0 \end{cases}
\gamma^{\mathrm{d}} [\mathrm{meV}] \approx \frac{n_i|U_0|^2E_{\mathrm{L}}}{2(\hbar \nu_F)^2} \sim 10n_\mathrm{i} [10^{12} \mathrm{cm}^{-2}]
Raman Spectroscopy can provide detailed information about the elastic
scattering potential due to impurities, allowing to identify the nature of defects
by using the laser energy dependence of the D and D’ bands, or the ID/ID’ ratio.
Several experiments can be used to test our predictions, such as correlations
with transport measurements or doping effects. Further computational work is
required to model more accurately the scattering potential introduced by the
different types of defects.
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VOL. 113 · NO. 3 | 15 June 2002
Duke Math. J. 113 (3), 399-419, (15 June 2002) DOI: 10.1215/S0012-7094-02-11331-3
KEYWORDS: 11P70, 11B13, 11B25
In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1).
Let be a finite set such that . Then A is contained in a proper d-dimensional progression P, where and .
Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.
Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5
KEYWORDS: 14G05, 11G35, 11G50, 14G25, 14J45
Nous établissons la conjecture de Manin dans le cas particulier des surfaces V de del Pezzo déployées de degré 5 sur ℚ. Autrement dit, nous montrons que, pour un ouvert , on a
{N}_{U\left(Q\right)}\left(B\right):=\text{card}\left\{P\in U\left(Q\right):h\left(P\right)\le B\right\}\sim CB{\left(\mathrm{log}B\right)}^{4}\text{ }\left(B\to +\infty \right)
La constante C est conforme à l'expression conjecturée par E. Peyre.
We state Manin's conjecture in the particular case of the split del Pezzo's surfaces of degree 5 over
Q
. We show that, for an open set ,
{N}_{U\left(Q\right)}\left(B\right):=\text{card}\left\{P\in U\left(Q\right):h\left(P\right)\le B\right\}\sim CB{\left(\mathrm{log}B\right)}^{4}\text{ }\left(B\to +\infty \right)
The constant C is the one conjectured by E. Peyre.
Geometric branched covers between generalized manifolds
Juha Heinonen, Seppo Rickman
KEYWORDS: 57M12, 30C65, 57P99
We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension.
Depth preservation in local theta correspondence
In this paper, we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type I reductive dual pairs over a nonarchimedean local field.
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\textcolor[rgb]{0.470588235294118,0,0.0549019607843137}{\mathbf{ω}}
\mathrm{π}:E→M
be a fiber bundle, with base dimension
m
{\mathrm{π}}^{\mathrm{∞}}:{J}^{\mathrm{∞}}\left(E\right) → M
E
({x}^{i}, {u}^{\mathrm{α}}, {u}_{{i}_{}}^{\mathrm{α}}, {u}_{{i}_{}j}^{\mathrm{α}}
{u}_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{α}}, ....)
{J}^{\infty }\left(E\right)
{\mathrm{dx}}^{i}
M
{\mathrm{Θ}}^{\mathrm{α}} = {\mathrm{du}}^{\mathrm{α}}-{u}_{\mathrm{ℓ}}^{\mathrm{α}}{\mathrm{dx}}^{\mathrm{ℓ}},
{\mathrm{Θ}}_{i}^{\mathrm{α}} = {\mathrm{du}}_{i}^{\mathrm{α}}-{u}_{i\mathrm{ℓ}}^{\mathrm{α}}{\mathrm{dx}}^{\mathrm{ℓ}\mathit{ }}, .... ,
{\mathrm{Θ}}_{\mathrm{ij}\cdot \cdot \cdot k}^{\mathrm{α}} = {\mathrm{du}}_{\mathrm{ij}\cdot \cdot \cdot k}^{\mathrm{α}}-{u}_{\mathrm{ij}\cdot \cdot \cdot \mathrm{kℓ}}^{\mathrm{α}} {\mathrm{dx}}^{\mathrm{ℓ}} , ....
{\mathrm{dΘ}}^{\mathrm{α}} = {\mathrm{dx}}^{\mathrm{ℓ}} ∧ {\mathrm{Θ}}_{\mathrm{ℓ}\mathit{ }}^{\mathrm{\alpha }},
{\mathrm{Θ}}_{i}^{\mathrm{α}} = {\mathrm{dx}}^{\mathrm{ℓ}} ∧ {\mathrm{\Theta }}_{\mathrm{iℓ}\mathit{ }}^{\mathrm{α}}, .... ,
{\mathrm{dΘ}}_{\mathrm{ij}\cdot \cdot \cdot k}^{\mathrm{α}} = {\mathrm{dx}}^{\mathrm{ℓ}} ∧ {\mathrm{\Theta }}_{\mathrm{ij}\cdot \cdot \cdot \mathrm{kℓ}\mathit{ }}^{\mathrm{α}}.
p-
\mathrm{ω} ∈ {\mathrm{Ω}}^{p}\left({J}^{\mathrm{∞}}\right)
\left(r,s\right)
r 1
M
s
\mathrm{ω} = {A}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{r} {a}_{1} \cdot \cdot \cdot {a}_{s}}^{ }{\mathrm{dx}}^{{i}_{1}}∧{\mathrm{dx}}^{{i}_{2} }∧ \cdot \cdot \cdot ∧{\mathrm{dx}}^{{i}_{r}} ∧ {C}^{{a}_{1}}∧{C}^{{a}_{2}} \cdot \cdot \cdot ∧{C}^{{a}_{s}},
{C}^{{a}_{k}}
p
{\mathrm{Ω}}^{p}\left({J}^{\mathrm{∞}} \right) = \underset{r+s =p}{\overset{}{⨁}} {\mathrm{Ω}}^{\left(r,s\right)}\left({J}^{\mathrm{∞}}\left(E\right)\right)
d:{\mathrm{\Omega }}^{\left(r,s\right)}\left({J}^{\infty }\left(E\right)\right)→ {\mathrm{\Omega }}^{\left(r+1,s\right)}\left({J}^{\infty }\left(E\right)\right) ⊕{\mathrm{\Omega }}^{\left(r,s+1\right)}\left({J}^{\infty }\left(E\right)\right)
d = {d}_{H } + {d}_{V},
{d}_{H }:{\mathrm{\Omega }}^{\left(r,s\right)}\left({J}^{\infty }\left(E\right)\right)→ {\mathrm{\Omega }}^{\left(r+1,s\right)}\left({J}^{\infty }\left(E\right)\right)
{d}_{V }:{\mathrm{\Omega }}^{\left(r,s\right)}\left({J}^{\infty }\left(E\right)\right)→ {\mathrm{\Omega }}^{\left(r,s+1\right)}\left({J}^{\infty }\left(E\right)\right).
{d}_{H }
{d}_{V}
{d}_{H}∘{d}_{H} =0, {d}_{H}∘{d}_{V} + {d}_{V}∘{d}_{H} =0,
{d}_{V}∘{d}_{V} =0
{d}_{H}\left({x}^{i}\right) = {\mathrm{dx}}^{i }, {d}_{H}\left({u}_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\alpha }}\right) = {u}_{\mathrm{ij} \cdot \cdot \cdot k\mathrm{ℓ}}^{\mathrm{α}} {\mathrm{dx}}^{\mathrm{ℓ}}, {d}_{H}\left({\mathrm{dx}}^{i}\right) = 0, {d}_{H}\left({\mathrm{Θ}}_{\mathrm{ij}\cdot \cdot \cdot k}^{\mathrm{\alpha }}\right) = {\mathrm{dx}}^{\mathrm{ℓ}} ∧ {\mathrm{Θ}}_{\mathrm{ij}\cdot \cdot \cdot k\mathrm{ℓ}}^{\mathrm{\alpha }}
{d}_{V}\left({x}^{i}\right) =0, {d}_{V}\left({u}_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\alpha }}\right) = {\mathrm{Θ}}_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{α}} , {d}_{V}\left({\mathrm{dx}}^{i}\right) = 0, {d}_{V}\left({\mathrm{Θ}}_{\mathrm{ij}\cdot \cdot \cdot k}^{\mathrm{\alpha }}\right) = 0.
\textcolor[rgb]{0.470588235294118,0,0.0549019607843137}{\mathrm{ω}}
{d}_{H}\left(\mathrm{ω}\right)
\mathrm{ω}
M
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{JetCalculus}\right):
{J}^{2}\left(E\right)
E
\left(x, y, u, v\right) → \left(x, y\right)
\mathrm{DGsetup}\left([x,y],[u,v],E,2\right):
F≔f\left(x,y,u[],u[1],u[2]\right):
\mathrm{PDEtools}[\mathrm{declare}]\left(F,\mathrm{quiet}\right):
\mathrm{HorizontalExteriorDerivative}\left(F\right)
\left({\textcolor[rgb]{0,0,1}{f}}_{{\textcolor[rgb]{0,0,1}{u}}_{[]}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{f}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{f}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{f}}_{\textcolor[rgb]{0,0,1}{x}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{f}}_{{\textcolor[rgb]{0,0,1}{u}}_{[]}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{f}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{f}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{f}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}
\mathrm{ω1}≔A\left(x,y,u[],u[1],u[2]\right)\mathrm{Dx}+B\left(x,y,u[],u[1],u[2]\right)\mathrm{Dy}
\textcolor[rgb]{0,0,1}{\mathrm{ω1}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{[}\textcolor[rgb]{0,0,1}{]}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{[}\textcolor[rgb]{0,0,1}{]}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}
\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω1}\right)
\textcolor[rgb]{0,0,1}{-}\left({\textcolor[rgb]{0,0,1}{A}}_{{\textcolor[rgb]{0,0,1}{u}}_{[]}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{A}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{A}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{B}}_{{\textcolor[rgb]{0,0,1}{u}}_{[]}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{B}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{B}}_{{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{A}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{B}}_{\textcolor[rgb]{0,0,1}{x}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}
\mathrm{ω2}≔\mathrm{Cu}[2]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Cv}[2]
\textcolor[rgb]{0,0,1}{\mathrm{ω2}}\textcolor[rgb]{0,0,1}{:=}{\textcolor[rgb]{0,0,1}{\mathrm{Cu}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cv}}}_{\textcolor[rgb]{0,0,1}{2}}
\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cu}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cv}}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{Dx}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cv}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cu}}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cu}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cv}}}_{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{Dy}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cv}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{Cu}}}_{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}
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Uranyl_nitrate Knowpia
Uranyl nitrate is a water-soluble yellow uranium salt with the formula UO2(NO3)2 · n H2O. The hexa-, tri-, and dihydrates are known.[3] The compound is mainly of interest because it is an intermediate in the preparation of nuclear fuels.
(T-4)-bis(nitrato-κO)dioxouranium
Uranium nitrate, Yellow salt
anhydrous: 10102-06-4 Y
hexahydrate: 13520-83-7
hexahydrate: Interactive image
anhydrous: 22177973 Y
hexahydrate: 55548
anhydrous: 24933
dihydrate: 22763670
hexahydrate: YR3850000
anhydrous: 0C0WI17JYF N
anhydrous: InChI=1S/2NO3.2O.U/c2*2-1(3)4;;;/q2*-1;;; Y
Key: QWDZADMNIUIMTC-UHFFFAOYSA-N Y
dihydrate: InChI=1S/2NO3.2H2O.2O.U/c2*2-1(3)4;;;;;/h;;2*1H2;;;/q2*-1;;;;;+2
Key: SUFYIOKRRLBZBE-UHFFFAOYSA-N
hexahydrate: InChI=1S/2HNO3.6H2O.2O.U/c2*2-1(3)4;;;;;;;;;/h2*(H,2,3,4);6*1H2;;;
Key: WRIBVRZWDPGVQH-UHFFFAOYSA-N
anhydrous: [N+](=O)([O-])[O-].O=[U+2]=O.[O-][N+](=O)[O-]
dihydrate: [N+](=O)([O-])[O-].[N+](=O)([O-])[O-].O.O.O=[U+2]=O
hexahydrate: [N+](=O)(O)[O-].[N+](=O)(O)[O-].O.O.O.O.O.O.O=[U]=O
Appearance yellow-green solid
Density 3.5 g/cm3 (dihydrate)[1]
Boiling point 118 °C (244 °F; 391 K) (decomposition)
g/100g H2O: 98 (0°C), 122 (20°C), 474 (100°C)<
Solubility in tributyl phosphate soluble
12 mg/kg (dog, oral)
238 (cat, oral)[2]
Uranyl nitrate can be prepared by reaction of uranium salts with nitric acid. It is soluble in water, ethanol, and acetone. As determined by neutron diffraction, the uranyl center is characteristically linear with short U=O distances. In the equatorial plane of the complex are six U-O bonds to bidentate nitrate and two water ligands. At 245 pm, these U-O bonds are much longer than the U=O bonds of the uranyl center.[1]
Processing of nuclear fuelsEdit
Uranyl nitrate is important for nuclear reprocessing. It is the compound of uranium that results from dissolving the decladded spent nuclear fuel rods or yellowcake in nitric acid, for further separation and preparation of uranium hexafluoride for isotope separation for preparing of enriched uranium. A special feature of uranyl nitrate is its solubility in tributyl phosphate (
{\displaystyle {\ce {PO(OC4H9)3}}}
), which allows uranium to be extracted from the nitric acid solution. Its high solubility is attributed to the formation of the lipophilic adduct UO2(NO3)2(OP(OBu)3)2.
Archaic photographyEdit
During the first half of the 19th century, many photosensitive metal salts had been identified as candidates for photographic processes, among them uranyl nitrate. The prints thus produced were called uranium prints or uranotypes. The first uranium printing processes were invented by Scotsman J. Charles Burnett between 1855 and 1857, and used this compound as the sensitive salt. Burnett authored a 1858 article comparing "Printing by the Salts of the Uranic and Ferric Oxides" The process employs the ability of the uranyl ion to pick up two electrons and reduce to the lower oxidation state of uranium(IV) under ultraviolet light. Uranotypes can vary from print to print from a more neutral, brown russet to strong Bartolozzi red, with a very long tone grade. Surviving prints are slightly radioactive, a property which serves as a means of non-destructively identifying them. Several other more elaborate photographic processes employing the compound appeared and vanished during the second half of the 19th century with names like Wothlytype, Mercuro-Uranotype and the Auro-Uranium process. Uranium papers were manufactured commercially at least until the end of the 19th century, vanishing due to the superior sensitivity and practical advantages of silver halides. From the 1930s through the 1950s Kodak Books described a uranium toner (Kodak T-9) using uranium nitrate hexahydrate. Some alternative process photographers including Blake Ferris and Robert Schramm continue to make uranotype prints today.
Stain for microscopyEdit
Along with uranyl acetate it is used as a negative stain for viruses in electron microscopy; in tissue samples it stabilizes nucleic acids and cell membranes.
As a reagentEdit
Uranyl nitrates are common starting materials for the synthesis of other uranyl compounds because the nitrate ligand is easily replaced by other anions. It reacts with oxalate to give uranyl oxalate. Treatment with hydrochloric acid gives uranyl chloride.[4]
Health and environmental issuesEdit
Uranyl nitrate is an oxidizing and highly toxic compound. When ingested, it causes severe chronic kidney disease and acute tubular necrosis and is a lymphocyte mitogen. Target organs include the kidneys, liver, lungs and brain. It also represents a severe fire and explosion risk when heated or subjected to shock in contact with oxidizable substances.
URANIUM DAYS: Notes On Uranium Photography (2007 archive from archive.org)
Chemical Database – Uranyl nitrate, solid
^ a b Mueller, Melvin Henry; Dalley, N. Kent; Simonsen, Stanley H. (1971). "Neutron Diffraction Study of Uranyl Nitrate Dihydrate". Inorganic Chemistry. 10 (2): 323–328. doi:10.1021/ic50096a021.
^ "Uranium (soluble compounds, as U)". Immediately Dangerous to Life or Health Concentrations (IDLH). National Institute for Occupational Safety and Health (NIOSH).
^ Peehs, Martin; Walter, Thomas; Walter, Sabine; Zemek, Martin (2007). "Uranium, Uranium Alloys, and Uranium Compounds". Ullmann's Encyclopedia of Industrial Chemistry. Weinheim: Wiley-VCH. doi:10.1002/14356007.a27_281.pub2.
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Digital Humanities & German Periodicals / Benjamin R. Bray
Image @FontaneArchiv
Digital Humanities & German Periodicals
nlp topic-models machine-learning python flask gensim javascript
DHGP Browser
As an undergraduate research assistant, I spent three years as the primary developer for an NLP-driven web application built to assist a humanities professor (Dr. Peter McIsaac, University of Michigan) with his research on 19th-century German literature. The application allowed him to run statistical topic models (LDA, HDP, DTM, etc.) on a large corpus of text and displayed helpful visualizations of the results. The application was built using Python / Flask / Bootstrap and also supported toponym detection and full-text search. We used gensim for topic modeling.
Using the web application I built, my supervisor was able to effectively detect cultural and historical trends in a large corpus of previously unstudied documentsThis is a cheeky remark!. Our efforts led to a number of publications in humanities journals and conferences, including [McIsaac 2014]:
McIsaac, Peter M. “Rethinking Nonfiction: Distant Reading the Nineteenth-Century Science-Literature Divide.” Distant Readings: Topologies of German Culture in the Long Nineteenth Century, edited by Matt Erlin and Lynne Tatlock, ed., Boydell and Brewer, 2014, pp. 185–208.
Our analysis focused on a corpus of widely-circulated periodicals, published in Germany during the 19th-century around the time of the administrative unification of Germany in 1871. Through HathiTrust and partnerships with university libraries, we obtained digital scans of the following periodicals:
Westermann's Illustrirte Monatshefte (1856-1987)
These periodicals, published weekly or monthly, were among Germany's most widely-read print material in the latter half of the nineteenth century, and served as precursors This is a longer remark that provides more detail about something in the main article.to the modern magazine. Scholars have long recognized the cultural significance of these publications (c.f. [Belgum 2002]), but their enormous volume had so far precluded comprehensive study.
Cover of Westermann's Monatshefte and front page of Die Gartenlaube . Courtesy of HathiTrust.
Using statistical methods, including topic models, we aimed to study the development of a German national identity following the 1848 revolutions, through the 1871 unification, and leading up to the world wars of the twentieth century. This approach is commonly referred to as digital humanities or distant reading (in contrast to close reading).
Initially, we only had access to digital scans of books printed in a difficult-to-read blackletter font. In order to convert our scanned images to text, I used Google Tesseract to train a custom optical character recognition (OCR) model specialized to fonts from our corpus. Tesseract performed quite well, but our scans exhibited a number of characteristics that introduced errors into the OCR process:
Poor scan quality (causing speckles, erosion, dilation, etc.)
Orthographic differences from modern German, including ligatures and the long s
Inconsistent layouts (floating images, multiple columns per page, etc.)
Blackletter fonts which are difficult to read, even for humans
The use of fonts such as Antiqua for dates and foreign words
Headers, footers, page numbers, illustrations, and hyphenation
The examples below highlight some of the challenges we faced during the OCR phase.
From Deutsche Rundschau , courtesy of HathiTrust.
Wikipedia, Antiqua-Fraktur Dispute
From Die Gartenlaube , courtesy of HathiTrust.
As a result, significant pre- and post-processing of OCR results was necessary. We combined a number of approaches in order to reduce the error rate to an acceptable level:
I used Processing to remove noise and other scanning artifacts from our images.
I wrote code to automatically remove running headers, text decorations, and page numbers.
Through manual inspection of a small number of documents, we compiled a list of common OCR mistakes. I developed scripts to automatically propagate these corrections across the entire corpus.
I experimented with several custom OCR-correction schemes to correct as many mistakes as possible and highlight ambiguities. Our most successful approach used a Hidden Markov Model to correct sequences of word fragments. Words were segmented using Letter Successor Entropy.
With these improvements, we found that our digitized texts were good enough for the type of exploratory analysis we had in mind. By evaluating our OCR pipeline on a synthetic dataset of "fake" scans with known text and a configurable amount of noise (speckles, erosion, dilation, etc.), we found that our OCR correction efforts improved accuracy from around 80% to 95% or higher.
In natural language processing, topic modeling is a form of statistical analysis used to help index and explore large collections of text documents. The output of a topic model typically includes:
A list of topics, each represened by a list of related words. Each word may also have an associated weight, indicating how strongly a word relates to this topic. For example:
(Topic 1) sport, team, coach, ball, coach, team, race, bat, run, swim...
(Topic 2) country, government, official, governor, tax, approve, law...
(Topic 3) train, bus, passenger, traffic, bicycle, pedestrian...
A topic probability vector for each document, representing the importance of each topic to this document. For example, a document about the Olympics may be 70% sports, 20% government, and 10% transportation.
The most popular topic model is Latent Dirichlet Allocation (LDA), which is succinctly described by the following probabilistic graphical model. There are
T
topics,
M
N
words per document, and
V
words in the vocabulary.
\begin{aligned} \text{hyperparameters} &&& \alpha \in \mathbb{R}^{T}, \eta \in \mathbb{R}^{V}\\ \text{topics} && \beta_t \mid \eta & \stackrel{iid}{\sim} \mathrm{Dirichlet}(\eta) \\ \text{topic mixtures} && \theta_m \mid \alpha &\stackrel{iid}{\sim} \mathrm{Dirichlet}(\alpha) \\ \text{topic indicators} && z_{mn} \mid \theta_m &\stackrel{iid}{\sim} \mathrm{Categorical}(\theta_m) \\ \text{word indicators} && w_{mn} \mid z_{mn} &\stackrel{iid}{\sim} \mathrm{Categorical}(\beta_{z_{mn}}) \end{aligned}
Each topic
t
is represented by a probability distribution
\beta_t
over the vocabulary, indicating how likely each word is to appear under topic
t
. LDA posits that documents are written using the following generative process:
d_{m}
Decide in what proportions
\theta_m = (\theta_{m1},\dots,\theta_{mt})
each topic will appear.
To choose each each word
w_{mn}
, 1. According to
\theta_m
, randomly decide which topic to use for this word. 2. Randomly sample a word according to the chosen topic.
Of course, this is not how humans actually write. LDA represents documents as bags-of-words, ignoring word order and sentence structure. When topic models are used to index or explore large corpora, as was our goal, this is an acceptable compromise. Given a collection of documents, LDA attempts to "invert" the generative process by computing a maximum likelihood estimate of the topics
\beta_t
and topic mixtures
\theta_m
. These estimates are typically computed using variational expectation-maximziation.
Using Python / Flask / Bootstrap, I built a web application enabling humanities researchers to train, visualize, and save topic models. Features:
Support for several popular topic models:
Online Latent Dirichlet Allocation (via gensim)
Online Hierarchical Dirichlet Process (via gensim)
Dynamic Topic Models (custom implementation based [Blei 2006])
Toponym Resolution for identifying and mapping place names mentioned in our texts
Full-text / metadata search using ElasticSearch
Support for any corpus with metadata saved in JSON format.
I no longer have access to the most up-to-date version of dhgp-browser, but here are some screenshots from mid-2015:
The poster below summarizes the progress made during my first year on the project, which I initially joined through the UROP Program at UM. After my first year, I was hired to continue working on the project as an undergraduate research assistant.
[McIsaac 2014] McIsaac, Peter M. “Rethinking Nonfiction: Distant Reading the Nineteenth-Century Science-Literature Divide.” Distant Readings: Topologies of German Culture in the Long Nineteenth Century, edited by Matt Erlin and Lynne Tatlock, ed., Boydell and Brewer, 2014, pp. 185–208.
[Belgum 2002] Belgum, Kirsten. Popularizing the Nation: Audience, Representation, and the Production of Identity in Die Gartenlaube, 1853-1900. U of Nebraska Press, 1998.
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Rectangular Coordinates - MATLAB & Simulink - MathWorks Switzerland
Definitions of Coordinates
Notation for Vectors and Points
Orthogonal Basis and Euclidean Norm
Orientation of Coordinate Axes
Rotations and Rotation Matrices
Construct a rectangular, or Cartesian, coordinate system for three-dimensional space by specifying three mutually orthogonal coordinate axes. The following figure shows one possible specification of the coordinate axes.
Rectangular coordinates specify a position in space in a given coordinate system as an ordered 3-tuple of real numbers, (x,y,z), with respect to the origin (0,0,0). Considerations for choosing the origin are discussed in Global and Local Coordinate Systems.
You can view the 3-tuple as a point in space, or equivalently as a vector in three-dimensional Euclidean space. Viewed as a vector space, the coordinate axes are basis vectors and the vector gives the direction to a point in space from the origin. Every vector in space is uniquely determined by a linear combination of the basis vectors. The most common set of basis vectors for three-dimensional Euclidean space are the standard unit basis vectors:
\left\{\left[1\text{\hspace{0.17em}}\text{ }\text{ }0\text{\hspace{0.17em}}0\right],\left[0\text{\hspace{0.17em}}1\text{\hspace{0.17em}}0\right],\left[0\text{\hspace{0.17em}}0\text{\hspace{0.17em}}1\right]\right\}
In Phased Array System Toolbox™ software, you specify both coordinate axes and points as column vectors.
In this software, all coordinate vectors are column vectors. For convenience, the documentation represents column vectors in the format [x y z] without transpose notation.
Both the vector notation [x y z] and point notation (x,y,z) are used interchangeably. The interpretation of the column vector as a vector or point depends on the context. If the column vector specifies the axes of a coordinate system or direction, it is a vector. If the column vector specifies coordinates, it is a point.
Any three linearly independent vectors define a basis for three-dimensional space. However, this software assumes that the basis vectors you use are orthogonal.
The standard distance measure in space is the l2 norm, or Euclidean norm. The Euclidean norm of a vector [x y z] is defined by:
\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}
The Euclidean norm gives the length of the vector measured from the origin as the hypotenuse of a right triangle. The distance between two vectors [x0 y0 z0] and [x1 y1 z1] is:
\sqrt{{\left({x}_{0}-{x}_{1}\right)}^{2}+{\left({y}_{0}-{y}_{1}\right)}^{2}+{\left({z}_{0}-{z}_{1}\right)}^{2}}
Given an orthonormal set of basis vectors representing the coordinate axes, there are multiple ways to orient the axes. The following figure illustrates one such orientation, called a right-handed coordinate system. The arrows on the coordinate axes indicate the positive directions.
If you take your right hand and point it along the positive x-axis with your palm facing the positive y-axis and extend your thumb, your thumb indicates the positive direction of the z-axis.
{v}^{\prime }=Av={R}_{z}\left(\gamma \right){R}_{y}\left(\beta \right){R}_{x}\left(\alpha \right)v
{R}_{x}\left(\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]
{R}_{y}\left(\beta \right)=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]
{R}_{z}\left(\gamma \right)=\left[\begin{array}{ccc}\mathrm{cos}\gamma & -\mathrm{sin}\gamma & 0\\ \mathrm{sin}\gamma & \mathrm{cos}\gamma & 0\\ 0& 0& 1\end{array}\right]
{A}^{-1}A=1
{R}_{x}^{-1}\left(\alpha \right)={R}_{x}\left(-\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & \mathrm{sin}\alpha \\ 0& -\mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]={R}_{x}^{\prime }\left(\alpha \right)
i,j,k
{i}^{\prime },j{,}^{\prime }{k}^{\prime }
\begin{array}{ll}{i}^{\prime }\hfill & =Ai\hfill \\ {j}^{\prime }\hfill & =Aj\hfill \\ {k}^{\prime }\hfill & =Ak\hfill \end{array}
\left[\begin{array}{c}{i}^{\prime }\\ {j}^{\prime }\\ {k}^{\prime }\end{array}\right]={A}^{\prime }\left[\begin{array}{c}i\\ j\\ k\end{array}\right]
v={v}_{x}i+{v}_{y}j+{v}_{z}k={{v}^{\prime }}_{x}{i}^{\prime }+{{v}^{\prime }}_{y}{j}^{\prime }+{{v}^{\prime }}_{z}{k}^{\prime }
\left[\begin{array}{c}{{v}^{\prime }}_{x}\\ {{v}^{\prime }}_{y}\\ {{v}^{\prime }}_{z}\end{array}\right]={A}^{-1}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]={A}^{\prime }\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]
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GO_3D_OBS: the multi-parameter benchmark geomodel for seismic imaging method...
Górszczyk, Andrzej; Operto, Stéphane
Detailed reconstruction of deep crustal targets by seismic methods remains a long-standing challenge. One key to address this challenge is the joint development of new seismic acquisition systems and leading-edge processing techniques. In marine environments, controlled-source seismic surveys at a regional scale are typically carried out with sparse arrays of ocean bottom seismometers (OBSs), which provide incomplete and down-sampled subsurface illumination. To assess and minimize the acquisition footprint in high-resolution imaging process such as full waveform inversion, realistic crustal-scale benchmark models are clearly required. The deficiency of such models prompts us to build one and release it freely to the geophysical community. Here, we introduce GO_3D_OBS – a 3D high-resolution geomodel representing a subduction zone, inspired by the geology of the Nankai Trough. The inline-formula
M1inlinescrollmathmlnormal 175\phantom{\rule{0ex}{0ex}}unit\mathrm{normal km}×normal 100\phantom{\rule{0ex}{0ex}}unit\mathrm{normal km}×normal 30\phantom{\rule{0ex}{0ex}}unit\mathrm{normal km}
121pt10ptsvg-formulamathimge6ce5260b124ada18d8542d13b8811df gmd-14-1773-2021-ie00001.svg121pt10ptgmd-14-1773-2021-ie00001.png model integrates complex geological structures with a viscoelastic isotropic parameterization. It is defined in the form of a uniform Cartesian grid containing inline-formula∼33.6e9 degrees of freedom for a grid interval of 25 m. The size of the model raises significant high-performance computing challenges to tackle large-scale forward propagation simulations and related inverse problems. We describe the workflow designed to implement all the model ingredients including 2D structural segments, their projection into the third dimension, stochastic components, and physical parameterization. Various wavefield simulations that we present clearly reflect in the seismograms the structural complexity of the model and the footprint of different physical approximations. This benchmark model is intended to help to optimize the design of next-generation 3D academic surveys – in particular, but not only, long-offset OBS experiments – to mitigate the acquisition footprint during high-resolution imaging of the deep crust.
Górszczyk, Andrzej / Operto, Stéphane: GO_3D_OBS: the multi-parameter benchmark geomodel for seismic imaging method assessment and next-generation 3D survey design (version 1.0). 2021. Copernicus Publications.
Rechteinhaber: Andrzej Górszczyk
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VOL. 115 · NO. 1 | 1 October 2002
Special Lagrangian m-folds in ℂm with symmetries
Duke Math. J. 115 (1), 1-51, (1 October 2002) DOI: 10.1215/S0012-7094-02-11511-7
This the first in a series of papers on special Lagrangian submanifolds in ℂm. We study special Lagrangian submanifolds in ℂm with large symmetry groups, and we give a number of explicit constructions. Our main results concern special Lagrangian cones in ℂm invariant under a subgroup G in SU(m) isomorphic to U(1)m−2. By writing the special Lagrangian equation as an ordinary differential equation (ODE) in G-orbits and solving the ODE, we find a large family of distinct, G-invariant special Lagrangian cones on Tm−2 in ℂm. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models are needed to understand mirror symmetry and the Strominger-Yau-Zaslow (SYZ) conjecture.
Logarithm-free A-hypergeometric series
Duke Math. J. 115 (1), 53-73, (1 October 2002) DOI: 10.1215/S0012-7094-02-11512-9
KEYWORDS: 16S32, 13N10, 14M25, 33C70
We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a Gel’fand-Kapranov-Zelevinskiĭ (GKZ) hypergeometric) system. In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set. Furthermore, we classify A-hypergeometric systems as analytic
\mathcal{D}
Duke Math. J. 115 (1), 75-103, (1 October 2002) DOI: 10.1215/S0012-7094-02-11513-0
KEYWORDS: 14C35, 05E10, 19E08
Duke Math. J. 115 (1), 105-169, (1 October 2002) DOI: 10.1215/S0012-7094-02-11514-2
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Unraveling the role of silicon in atmospheric aerosol secondary formation: a new conservative tracer for aerosol chemistry
Unraveling the role of silicon in atmospheric aerosol secondary formation:...
Lu, Dawei; Tan, Jihua; Yang, Xuezhi; Sun, Xu; Liu, Qian; Jiang, Guibin
Aerosol particles are ubiquitous in the atmosphere and affect the quality of human life through their climatic and health effects. The formation and growth of aerosol particles involve extremely complex reactions and processes. Due to limited research tools, the sources and chemistry of aerosols are still not fully understood, and until now have normally been investigated by using chemical species of secondary aerosols (e.g., inline-formula
M1inlinescrollmathmlchem{\mathrm{normal NH}}_{normal 4}^{+}
24pt15ptsvg-formulamathimg8aeb386a576ed6c8280ae774099f80e4 acp-19-2861-2019-ie00001.svg24pt15ptacp-19-2861-2019-ie00001.png , inline-formula
M2inlinescrollmathmlchem{\mathrm{normal NO}}_{normal 3}^{-}
25pt16ptsvg-formulamathimg4c315b3ea451cf26923ad12993612b33 acp-19-2861-2019-ie00002.svg25pt16ptacp-19-2861-2019-ie00002.png , inline-formula
M3inlinescrollmathmlchem{\mathrm{normal SO}}_{normal 4}^{normal 2-}
29pt17ptsvg-formulamathimg6060a0eb6022af681aa55d19b3180df9 acp-19-2861-2019-ie00003.svg29pt17ptacp-19-2861-2019-ie00003.png , SOC) as tracers. Here we investigated the role of silicon (Si), an ubiquitous but relatively inert element, during the secondary aerosol formation process. We analyzed the correlation of Si in airborne fine particles (inline-formulaPM2.5) collected in Beijing – a typical pollution region – with the secondary chemical species and secondary particle precursors (e.g., inline-formulaSO2 and inline-formulaNOx). The total mass of Si in inline-formulaPM2.5 was found to be uncorrelated with the secondary aerosol formation process, which suggested that Si is a new conservative tracer for the amount of primary materials in inline-formulaPM2.5 and can be used to estimate the relative amount of secondary and primary compounds in inline-formulaPM2.5. This finding enables the accurate estimation of secondary aerosol contribution to inline-formulaPM2.5 by using Si as a single tracer rather than the commonly used multiple chemical tracers. In addition, we show that the correlation analysis of secondary aerosols with the Si isotopic composition of inline-formulaPM2.5 can further reveal the sources of the precursors of secondary aerosols. Therefore, Si may provide a new tool for aerosol chemistry studies.
Lu, Dawei / Tan, Jihua / Yang, Xuezhi / et al: Unraveling the role of silicon in atmospheric aerosol secondary formation: a new conservative tracer for aerosol chemistry. 2019. Copernicus Publications.
Rechteinhaber: Dawei Lu et al.
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Did you know that the Statue of Liberty was a gift from France? It was shipped to New York and reassembled on an island in New York Harbor. It was finished in
1886
. The distance from the base to the torch is
152
feet. The gift store sells a scale model of the statue measuring
18
inches (
1.5
feet) tall.
If the length of the index finger on the real statue is eight feet, what is its length on the scale model?
Set up a proportion using the information in the problem.
\frac{\text{statue}}{\text{model}} = \frac{152}{1.5} = \frac{8}{x}
x
x=0.08
ft or about
1
Alex wanted to know the length of the right arm on the statue. He measured the model, and the right arm was five inches long. What is the length of the arm on the statue?
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Breathers and Soliton Solutions for a Generalization of the Nonlinear Schrödinger Equation
Hai-Feng Zhang, Hui-Qin Hao, Jian-Wen Zhang, "Breathers and Soliton Solutions for a Generalization of the Nonlinear Schrödinger Equation", Mathematical Problems in Engineering, vol. 2013, Article ID 456864, 5 pages, 2013. https://doi.org/10.1155/2013/456864
Hai-Feng Zhang,1 Hui-Qin Hao,1 and Jian-Wen Zhang1
Academic Editor: Farzad Khani
A generalized nonlinear Schrödinger equation, which describes the propagation of the femtosecond pulse in single mode optical silica fiber, is analytically investigated. By virtue of the Darboux transformation method, some new soliton solutions are generated: the bright one-soliton solution on the zero background, the dark one-soliton solution on the continuous wave background, the Akhmediev breather which delineates the modulation instability process, and the breather evolving periodically along the straight line with a certain angle of -axis and -axis. Those results might be useful in the study of the femtosecond pulse in single mode optical silica fiber.
Investigations on the dynamic features of solitons have attracted certain interest in nonlinear optics [1–4]. Optical solitons have been regarded as a candidate for the optical communication networks [5–8]. On the basis of the balance between the group velocity dispersion and self-phase modulation [9, 10], the propagation of optical soliton is usually governed by the nonlinear Schrödinger (NLS) equation [11–14]: However, when optical pulses are shorter, the NLS equation becomes inadequate, and it is necessary to include additional terms [6, 7]. For example, in single mode optical silica fiber, in order to describe the propagation of femtosecond pulse, the higher order asymptotic terms should be retained [15]; to understand such phenomena, we consider the following generalization of the NLS equation [16]: Analogous to the circumstance that the Camassa-Holm equation provides a better approximation of the KdV equation [15], (2) is related to the NLS equation, provided that one retains terms of the next asymptotic order. Under the transformation , (2) can be converted into the following equation [16, 17]: where denotes the complex field envelope and the subscripts and are the longitudinal distance and retarded time, respectively. In recent years, some results have been obtained for (1): (1) Reference [15] has analyzed the dynamic features of the rogue wave solutions; (2) Reference [16] has analyzed the conservation laws, bi-Hamiltonian structure, Lax pair, and initial-value problem; (3) Reference [17] has derived some soliton solutions by using the bilinear method. The aim of this paper is mainly to derive some new soliton solutions for (3) using the Darboux transformation (DT) method and analyze the dynamic features of soliton solutions.
This paper will be organized as follows. In Section 2, we will give the Lax pair and construct the DT for (3). In Section 3, we will obtain bright one-soliton, dark one-soliton, and breather solutions and analyze the dynamic features of soliton solutions by using some figures. Finally, our conclusions will be addressed in Section 4.
Employing the Ablowitz-Kaup-Newell-Segur formalism [18], [15, 16] has given the Lax pair associated with (3) as where ( denotes the transpose of a matrix), and the matrices and have the form where is a spectral parameter and
Through direct computations, it can be verified that the zero curvature equation exactly gives rise to (3).
Next, based on Lax pair (4a) and (4b), we will give the DT [19–22] formalism for (3). Define where denotes the identity matrix and with One can verify that if is the solution of Lax pair (4a) and (4b) with , then is also the solution of Lax pair (4a) and (4b) corresponding to .
Through direct computation, we can obtain
So, if is a seed solution of (3), is also a solution of (3).
3. One and Breather Solutions (3)
In this section, we will apply the DT constructed to obtain one and breather solutions for (3). Now we take the nonzero continuous wave (cw) solution as the initial seed for (3), where , , and are all real parameters. Equation (3) requires that the frequency satisfies the nonlinear dispersion relation:
Solving (4a) and (4b) and setting , , one can obtain where
Through tedious computations, one can arrive at where and , , , and are complex constants satisfying: with with , , , , , and as real numbers, and we can derive
Now, substituting (14) into (10), we can obtain the solution for (1) as where with
Next according to different values of those parameters in solution (19), we will analyse the novel properties of solitons.
3.1. One-Soliton Solutions for (3)
When , that is to say, the initial seed for (3) is zero, solution (19) reduces to one-soliton solution as with Solution (22) represents a bright soliton whose dynamic features are delineated in Figures 1. Through symbolic computation, we can conclude the following physical quantities for solution (22): the maximum amplitude , the width , the envelope velocity , and the energy of the one-soliton solution So, the amplitude and envelope velocity will increase when the value of becomes bigger. As shown in Figures 1, the amplitude is higher and the envelope velocity is bigger in Figure 1(a) than in Figure 1(b).
Evolution of the one-soliton solution of (22). The parameters are (a) , ; (b) , .
3.2. Breather and Dark One-Soliton Solutions for (1)
When , the nonzero initial seed describes the nonvanishing boundary conditions. For simplicity, taking , , we have the following relations:With the previous conclusions, solution (19) can be converted into: where with Figure 2 display the propagation characteristics of solitons via solutions (26). Figures 2(a) and 2(b) depict the dynamic features of breathers; as shown in Figure 2(a), the main feature is the propagation of the breather that is periodic in the space coordinate and aperiodic in the time coordinate; that is to say, we can obtain the Akhmediev breather [23] via solutions (26) under suitable parameters chosen. In addition, the Akhmediev breather can be regarded as a modulation instability process. Figure 2(b) portrays the propagation of the breather evolving periodically along the straight line with a certain angle of -axis and -axis. Figure 2(c) describes the dynamic feature of the dark one-soliton solutions via solutions (26) on the continuous wave background, which is different with Figures 1 via Solutions (22) on the zero background.
Evolution of the soliton solutions of (26). The parameters are (a) , ; (b) , ; (c) , .
Our main attention has been focused on (3), which can describe the propagation of femtosecond pulse in single mode optical silica fiber. By using the Darboux transformation method, we have obtained (1) bright one soliton on the zero background; (2) two types of breathers: the Akhmediev soliton which delineates the modulation instability process and the breather evolving periodically along the straight line with a certain angle of -axis and -axis; (3) the dark one-soliton solution on the continuous wave background.
The express their sincere thanks to each member of their discussion group for their suggestions. This work has been supported by the National Natural Science Foundation of China under Grant nos. 11172194 and 61250011 and by the Natural Science Foundation of Shanxi Province under Grant no. 2012011004-3.
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N
-fold Darboux transformation,” Nonlinear Dynamics, vol. 69, no. 4, pp. 2009–2020, 2012. View at: Publisher Site | Google Scholar | MathSciNet
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Differentiate cfit or sfit object - MATLAB differentiate - MathWorks
Find the Derivatives of a Curve Using the differentiate Function
Find the Derivatives of a Surface Using the differentiate Function
Differentiate cfit or sfit object
fx = differentiate(FO, X)
[fx, fxx] = differentiate(FO, X)
[fx, fy] = differentiate(FO, X, Y)
[fx, fy] = differentiate(FO, [X, Y])
[fx, fy, fxx, fxy, fyy] = differentiate(FO, ...)
Use these syntaxes for cfit objects.
fx = differentiate(FO, X) differentiates the cfit object FO at the points specified by the vector X and returns the result in fx.
[fx, fxx] = differentiate(FO, X) differentiates the cfit object FO at the points specified by the vector X and returns the result in fx and the second derivative in fxx.
Use these syntaxes for sfit objects.
[fx, fy] = differentiate(FO, X, Y) differentiates the surface FO at the points specified by X and Y and returns the result in fx and fy.
FO is a surface fit (sfit) object generated by the fit function.
X and Y must be double-precision arrays and the same size and shape as each other.
All return arguments are the same size and shape as X and Y.
If FO represents the surface
z=f\left(x,y\right)
, then FX contains the derivatives with respect to x, that is,
\frac{df}{dx}
, and FY contains the derivatives with respect to y, that is,
\frac{df}{dy}
[fx, fy] = differentiate(FO, [X, Y]), where X and Y are column vectors, allows you to specify the evaluation points as a single argument.
[fx, fy, fxx, fxy, fyy] = differentiate(FO, ...) computes the first and second derivatives of the surface fit object FO.
fxx contains the second derivatives with respect to x, that is,
\frac{{\partial }^{2}f}{\partial {x}^{2}}
fxy contains the mixed second derivatives, that is,
\frac{{\partial }^{2}f}{\partial x\partial y}
fyy contains the second derivatives with respect to y, that is,
\frac{{\partial }^{2}f}{\partial {y}^{2}}
Create a baseline sinusoidal signal.
Add response-dependent Gaussian noise to the signal.
noise = 2*y0.*randn(size(y0));
Fit the noisy data with a custom sinusoidal model.
Find the derivatives of the fit at the predictors.
Plot the data, the fit, and the derivatives.
You can also compute and plot derivatives directly with the cfit plot method, as follows:
The plot method, however, does not return data on the derivatives, unlike the differentiate method.
You can use the differentiate method to compute the gradients of a fit and then use the quiver function to plot these gradients as arrows. This example plots the gradients over the top of a contour plot.
Create the derivation points and fit the data.
x = [0.64;0.95;0.21;0.71;0.24;0.12;0.61;0.45;0.46;...
y = [0.42;0.84;0.83;0.26;0.61;0.58;0.54;0.87;0.26;...
z = [0.49;0.051;0.27;0.59;0.35;0.41;0.3;0.084;0.6;...
fo = fit( [x, y], z, 'poly32', 'normalize', 'on' );
[xx, yy] = meshgrid( 0:0.04:1, 0:0.05:1 );
Compute the gradients of the fit using the differentiate function.
[fx, fy] = differentiate( fo, xx, yy );
Use the quiver function to plot the gradients.
plot( fo, 'Style', 'Contour' );
h = quiver( xx, yy, fx, fy, 'r', 'LineWidth', 2 );
colormap( copper )
If you want to use derivatives in an optimization, you can, for example, implement an objective function for fmincon as follows.
function [z, g, H] = objectiveWithHessian( xy )
% The input xy represents a single evaluation point
z = f( xy );
[fx, fy, fxx, fxy, fyy] = differentiate( f, xy );
g = [fx, fy];
H = [fxx, fxy; fxy, fyy];
FO — cfit function
Function to differentiate, specified as a cfit object for curves or as a sfit object for surfaces.
X — Differentiation points
Points at which to differentiate the function, specified as a vector. For surfaces, this argument must have the same size and shape of Y.
Y — Differentiation points
Points at which to differentiate the function, specified as a vector. For surfaces, this argument must have the same size and shape of X.
fx — First derivative with respect to x
First derivative of the function, returned as a vector of the same size and shape of X and Y.
If FO is a surface,
z=f\left(x,y\right)
, then fx contains the derivatives with respect to x.
fxx — Second derivative with respect to x
Second derivative of the function, returned as a vector of the same size and shape of X and Y.
z=f\left(x,y\right)
, then fxx contains the second derivatives with respect to x.
fy — First derivative with respect to y
z=f\left(x,y\right)
, then fy contains the derivatives with respect to y.
fyy — Second derivative with respect to y
z=f\left(x,y\right)
, then fyy contains the second derivatives with respect to y.
fxy — Mixed second derivative
Mixed second derivative of the function, returned as a vector of the same size and shape of X and Y.
For library models with closed forms, the toolbox calculates derivatives analytically. For all other models, the toolbox calculates the first derivative using the centered difference quotient
\frac{df}{dx}=\frac{f\left(x+\Delta x\right)-f\left(x-\Delta x\right)}{2\Delta x}
where x is the value at which the toolbox calculates the derivative,
\Delta x
is a small number (on the order of the cube root of eps),
f\left(x+\Delta x\right)
is fun evaluated at
x+\Delta x
f\left(x-x\Delta \right)
x-\Delta x
The toolbox calculates the second derivative using the expression
\frac{{d}^{2}f}{d{x}^{2}}=\frac{f\left(x+\Delta x\right)+f\left(x-\Delta x\right)-2f\left(x\right)}{{\left(\Delta x\right)}^{2}}
The toolbox calculates the mixed derivative for surfaces using the expression
\frac{{\partial }^{2}f}{\partial x\partial y}\left(x,y\right)=\frac{f\left(x+\Delta x,y+\Delta y\right)-f\left(x-\Delta x,y+\Delta y\right)-f\left(x+\Delta x,y-\Delta y\right)+f\left(x-\Delta x,y-\Delta y\right)}{4\Delta x\Delta y}
fit | plot | integrate
|
Numerical Investigation of Forced Convective Heat Transfer Around and Through a Porous Circular Cylinder With Internal Heat Generation | J. Heat Transfer | ASME Digital Collection
Mohammad Sadegh Valipour,
,3519645399 Semnan,
e-mail: msvalipour@semnan.ac.ir
Ariyan Zare Ghadi
e-mail: zare.mech@yahoo.com
Sadegh Valipour, M., and Zare Ghadi, A. (April 26, 2012). "Numerical Investigation of Forced Convective Heat Transfer Around and Through a Porous Circular Cylinder With Internal Heat Generation." ASME. J. Heat Transfer. June 2012; 134(6): 062601. https://doi.org/10.1115/1.4005741
In this study, convective heat transfer around and through a porous circular cylinder together with internal heat generation has been investigated numerically. Governing equations containing continuity, momentum, and energy equations have been developed in polar coordinate system in both porous and nonporous media based on single-domain approach. However, governing equations in porous medium are derived using intrinsic volume averaging method. The equations are solved numerically based on finite volume method over staggered grid arrangement. Also, pressure correction-based iterative algorithm, SIMPLE, is applied for solving the pressure linked equations. Reynolds and Peclet numbers (based on cylinder diameter and velocity of free stream) are from 1 to 40. Also, Darcy number (Da) varies within the range of
10-6≤Da≤10-2
and porosity is considered 0.9 for all calculations. The influence of Da and Re numbers on local and average Nu numbers has been investigated. It is found that the local and average Nu numbers increase with any increase in Da number. Two correlations of average Nu number are presented for high and low Da numbers.
porous circular cylinder, porous media, convection in porous media, single-domain approach, heat transfer
Circular cylinders, Convection, Cylinders, Heat, Porous materials, Temperature, Fluids, Heat transfer, Porosity, Momentum, Boundary-value problems
An Analysis of Combined Free and Forced Convection Heat Transfer From a Horizontal Circular Cylinder to a Transverse Flow
“Analysis of Mixed Convection About a Horizontal Cylinder,”
“Mixed Convection From Horizontal Circular Cylinder,”
“Laminar Combined Convection From a Horizontal Cylinder-Parallel and Contra Flow Regime
“A Numerical Study of the Steady Forced Convection Heat Transfer From an Unconfined Circular Cylinder,”
Study of Heat-Transfer on the Surface of a Circular Cylinder in Flow Using an Immersed-Boundary Method
Mixed Convection About a Horizontal Cylinder and a Sphere in a Fluid Saturated Porous Medium
About Enhancement of Heat Transfer Over a Circular Cylinder Embedded in Porous Medium
Layeghi
Darcy Model for the Study of Fluid Flow and Heat Transfer Around a Cylinder Embedded in Porous Media
Boundary Condition at a Naturally Permeable Wall
On the Boundary Condition at the Interface of a Porous Medium,”
Numerical Investigation of Fluid Flow and Heat Transfer Around a Solid Circular Cylinder Utilizing Nanofluid
Double‐Diffusive Convection during Dendritic Solidification of a Binary Mixture
PhysicoChemical Hydrodynamics.
Forced Convection in a Channel Filled With Porous Medium: An Exact Solution
Computation of Flow Through a Fluid-Sediment Interface in a Benthic Chamber
Numerical Study and Physical Analysis of the Pressure and Velocity Fields in the Near Wake of a Circular Cylinder
Numerical Simulation of Laminar Flow Past a Circular Cylinder
Optimization of the Axial Porosity Distribution of Porous Inserts in a Liquid-Piston Gas Compressor Using a One-Dimensional Formulation
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Flat-field correction – terracotta INFO
Flat-field correction (FFC) is a technique used to improve quality in digital imaging. It cancels the effects of image artifacts caused by variations in the pixel-to-pixel sensitivity of the detector and by distortions in the optical path. It is a standard calibration procedure in everything from personal digital cameras to large telescopes.
Not to be confused with Petzval field curvature, which refers to focus uniformity.
The brightness variation due to vignetting, as shown here, can be corrected by selectively brightening the perimeter of the image.
. . . Flat-field correction . . .
Flat fielding refers to the process of compensating for different gains and dark currents in a detector. Once a detector has been appropriately flat-fielded, a uniform signal will create a uniform output (hence flat-field). This then means any further signal is due to the phenomenon being detected and not a systematic error.
A flat-field image is acquired by imaging a uniformly-illuminated screen, thus producing an image of uniform color and brightness across the frame. For handheld cameras, the screen could be a piece of paper at arm’s length, but a telescope will frequently image a clear patch of sky at twilight, when the illumination is uniform and there are few, if any, stars visible.[1] Once the images are acquired, processing can begin.
A flat-field consists of two numbers for each pixel, the pixel’s gain and its dark current (or dark frame). The pixel’s gain is how the amount of signal given by the detector varies as a function of the amount of light (or equivalent). The gain is almost always a linear variable, as such the gain is given simply as the ratio of the input and output signals. The dark-current is the amount of signal given out by the detector when there is no incident light (hence dark frame). In many detectors this can also be a function of time, for example in astronomical telescopes it is common to take a dark-frame of the same time as the planned light exposure. The gain and dark-frame for optical systems can also be established by using a series of neutral density filters to give input/output signal information and applying a least squares fit to obtain the values for the dark current and gain.
{displaystyle C={{(R-D)*m} over {(F-D)}}={(R-D)*G}}
C = corrected image
R = raw image
F = flat field image
D = dark field or dark frame
m = image-averaged value of (F-D)
G = Gain =
{displaystyle m over (F-D)}
In this equation, capital letters are 2D matrices, and lowercase letters are scalars. All matrix operations are performed element-by-element.
In order for an astrophotographer to capture a light frame, he or she must place a light source over the imaging instrument’s objective lens such that the light source emanates evenly through the users optics. The photographer must then adjust the exposure of their imaging device (charge-coupled device (CCD) or digital single-lens reflex camera (DSLR) ) so that when the histogram of the image is viewed, a peak reaching about 40–70% of the dynamic range (maximum range of pixel values) of the imaging device is seen. The photographer typically takes 15–20 light frames and performs median stacking. Once the desired light frames are acquired, the objective lens is covered so that no light is allowed in, then 15–20 dark frames are taken, each of equal exposure time as a light frame. These are called Dark-Flat frames.
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Training and Testing a Neural Network for LLR Estimation - MATLAB & Simulink - MathWorks Deutschland
{l}_{i}\triangleq \mathrm{log}\left(\frac{{P}_{r}\left({c}_{i}=0|\underset{}{\overset{ˆ}{s}}\right)}{{P}_{r}\left({c}_{i}=1|\underset{}{\overset{ˆ}{s}}\right)}\right)
i=1,...,k
\underset{}{\overset{ˆ}{s}}
{c}_{i}
{i}^{th}
{l}_{i}\triangleq \mathrm{log}\left(\frac{\sum _{s\in {c}_{i}^{0}}\mathrm{exp}\left(-\frac{‖\underset{}{\overset{ˆ}{s}}-s{‖}_{2}^{2}}{{\sigma }^{2}}\right)}{\sum _{s\in {c}_{i}^{1}}\mathrm{exp}\left(-\frac{‖\underset{}{\overset{ˆ}{s}}-s{‖}_{2}^{2}}{{\sigma }^{2}}\right)}\right)
{\sigma }^{2}
\mathit{x}
\mathrm{log}\left(\sum _{j}\mathrm{exp}\left(-{x}_{j}^{2}\right)\right)\approx \underset{j}{\mathrm{max}}\left(-{x}_{j}^{2}\right)
{l}_{i}\approx \frac{1}{{\sigma }^{2}}\left(\underset{s\in {C}_{i}^{1}}{\mathrm{min}}‖\underset{}{\overset{ˆ}{s}}-s{‖}_{2}^{2}-\underset{s\in {C}_{i}^{0}}{\mathrm{min}}‖\underset{}{\overset{ˆ}{s}}-s{‖}_{2}^{2}\right)
±3\sigma
±3\sigma
\left[\underset{s\in C}{\mathrm{max}}\left(Re\left(s\right)+3\sigma \right)\underset{s\in C}{\mathrm{min}}\left(Re\left(s\right)-3\sigma \right)\right]+i\left[\underset{s\in C}{\mathrm{max}}\left(Im\left(s\right)+3\sigma \right)\underset{s\in C}{\mathrm{min}}\left(Im\left(s\right)-3\sigma \right)\right]
. Generate uniformly distributed I/Q symbols over this space and use qamdemod (Communications Toolbox) function to calculate exact LLR and approximate LLR values.
\mathit{k}×\mathit{N}
\mathit{k}
\mathit{N}
DVB-S.2 system uses a soft demodulator to generate inputs for the LDPC decoder. Simulate the packet error rate (PER) of a DVB-S.2 system with 16-APSK modulation and 2/3 LDPC code using exact LLR, approximate LLR, and LLRNet using llrNetDVBS2PER function. This function uses the comm.PSKDemodulator (Communications Toolbox) System object™ and the dvbsapskdemod (Communications Toolbox) function to calculate exact and approximate LLR values and the comm.AWGNChannel (Communications Toolbox) System object to simulate the channel.
For more information on the DVB-S.2 PER simulation, see the DVB-S.2 Link, Including LDPC Coding in Simulink (Communications Toolbox) example. For more information on training the network, refer to the llrnetTrainDVBS2LLRNetwork function and [1].
Try different modulation and coding schemes for the DVB-S.2 system. The full list of modulation types and coding rates are given in the DVB-S.2 Link, Including LDPC Coding in Simulink (Communications Toolbox) example. You can also try different sizes for the hidden layer of the network to reduce the number of operations and measure the performance loss as compared to exact LLR.
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LALR parser - Wikipedia
In computer science, an LALR parser[a] or Look-Ahead LR parser is a simplified version of a canonical LR parser, to parse a text according to a set of production rules specified by a formal grammar for a computer language. ("LR" means left-to-right, rightmost derivation.)
The LALR parser was invented by Frank DeRemer in his 1969 PhD dissertation, Practical Translators for LR(k) languages,[1] in his treatment of the practical difficulties at that time of implementing LR(1) parsers. He showed that the LALR parser has more language recognition power than the LR(0) parser, while requiring the same number of states as the LR(0) parser for a language that can be recognized by both parsers. This makes the LALR parser a memory-efficient alternative to the LR(1) parser for languages that are LALR. It was also proven that there exist LR(1) languages that are not LALR. Despite this weakness, the power of the LALR parser is sufficient for many mainstream computer languages,[2] including Java,[3] though the reference grammars for many languages fail to be LALR due to being ambiguous.[2]
The original dissertation gave no algorithm for constructing such a parser given a formal grammar. The first algorithms for LALR parser generation were published in 1973.[4] In 1982, DeRemer and Tom Pennello published an algorithm that generated highly memory-efficient LALR parsers.[5] LALR parsers can be automatically generated from a grammar by an LALR parser generator such as Yacc or GNU Bison. The automatically generated code may be augmented by hand-written code to augment the power of the resulting parser.
3 Relation to other parsers
3.2 LL parsers
In 1965, Donald Knuth invented the LR parser (Left to Right, Rightmost derivation). The LR parser can recognize any deterministic context-free language in linear-bounded time.[6] Rightmost derivation has very large memory requirements and implementing an LR parser was impractical due to the limited memory of computers at that time. To address this shortcoming, in 1969, Frank DeRemer proposed two simplified versions of the LR parser, namely the Look-Ahead LR (LALR)[1] and the Simple LR parser that had much lower memory requirements at the cost of less language-recognition power, with the LALR parser being the most-powerful alternative.[1] In 1977, memory optimizations for the LR parser were invented[7] but still the LR parser was less memory-efficient than the simplified alternatives.
In 1979, Frank DeRemer and Tom Pennello announced a series of optimizations for the LALR parser that would further improve its memory efficiency.[8] Their work was published in 1982.[5]
Generally, the LALR parser refers to the LALR(1) parser,[b] just as the LR parser generally refers to the LR(1) parser. The "(1)" denotes one-token lookahead, to resolve differences between rule patterns during parsing. Similarly, there is an LALR(2) parser with two-token lookahead, and LALR(k) parsers with k-token lookup, but these are rare in actual use. The LALR parser is based on the LR(0) parser, so it can also be denoted LALR(1) = LA(1)LR(0) (1 token of lookahead, LR(0)) or more generally LALR(k) = LA(k)LR(0) (k tokens of lookahead, LR(0)). There is in fact a two-parameter family of LA(k)LR(j) parsers for all combinations of j and k, which can be derived from the LR(j + k) parser,[9] but these do not see practical use.
As with other types of LR parsers, an LALR parser is quite efficient at finding the single correct bottom-up parse in a single left-to-right scan over the input stream, because it does not need to use backtracking. Being a lookahead parser by definition, it always uses a lookahead, with LALR(1) being the most-common case.
Relation to other parsers[edit]
LR parsers[edit]
The LALR(1) parser is less powerful than the LR(1) parser, and more powerful than the SLR(1) parser, though they all use the same production rules. The simplification that the LALR parser introduces consists in merging rules that have identical kernel item sets, because during the LR(0) state-construction process the lookaheads are not known. This reduces the power of the parser because not knowing the lookahead symbols can confuse the parser as to which grammar rule to pick next, resulting in reduce/reduce conflicts. All conflicts that arise in applying a LALR(1) parser to an unambiguous LR(1) grammar are reduce/reduce conflicts. The SLR(1) parser performs further merging, which introduces additional conflicts.
The standard example of an LR(1) grammar that cannot be parsed with the LALR(1) parser, exhibiting such a reduce/reduce conflict, is:[10][11]
S → a E c
→ a F d
→ b F c
→ b E d
In the LALR table construction, two states will be merged into one state and later the lookaheads will be found to be ambiguous. The one state with lookaheads is:
E → e. {c,d}
F → e. {c,d}
An LR(1) parser will create two different states (with non-conflicting lookaheads), neither of which is ambiguous. In an LALR parser this one state has conflicting actions (given lookahead c or d, reduce to E or F), a "reduce/reduce conflict"; the above grammar will be declared ambiguous by a LALR parser generator and conflicts will be reported.
To recover, this ambiguity is resolved by choosing E, because it occurs before F in the grammar. However, the resultant parser will not be able to recognize the valid input sequence b e c, since the ambiguous sequence e c is reduced to (E → e) c, rather than the correct (F → e) c, but b E c is not in the grammar.
LL parsers[edit]
The LALR(j) parsers are incomparable with LL(k) parsers: for any j and k both greater than 0, there are LALR(j) grammars that are not LL(k) grammars and conversely. In fact, it is undecidable whether a given LL(1) grammar is LALR(k) for any
{\displaystyle k>0}
Depending on the presence of empty derivations, a LL(1) grammar can be equal to a SLR(1) or a LALR(1) grammar. If the LL(1) grammar has no empty derivations it is SLR(1) and if all symbols with empty derivations have non-empty derivations it is LALR(1). If symbols having only an empty derivation exist, the grammar may or may not be LALR(1).[12]
^ "LALR" is pronounced as the initialism "el-ay-el-arr"
^ "LALR(1)" is pronounced as the initialism "el-ay-el-arr-one"
^ a b c DeRemer 1969.
^ a b c LR Parsing: Theory and Practice, Nigel P. Chapman, p. 86–87
^ "Generate the Parser". Eclipse JDT Project. Retrieved 29 June 2012.
^ Anderson, T.; Eve, J.; Horning, J. (1973). "Efficient LR(1) parsers". Acta Informatica (2): 2–39.
^ a b DeRemer, Frank; Pennello, Thomas (October 1982). "Efficient Computation of LALR(1) Look-Ahead Sets" (PDF). ACM Transactions on Programming Languages and Systems. 4 (4): 615–649. doi:10.1145/69622.357187.
^ Pager, D. (1977), "A Practical General Method for Constructing LR(k) Parsers", Acta Informatica 7, vol. 7, no. 3, pp. 249–268, doi:10.1007/BF00290336
^ Frank DeRemer, Thomas Pennello (1979), "Efficient Computation of LALR(1) Look-Ahead Sets", Sigplan Notices - SIGPLAN, vol. 14, no. 8, pp. 176–187
^ Parsing Techniques: A Practical Guide, by Dick Grune and Ceriel J. H. Jacobs, "9.7 LALR(1)", p. 302
^ "7.9 LR(1) but not LALR(1) Archived 4 August 2010 at the Wayback Machine", CSE 756: Compiler Design and Implementation Archived 23 July 2010 at the Wayback Machine, Eitan Gurari, Spring 2008
^ "Why is this LR(1) grammar not LALR(1)?"
^ (Beatty 1982)
DeRemer, Franklin L. (1969). Practical Translators for LR(k) languages (PDF) (PhD). MIT. Archived from the original (PDF) on 19 August 2013. Retrieved 13 November 2012.
Beatty, J. C. (1982). "On the relationship between LL(1) and LR(1) grammars" (PDF). Journal of the ACM. 29 (4 (Oct)): 1007–1022. doi:10.1145/322344.322350.
Parsing Simulator This simulator is used to generate parsing tables LALR and resolve the exercises of the book.
JS/CC JavaScript based implementation of a LALR(1) parser generator, which can be run in a web-browser or from the command-line.
LALR(1) tutorial at the Wayback Machine (archived May 7, 2021), a flash card-like tutorial on LALR(1) parsing.
Retrieved from "https://en.wikipedia.org/w/index.php?title=LALR_parser&oldid=1078075749"
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Understanding balloon-borne frost point hygrometer measurements after contamination...
Jorge, Teresa; Brunamonti, Simone; Poltera, Yann; Wienhold, Frank G.; Luo, Bei P.; Oelsner, Peter; Hanumanthu, Sreeharsha; Singh, Bhupendra B.; Körner, Susanne; Dirksen, Ruud; Naja, Manish; Fadnavis, Suvarna; Peter, Thomas
Balloon-borne water vapour measurements in the upper troposphere and lower stratosphere (UTLS) by means of frost point hygrometers provide important information on air chemistry and climate. However, the risk of contamination from sublimating hydrometeors collected by the intake tube may render these measurements unusable, particularly after crossing low clouds containing supercooled droplets. A large set of (sub)tropical measurements during the 2016–2017 StratoClim balloon campaigns at the southern slopes of the Himalayas allows us to perform an in-depth analysis of this type of contamination. We investigate the efficiency of wall contact and freezing of supercooled droplets in the intake tube and the subsequent sublimation in the UTLS using computational fluid dynamics (CFD). We find that the airflow can enter the intake tube with impact angles up to 60inline-formula∘, owing to the pendulum motion of the payload. Supercooled droplets with radii inline-formula> 70 inline-formulaµm, as they frequently occur in mid-tropospheric clouds, typically undergo contact freezing when entering the intake tube, whereas only about 50 % of droplets with 10 inline-formulaµm radius freeze, and droplets inline-formula< 5 inline-formulaµm radius mostly avoid contact. According to CFD, sublimation of water from an icy intake can account for the occasionally observed unrealistically high water vapour mixing ratios (inline-formula
M7inlinescrollmathml{\mathrm{italic \chi }}_{chem{\mathrm{normal H}}_{normal 2}\mathrm{normal O}}
24pt12ptsvg-formulamathimg51c17c5bd0e3115d2cc798564bb0460f amt-14-239-2021-ie00001.svg24pt12ptamt-14-239-2021-ie00001.png inline-formula> 100 inline-formulappmv) in the stratosphere. Furthermore, we use CFD to differentiate between stratospheric water vapour contamination by an icy intake tube and contamination caused by outgassing from the balloon and payload, revealing that the latter starts playing a role only during ascent at high altitudes (inline-formulap inline-formula< 20 inline-formulahPa).
Jorge, Teresa / Brunamonti, Simone / Poltera, Yann / et al: Understanding balloon-borne frost point hygrometer measurements after contamination by mixed-phase clouds. 2021. Copernicus Publications.
Rechteinhaber: Teresa Jorge et al.
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Big High War God shrine - The RuneScape Wiki
For the scenery, see Big High War God shrine (scenery).
A shrine to Da Big High War God.
Big High War God shrines are excavation hotspots at Warforge - North goblin tunnels that players can excavate with level 89 Archaeology.
The hotspots initially appear as earthen clay, requiring uncovering to become usable. Uncovering the hotspots yields a one-time reward of 288 Archaeology experience.
High priest crozier 10,500 3 N/A • Chief Tess (Hitty Fings)
• General Wartface (Green Gobbo Goodies II)
High priest mitre 10,500 4 N/A • Wise Old Man (Hat Problem)
High priest orb 10,500 3 N/A • Chief Tess (Hitty Fings)
High priest crozier (damaged) 1 1/3 Not sold Not alchemisable
High priest mitre (damaged) 1 1/3 Not sold Not alchemisable
High priest orb (damaged) 1 1/3 Not sold Not alchemisable
Saragorgak key 1 1/225[d 1] 1 1
^ Only obtained once. Guaranteed to drop after a success on the Big High War God Shrine with 90+ Archaeology.
{\displaystyle {\frac {L+E}{250{,}000}}}
{\displaystyle L}
{\displaystyle E}
{\displaystyle {\frac {1}{125{,}000}}}
{\displaystyle {\frac {1}{1{,}042}}}
template = Archaeology Hotspot Calculator Lite/Results form = Big_High_War_God_shrineArchForm result = ArchResult param=name|Hotspot Name|Big High War God shrine|combobox|Acropolis debris,Administratum debris,Aetherium forge,Amphitheatre debris,Ancient magick munitions,Animal trophies,Armarium debris,Aughra remains,Autopsy table,Bandos sanctum debris,Bibliotheke debris,Big High War God shrine,Byzroth remains,Carcerem debris,Castra debris,Ceramics studio debris,Chthonian trophies,Crucible stands debris,Culinarum debris,Cultist footlocker,Destroyed golem,Dis dungeon debris,Dis overspill,Dominion Games podium,Dragonkin coffin,Dragonkin reliquary,Experiment workbench,Flight research debris,Gladitorial goblin remains,Goblin dorm debris,Goblin trainee remains,Gravitron research debris,Hellfire forge,Howls workshop debris,Icyene weapon rack,Ikovian memorial,Infernal art,Keshik ger,Keshik weapon rack,Kharid-et chapel debris,Kyzaj champions boudoir,Legionary remains,Lodge art storage,Lodge bar storage,Makeshift pie oven,Moksha device,Monoceros remains,Oikos fishing hut remnants,Oikos studio debris,Optimatoi remains,Orcus altar,Pontifex remains,Praesidio remains,Praetorian remains,Prodromoi remains,Sacrificial altar,Saurthen debris,Stadio debris,Stockpiled art,Tailory debris,Tsutsaroth remains,Venator remains,Varanusaur remains,War table debris,Warforge scrap pile,Warforge weapon rack,Weapons research debris,Xolo mine,Xolo remains,Yubiusk animal pen param=method|Method|High intensity|buttonselect|High intensity,Medium intensity,AFK param=mattock|Mattock|Necronium|select|Bronze,Iron,Steel,Mithril,Adamant,Rune,Orikalkum,Dragon,Necronium,Crystal,Bane,Imcando,Elder Rune,Time and Space,Guildmaster Tony
Retrieved from ‘https://runescape.wiki/w/Big_High_War_God_shrine?oldid=35700600’
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Machine learning estimates of eddy covariance carbon flux in a scrub in...
Guevara-Escobar, Aurelio; González-Sosa, Enrique; Cervantes-Jiménez, Mónica; Suzán-Azpiri, Humberto; Queijeiro-Bolaños, Mónica Elisa; Carrillo-Ángeles, Israel; Cambrón-Sandoval, Víctor Hugo
Arid and semiarid ecosystems contain relatively high species diversity and are subject to intense use, in particular extensive cattle grazing, which has favored the expansion and encroachment of perennial thorny shrubs into the grasslands, thus decreasing the value of the rangeland. However, these environments have been shown to positively impact global carbon dynamics. Machine learning and remote sensing have enhanced our knowledge about carbon dynamics, but they need to be further developed and adapted to particular analysis. We measured the net ecosystem exchange (NEE) of C with the eddy covariance (EC) method and estimated gross primary production (GPP) in a thorny scrub at Bernal in Mexico. We tested the agreement between EC estimates and remotely sensed GPP estimates from the Moderate Resolution Imaging Spectroradiometer (MODIS), and also with two alternative modeling methods: ordinary-least-squares (OLS) regression and ensembles of machine learning algorithms (EMLs). The variables used as predictors were MODIS spectral bands, vegetation indices and products, and gridded environmental variables. The Bernal site was a carbon sink even though it was overgrazed, the average NEE during 15 months of 2017 and 2018 was inline-formula−0.78 inline-formula
M2inlinescrollmathmlunit\mathrm{normal g}\phantom{\rule{0ex}{0ex}}\mathrm{normal C}\phantom{\rule{0ex}{0ex}}{\mathrm{normal m}}^{-normal 2}\phantom{\rule{0ex}{0ex}}{\mathrm{normal d}}^{-normal 1}
56pt15ptsvg-formulamathimg15ac761ab043ccc04915a8227df2339e bg-18-367-2021-ie00001.svg56pt15ptbg-18-367-2021-ie00001.png , and the flux was negative or neutral during the measured months. The probability of agreement (inline-formulaθs) represented the agreement between observed and estimated values of GPP across the range of measurement. According to the mean value of inline-formulaθs, agreement was higher for the EML (0.6) followed by OLS (0.5) and then MODIS (0.24). This graphic metric was more informative than inline-formular2 (0.98, 0.67, 0.58, respectively) to evaluate the model performance. This was particularly true for MODIS because the maximum inline-formulaθs of 4.3 was for measurements of 0.8 inline-formula
M7inlinescrollmathmlunit\mathrm{normal g}\phantom{\rule{0ex}{0ex}}\mathrm{normal C}\phantom{\rule{0ex}{0ex}}{\mathrm{normal m}}^{-normal 2}\phantom{\rule{0ex}{0ex}}{\mathrm{normal d}}^{-normal 1}
56pt15ptsvg-formulamathimg1c7ec7db4a4b0be66e19c59f4e1acc18 bg-18-367-2021-ie00002.svg56pt15ptbg-18-367-2021-ie00002.png and then decreased steadily below 1 inline-formulaθs for measurements above 6.5 inline-formula
M9inlinescrollmathmlunit\mathrm{normal g}\phantom{\rule{0ex}{0ex}}\mathrm{normal C}\phantom{\rule{0ex}{0ex}}{\mathrm{normal m}}^{-normal 2}\phantom{\rule{0ex}{0ex}}{\mathrm{normal d}}^{-normal 1}
56pt15ptsvg-formulamathimgb8e1ecc86fefb11b9db1227eb7813bb1 bg-18-367-2021-ie00003.svg56pt15ptbg-18-367-2021-ie00003.png for this scrub vegetation. In the case of EML and OLS, the inline-formulaθs was stable across the range of measurement. We used an EML for the Ameriflux site US-SRM, which is similar in vegetation and climate, to predict GPP at Bernal, but inline-formulaθs was low (0.16), indicating the local specificity of this model. Although cacti were an important component of the vegetation, the nighttime flux was characterized by positive NEE, suggesting that the photosynthetic dark-cycle flux of cacti was lower than ecosystem respiration. The discrepancy between MODIS and EC GPP estimates stresses the need to understand the limitations of both methods.
Guevara-Escobar, Aurelio / González-Sosa, Enrique / Cervantes-Jiménez, Mónica / et al: Machine learning estimates of eddy covariance carbon flux in a scrub in the Mexican highland. 2021. Copernicus Publications.
Rechteinhaber: Aurelio Guevara-Escobar et al.
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Block LU decomposition - Wikipedia
Find sources: "Block LU decomposition" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this template message)
In linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula.
Block LDU decomposition[edit]
{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}I&0\\CA^{-1}&I\end{pmatrix}}{\begin{pmatrix}A&0\\0&D-CA^{-1}B\end{pmatrix}}{\begin{pmatrix}I&A^{-1}B\\0&I\end{pmatrix}}}
Block Cholesky decomposition[edit]
Consider a block matrix:
{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}I\\CA^{-1}\end{pmatrix}}\,A\,{\begin{pmatrix}I&A^{-1}B\end{pmatrix}}+{\begin{pmatrix}0&0\\0&D-CA^{-1}B\end{pmatrix}},}
{\displaystyle {\begin{matrix}A\end{matrix}}}
is assumed to be non-singular,
{\displaystyle {\begin{matrix}I\end{matrix}}}
is an identity matrix with proper dimension, and
{\displaystyle {\begin{matrix}0\end{matrix}}}
is a matrix whose elements are all zero.
We can also rewrite the above equation using the half matrices:
{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}A^{\frac {1}{2}}\\CA^{-{\frac {*}{2}}}\end{pmatrix}}{\begin{pmatrix}A^{\frac {*}{2}}&A^{-{\frac {1}{2}}}B\end{pmatrix}}+{\begin{pmatrix}0&0\\0&Q^{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&0\\0&Q^{\frac {*}{2}}\end{pmatrix}},}
where the Schur complement of
{\displaystyle {\begin{matrix}A\end{matrix}}}
in the block matrix is defined by
{\displaystyle {\begin{matrix}Q=D-CA^{-1}B\end{matrix}}}
and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition. The half matrices satisfy that
{\displaystyle {\begin{matrix}A^{\frac {1}{2}}\,A^{\frac {*}{2}}=A;\end{matrix}}\qquad {\begin{matrix}A^{\frac {1}{2}}\,A^{-{\frac {1}{2}}}=I;\end{matrix}}\qquad {\begin{matrix}A^{-{\frac {*}{2}}}\,A^{\frac {*}{2}}=I;\end{matrix}}\qquad {\begin{matrix}Q^{\frac {1}{2}}\,Q^{\frac {*}{2}}=Q.\end{matrix}}}
{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=LU,}
{\displaystyle LU={\begin{pmatrix}A^{\frac {1}{2}}&0\\CA^{-{\frac {*}{2}}}&0\end{pmatrix}}{\begin{pmatrix}A^{\frac {*}{2}}&A^{-{\frac {1}{2}}}B\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&Q^{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&0\\0&Q^{\frac {*}{2}}\end{pmatrix}}.}
{\displaystyle {\begin{matrix}LU\end{matrix}}}
can be decomposed in an algebraic manner into
{\displaystyle L={\begin{pmatrix}A^{\frac {1}{2}}&0\\CA^{-{\frac {*}{2}}}&Q^{\frac {1}{2}}\end{pmatrix}}\mathrm {~~and~~} U={\begin{pmatrix}A^{\frac {*}{2}}&A^{-{\frac {1}{2}}}B\\0&Q^{\frac {*}{2}}\end{pmatrix}}.}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Block_LU_decomposition&oldid=1062012844"
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Unboggler / Benjamin R. Bray
Unboggler
algorithms satisfiability games javascript
The Unboggler
Boggle is a classic board game in which players search for words constructed by connecting adjacent tiles in a grid of letters. Letters can be used at most once, and are allowed to connect horizontally, vertically, or diagonally. Listing all words in a Boggle board is a common interview question, easily achieved by querying a trie data structure assembled from a list of dictionary words. Use the widget below to see it in action!
Boggle Dice Frequency Uniform Long Word
PYBO ALGG MELW HTIE
Boggle was easy enough, so let's try Unboggle, or reverse boggle, where the goal is to reconstruct a Boggle board from a given list of words. There is no obvious way to divide-and-conquer the problem and no obvious greedy algorithm guaranteed to use all the words in the list. Instead, I encode Unboggle as a boolean satisfiability problem using logic-solver.js, a convenient wrapper around the general-purpose minisat solver, which has been compiled to JavaScript using emscripten.
Try it out below! For a 4x4 grid and about 10 words, you can expect the Unboggle search to take about 30 seconds. Since the solver isn't designed to detect unsatisfiability, the Unboggler may hang for much longer if no satisfying board exists!
THE- UN-- BOGG LER-
ANTIQUE EQUITY QUALITY EQUIVALENT ANTIQUITY DIVIDE QUILT
UNSOLVE?
SAT solvers search for a satisfying assignment of true and false values to a boolean formula containing literals, conjunction, disjunction, and negation. For example, the formula
(x_1 \vee x_2 \vee \neg x_3) \wedge (\neg x_1 \vee x_2 \vee \neg x_4) \wedge (\neg x_3 \vee \neg x_4 \vee x_5)
is satisfied by the assignment
x_1 = x_2 = \mathtt{true}
x_3 = x_4 = \mathtt{false}
. Integer constraints like
y=11
can be encoded as binary constraints on the individual binary digits
y = y_4 y_3 y_2 y_1
, like so:
y_4 \wedge \neg y_3 \wedge y_2 \wedge y_1
Thankfully, logic-solver.js takes care of binary encodings for us and supports basic arithmetic constraints like
(=, <, >, \leq, \geq)
, together with addition and subtraction.
To encode Unboggle as a satisfiability problem, suppose we wish to fit
m
words on an
n \times n
board. Assume an average word length
\ell
For each letter of every word on the list, we use two integer variables to represent the letter's grid coordinates, requiring
L = 2 m \ell \lceil \log_2 n \rceil
binary literals.
When the board size is not a power of two, each coordinate needs an upper bound.
Adjacent letters within the same word must be adjacent on the board.
A word cannot overlap itself; its coordinate pairs must all be distinct.
Two different words can only intersect at a common letter; that is, two different letters from distinct words cannot occupy the same space.
The number of constraints increases roughly quadratically with the number of words, so Unboggling more than ten words may take quite a while! The code below uses logic-solver.js to represent these constraints:
function encode(numRows, numCols, words){
let solver = new Logic.Solver();
let numWords = words.length;
let numRowBits = Math.ceil(Math.log2(numRows));
let numColBits = Math.ceil(Math.log2(numCols));
// bit constants
let const_1 = Logic.constantBits(1);
let const_numRows = Logic.constantBits(numRows);
let const_numCols = Logic.constantBits(numCols);
// word constraints
let wordPaths = [];
for(let w = 0; w < numWords; w++){
// translate word
let word = words[w];
let wordPath = [];
for(let i = 0; i < word.length; i++){
// location of ith letter stored in variables:
// p_(w)_(i)_r and p_(w)_(i)_c
let pr = Logic.variableBits(`p_${w}_${i}_r`, numRowBits);
let pc = Logic.variableBits(`p_${w}_${i}_c`, numColBits);
wordPath.push([pr,pc]);
// enforce board boundaries on paths (coordinates zero-indexed)
solver.require(Logic.lessThan(pr, const_numRows));
solver.require(Logic.lessThan(pc, const_numCols));
// adjacent letters in word must be adjacent on board
// x difference
solver.require(Logic.lessThanOrEqual(wordPath[i-1][0], Logic.sum(wordPath[i][0], const_1)));
solver.require(Logic.lessThanOrEqual(wordPath[i][0], Logic.sum(wordPath[i-1][0], const_1)));
// y difference
// path cannot overlap itself
for(let j = i+1; j < word.length; j++){
solver.require(Logic.atMostOne(
Logic.equalBits(wordPath[i][0],wordPath[j][0]),
Logic.equalBits(wordPath[i][1],wordPath[j][1])
wordPaths.push(wordPath);
// now, ensure word paths are all compatible
for(let w1 = 0; w1 < numWords; w1++){
for(let w2 = w1+1; w2 < numWords; w2++){
let word1 = words[w1];
for(let i = 0; i < word1.length; i++){
for(let j = 0; j < word2.length; j++){
// no constraint if words have letter in common
if(word1[i] == word2[j]){ continue; }
// prevent different letters from occupying same space
Logic.equalBits( wordPaths[w1][i][0], wordPaths[w2][j][0] ),
Logic.equalBits( wordPaths[w1][i][1], wordPaths[w2][j][1] )
solution: solver.solve(),
wordPaths: wordPaths
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Fourier-transform ion cyclotron resonance - Wikipedia
(Redirected from Fourier transform ion cyclotron resonance)
Instrument in mass spectrometry
Fourier-transform ion cyclotron resonance mass spectrometry is a type of mass analyzer (or mass spectrometer) for determining the mass-to-charge ratio (m/z) of ions based on the cyclotron frequency of the ions in a fixed magnetic field.[1] The ions are trapped in a Penning trap (a magnetic field with electric trapping plates), where they are excited (at their resonant cyclotron frequencies) to a larger cyclotron radius by an oscillating electric field orthogonal to the magnetic field. After the excitation field is removed, the ions are rotating at their cyclotron frequency in phase (as a "packet" of ions). These ions induce a charge (detected as an image current) on a pair of electrodes as the packets of ions pass close to them. The resulting signal is called a free induction decay (FID), transient or interferogram that consists of a superposition of sine waves. The useful signal is extracted from this data by performing a Fourier transform to give a mass spectrum.
3.2 Stored-waveform inverse Fourier transform
FT-ICR was invented by Melvin B. Comisarow[2] and Alan G. Marshall at the University of British Columbia. The first paper appeared in Chemical Physics Letters in 1974.[3] The inspiration was earlier developments in conventional ICR and Fourier-transform nuclear magnetic resonance (FT-NMR) spectrometry. Marshall has continued to develop the technique at The Ohio State University and Florida State University.
Linear ion trap – Fourier-transform ion cyclotron resonance mass spectrometer (panels around magnet are missing)
The physics of FTICR is similar to that of a cyclotron at least in the first approximation.
In the simplest idealized form, the relationship between the cyclotron frequency and the mass-to-charge ratio is given by
{\displaystyle f={\frac {qB}{2\pi m}},}
where f = cyclotron frequency, q = ion charge, B = magnetic field strength and m = ion mass.
This is more often represented in angular frequency:
{\displaystyle \omega _{\text{c}}={\frac {qB}{m}},}
{\displaystyle \omega _{\text{c}}}
is the angular cyclotron frequency, which is related to frequency by the definition
{\displaystyle f={\frac {\omega }{2\pi }}}
Because of the quadrupolar electrical field used to trap the ions in the axial direction, this relationship is only approximate. The axial electrical trapping results in axial oscillations within the trap with the (angular) frequency
{\displaystyle \omega _{\text{t}}={\sqrt {\frac {q\alpha }{m}}},}
{\displaystyle \alpha }
is a constant similar to the spring constant of a harmonic oscillator and is dependent on applied voltage, trap dimensions and trap geometry.
The electric field and the resulting axial harmonic motion reduces the cyclotron frequency and introduces a second radial motion called magnetron motion that occurs at the magnetron frequency. The cyclotron motion is still the frequency being used, but the relationship above is not exact due to this phenomenon. The natural angular frequencies of motion are
{\displaystyle \omega _{\pm }={\frac {\omega _{\text{c}}}{2}}\pm {\sqrt {\left({\frac {\omega _{\text{c}}}{2}}\right)^{2}-{\frac {\omega _{\text{t}}^{2}}{2}}}},}
{\displaystyle \omega _{\text{t}}}
is the axial trapping frequency due the axial electrical trapping and
{\displaystyle \omega _{+}}
is the reduced cyclotron (angular) frequency and
{\displaystyle \omega _{-}}
is the magnetron (angular) frequency. Again,
{\displaystyle \omega _{+}}
is what is typically measured in FTICR. The meaning of this equation can be understood qualitatively by considering the case where
{\displaystyle \omega _{\text{t}}}
is small, which is generally true. In that case the value of the radical is just slightly less than
{\displaystyle \omega _{\text{c}}/2}
{\displaystyle \omega _{+}}
is just slightly less than
{\displaystyle \omega _{\text{c}}}
(the cyclotron frequency has been slightly reduced). For
{\displaystyle \omega _{-}}
the value of the radical is the same (slightly less than
{\displaystyle \omega _{\text{c}}/2}
), but it is being subtracted from
{\displaystyle \omega _{\text{c}}/2}
, resulting in a small number equal to
{\displaystyle \omega _{\text{c}}-\omega _{+}}
(i.e. the amount that the cyclotron frequency was reduced by).
FTICR-MS differs significantly from other mass spectrometry techniques in that the ions are not detected by hitting a detector such as an electron multiplier but only by passing near detection plates. Additionally the masses are not resolved in space or time as with other techniques but only by the ion cyclotron resonance (rotational) frequency that each ion produces as it rotates in a magnetic field. Thus, the different ions are not detected in different places as with sector instruments or at different times as with time-of-flight instruments, but all ions are detected simultaneously during the detection interval. This provides an increase in the observed signal-to-noise ratio owing to the principles of Fellgett's advantage.[1] In FTICR-MS, resolution can be improved either by increasing the strength of the magnet (in teslas) or by increasing the detection duration.[4]
A cylindrical ICR cell. The walls of the cell are made of copper, and ions enter the cell from the right, transmitted by the octopole ion guides.
A review of different cell geometries with their specific electric configurations is available in the literature.[5] However, ICR cells can belong to one of the following two categories: closed cells or open cells.
Several closed ICR cells with different geometries were fabricated and their performance has been characterized. Grids were used as end caps to apply an axial electric field for trapping ions axially (parallel to the magnetic field lines). Ions can be either generated inside the cell or can be injected to the cell from an external ionization source. Nested ICR cells with double pair of grids were also fabricated to trap both positive and negative ions simultaneously.
The most common open cell geometry is a cylinder, which is axially segmented to produce electrodes in the shape of a ring. The central ring electrode is commonly used for applying radial excitation electric field and detection. DC electric voltage is applied on the terminal ring electrodes to trap ions along the magnetic field lines.[6] Open cylindrical cells with ring electrodes of different diameters have also been designed.[7] They proved not only capable in trapping and detecting both ion polarities simultaneously, but also they succeeded to separate positive from negative ions radially. This presented a large discrimination in kinetic ion acceleration between positive and negative ions trapped simultaneously inside the new cell. Several ion axial acceleration schemes were recently written for ion–ion collision studies.[8]
Stored-waveform inverse Fourier transform[edit]
Stored-waveform inverse Fourier transform (SWIFT) is a method for the creation of excitation waveforms for FTMS.[9] The time-domain excitation waveform is formed from the inverse Fourier transform of the appropriate frequency-domain excitation spectrum, which is chosen to excite the resonance frequencies of selected ions. The SWIFT procedure can be used to select ions for tandem mass spectrometry experiments.
Fourier-transform ion cyclotron resonance (FTICR) mass spectrometry is a high-resolution technique that can be used to determine masses with high accuracy. Many applications of FTICR-MS use this mass accuracy to help determine the composition of molecules based on accurate mass. This is possible due to the mass defect of the elements. FTICR-MS is able to achieve higher levels of mass accuracy than other forms of mass spectrometer, in part, because a superconducting magnet is much more stable than radio-frequency (RF) voltage.[10]
Another place that FTICR-MS is useful is in dealing with complex mixtures, such as biomass or waste liquefaction products, [11][12] since the resolution (narrow peak width) allows the signals of two ions with similar mass-to-charge ratios (m/z) to be detected as distinct ions.[13][14][15] This high resolution is also useful in studying large macromolecules such as proteins with multiple charges, which can be produced by electrospray ionization. For example, attomole level of detection of two peptides has been reported.[16] These large molecules contain a distribution of isotopes that produce a series of isotopic peaks. Because the isotopic peaks are close to each other on the m/z axis, due to the multiple charges, the high resolving power of the FTICR is extremely useful. FTICR-MS is very useful in other studies of proteomics as well. It achieves exceptional resolution in both top-down and bottom-up proteomics. Electron-capture dissociation (ECD), collisional-induced dissociation (CID), and infrared multiphoton dissociation (IRMPD) are all utilized to produce fragment spectra in tandem mass spectrometry experiments.[17] Although CID and IRMPD use vibrational excitation to further dissociate peptides by breaking the backbone amide linkages, which are typically low in energy and weak, CID and IRMPD may also cause dissociation of post-translational modifications. ECD, on the other hand, allows specific modifications to be preserved. This is quite useful in analyzing phosphorylation states, O- or N-linked glycosylation, and sulfating.[17]
^ a b Marshall, A. G.; Hendrickson, C. L.; Jackson, G. S. (1998). "Fourier transform ion cyclotron resonance mass spectrometry: a primer". Mass Spectrom. Rev. 17 (1): 1–35. Bibcode:1998MSRv...17....1M. doi:10.1002/(sici)1098-2787(1998)17:1<1::aid-mas1>3.0.co;2-k. PMID 9768511.
^ "UBC Chemistry Personnel: Melvin B. Comisarow". University of British Columbia. Retrieved 2009-11-05.
^ Comisarow, Melvin B. (1974). "Fourier transform ion cyclotron resonance spectroscopy". Chemical Physics Letters. 25 (2): 282–283. Bibcode:1974CPL....25..282C. doi:10.1016/0009-2614(74)89137-2.
^ Marshall, A. (2002). "Fourier transform ion cyclotron resonance detection: principles and experimental configurations". International Journal of Mass Spectrometry. 215 (1–3): 59–75. Bibcode:2002IJMSp.215...59M. doi:10.1016/S1387-3806(01)00588-7.
^ Guan, Shenheng; Marshall, Alan G. (1995). "Ion traps for Fourier transform ion cyclotron resonance mass spectrometry: principles and design of geometric and electric configurations". International Journal of Mass Spectrometry and Ion Processes. 146–147: 261–296. Bibcode:1995IJMSI.146..261G. doi:10.1016/0168-1176(95)04190-V.
^ Marshall, Alan G.; Hendrickson, Christopher L.; Jackson, George S. (1998). "Fourier transform ion cyclotron resonance mass spectrometry: A primer". Mass Spectrometry Reviews. 17 (1): 1–35. Bibcode:1998MSRv...17....1M. doi:10.1002/(SICI)1098-2787(1998)17:1<1::AID-MAS1>3.0.CO;2-K. ISSN 0277-7037. PMID 9768511.
^ Kanawati, B.; Wanczek, K. P. (2007). "Characterization of a new open cylindrical ion cyclotron resonance cell with unusual geometry". Review of Scientific Instruments. 78 (7): 074102–074102–8. Bibcode:2007RScI...78g4102K. doi:10.1063/1.2751100. PMID 17672776.
^ Kanawati, B.; Wanczek, K. (2008). "Characterization of a new open cylindrical ICR cell for ion–ion collision studies☆". International Journal of Mass Spectrometry. 269 (1–2): 12–23. Bibcode:2008IJMSp.269...12K. doi:10.1016/j.ijms.2007.09.007.
^ Cody, R. B.; Hein, R. E.; Goodman, S. D.; Marshall, Alan G. (1987). "Stored waveform inverse fourier transform excitation for obtaining increased parent ion selectivity in collisionally activated dissociation: Preliminary results". Rapid Communications in Mass Spectrometry. 1 (6): 99–102. Bibcode:1987RCMS....1...99C. doi:10.1002/rcm.1290010607.
^ Shi, S; Drader, Jared J.; Freitas, Michael A.; Hendrickson, Christopher L.; Marshall, Alan G. (2000). "Comparison and interconversion of the two most common frequency-to-mass calibration functions for Fourier transform ion cyclotron resonance mass spectrometry". International Journal of Mass Spectrometry. 195–196: 591–598. Bibcode:2000IJMSp.195..591S. doi:10.1016/S1387-3806(99)00226-2.
^ Leonardis, Irene; Chiaberge, Stefano; Fiorani, Tiziana; Spera, Silvia; Battistel, Ezio; Bosetti, Aldo; Cesti, Pietro; Reale, Samantha; De Angelis, Francesco (8 November 2012). "Characterization of Bio-oil from Hydrothermal Liquefaction of Organic Waste by NMR Spectroscopy and FTICR Mass Spectrometry". ChemSusChem. 6 (2): 160–167. doi:10.1002/cssc.201200314. PMID 23139164.
^ Sudasinghe, Nilusha; Cort, John; Hallen, Richard; Olarte, Mariefel; Schmidt, Andrew; Schaub, Tanner (1 December 2014). "Hydrothermal liquefaction oil and hydrotreated product from pine feedstock characterized by heteronuclear two-dimensional NMR spectroscopy and FT-ICR mass spectrometry". Fuel. 137: 60–69. doi:10.1016/j.fuel.2014.07.069.
^ Sleno L., Volmer D. A., Marshall A. G. (February 2005). "Assigning product ions from complex MS/MS spectra: the importance of mass uncertainty and resolving power". J. Am. Soc. Mass Spectrom. 16 (2): 183–98. doi:10.1016/j.jasms.2004.10.001. PMID 15694769. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ Bossio R. E., Marshall A. G. (April 2002). "Baseline resolution of isobaric phosphorylated and sulfated peptides and nucleotides by electrospray ionization FTICR ms: another step toward mass spectrometry-based proteomics". Anal. Chem. 74 (7): 1674–9. doi:10.1021/ac0108461. PMID 12033259. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ He F., Hendrickson C. L., Marshall A. G. (February 2001). "Baseline mass resolution of peptide isobars: a record for molecular mass resolution". Anal. Chem. 73 (3): 647–50. doi:10.1021/ac000973h. PMID 11217775. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ Solouki T., Marto J. A., White F. M., Guan S., Marshall A. G. (November 1995). "Attomole biomolecule mass analysis by matrix-assisted laser desorption/ionization Fourier transform ion cyclotron resonance". Anal. Chem. 67 (22): 4139–44. doi:10.1021/ac00118a017. PMID 8633766. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ a b Scigelova, M.; Hornshaw, M.; Giannakopulos, A.; Makarov, A. (2011). "Fourier Transform Mass Spectrometry". Molecular & Cellular Proteomics. 10 (7): M111.009431. doi:10.1074/mcp.M111.009431. ISSN 1535-9476. PMC 3134075. PMID 21742802.
What's in an Oil Drop? An Introduction to Fourier Transform Ion Cyclotron Resonance (FT-ICR) for Non-scientists National High Magnetic Field Laboratory
Scottish Instrumentation Resource Centre for Advanced Mass Spectrometry
Fourier-transform Ion Cyclotron Resonance (FT-ICR) FT-ICR Introduction University of Bristol
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fourier-transform_ion_cyclotron_resonance&oldid=1070313346"
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Complex Domain Coloring / Benjamin R. Bray
math complex-analysis image-processing javascript
f : \R \rightarrow \R
with real inputs can be easily visualized on a two-dimensional graph. Real-valued complex functions
f : \C \rightarrow \R
have two input dimensions and one output dimension, so can be visualized as a three-dimensional surface. Functions
f : \C \rightarrow \C
with both complex inputs and complex outputs have four dimensions to consider, making them difficult to visualize directly. One popular visualization technique for complex functions is domain coloring, which uses color to represent the value a function takes at each point in the complex plane. Domain coloring can help us build visual intuition about complex analysis.
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Uplink perfect channel estimation - MATLAB lteULPerfectChannelEstimate - MathWorks India
lteULPerfectChannelEstimate
Perform Uplink Perfect Channel Estimation
Perform Uplink Perfect Channel Estimation on Time Offset Waveform
Perform Uplink Perfect Channel Estimation for NB-IoT configuration
Uplink perfect channel estimation
hest = lteULPerfectChannelEstimate(ue,channel)
hest = lteULPerfectChannelEstimate(ue,channel,offset)
hest = lteULPerfectChannelEstimate(ue,chs,channel)
hest = lteULPerfectChannelEstimate(ue,chs,channel,offset)
hest = lteULPerfectChannelEstimate(ue,channel) performs perfect channel estimation for a system configuration given user-equipment-specific (UE-specific) settings ue and propagation channel configuration channel. The perfect channel estimates are produced only for fading channel models created using the lteFadingChannel function.
This function provides a perfect multiple-input-multiple-output (MIMO) channel estimate after single-carrier frequency-division multiple access (SC-FDMA) modulation. To obtain this estimate, the function sets the channel with the specified configuration and sends a set of known symbols through that channel for each transmit antenna in turn.
hest = lteULPerfectChannelEstimate(ue,channel,offset) performs perfect channel estimation for the timing and frequency offset specified by offset. Specifying offset guarantees that hest is the channel that results when the receiver is precisely synchronized.
hest = lteULPerfectChannelEstimate(ue,chs,channel) performs perfect channel estimation for channel transmission configuration chs. This syntax supports SC-FDMA for LTE, single-tone narrowband Internet of Things (NB-IoT), and multitone NB-IoT.
hest = lteULPerfectChannelEstimate(ue,chs,channel,offset) performs perfect channel estimation for the channel transmission configuration and the specified timing and frequency offset.
Perform uplink perfect channel estimation for a chosen propagation channel configuration.
Initialize UE-specific settings, specifying fields appropriate for an LTE uplink configuration.
Specify propagation channel conditions.
channel.DelayProfile = 'EPA';
channel.DopplerFreq = 5.0;
channel.InitTime = 0.0;
Perform uplink perfect channel estimation and display the dimension of the channel estimate array.
hest = lteULPerfectChannelEstimate(ue,channel);
disp(size(hest));
Perform uplink perfect channel estimation on a time offset waveform passed through a fading channel.
Initialize UE-specific settings by specifying fields appropriate for an LTE uplink configuration.
ue = lteRMCUL('A1-1','FDD',1);
Specify the propagation channel configuration.
channel.MIMOCorrelation = 'UplinkMedium';
Create a waveform and add samples for channel delay.
[txWaveform,txgrid,rmcCfg] = lteRMCULTool(ue,[1;0;0;1]);
txWaveform = [txWaveform; zeros(25,4)];
channel.SamplingRate = rmcCfg.SamplingRate;
Pass the waveform through a fading channel, generating time-domain receiver samples.
rxWaveform = lteFadingChannel(channel,txWaveform);
Use the lteULFrameOffset function to estimate time offset.
offset = lteULFrameOffset(ue,ue.PUSCH,rxWaveform);
disp(offset);
Modify the received waveform to account for the timing offset.
Demodulation and Uplink Perfect Channel Estimation
Generate frequency-domain receiver data by demodulating the received time-domain waveform.
grid = lteSCFDMADemodulate(ue,rxWaveform);
Perform uplink perfect channel estimation with the specified time offset.
hest = lteULPerfectChannelEstimate(ue,channel,offset);
120 14 2 4
Visualize Effect of Fading Channel
Plot resource element grids to show the impact of the fading channel on the transmitted signal and recovery of the signal using the perfect channel estimate.
The output channel estimate is a 4-D array. The input specified ten resource blocks leading to 120 subcarriers per symbol. Normal cyclic prefix results in 14 symbols per subframe. The third and fourth dimensions represent the two receive and four transmit antennas specified in the input configuration structures.
Comparing the transmitted grid to the recovered grid shows how equalization of the received grid with the perfect channel estimate recovers the transmission.
recoveredgrid = grid./hest;
surf(abs(txgrid(:,:,1,1)))
title('Transmitted Grid')
surf(abs(grid(:,:,1,1)))
surf(abs(hest(:,:,1,1)))
title('Perfect Channel Estimate')
surf(abs(recoveredgrid(:,:,1,1)))
title('Recovered Grid')
Initialize UE-specific settings, specifying fields appropriate for an NB-IoT uplink configuration.
ue.TotSlots = 10;
Specify NPUSCH configuration information.
chs.NBULSubcarrierSet = 0;
chs.NULSlots = 2;
chs.NRU = 2;
chs.NRep = 1;
hest = lteULPerfectChannelEstimate(ue,chs,channel);
UE-specific settings, specified as a structure. The fields you specify in ue determine whether the function performs channel estimation for an LTE or NB-IoT configuration. To indicate an LTE configuration, specify the NULRB field. To indicate an NB-IoT configuration, specify the NBULSubcarrierSpacing field. The NTxAnts field is required for both LTE and NB-IoT configurations. The other fields in ue are optional. The CyclicPrefixUL and TotSubframes fields are applicable only for an LTE configuration. The TotSlots field is applicable only for an NB-IoT configuration.
{N}_{\text{RB}}^{\text{UL}}
, specified as an integer in the interval [6, 110]. To perform channel estimation for an LTE configuration, you must specify this field.
NTxAnts — Number of transmit antennas
Number of transmit antennas, NTX, specified as 1, 2, or 4.
TotSubframes — Total number of subframes to generate
Total number of subframes to generate, specified as a nonnegative integer.
To perform channel estimation for an NB-IoT configuration, you must specify this field. To indicate an LTE configuration, omit this field.
TotSlots — Total number of slots to generate
Total number of slots to generate, specified as a nonnegative integer.
channel — Propagation channel configuration structure
Propagation channel configuration, specified as a structure. This argument must contain all the fields required to parameterize the fading channel model, that is, to call the lteFadingChannel function.
Before execution of the channel, lteULPerfectChannelEstimate sets the SamplingRate field internally to the sampling rate of the time domain waveform passed to the lteFadingChannel function for filtering. Therefore, this channel input does not require the SamplingRate field. If one is included, it is not used.
NRxAnts — Number of receive antennas
Number of receive antennas, NRX, specified as a positive integer.
MIMOCorrelation — Correlation between UE and eNodeB antennas
'Low' | 'Medium' | 'UplinkMedium' | 'High' | 'Custom'
Correlation between UE and Evolved Node B (eNodeB) antennas, specified as one of these values:
'Low' – No correlation between antennas
'Medium' – Correlation level is applicable to tests defined in TS 36.101 [1]
'UplinkMedium' – Correlation level is applicable to tests defined in TS 36.104 [2]
'High' – Strong correlation between antennas
'Custom' – Apply user-defined TxCorrelationMatrix and RxCorrelationMatrix
NormalizeTxAnts — Transmit antenna number normalization
Transmit antenna number normalization, specified as 'On' or 'Off'. If you specify NormalizeTxAnts as 'On', lteULPerfectChannelEstimate normalizes the model output by 1/√NTX. Normalization by the number of transmit antennas ensures that the output power per receive antenna is unaffected by the number of transmit antennas. If you specify NormalizeTxAnts as 'Off', lteULPerfectChannelEstimate does not perform normalization. This field is optional.
'EPA' | 'EVA' | 'ETU' | 'Custom' | 'Off'
Delay profile model, specified as 'EPA', 'EVA', 'ETU', 'Custom', or 'Off'. For more information, see Propagation Channel Models.
Setting DelayProfile to 'Off' switches off fading completely and implements a static MIMO channel model. In this case, the antenna geometry corresponds to the MIMOCorrelation and NRxAnts fields, and the number of transmit antennas. The temporal part of the model for each link between transmit and receive antennas consists of a single path with zero delay and constant unit gain.
DopplerFreq — Maximum Doppler frequency
Maximum Doppler frequency, in Hz, specified as a nonnegative scalar.
This field applies only when you specify the DelayProfile field as a value other than 'Off'.
SamplingRate — Sampling rate of input signal
Sampling rate of input signal, specified as a nonnegative scalar.
InitTime — Fading process time offset
Fading process time offset, in seconds, specified as a nonnegative scalar.
NTerms — Number of oscillators used in fading path modeling
16 (default) | power of two
Number of oscillators used in fading path modeling, specified as a power of two. This field is optional
ModelType — Rayleigh fading model type
'GMEDS' (default) | 'Dent'
Rayleigh fading model type, specified as 'GMEDS' or 'Dent'. To model Rayleigh fading using the generalized method of exact Doppler spread (GMEDS) described in [4], specify ModelType as 'GMEDS'. To model Rayleigh fading using the modified Jakes fading model described in [3], specify ModelType as 'Dent'. This field is optional.
Specifying ModelType as 'Dent' is not recommended.
NormalizePathGains — Model output normalization indicator
Model output normalization indicator, specified as 'On' or 'Off'. To normalize the model output such that the average power is unity, specify NormalizePathGains as 'On'. To return the average output power as the sum of the powers of the taps of the delay profile, specify NormalizePathGains as 'Off'. This field is optional.
InitPhase — Phase initialization for sinusoidal components of model
'Random' (default) | real-valued scalar | 4-D array
Phase initialization for the sinusoidal components of the model, specified as one of these values:
'Random' – Randomly initialize the phases according to the value you specify in the Seed field
A real-valued scalar – Specify the single initial value of the phases of all components, in radians
An N-by-L-by-NTX-by-NRX array – Explicitly initialize the phase, in radians, of each component. In this case, N is the number of phase initialization values per path and L is the number of paths
When you specify ModelType as 'GMEDS', N = 2×NTerms.
When you specify ModelType as 'Dent', N = NTerms.
Random number generator seed, specified as a real-valued scalar. To use a random seed, specify Seed as 0.
Seed values in the interval [0, 231 – 1 – (K(K – 1)/2)], where K = NTX × NRX and is the product of the number of transmit and receive antennas, are recommended. Seed values outside of this interval are not guaranteed to give distinct results.
This field applies only when you specify the DelayProfile field as a value other than 'Off' and the InitPhase field as 'Random'.
AveragePathGaindB — Average gains of the discrete paths
Average gains of the discrete paths, in dB, specified as a real-valued vector.
This field applies only when you specify the DelayProfile field as 'Custom'.
PathDelays — Delays of discrete paths
Delays of the discrete paths, in seconds, specified as a real-valued vector.
TxCorrelationMatrix — Correlation between each of the transmit antennas
NTX-by-NTX complex-valued matrix
Correlation between each of the transmit antennas, specified as an NTX-by-NTX complex-valued matrix.
This field applies only when you specify the MIMOCorrelation field as 'Custom'.
RxCorrelationMatrix — Correlation between each of the receive antennas
NRX-by-NRX complex-valued matrix
Correlation between each of the receive antennas, specified as an NRX-by-NRX complex-valued matrix.
Timing offset, in samples, specified as a nonnegative integer. The timing offset is specified from the start of the output of the channel to the estimated SC-FDMA demodulation starting point. Specify the timing offset, when known, to obtain the perfect channel estimate as seen by a synchronized receiver. Use the lteULFrameOffset function to derive the value for offset.
NPUSCH information, specified as a structure. For an NB-IoT configuration, you can set additional uplink-specific parameters by specifying the NB-IoT-specific fields in chs. Except for the NBULSubcarrierSet field, the fields in chs are applicable either when ue.NBULSubcarrierSpacing is '3.75kHz' or when ue.NBULSubcarrierSpacing is '15kHz' and length(NBULSubcarrierSet) is 1.
NB-IoT uplink subcarrier indices, specified as a vector of nonnegative integers in the interval [0, 11] or a nonnegative integer in the interval [0, 47]. The indices are in zero-based form. To use lteULPerfectChannelEstimate for a single-tone NB-IoT configuration, you must specify NBULSubcarrierSet as a scalar. If you do not specify NBULSubcarrierSet, lteULPerfectChannelEstimate returns an estimate for a multi-tone NB-IoT configuration by default.If you specify ue.NBULSubcarrierSpacing as '15kHz', this field is required.
Number of slots per resource unit (RU), specified as a positive integer. To use lteULPerfectChannelEstimate for a single-tone NB-IoT configuration, you must specify this field.
Number of RUs, specified as a positive integer. To use lteULPerfectChannelEstimate for a single-tone NB-IoT configuration, you must specify this field.
Number of repetitions for a codeword, specified as a nonnegative integer. To use lteULPerfectChannelEstimate for a single-tone NB-IoT configuration, you must specify this field.
SlotIdx — Relative slot index in an NPUSCH bundle
Relative slot index in an NPUSCH bundle, specified as a nonnegative integer. This field determines the zero-based relative slot index in a bundle of time slots for transmission of a transport block or control information bit. This field is optional.
complex-valued 4-D array
Perfect channel estimate, returned as an NSC-by-NSYM-by-NRX-by-NTX complex-valued array, where NSC is the number of subcarriers and NSYM is the number of SC-FDMA symbols.
[1] 3GPP TS 36.101. “User Equipment (UE) Radio Transmission and Reception.” 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA).
[3] Dent, P., Bottomley, G. E., and Croft, T. “Jakes Fading Model Revisited.” Electronics Letters. Vol. 29, Number 13, 1993, pp. 1162–1163.
[4] Pätzold, M., Wang, C., and Hogstad, B. O. “Two New Sum-of-Sinusoids-Based Methods for the Efficient Generation of Multiple Uncorrelated Rayleigh Fading Waveforms.” IEEE Transactions on Wireless Communications. Vol. 8, Number 6, 2009, pp. 3122–3131.
lteULChannelEstimate | lteULChannelEstimatePUCCH1 | lteULChannelEstimatePUCCH2 | lteULChannelEstimatePUCCH3 | lteDLPerfectChannelEstimate
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Table 1 Participant demographic and baseline characteristics
\overline{\mathit{x}}±\mathit{s}
Age, years 65.80 ± 7.45 64.55 ± 8.38 65.17 ± 7.89
Target knees, n (%)
1 knee 12 (21.82) 18 (32.73) 30 (27.27)
Both knees 43 (78.18) 37 (67.27) 80 (72.72)
Length of osteoarthritis diagnosis, n (%)
<5 years 30 (54.55) 37 (67.27) 67 (60.90)
6-10 years 18 (32.73) 12 (21.82) 30 (27.27)
>10 years 7 (12.73) 6 (1.090) 13 (11.82)
Weight, kg 64.06 ± 9.02 66.01 ± 5.21 65.04 ± 6.33
Height, cm 1.63 ± 5.28 1.62 ± 1.45 1.62 ± 7. 98
Body mass index 24.11 ± 1.08 25.15 ± 2.41 24.63 ± 5.52
Outcomes (baseline)
WOMAC pain score 6.73 ± 2.35 6.29 ± 2.70 6.51 ± 2.53
WOMAC function score 33.47.0 ± 15.37 30.99 ± 17.82 32.23 ± 16.61
There were no differences between the groups in age, gender, course of disease, or condition of the diseased knee (P >0.05). There were no differences between the groups in Western Ontario and McMaster Universities’ Osteoarthritis Index (WOMAC) pain or physical function scores (P >0.05).
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Perfect matching - Wikipedia
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G = (V, E), a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M.
A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used.
Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs:[1]
In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched.
A perfect matching is also a minimum-size edge cover. If there is a perfect matching, then both the matching number and the edge cover number equal |V | / 2.
A perfect matching can only occur when the graph has an even number of vertices. A near-perfect matching is one in which exactly one vertex is unmatched. This can only occur when the graph has an odd number of vertices, and such a matching must be maximum. In the above figure, part (c) shows a near-perfect matching. If, for every vertex in a graph, there is a near-perfect matching that omits only that vertex, the graph is also called factor-critical.
3 Perfect matching polytope
Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching.
The Tutte theorem provides a characterization for arbitrary graphs.
A perfect matching is a spanning 1-regular subgraph, a.k.a. a 1-factor. In general, a spanning k-regular subgraph is a k-factor.
A spectral characterization for a graph to have a perfect matching is given by Hassani Monfared and Mallik as follows: Let
{\displaystyle G}
be a graph on eve{\displaystyle n}
{\displaystyle \lambda _{1}>\lambda _{2}>\ldots >\lambda _{\frac {n}{2}}>0}
{\displaystyle {\frac {n}{2}}}
distinct nonzero purely imaginary numbers. Then
{\displaystyle G}
has a perfect matching if and only if there is a real skew-symmetric matrix
{\displaystyle A}
{\displaystyle G}
{\displaystyle \pm \lambda _{1},\pm \lambda _{2},\ldots ,\pm \lambda _{\frac {n}{2}}}
.[2] Note that the (simple) graph of a real symmetric or skew-symmetric matrix
{\displaystyle A}
{\displaystyle n}
{\displaystyle n}
vertices and edges given by the nonzero off-diagonal entries of
{\displaystyle A}
Deciding whether a graph admits a perfect matching can be done in polynomial time, using any algorithm for finding a maximum cardinality matching.
However, counting the number of perfect matchings, even in bipartite graphs, is #P-complete. This is because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix.
A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm.
The number of perfect matchings in a complete graph Kn (with n even) is given by the double factorial:
{\displaystyle (n-1)!!}
Perfect matching polytope[edit]
Main article: Matching polytope
The perfect matching polytope of a graph is a polytope in R|E| in which each corner is an incidence vector of a perfect matching.
Envy-free matching
Maximum-cardinality matching
Perfect matching in high-degree hypergraphs
Hall-type theorems for hypergraphs
^ Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5.
^ Keivan Hassani Monfared and Sudipta Mallik, Theorem 3.6, Spectral characterization of matchings in graphs, Linear Algebra and its Applications 496 (2016) 407–419, https://doi.org/10.1016/j.laa.2016.02.004
Retrieved from "https://en.wikipedia.org/w/index.php?title=Perfect_matching&oldid=1047395523"
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Mortar (masonry) - PiPiWiki
Enlarge Firestop Wikipedia:Citation needed
For any other use, see Mortar (disambiguation).
Mortar holding weathered bricks
Mortar is a workable paste which dries to bind building blocks such as stones, bricks, and concrete masonry units, to fill and seal the irregular gaps between them, and sometimes to add decorative colors or patterns to masonry walls. In its broadest sense, mortar includes pitch, asphalt, and soft mud or clay, as used between mud bricks. The word "mortar" comes from Latin mortarium, meaning crushed.
Cement mortar becomes hard when it cures, resulting in a rigid aggregate structure; however, the mortar functions as a weaker component than the building blocks and serves as the sacrificial element in the masonry, because mortar is easier and less expensive to repair than the building blocks. Bricklayers typically make mortars using a mixture of sand, a binder, and water. The most common binder since the early 20th century is Portland cement, but the ancient binder lime mortar is still used in some specialty new construction. Lime, lime mortar and gypsum in the form of plaster of Paris are used particularly in the repair and repointing of historic buildings and structures so that the repair materials will be similar in performance and appearance to the original materials. Several types of cement mortars and additives exist.
1 Ancient mortar
2 Ordinary Portland cement mortar
3 Polymer cement mortar
4 Lime mortar
5 Pozzolanic mortar
6 Firestop mortar
Ancient mortar
Roman mortar on display at Chetham's School of Music.
Workers prepare mortar in a trough. A 10th-century sculpture from the Korogho church, Georgia.
The first mortars were made of mud and clay,[1] as demonstrated in the 10th millennia BCE buildings of Jericho, and the 8th millennia BCE of Ganj Dareh.[1]
According to Roman Ghirshman, the first evidence of humans using a form of mortar was at the Mehrgarh of Baluchistan in the Indus Valley, Pakistan, built of sun-dried bricks in 6500 BCE.[2]
Gypsum mortar, also called plaster of Paris, was used in the construction of many ancient structures. It is made from gypsum, which requires a lower firing temperature. It is therefore easier to make than lime mortar and sets up much faster, which may be a reason it was used as the typical mortar in ancient, brick arch and vault construction. Gypsum mortar is not as durable as other mortars in damp conditions.[3]
In the Indian subcontinent, multiple cement types have been observed in the sites of the Indus Valley Civilization, with gypsum appearing at sites such as the Mohenjo-daro city-settlement that dates to earlier than 2600 BCE.
Gypsum cement that was "light grey and contained sand, clay, traces of calcium carbonate, and a high percentage of lime" was used in the construction of wells, drains, and on the exteriors of "important looking buildings." Bitumen mortar was also used at a lower-frequency, including in the Great Bath at Mohenjo-daro.[4][5]
In early Egyptian pyramids, which were constructed during the Old Kingdom (~2600–2500 BCE), the limestone blocks were bound by a mortar of mud and clay, or clay and sand.[6] In later Egyptian pyramids, the mortar was made of gypsum, or lime.[7] Gypsum mortar was essentially a mixture of plaster and sand and was quite soft.
2nd millennia BCE Babylonian constructions used lime or pitch for mortar.
Historically, building with concrete and mortar next appeared in Greece. The excavation of the underground aqueduct of Megara revealed that a reservoir was coated with a pozzolanic mortar 12 mm thick. This aqueduct dates back to c. 500 BCE.[8] Pozzolanic mortar is a lime based mortar, but is made with an additive of volcanic ash that allows it to be hardened underwater; thus it is known as hydraulic cement. The Greeks obtained the volcanic ash from the Greek islands Thira and Nisiros, or from the then Greek colony of Dicaearchia (Pozzuoli) near Naples, Italy. The Romans later improved the use and methods of making what became known as pozzolanic mortar and cement.[7] Even later, the Romans used a mortar without pozzolana using crushed terra cotta, introducing aluminum oxide and silicon dioxide into the mix. This mortar was not as strong as pozzolanic mortar, but, because it was denser, it better resisted penetration by water.[9]
Hydraulic mortar was not available in ancient China, possibly due to a lack of volcanic ash. Around 500 CE, sticky rice soup was mixed with slaked lime to make an inorganic−organic composite sticky rice mortar that had more strength and water resistance than lime mortar.[10][11]
It is not understood how the art of making hydraulic mortar and cement, which was perfected and in such widespread use by both the Greeks and Romans, was then lost for almost two millennia. During the Middle Ages when the Gothic cathedrals were being built, the only active ingredient in the mortar was lime. Since cured lime mortar can be degraded by contact with water, many structures suffered over the centuries from wind-blown rain.
Ordinary Portland cement mortar
Laying bricks with Portland cement mortar
Mortar mixed inside a 5-gallon bucket using clean water and mortar from a bag. When it's the right consistency, as in the photo (trowel stands up), it's ready to apply.
Ordinary Portland cement mortar, commonly known as OPC mortar or just cement mortar, is created by mixing powdered Ordinary Portland Cement, fine aggregate and water.
It was invented in 1794 by Joseph Aspdin and patented on 18 December 1824, largely as a result of efforts to develop stronger mortars. It was made popular during the late nineteenth century, and had by 1930 became more popular than lime mortar as construction material. The advantages of Portland cement is that it sets hard and quickly, allowing a faster pace of construction. Furthermore, fewer skilled workers are required in building a structure with Portland cement.
As a general rule, however, Portland cement should not be used for the repair or repointing of older buildings built in lime mortar, which require the flexibility, softness and breathability of lime if they are to function correctly.[12][13]
In the United States and other countries, five standard types of mortar (available as dry pre-mixed products) are generally used for both new construction and repair. Strengths of mortar change based on the mix ratio for each type of mortar, which are specified under the ASTM standards. These premixed mortar products are designated by one of the five letters, M, S, N, O, and K. Type M mortar is the strongest, and Type K the weakest. The mix ratio is always expressed by volume of
{\displaystyle {\text{Portland cement : lime : sand}}}
These type letters are apparently taken from the alternate letters of the words "MaSoN wOrK". [14]
Polymer cement mortars (PCM) are the materials which are made by partially replacing the cement hydrate binders of conventional cement mortar with polymers. The polymeric admixtures include latexes or emulsions, redispersible polymer powders, water-soluble polymers, liquid thermoset resins and monomers. Polymer mortar has low permeability that may be detrimental to moisture accumulation when used to repair a traditional brick, block or stone wall. It is mainly designed for repairing concrete structures.
Main article: Lime mortar
It would be problematic to use Portland cement mortars to repair older buildings originally constructed using lime mortar. Lime mortar is softer than cement mortar, allowing brickwork a certain degree of flexibility to adapt to shifting ground or other changing conditions. Cement mortar is harder and allows little flexibility. The contrast can cause brickwork to crack where the two mortars are present in a single wall.
Lime mortar is considered breathable in that it will allow moisture to freely move through and evaporate from the surface. In old buildings with walls that shift over time, cracks can be found which allow rain water into the structure. The lime mortar allows this moisture to escape through evaporation and keeps the wall dry. Re−pointing or rendering an old wall with cement mortar stops the evaporation and can cause problems associated with moisture behind the cement.
Pozzolanic mortar
Main article: Pozzolana
Pozzolana is a fine, sandy volcanic ash. It was originally discovered and dug at Pozzuoli, nearby Mount Vesuvius in Italy, and was subsequently mined at other sites, too. The Romans learned that pozzolana added to lime mortar allowed the lime to set relatively quickly and even under water. Vitruvius, the Roman architect, spoke of four types of pozzolana. It is found in all the volcanic areas of Italy in various colours: black, white, grey and red. Pozzolana has since become a generic term for any siliceous and/or aluminous additive to slaked lime to create hydraulic cement.[15]
Finely ground and mixed with lime it is a hydraulic cement, like Portland cement, and makes a strong mortar that will also set under water.
A Firestop is a kind of passive fire protection measure. Firestop mortars are mortars most typically used to firestop large openings in walls and floors required to have a fire-resistance rating. Firestop mortars differ in formula and properties from most other cementitious substances[citation needed] and cannot be substituted with generic mortars without violating the listing and approval use and compliance.
Firestop mortar is usually a combination of powder mixed with water, forming a cementitious stone which dries hard. It is sometimes mixed with lightweight aggregates, such as perlite or vermiculite[citation needed]. It is sometimes pigmented to distinguish it from generic materials[citation needed] in an effort to prevent unlawful substitution and to enable verification of the certification listing.
Mortar constituents.
Firestopped cable tray penetration. The cables and the tray are penetrants.
Cable tray cross barrier firestop test, full scale wall
As the mortar hardens, the current atmosphere is encased in the mortar and thus provides a sample for analysis. Various factors affect the sample and raise the margin of error for the analysis.[16][17][18][19] The possibility to use radiocarbon dating as a tool for mortar dating was introduced as early as the 1960s, soon after the method was established (Delibrias and Labeyrie 1964; Stuiver and Smith 1965; Folk and Valastro 1976). The very first data were provided by van Strydonck et al. (1983), Heinemeier et al.(1997) and Ringbom and Remmer (1995). Methodological aspects were further developed by different groups (an international team headed by Åbo Akademi University, and teams from CIRCE, CIRCe, ETHZ, Poznań, RICH and Milano-Bicocca laboratory. To evaluate the different anthropogenic carbon extraction methods for radiocarbon dating as well as to compare the different dating methods, i.e. radiocarbon and OSL, the first intercomparison study (MODIS) was set up and published in 2017.[20][21]
Thick bed mortar (technique)
What Is Mortar - Introduction To Mortar
Qualities Of Ideal Mortar List - Properties of Good Mortar List
Uses of Mortar List - Uses of Mortar In Civil Engineering
Types of Mortars Used In Civil Engineering
Technical data sheets, Mortar Industry Association, www.mortar.org.uk
^ a b Artioli, G. (2019). "The Vitruvian legacy: mortars and binders before and after the Roman world" (PDF). EMU Notes in Mineralogy. 20: 151–202.
^ Khan, Aurangzeb. "Ancient Bricks". Aurangzeb Khan. Retrieved 2013-02-16. Cite journal requires |journal= (help)
^ ""Introduction to Mortars" Cemex Corporation" (PDF). Archived from the original (PDF) on 2013-05-25. Retrieved 2014-04-03.
^ O. P. Jaggi (1969), History of science and technology in India, Volume 1, Atma Ram, 1969, ... In some of the important-looking buildings, gypsum cement of a light gray colour was used on the outside to prevent the mud mortar from crumbling down. In a very well constructed drain of the Intermediate period, the mortar which was used contains a high percentage of lime instead of gypsum. Bitumen was found to have been used only at one place in Mohenjo-daro. This was in the construction of the great bath ...
^ Abdur Rahman (1999), History of Indian science, technology, and culture, Oxford University Press, 1999, ISBN 978-0-19-564652-8, ... Gypsum cement was found to have been used in the construction of a well in Mohenjo-daro. The cement was light grey and contained sand, clay, traces of calcium carbonate, and a high percentage of lime ...
^ "Egypt: Egypt's Ancient, Small, Southern, Step Pyramids". Touregypt.net. 2011-06-21. Retrieved 2012-11-03.
^ a b "HCIA - 2004". Hcia.gr. Archived from the original on 2012-02-09. Retrieved 2012-11-03.
^ "American Scientist Online". Americanscientist.org. Retrieved 2012-11-03.
^ "Revealing the Ancient Chinese Secret of Sticky Rice Mortar". Science Daily. Retrieved 23 June 2010.
^ Yang Fuwei, Zhang Bingjian, Ma Qinglin (2010). "Study of Sticky Rice−Lime Mortar Technology for the Restoration of Historical Masonry Construction". Accounts of Chemical Research. 43 (6): 936–944. doi:10.1021/ar9001944. PMID 20455571. CS1 maint: multiple names: authors list (link)
^ Masonry: the best of Fine homebuilding.. Newtown, CT: Taunton Press, 1997. Print. 113.
^ "Information about Lime - LimeWorks.us". limeworks.us. Retrieved 2016-11-02.
^ "ASTM C 270-51T". ASTM International. Retrieved 27 September 2019.
^ "pozzolana." Collins English Dictionary - Complete & Unabridged 10th Edition. HarperCollins Publishers. 14 May. 2014. <Dictionary.com http://dictionary.reference.com/browse/pozzolana>
^ Folk RL, Valastro S (1979). Dating of lime mortar by 14C (Berger R, Suess H. ed.). Proceedings of the Ninth International Conference: Berkeley: University of California Press. pp. 721–730.
^ Hayen R, Van Strydonck M, Fontaine L, Boudin M, Lindroos A, Heinemeier J, Ringbom A, Michalska D, Hajdas I, Hueglin S, Marzaioli F, Terrasi F, Passariello I, Capano M, Maspero F, Panzeri L, Galli A, Artioli G, Addis A, Secco M, Boaretto E, Moreau C, Guibert P, Urbanova P, Czernik J, Goslar T, Caroselli M (2017). "Mortar dating methodology: intercomparison of available methods". Radiocarbon. 59 (6).
^ Hayen R, Van Strydonck M, Boaretto E, Lindroos A, Heinemeier J, Ringbom Å, Hueglin S, Michalska D, Hajdas I, Marzaoili F, Maspero F, Galli A, Artioli G, Moreau Ch, Guibert P, Caroselli M (2016). Absolute dating of mortars – integrating chemical and physical techniques to characterize and select the mortar samples. Proceedings of the 4th Historic Mortars Conference - HMC2016. pp. 656–667. CS1 maint: multiple names: authors list (link)
^ Dating Ancient Mortar - American Scientist Online vol. 91, 2003
^ Hajdas I, Lindroos A, Heinemeier J, Ringbom Å, Marzaioli F, Terrasi F, Passariello I, Capano M, Artioli G, Addis A, Secco M, Michalska D, Czernik J, Goslar T, Hayen R, Van Strydonck M, Fontaine L, Boudin M, Maspero F, Panzeri L, Galli A, Urbanova P, Guibert P (2017). "Preparation and dating of mortar samples—Mortar Dating Inter-comparison Study (MODIS)" (PDF). Radiocarbon. 59 (6): 1845–1858. doi:10.1017/RDC.2017.112.
^ Hayen R, Van Strydonck M, Fontaine L, Boudin M, Lindroos A, Heinemeier J, Ringbom A, Michalska D, Hajdas I, Hueglin S, Marzaioli F, Panzeri L, Galli A, Artioli G, Addis A, Secco M, Boaretto E, Moreau C, Guibert P, Urbanova P, Czernik J, Goslar T, Caroselli M (2017). "Mortar dating methodology: intercomparison of available methods". Radiocarbon. 59 (6).
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Mechanics of Three-Dimensional Printed Lattices for Biomedical Devices | J. Mech. Des. | ASME Digital Collection
ME North 201,
Lubbock, TX 79409 − 1021;
Institute for Biomechanics,
Building HPP,
Honggerbergring 64,
e-mails: paul.egan@ttu.edu;
paul.egan.phd@gmail.com
e-mail: baueri@student.ethz.ch
Building CLA,
e-mail: kshea@ethz.ch
e-mail: sferguson@ethz.ch
Contributed by the Design for Manufacturing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 1, 2018; final manuscript received December 5, 2018; published online January 14, 2019. Assoc. Editor: Carolyn Seepersad.
Egan, P. F., Bauer, I., Shea, K., and Ferguson, S. J. (January 14, 2019). "Mechanics of Three-Dimensional Printed Lattices for Biomedical Devices." ASME. J. Mech. Des. March 2019; 141(3): 031703. https://doi.org/10.1115/1.4042213
Advances in three-dimensional (3D) printing are enabling the design and fabrication of tailored lattices with high mechanical efficiency. Here, we focus on conducting experiments to mechanically characterize lattice structures to measure properties that inform an integrated design, manufacturing, and experiment framework. Structures are configured as beam-based lattices intended for use in novel spinal cage devices for bone fusion, fabricated with polyjet printing. Polymer lattices with
50%
70%
porosity were fabricated with beam diameters of
0.4–1.0mm
, with measured effective elastic moduli from
28MPa
213MPa
. Effective elastic moduli decreased with higher lattice porosity, increased with larger beam diameters, and were highest for lattices compressed perpendicular to their original build direction. Cages were designed with
50%
70%
lattice porosities and included central voids for increased nutrient transport, reinforced shells for increased stiffness, or both. Cage stiffnesses ranged from
4.1kN/mm
9.6kN/mm
with yielding after
0.36–0.48mm
displacement, thus suggesting their suitability for typical spinal loads of
1.65kN
. The
50%
porous cage with reinforced shell and central void was particularly favorable, with an
8.4kN/mm
stiffness enabling it to potentially function as a stand-alone spinal cage while retaining a large open void for enhanced nutrient transport. Findings support the future development of fully integrated design approaches for 3D printed structures, demonstrated here with a focus on experimentally investigating lattice structures for developing novel biomedical devices.
Additive manufacturing, Design, Elastic moduli, Stiffness, Porosity, Biomedicine, Manufacturing, Printing, Bone, Testing
Simulated Tissue Growth for 3D Printed Scaffolds
3D Printing Addit. Manuf.
Wahafu
X.-y
A Novel Open-Porous Magnesium Scaffold With Controllable Microstructures and Properties for Bone Regeneration
Scaffold Curvature-Mediated Novel Biomineralization Process Originates a Continuous Soft Tissue-to-Bone Interface
Design of Hierarchical 3D Printed Scaffolds Considering Mechanical and Biological Factors for Bone Tissue Engineering
Clinically Relevant Bioprinting Workflow and Imaging Process for Tissue Construct Design and Validation
A Mathematical Approach to Bone Tissue Engineering
Design and Development of Scaffolds for Tissue Engineering Using Three-Dimensional Printing for Bio-Based Applications
The Properties of Foams and Lattices
Investigating the Role of Geometric Dimensioning and Tolerancing in Additive Manufacturing
Numerical Investigation on Mechanical Properties of Cellular Lattice Structures Fabricated by Fused Deposition Modeling
Engensperger
Computationally Designed Lattices With Tuned Properties for Tissue Engineering Using 3D Printing
Biomechanical Comparison of a New Stand-Alone Anterior Lumbar Interbody Fusion Cage With Established Fixation Techniques—A Three-Dimensional Finite Element Analysis
Biomechanical Comparison of Anterior Lumbar Interbody Fusion: Stand-Alone Interbody Cage Versus Interbody Cage With Pedicle Screw Fixation—A Finite Element Analysis
Van de Kelft
Trabecular Metal Spacers as Standalone or With Pedicle Screw Augmentation, in Posterior Lumbar Interbody Fusion: A Prospective, Randomized Controlled Trial
Production of New 3D Scaffolds for Bone Tissue Regeneration by Rapid Prototyping
The D3 Methodology: Bridging Science and Design for Bio-Based Product Development
Perspectives on Iteration in Design and Development
Design Control for Clinical Translation of 3D Printed Modular Scaffolds
Mater. Sci. Eng.: C
Yield Strain Behavior of Trabecular Bone
Geometrical Aspects of Patient-specific Modelling of the Intervertebral Disc: Collagen Fibre Orientation and Residual Stress Distribution
Mech. Model. Mechanobiol.
Roohani-Esfahani
Mechanical Properties of Parts Fabricated With Inkjet 3D Printing Through Efficient Experimental Design
Alifui-Segbaya
Biocompatibility of Photopolymers in 3D Printing
Comparisons of Elasticity Moduli of Different Specimens Made Through Three Dimensional Printing
Ten Challenges in 3D Printing
Precision and Trueness of Dental Models Manufactured With Different 3-Dimensional Printing Techniques
Am. J. Orthod. Dentofacial Orthop.
Kadkhodapour
Montazerian
Limmahakhun
Sitthiseripratip
Finite Element Modelling of the Compressive Response of Lattice Structures Manufactured Using the Selective Laser Melting Technique
J. Appl. Biomater. Funct. Mater.
Design of Polymer Scaffolds for Tissue Engineering Applications
Bashkuev
Computational Analyses of Different Intervertebral Cages for Lumbar Spinal Fusion
Biomechanical Investigation Into the Structural Design of Porous Additive Manufactured Cages Using Numerical and Experimental Approaches
.https://imagescience.org/meijering/publications/download/bio2004.pdf
Comparative Anatomical Dimensions of the Complete Human and Porcine Spine
Finite Element Modeling Concepts and Linear Analyses of 3D Regular Open Cell Structures
The Effect of Anisotropy on the Optimization of Additively Manufactured Lattice Structures
, 155, pp. 220–232.https://www.sciencedirect.com/science/article/pii/S026412751830443X
Activities of Everyday Life With High Spinal Loads
Three-Dimensional Nano-Architected Scaffolds With Tunable Stiffness for Efficient Bone Tissue Growth
Osteogenic Cell Functionality on 3-Dimensional Nano-Scaffolds With Varying Stiffness
|
P
be a set of points in
p
q
P
\mathrm{dist}\left(p,q\right)
p
q
The Euclidean minimum spanning tree is simply a minimum-weight spanning tree for the complete weighted graph on
P
with the weight of the edge between points
p
q
defined to be
\mathrm{dist}\left(p,q\right)
\mathrm{\rho }
P
, the minimum spanning tree for norm
\mathrm{\rho }
is a minimum-weight spanning tree for the complete weighted graph on
P
p
q
\mathrm{\rho }\left(p-q\right)
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{GeometricGraphs}\right):
\mathrm{points}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(60,2,\mathrm{generator}=0..100.,\mathrm{datatype}=\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{\mathrm{points}}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{9.85017697341803}& \textcolor[rgb]{0,0,1}{82.9750304386195}\\ \textcolor[rgb]{0,0,1}{86.0670183749663}& \textcolor[rgb]{0,0,1}{83.3188659363996}\\ \textcolor[rgb]{0,0,1}{64.3746795546741}& \textcolor[rgb]{0,0,1}{73.8671607639673}\\ \textcolor[rgb]{0,0,1}{57.3670557294666}& \textcolor[rgb]{0,0,1}{2.34399775883031}\\ \textcolor[rgb]{0,0,1}{23.6234264844933}& \textcolor[rgb]{0,0,1}{52.6873367387328}\\ \textcolor[rgb]{0,0,1}{47.0027547350003}& \textcolor[rgb]{0,0,1}{22.2459488367552}\\ \textcolor[rgb]{0,0,1}{74.9213491558963}& \textcolor[rgb]{0,0,1}{62.0471820220718}\\ \textcolor[rgb]{0,0,1}{92.1513434709073}& \textcolor[rgb]{0,0,1}{96.3107262637080}\\ \textcolor[rgb]{0,0,1}{48.2319624355944}& \textcolor[rgb]{0,0,1}{63.7563267144141}\\ \textcolor[rgb]{0,0,1}{90.9441877431805}& \textcolor[rgb]{0,0,1}{33.8527464913022}\\ \textcolor[rgb]{0,0,1}{⋮}& \textcolor[rgb]{0,0,1}{⋮}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{60 × 2 Matrix}}\end{array}
\mathrm{EMST}≔\mathrm{EuclideanMinimumSpanningTree}\left(\mathrm{points}\right)
\textcolor[rgb]{0,0,1}{\mathrm{EMST}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected weighted graph with 60 vertices and 59 edge\left(s\right)}}
\mathrm{DrawGraph}\left(\mathrm{EMST}\right)
\mathrm{DrawGraph}\left(\mathrm{EuclideanMinimumSpanningTree}\left(\mathrm{points},\mathrm{method}=\mathrm{Prim}\right)\right)
\mathrm{RMST}≔\mathrm{GeometricMinimumSpanningTree}\left(\mathrm{points},1\right)
\textcolor[rgb]{0,0,1}{\mathrm{RMST}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected weighted graph with 60 vertices and 59 edge\left(s\right)}}
\mathrm{DrawGraph}\left(\mathrm{RMST}\right)
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NCERT Solutions for Class 11 Science Chemistry Chapter 5 - States Of Matter
NCERT Solutions for Class 11 Science Chemistry Chapter 5 States Of Matter are provided here with simple step-by-step explanations. These solutions for States Of Matter are extremely popular among Class 11 Science students for Chemistry States Of Matter Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the NCERT Book of Class 11 Science Chemistry Chapter 5 are provided here for you for free. You will also love the ad-free experience on Meritnation’s NCERT Solutions. All NCERT Solutions for class Class 11 Science Chemistry are prepared by experts and are 100% accurate.
Replacing n with , we have
But, (d = density of gas)
Molar mass (M) of a gas is always constant and therefore, at constant temperature = constant.
0.15 g Al gives i.e., 186.67 mL of H2.
Let the volume of dihydrogen be at p2 = 0.987 atm (since 1 bar = 0.987 atm) and T2 = 20°C = (273.15 + 20) K = 293.15 K..
Let the partial pressure of H2 in the vessel be .
Now, let the partial pressure of O2 in the vessel be .
\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{ }\phantom{\rule{0ex}{0ex}}{\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{p}}}_{1}{\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{V}}}_{1}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{=}{\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{p}}}_{2}{\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{V}}}_{2}\phantom{\rule{0ex}{0ex}}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{⇒}{\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{p}}}_{2}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{=}\frac{{\mathrm{p}}_{1}{\mathrm{V}}_{1}}{{\mathrm{V}}_{2}}\phantom{\rule{0ex}{0ex}}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{⇒}{\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{p}}}_{{\mathrm{O}}_{2}}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{=}\frac{0.7×2.0}{1}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{=}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{ }\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{1}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{.}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{4}\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{ }\textcolor[rgb]{0.0980392156862745,0.0980392156862745,0.0980392156862745}{\mathrm{bar}}\phantom{\rule{0ex}{0ex}}
Hence, the total pressure of the gaseous mixture in the vessel is .
Therefore, molar mass of phosphorus = 1247.5 g mol–1
Now, 1 molecule of contains 14 electrons.
Hence, the time taken would be .
Then, the number of moles of dihydrogen, and the number of moles of dioxygen, .
Hence, the partial pressure of dihydrogen is .
Therefore, the SI unit for quantity is given by,
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Below are two of the special angles that are used in the unit circle. Identify the radian measure for the angles shown. Remember counter-clockwise angles are positive and clockwise angles are negative.
-\frac{2\pi}{3}
It went around the entire unit circle once before going again.
\frac{17\pi}{6}
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Contrasting exhumation histories and relief development within the Three...
Ou, Xiong; Replumaz, Anne; van der Beek, Peter
The Three Rivers Region in south-east Tibet represents a transition between the strongly deformed zone around the Eastern Himalayan Syntaxis (EHS) and the less deformed south-east Tibetan Plateau margin in Yunnan and Sichuan. In this study, we compile and model published thermochronometric ages for two massifs facing each other across the Mekong River in the core of the Three Rivers Region (TRR), using the thermo-kinematic code Pecube to constrain their exhumation and relief history. Modelling results for the low-relief (inline-formula< 600 m), moderate-elevation (inline-formula∼ 4500 m) Baima Xueshan massif, east of the Mekong River, suggest regional rock uplift at a rate of 0.25 km/Myr since inline-formula∼ 10 Ma, following slow exhumation at a rate of 0.01 km/Myr since at least 22 Ma. Estimated Mekong River incision accounts for 30 % of the total exhumation since 10 Ma. We interpret exhumation of the massif as a response to regional uplift around the EHS and conclude that the low relief of the massif was acquired at high elevation (inline-formula> 4500 m), probably in part due to glacial “buzzsaw-like” processes active at such high elevation and particularly efficient during Quaternary glaciations. Exhumation of the Baima Xueshan is significantly higher (2.5 km since inline-formula∼ 10 Ma) than that estimated for the most emblematic low-relief “relict” surfaces of eastern Tibet, where apatite (U–Th) inline-formula
M6inlinescrollmathml/
8pt14ptsvg-formulamathimg073414a2b77546d8d5847ae97897d626 se-12-563-2021-ie00001.svg8pt14ptse-12-563-2021-ie00001.png He (AHe) ages inline-formula> 50 Ma imply only a few hundreds of metres of exhumation since the onset of the India–Asia collision. The low-relief Baima Xueshan massif, with its younger AHe ages (inline-formula< 50 Ma) that record significant rock uplift and exhumation, thus cannot be classified as a relict surface. Modelling results for the high-relief, high-elevation Kawagebo massif, to the west of the Mekong, imply a similar contribution of Mekong River incision (25 %) to exhumation but much stronger local rock uplift at a rate of 0.45 km/Myr since at least 10 Ma, accelerating to 1.86 km/Myr since 1.6 Ma. We show that the thermochronometric ages are best reproduced by a model of rock uplift on a kinked westward-dipping thrust striking roughly parallel to the Mekong River, with a steep shallow segment flattening out at depth. Thus, the strong differences in elevation and relief of two massifs are linked to variable exhumation histories due to strongly differing tectonic imprint.
Ou, Xiong / Replumaz, Anne / van der Beek, Peter: Contrasting exhumation histories and relief development within the Three Rivers Region (south-east Tibet). 2021. Copernicus Publications.
Rechteinhaber: Xiong Ou et al.
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Tiffani has a craft business that is expanding and she plans to cut down on expenses by purchasing craft paints in bulk. Most of them are straight mixtures, except for vibrant green, which she mixes specially. To create the mixture, she mixes one pint of the standard green paint with one teaspoon of red and two teaspoons of yellow. She has one pint of red and two pints of yellow paint that she wishes to add to the green paint to create the special mix. How many gallons of the green paint will she need in order to use all of her red and yellow paint? Below is a list of the equivalent measures.
\text{one gallon}=128\ \text{ounces}
\text{one ounce}=6\ \text{teaspoons}
\text{one pint}=16\ \text{ounces}
\text{one pint}=2\ \text{cups}
Calculate the number of batches of paint one pint of red could make.
What about the yellow paint?
Now, calculate the number of batches one gallon of green paint will make.
\left(\frac{\text{6 teaspons}}{\text{ounce}}\right)\left(\frac{\text{16 ounces}}{\text{pint}}\right)=96\text{ batches}
It would be the same number of batches since you have twice as much yellow paint but the amount needed is twice as much as the red.
There are 4 quarts to a gallon. Each quart has 2 pints. So you can make 8 batches of the special green with one gallon of the standard green.Now set up a proportion and solve.
\frac{1\ \text{gallon\ green}}{8\ \text{batches}}= \frac{x\text{\ number\ of\ gallons\ of\ green}}{96\text{\ batches}}
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RunWorksheet - Maple Help
Home : Support : Online Help : Programming : Document Tools : RunWorksheet
execute a worksheet as a procedure
Example Using RunWorksheet
RunWorksheet(ws, var_init)
string ; name of the file containing the worksheet to run
(optional) list(symbol=anything) ; list of equations of the form symbol = expression, specifying the initial values for the corresponding variables in ws
(optional) list({symbol,string}) ; list of variables and/or command strings
inheritlibname
(optional,cmaple only) truefalse ; indicate whether the worksheet process should inherit the current value of libname. The default is true.
The RunWorksheet function invokes the worksheet ws as if it were a procedure. The variables appearing on the left-hand sides of equations in the var_init parameter are initialized to their corresponding right-hand side values. Names prefixed with % in var_init represent embedded-components. For example, %TextArea0 = 5 as part of var_init, will initialize the TextArea0 component value to 5.
The worksheet filename ws can be either fully qualified or be relative to the value of currentdir.
Values can be extracted from the worksheet by specifying outputs=[list of names]. In addition to the names specified in outputs, command strings can be given in the outputs list. These commands will be evaluated as a post-processing step after the worksheet is finished executing but while the worksheet state is still active.
The invoked worksheet ws must identify in its Document Properties a section which initializes via assignment statements the variables given in var_init. Specifically, within the Document Properties there must be an attribute with Attribute Name InputSectionTitle and whose value is the section name.
This section of the worksheet ws must have at least one assignment statement for each variable appearing on the left-hand side of an equation in var_init. These assignment statements must use the assignment operator (:=) and should be the only statements appearing in their respective execution groups. (There can be other assignments or other Maple computations in this section, as well.)
If the invoked worksheet ws includes a return statement at the top level (not inside a procedure), the expression given in that return statement will be returned as the output of the RunWorksheet command. Execution of the invoked worksheet ws stops when a top-level return is evaluated.
If the invoked worksheet ws does not include such a top-level return statement, the RunWorksheet command will have no output. That is, the value of the RunWorksheet call will be NULL.
Multiple values can be returned in an expression sequence.
The invoked worksheet runs "headless", meaning that it will not appear with a user interface.
The invoked worksheet runs in a new engine, so expressions whose subsequent evaluation may depend on the state of the engine in the calling worksheet cannot be passed in var_init. This includes procedures and modules which are not lexically closed (for example, which use global or environment variables in their bodies). To pass a procedure, table, matrix, vector, or array defined elsewhere in the calling worksheet, it is necessary to apply eval to the expression first. For example, use g = eval(f) in var_init, where f is a user-defined procedure.
RunWorksheet can be used in a workbook together with a worksheet with password protection. For an example, see Password-Protected Workbook Example. For more information, see The Workbook File Format and Password Protecting Workbook Pages.
Create a worksheet with a section entitled Inputs. In that section, enter these assignments (each in its own execution group):
Create another section entitled Calculation with this line:
return a+b^2+c^3;
Under the File menu, open Document Properties. Add a new attribute InputSectionTitle. For its value, enter Inputs.
Save this worksheet as test.mw in the current directory.
In another worksheet, execute the following command to run the worksheet:
DocumentTools[RunWorksheet]( "test.mw", [a=-1, b=4] );
\textcolor[rgb]{0,0,1}{42}
Note that it is not necessary to provide values for all (or any) of the variables appearing in the invoked worksheet's input section (but it is an error to try to provide a value for a variable which does not appear as the left-hand side of an assignment statement in that input section).
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Investment Multiplier Definition
The term investment multiplier refers to the concept that any increase in public or private investment spending has a more than proportionate positive impact on aggregate income and the general economy. It is rooted in the economic theories of John Maynard Keynes.
The multiplier attempted to quantify the additional effects of investment spending beyond those immediately measurable. The larger an investment’s multiplier, the more efficient it is in creating and distributing wealth throughout the economy.
The investment multiplier refers to the stimulative effects of public or private investments.
It is rooted in the economic theories of John Maynard Keynes.
The extent of the investment multiplier depends on two factors: the marginal propensity to consume (MPC) and the marginal propensity to save (MPS).
A higher investment multiplier suggests that the investment will have a larger stimulative effect on the economy.
Understanding the Investment Multiplier
The investment multiplier tries to determine the economic impact of public or private investment. For instance, extra government spending on roads can increase the income of construction works, as well as the income of materials suppliers. These people may spend the extra income in the retail, consumer goods, or service industries, boosting the income of the workers in those sectors.
As you can see, this cycle can repeat itself through several iterations; what began as an investment in roads quickly multiplied into an economic stimulus benefiting workers across a wide range of industries.
Mathematically, the investment multiplier is a function of two main factors: the marginal propensity to consume (MPC) and the marginal propensity to save (MPS).
Real World Example of the Investment Multiplier
Consider the road construction workers in our previous example. If the average worker has an MPC of 70%, that means they consume $0.70 out of every dollar they earn, on average. In practice, they might spend that $0.70 on items such as rent, gasoline, groceries, and entertainment. If that same worker has an MPS of 30%, that means they would save $0.30 out of every dollar earned, on average.
These concepts also apply to businesses. Like individuals, businesses must “consume” a significant portion of their income by paying for expenditures such as employees’ wages, facilities’ rents, and the leases and repairs of equipment. A typical company might consume 90% of their income on such payments, meaning that its MPS—the profits earned by its shareholders—would be only 10%.
The formula for calculating the investment multiplier of a project is simply:
1 / (1 - MPC)
1/(1−MPC)
Therefore, in our above examples, the investment multipliers would be 3.33 and 10 for the workers and the businesses, respectively. The reason the businesses are associated with a higher investment multiple is that their MPC is higher than that of the workers. In other words, they spend a greater percentage of their income on other parts of the economy, thereby spreading the economic stimulus caused by the initial investment more widely.
Marginal propensity to save (MPS) refers to the proportion of a pay raise that a consumer saves rather than spends on immediate consumption.
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Because of natural variability in manufacturing, a
12
-ounce can of soda does not usually hold exactly
12
ounces of soda. A can is permitted to hold a little more or a little less. The specifications for the soda-filling machine are that it needs to fill each can with
12\pm0.25
ounces of soda. If a can of soda is filled with
11.97
ounces of soda, is the filling machine operating within specifications?
11.97
12-0.25
12+0.25?
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A box spread, or long box, is an options arbitrage strategy that combines buying a bull call spread with a matching bear put spread. A box spread can be thought of as two vertical spreads that each has the same strike prices and expiration dates.
Box spreads are used for borrowing or lending at implied rates that are more favorable than a trader going to their prime broker, clearing firm, or bank. Because the price of a box at its expiration will always be the distance between the strikes involved (e.g., a 100-pt box might utilize the 25 and 125 strikes and would be worth $100 at expiration), the price paid for today can be thought of as that of a zero-coupon bond. The lower the initial cost of the box, the higher its implied interest rate. This concept is known as a synthetic loan.
A box spread's ultimate payoff will always be the difference between the two strike prices.
The longer the time to expiration, the lower the market price of the box spread today.
The cost to implement a box spread—specifically, the commissions charged—can be a significant factor in its potential profitability.
Traders use box spreads to synthetically borrow or lend for cash management purposes.
Understanding a Box Spread
A box spread is optimally used when the spreads themselves are underpriced with respect to their expiration values. When the trader believes the spreads are overpriced, they may employ a short box, which uses the opposite options pairs, instead. The concept of a box comes to light when one considers the purpose of the two vertical, bull call and bear put, spreads involved.
A bullish vertical spread maximizes its profit when the underlying asset closes at the higher strike price at expiration. The bearish vertical spread maximizes its profit when the underlying asset closes at the lower strike price at expiration.
By combining both a bull call spread and a bear put spread, the trader eliminates the unknown, namely where the underlying asset closes at expiration. This is so because the payoff is always going to be the difference between the two strike prices at expiration.
If the cost of the spread, after commissions, is less than the difference between the two strike prices, then the trader locks in a riskless profit, making it a delta-neutral strategy. Otherwise, the trader has realized a loss comprised solely of the cost to execute this strategy.
Box spreads effectively establish synthetic loans. Like a zero-coupon bond, they are initially bought at a discount and the price steadily rises over time until expiration where it equals the distance between strikes.
\begin{aligned} &\text{BVE}=\text{ HSP }-\text{ LSP}\\ &\text{MP}=\text{ BVE }-\text{ (NPP} + \text{ Commissions)}\\ &\text{ML }= \text{ NPP }+ \text{ Commissions}\\ &\textbf{where:}\\ &\text{BVE}=\text{ Box value at expiration}\\ &\text{HSP}=\text{ Higher strike price}\\ &\text{LSP}=\text{ Lower strike price}\\ &\text{MP}=\text{ Max profit}\\ &\text{NPP}=\text{ Net premium paid}\\ &\text{ML}=\text{ Max Loss} \end{aligned}
BVE= HSP − LSPMP= BVE − (NPP+ Commissions)ML = NPP + Commissionswhere:BVE= Box value at expirationHSP= Higher strike priceLSP= Lower strike priceMP= Max profitNPP= Net premium paidML= Max Loss
To construct a box spread, a trader buys an in-the-money (ITM) call, sells an out-of-the-money(OTM) call, buys an ITM put, and sells an OTM put. In other words, buy an ITM call and put and then sell an OTM call and put.
Given that there are four options in this combination, the cost to implement this strategy—specifically, the commissions charged—can be a significant factor in its potential profitability. Complex option strategies, such as these, are sometimes referred to as alligator spreads.
There will be times when the box costs more than the spread between the strikes. Should this be the case, the long box would not work but a short box might. This strategy reverses the plan, selling the ITM options and buying the OTM options.
Company A stock trades for $51.00. Each options contract in the four legs of the box controls 100 shares of stock. The plan is to:
Buy the 49 call for 3.29 (ITM) for $329 debit per options contract
Sell the 53 call for 1.23 (OTM) for $123 credit
Buy the 53 put for 2.69 (ITM) for $269 debit
Sell the 49 put for 0.97 (OTM) for $97 credit
The total cost of the trade before commissions would be $329 - $123 + $269 - $97 = $378. The spread between the strike prices is 53 - 49 = 4. Multiply by 100 shares per contract = $400 for the box spread.
In this case, the trade can lock in a profit of $22 before commissions. The commission cost for all four legs of the deal must be less than $22 to make this profitable. That is a razor-thin margin, and this is only when the net cost of the box is less than the expiration value of the spreads, or the difference between the strikes.
Hidden Risks in Box Spreads
While box spreads are commonly used for cash management and are seen as a way to arbitrage interest rates with low risk, there are some hidden risks. The first is that interest rates may move strongly against you, causing losses like they would on any other fixed-income investments that are sensitive to rates.
A second potential danger, which is perhaps less obvious, is the risk of early exercise. American style options, such as those options listed on most U.S. stocks may be exercised early (i.e., before expiration), and so it is possible that a short option that becomes deep in-the-money can be assigned. In the normal construction of a box, this is unlikely, since you would own the deep call and put, but the stock price can move significantly and then find yourself in a situation where you might be assigned.
This risk increases for short boxes written on single stock options, as was the infamous case of a Robinhood trader who lost more than 2,000% on a short box when the deep puts that were sold were subsequently assigned, causing Robinhood to exercise the long calls in an effort to come up with the shares needed to satisfy the assignment. This debacle was posted online including on various subreddits, where it has become a cautionary tale (especially after said trader boasted that it was a virtually riskless strategy).
The lesson here is to avoid short boxes, or to only write short boxes on indexes (or similar) that instead use European options, which do not allow for early exercise.
When should one use a box strategy?
A box strategy is best-suited for taking advantage of more favorable implied interest rates than can be obtained through usual credit channels (e.g., a bank). It is therefore most often used for purposes of cash management.
Are box spreads risk-free?
A long box is, in theory, a low-risk strategy that is sensitive primarily to interest rates. A long box will always expire at a value worth the distance between the two strike prices utilized. A short box, however, may be subject to early assignment risk when using American options.
What is a short box spread?
A short box, in contrast to a standard long box, involves selling deep ITM calls and puts and buying OTM ones. This would be done if the price of the box is trading at higher than the distance between strikes (which can be caused for several reasons, including a low interest rate environment or pending dividend payments for single stock options).
A vertical spread involves the simultaneous buying and selling of options of the same type (puts or calls) and expiry, but at different strike prices.
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Energies | Special Issue : Integrated Energy Systems and Transportation Electrification
Integrated Energy Systems and Transportation Electrification
Special Issue "Integrated Energy Systems and Transportation Electrification"
Prof. Dr. Andrey V. Savkin
School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia
Interests: robot navigation; deployment of drones; control of power systems; robust control and filtering; hybrid dynamical systems; control engineering; biomedical engineering
Special Issue in Energies: Advanced Control in Microgrid Systems
Special Issue in Sensors: Deployment and Navigation of Aerial Drones for Surveillance and Monitoring
Special Issue in Sensors: Deployment and Navigation of Aerial Drones and Unmanned Marine Vehicles for Monitoring, Communication and Delivery
Special Issue in Energies: Advanced Control in Microgrid Systems 2021
We are inviting submissions to a Special Issue of Energies on the subject area of “Integrated Energy Systems and Transportation Electrification”. Electrification is considered to be an effective solution to the environmental impact of transportation, helping to reduce fossil fuel dependency. It is widely believed that electrifying transportation brings enormous economic and environmental benefits and greatly improves quality of life. With the increasing integration of renewable energy and the development of a smart grid, the topic of transportation electrification has attracted a lot of attention in recent years.
Researchers and engineers worldwide are working together to develop novel and efficient tools for integrated energy systems and transportation electrification. This Special Issue is focused on new developments in the field of integrated energy systems and transportation electrification.
Electric drives for transportation applications
Design, modelling and control of electric vehicles
All types of electric vehicles, including on-road vehicles, off-road vehicles, rail vehicles, aerial drones, surface marine vehicles, and underwater vehicles
Applications and control of solar powered vehicles
Energy-efficient drone delivery
Optimization and control of charging and charging station deployment for electric vehicles
Electrical systems in transportation
On-off charging
Control of charging
The Development of Decarbonisation Strategies: A Three-Step Methodology for the Suitable Analysis of Current EVCS Locations Applied to Istanbul, Turkey
Kadir Diler Alemdar
Merve Kayaci Çodur
One of the solutions to reduce environmental emissions is related to the deployment of electric vehicles (EVs) with sustainable energy. In order to be able to increase the number of electric vehicles in circulation, it is important to implement optimal planning and design [...] Read more.
One of the solutions to reduce environmental emissions is related to the deployment of electric vehicles (EVs) with sustainable energy. In order to be able to increase the number of electric vehicles in circulation, it is important to implement optimal planning and design of the infrastructure, with particular reference to areas equipped with charging stations. The suitable analysis of the location of current electric vehicle charging stations (EVCSs) is the central theme of this document. The research focused on the actual location of the charging stations of five major EVCS companies in the province by selecting Istanbul as the study area. The study was conducted through a three-step approach and specifically (i) the application of the analytical hierarchy process (AHP) method for creating the weights of the 6 main and 18 secondary criteria that influence the location of EVCSs; (ii) a geospatial analysis using GIS considering each criterion and developing the suitability map for the locations of EVCSs, and (iii) application of the technique for order preference by similarity to ideal solution (TOPSIS) to evaluate the location performance of current EVCSs. The results show that the ratio between the most suitable and unsuitable areas for the location of EVCSs in Istanbul and the study area is about 5% and 4%, respectively. The results achieved means of improving sustainable urban planning and laying the basis for an assessment of other areas where EVCSs could be placed. Full article
(This article belongs to the Special Issue Integrated Energy Systems and Transportation Electrification)
Methodology for Estimating the Spatial and Temporal Power Demand of Private Electric Vehicles for an Entire Urban Region Using Open Data
Simon Streppel
With continuous proliferation of private battery electric vehicles (BEVs) in urban regions, the demand for electrical energy and power is constantly increasing. Electrical grid infrastructure operators are facing the question of where and to what extent they need to expand their infrastructure in [...] Read more.
With continuous proliferation of private battery electric vehicles (BEVs) in urban regions, the demand for electrical energy and power is constantly increasing. Electrical grid infrastructure operators are facing the question of where and to what extent they need to expand their infrastructure in order to meet the additional demand. Therefore, the aim of this paper is to develop an activity-based mobility model that supports electrical grid operators in detecting and evaluating possible overloads within the electrical grid, deriving from the aforementioned electrification. We apply our model, which fully relies on open data, to the urban area of Berlin. In addition to a household travel survey, statistics on the population density, the degree of motorisation, and the household income in fine spatial resolution are key data sources for generation of the model. The results show that the spatial distribution of the BEV charging energy demand is highly heterogeneous. The demand per capita is higher in peripheral areas of the city, while the demand per m
{}^{2}
area is higher in the inner city. For reference areas, we analysed the temporal distribution of the BEV charging power demand, by assuming that the vehicles are solely charged at their residential district. We show that the households’ power demand peak in the evening coincide with the BEV power demand peak while the total power demand can increase up to 77.9%. Full article
Path Planning for a Solar-Powered UAV Inspecting Mountain Sites for Safety and Rescue
This paper focuses on the application using a solar-powered unmanned aerial vehicle (UAV) to inspect mountain sites for the purpose of safety and rescue. An inspection path planning problem is formulated, which looks for the path for an UAV to visit a set [...] Read more.
This paper focuses on the application using a solar-powered unmanned aerial vehicle (UAV) to inspect mountain sites for the purpose of safety and rescue. An inspection path planning problem is formulated, which looks for the path for an UAV to visit a set of sites where people may appear while avoiding collisions with mountains and maintaining positive residual energy. A rapidly exploring random tree (RRT)-based planning method is proposed. This method firstly finds a feasible path that satisfies the residual energy requirement and then shortens the path if there is some abundant residual energy at the end. Computer simulations are conducted to demonstrate the performance of the proposed method. Full article
Mathijs M. de Weerdt
Due to increasing numbers of intermittent and distributed generators in power systems, there is an increasing need for demand responses to maintain the balance between electricity generation and use at all times. For example, the electrification of transportation significantly adds to the amount [...] Read more.
Due to increasing numbers of intermittent and distributed generators in power systems, there is an increasing need for demand responses to maintain the balance between electricity generation and use at all times. For example, the electrification of transportation significantly adds to the amount of flexible electricity demand. Several methods have been developed to schedule such flexible energy consumption. However, an objective way of comparing these methods is lacking, especially when decisions are made based on incomplete information which is repeatedly updated. This paper presents a new benchmarking framework designed to bridge this gap. Surveys that classify flexibility planning algorithms were an input to define this benchmarking standard. The benchmarking framework can be used for different objectives and under diverse conditions faced by electricity production stakeholders interested in flexibility scheduling algorithms. Our contribution was implemented in a software toolbox providing a simulation environment that captures the evolution of look-ahead information, which enables comparing online planning and scheduling algorithms. This toolbox includes seven planning algorithms. This paper includes two case studies measuring the performances of these algorithms under uncertain market conditions. These case studies illustrate the importance of online decision making, the influence of data quality on the performance of the algorithms, the benefit of using robust and stochastic programming approaches, and the necessity of trustworthy benchmarking. Full article
Day-Ahead and Intra-Day Collaborative Optimized Operation among Multiple Energy Stations
Jingjing Zhai
Xiaobei Wu
An integrated energy system (IES) shows great potential in reducing the terminal energy supply cost and improving energy efficiency, but the operation scheduling of an IES, especially integrated with inter-connected multiple energy stations, is rather complex since it is affected by various factors. [...] Read more.
An integrated energy system (IES) shows great potential in reducing the terminal energy supply cost and improving energy efficiency, but the operation scheduling of an IES, especially integrated with inter-connected multiple energy stations, is rather complex since it is affected by various factors. Toward a comprehensive operation scheduling of multiple energy stations, in this paper, a day-ahead and intra-day collaborative operation model is proposed. The targeted IES consists of electricity, gas, and thermal systems. First, the energy flow and equipment composition of the IES are analyzed, and a detailed operation model of combined equipment and networks is established. Then, with the objective of minimizing the total expected operation cost, a robust optimization of day-ahead and intra-day scheduling for energy stations is constructed subject to equipment operation constraints, network constraints, and so on. The day-ahead operation provides start-up and shut-down scheduling of units, and in the operating day, the intra-day rolling operation optimizes the power output of equipment and demand response with newly evolved forecasting information. The photovoltaic (PV) uncertainty and electric load demand response are also incorporated into the optimization model. Eventually, with the piecewise linearization method, the formulated optimization model is converted to a mixed-integer linear programming model, which can be solved using off-the-shelf solvers. A case study on an IES with five energy stations verifies the effectiveness of the proposed day-ahead and intra-day collaborative robust operation strategy. Full article
In this paper, we consider the navigation of a group of solar-powered unmanned aerial vehicles (UAVs) for periodical monitoring of a set of mobile ground targets in urban environments. We consider the scenario where the number of targets is larger than that of [...] Read more.
In this paper, we consider the navigation of a group of solar-powered unmanned aerial vehicles (UAVs) for periodical monitoring of a set of mobile ground targets in urban environments. We consider the scenario where the number of targets is larger than that of the UAVs, and the targets spread in the environment, so that the UAVs need to carry out a periodical surveillance. The existence of tall buildings in urban environments brings new challenges to the periodical surveillance mission. They may not only block the Line-of-Sight (LoS) between a UAV and a target, but also create some shadow region, so that the surveillance may become invalid, and the UAV may not be able to harvest energy from the sun. The periodical surveillance problem is formulated as an optimization problem to minimize the target revisit time while accounting for the impact of the urban environment. A nearest neighbour based navigation method is proposed to guide the movements of the UAVs. Moreover, we adopt a partitioning scheme to group targets for the purpose of narrowing UAVs’ moving space, which further reduces the target revisit time. The effectiveness of the proposed method is verified via computer simulations. Full article
Regional Integrated Energy Site Layout Optimization Based on Improved Artificial Immune Algorithm
Regional integrated energy site layout optimization involves multi-energy coupling, multi-data processing and multi-objective decision making, among other things. It is essentially a kind of non-convex multi-objective nonlinear programming problem, which is very difficult to solve by traditional methods. This paper proposes a decentralized [...] Read more.
Regional integrated energy site layout optimization involves multi-energy coupling, multi-data processing and multi-objective decision making, among other things. It is essentially a kind of non-convex multi-objective nonlinear programming problem, which is very difficult to solve by traditional methods. This paper proposes a decentralized optimization and comprehensive decision-making planning strategy and preprocesses the data information, so as to reduce the difficulty of solving the problem and improve operational efficiency. Three objective functions, namely the number of energy stations to be built, the coverage rate and the transmission load capacity of pipeline network, are constructed, normalized by linear weighting method, and solved by the improved p-median model to obtain the optimal value of comprehensive benefits. The artificial immune algorithm was improved from the three aspects of the initial population screening mechanism, population updating and bidirectional crossover-mutation, and its performance was preliminarily verified by test function. Finally, an improved artificial immune algorithm is used to solve and optimize the regional integrated energy site layout model. The results show that the strategies, models and methods presented in this paper are feasible and can meet the interest needs and planning objectives of different decision-makers. Full article
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Physics-based SNOWPACK model improves representation of near-surface Antarctic...
Keenan, Eric; Wever, Nander; Dattler, Marissa; Lenaerts, Jan T. M.; Medley, Brooke; Kuipers Munneke, Peter; Reijmer, Carleen
Estimates of snow and firn density are required for satellite-altimetry-based retrievals of ice sheet mass balance that rely on volume-to-mass conversions. Therefore, biases and errors in presently used density models confound assessments of ice sheet mass balance and by extension ice sheet contribution to sea level rise. Despite this importance, most contemporary firn densification models rely on simplified semi-empirical methods, which are partially reflected by significant modeled density errors when compared to observations. In this study, we present a new drifting-snow compaction scheme that we have implemented into SNOWPACK, a physics-based land surface snow model. We show that our new scheme improves existing versions of SNOWPACK by increasing simulated near-surface (defined as the top 10 inline-formulam) density to be more in line with observations (near-surface bias reduction from inline-formula−44.9 to inline-formula−5.4 inline-formulakg m−3). Furthermore, we demonstrate high-quality simulation of near-surface Antarctic snow and firn density at 122 observed density profiles across the Antarctic ice sheet, as indicated by reduced model biases throughout most of the near-surface firn column when compared to two semi-empirical firn densification models (SNOWPACK inline-formula
M5inlinescrollmathml\text{mean bias}=-normal 9.7
88pt10ptsvg-formulamathimgd4563868c957e632f059fa36e2231f24 tc-15-1065-2021-ie00001.svg88pt10pttc-15-1065-2021-ie00001.png inline-formulakg m−3, IMAU-FDM inline-formula
M7inlinescrollmathml\text{mean bias}=-normal 32.5
94pt10ptsvg-formulamathimg69a418015481211ba8e90f0255d99d5f tc-15-1065-2021-ie00002.svg94pt10pttc-15-1065-2021-ie00002.png inline-formulakg m−3, GSFC-FDM inline-formulamean bias=15.5 inline-formulakg m−3). Notably, our analysis is restricted to the near surface where firn density is most variable due to accumulation and compaction variability driven by synoptic weather and seasonal climate variability. Additionally, the GSFC-FDM exhibits lower mean density bias from 7–10 inline-formulam (SNOWPACK inline-formula
M12inlinescrollmathml\text{bias}=-normal 22.5
64pt10ptsvg-formulamathimge6c05470f8e77a22a94e434af6f2a97a tc-15-1065-2021-ie00003.svg64pt10pttc-15-1065-2021-ie00003.png inline-formulakg m−3, GSFC-FDM inline-formulabias=10.6 inline-formulakg m−3) and throughout the entire near surface at high-accumulation sites (SNOWPACK inline-formula
M16inlinescrollmathml\text{bias}=-normal 31.4
64pt10ptsvg-formulamathimg4ad179be01fb3b38b331d9a69c2e79e6 tc-15-1065-2021-ie00004.svg64pt10pttc-15-1065-2021-ie00004.png inline-formulakg m−3, GSFC-FDM inline-formula
M18inlinescrollmathml\text{bias}=-normal 4.7
58pt10ptsvg-formulamathimge8f3edc2b3728534494a59661e6445e7 tc-15-1065-2021-ie00005.svg58pt10pttc-15-1065-2021-ie00005.png inline-formulakg m−3). However, we found that the performance of SNOWPACK did not degrade when applied to sites that were not included in the calibration of semi-empirical models. This suggests that SNOWPACK may possibly better represent firn properties in locations without extensive observations and under future climate scenarios, when firn properties are expected to diverge from their present state.
Keenan, Eric / Wever, Nander / Dattler, Marissa / et al: Physics-based SNOWPACK model improves representation of near-surface Antarctic snow and firn density. 2021. Copernicus Publications.
Rechteinhaber: Eric Keenan et al.
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General - Beefy.com
Is Beefy audited?
Our first auditor was DefiYield, which audited $BIFI token, the RewardPool and all the timelocks.
Beefy is also audited by Certik, which guarantees the robustness of our smart contracts and the safety of funds invested through Beefy.
Certik has audited some of the most complex and reusable investment strategies used within the platform. This ensures the safety and sturdiness of important smart contract aspects that the majority of our users interact with.
All Beefy audits can be found here.
A yield optimizer is an automated service that seeks to gain the maximum possible return on crypto-investments, much more efficiently than attempting to maximize yield through manual means.
Each vault has its own unique strategy for farming, which normally involves the reinvestment of crypto assets staked in liquidity pools. At the most simple level, it farms the rewards given from staked assets and reinvests them back into the liquidity pool. This compounds the amount of interest received and increases the amount staked that the yield is based on. A yield optimizer can repeat this up to process up to thousands of times a day.
This fairly simple method is the principle reason behind the large APYs found on Beefy. Compounding fees are amortized among all vault participants, making it cheaper for the user.
APR (Annual Percentage Rate) is the yearly interest, minus fees. This does not include compounding effects that occur from reinvesting profits. If you were to invest $100 with 100% APR, you would make $100 in profit in a year time.
Large APYs in the percentage of thousands are possible with investments that provide daily yields of 1% or more. Due to your liquidity pool rewards being constantly farmed and reinvested, the interest compounds on larger and larger amounts.
What do Vault Daily and Trading Daily mean?
Trading Daily means how much your liquidity tokens will increase in value. Liquidity pools share trading fees amongst all liquidity providers, as introduced by the Uniswap liquidity model. Trading Daily is affected by trading volume and the percentage of swap fees allocated to liquidity providers.
Vault Daily means how much your token will increase in number. Due to the vault constantly farming rewards, and reinvesting that, your deposited token amount will increase. Vault Daily is affected by the yield farm rewards (i.e. additional incentives besides trading fees), such as CAKE on Pancakeswap.
Trading Daily and Vault Daily can be multiplied by 365 to compute Trading APR and Vault APR. Vault APR is then converted to Vault APY to factor in compound interest. The displayed total APY percentage is calculated as follows:
APY = (1 + vaultAPY) * (1 + tradingAPR) - 1
To calculate the Trading APR, Beefy uses on-chain data and a 24 hour period to determine the trading volume and subsequent fees, whereas most DEXes use a 7 day period. This may lead to differences in the displayed APY when compared with a DEX, but know that it is due to the calculation method. In fact, we argue that Beefy is more accurate because it uses a shorter time span which reflects changes in Trading APR sooner.
A handy tool to convert APR to APY is: APRtoAPY.com
How do I contribute to Beefy?
Beefy Finance is a community powered project from day one. If you want to join the ever growing pool of contributors, it depends what you would like to work on. We have open places for Solidity devs, or devs wanting to start a career in Solidity, to join as strategist and deploy vaults (and earn passive income from the strategist fees). Beefy is on a lot of chains and there are often opportunities for simple and complex vaults. You can start with simple ones and then progress to the harder ones as your knowledge of Solidity grows. You don't have to be the best right at the start, and rest assured that there is a rigorous review process in place to ensure safety and quality. You can reach out to our lead strategists in Beefy's Discord in #strategy-devs.
Beefy would also like people to work on non-strategy projects; pretty much anything you can think of can be formulated into a grant. Speak with others in the cowmoonity about projects and join one of the teams or lead one up yourself, you can be paid for any work you do to make Beefy better. A quick list of previous grants: here and here. Beefy V2 is an ongoing project that requires all kinds of devs, not just technical ones; design input is crucial to improve the UI/UX.
Beefy's GitHub embraces the idea of open collaboration, hence many of the repositories are open-source. We use CONTRIBUTING.md files to allow people to just make contributions or recommendations by means of Pull-Requests. You can get started even just by updating the Git docs or fixing a typo, it helps you get closer to the team of contributors.
If you have an interest in business development you could help with partnerships and proposing business decisions to the DAO. Beefy is still a relatively new business that can use talented people to help advise the core team.
There is marketing that you can contribute to too, if you can write a decent tweet then you can help out in #tweet-development. The Discord has a #social-watch channel where links to Beefy mentions on social media are posted, you can help out with user queries there or in the Discord or Telegram itself. Moderators of Discord and Telegram are (variably) paid positions too and are usually the first line of customer support.
The best way to get involved is to just go ahead and get started, help where you can, contribute to discussions and collaborate with everyone.
What is the difference between a Vault and an Earnings Pool?
In a Vault you earn more of what you deposited into it, with compound interest (APY). In an Earnings Pool you earn a different token than the asset you deposited, with linear interest (APR).
An example is the BIFI Maxi Vault, in which you earn more BIFI exponentially, and the many BIFI Earnings Pools, in which you earn linear interest in the form of $ETH, $BNB, $AVAX and more.
Why does it cost so much gas to deposit into a Beefy vault?
Many of Beefy's vaults "Harvest on Deposit". This means that when you deposit into the vault, you are also calling the harvest function. Calling the Harvest function is more complex than a simple deposit and thus has a higher gas limit/fee. Beefy does this so that it is impossible for malicious actors to steal yield so a withdrawal fee is not required. This greatly benefits long-term investors.
Almost all of the vaults on more inexpensive chains like Fantom and Polygon harvest on deposit. You can also tell if a vault harvests on deposit if there is no withdrawal fee.
As the Harvest Caller, you will also receive some of the wrapped native chain token in as a reward for calling the harvest. See Beefy Finance Fees Breakdown for more information on the Harvest Caller.
How can I find out how much earnings I have accumulated?
Your rewards are added to your deposited token amount on each harvest and compound cycle. You can use a DeFi dashboard that will be able to calculate exactly how much profit you have made on your investments. External tools such as TopDeFi will read your wallet address and give you an accurate picture of your initial investment and current earnings.
|
Write expressions to represent the quantities described below.
Geraldine is
4
years younger than Tom. If Tom is
t
years old, how old is Geraldine? Also, if Steven is twice as old as Geraldine, how old is he?
4
years younger than Tom, so Tom's age subtracted by
4
is Geraldine's age:
(t−4)
Steven is twice as old as Geraldine:
2(t−4)
. Remember that Geraldine's age is equal to
(t−4)
150
people went to see “Ode to Algebra” performed in the school auditorium. If the number of children that attended the performance was
c
, how many adults attended?
We know that a total of
150
people attended the Ode to Algebra. There were two main groups of people
(the children and the adults) so:
(150−c)
yields the amount of adults.
The cost of a new CD is
\$14.95
, and the cost of a video game is
\$39.99
c
CDs and
video games cost?
Refer to both problems (a) and (b) for help. Use what you know about the problem to create a
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Time Table - MapleSim Help
Home : Support : Online Help : MapleSim : MapleSim Component Library : Signal Blocks : Interpolation Tables : Time Table
Generate output from a time-based lookup table
The Time Lookup Table (or Time Table) component generates an output signal by interpolation using time as the input. The input values of the data table (that is, the first column of values) must be increasing; however, a discontinuous function can be created by repeating an input value.
The start time parameter is subtracted from the actual simulation time and the resulting time is used as the interpolation time.
The offset parameter is added to the computed interpolation value. The interpolation value is 0 if the interpolation time is less than the first time point.
The data source parameter selects the source for the data. It can be either file, attachment, or inline.
file: the data is saved in an Excel or a CSV file on the hard drive. Use the data parameter to browse to and select the data file.
attachment: the data set is attached to the MapleSim model. Use the data parameter to select the attachment that contains the data set.
Note: In the Attachments pane, your data set file must be attached in the Data Sets category. For more information, see Attaching a File to a Model.
Attach a Microsoft® Excel® (.xls/.xlsx) or comma separated value (.csv) file containing the data values to the model.
Generate a data set in the Apps Manager tab using either the Data Generation app or Random Data Generation app. Data sets that you generate have the .csv file extension. For more information about MapleSim apps, see Opening MapleSim Apps and Templates.
inline: enter the data table in the table parameter as an m (rows) by n (columns) matrix.
To change the dimensions of table, right-click (Control-click for Mac®) the parameter field and select Edit Matrix Dimensions. In the Matrix Dimensions dialog, enter values for the number or rows and columns and then click OK.
For all data source options, the first column in the data table represents the time. The other columns,
2
through n, represent the output data.
The columns parameter selects which data column to output if your data table has more than two columns. By default, the second column,
[2]
, is output.
You can output an array by entering an array in columns. For example, set columns to
[2,3,4]
to output a three-element array corresponding to the data in the second, third, and fourth columns of your data table.
y
Real output of dimension
, the length of the
\mathrm{columns}
\mathrm{data source}
\mathrm{inline}
Specifies data source; see section (above)
\mathrm{table}
\left[\begin{array}{cc}0& 1\end{array}\right]
\mathrm{file name}
\mathrm{DataFileName}
The path to the external file
\mathrm{data}
\mathrm{columns}
\left[2\right]
Data columns used. In a spreadsheet, column A corresponds to 1.
\mathrm{offset}
0
Offset of output signal
y
\mathrm{start time}
0
s
Output y = offset for time < startTime
\mathrm{skip rows}
0
\mathrm{smoothness}
Selects interpolation: linear, cubic spline, or none
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Rotary transformer that measures angle of rotation - MATLAB - MathWorks í•œêµ
Equations when Omitting Dynamics
Equations when Including Dynamics
Stator reactance
Frequency at which reactances and transformation ratio are specified
Peak coefficient of coupling
Rotary transformer that measures angle of rotation
The Resolver block models a generic resolver, which measures the electrical phase angle of a signal through electromagnetic coupling. The resolver consists of a rotary transformer that couples an AC voltage applied to the primary winding to two secondary windings. These secondary windings are physically oriented at 90 degrees to each other. As the rotor angle changes, the relative coupling between the primary and the two secondary windings varies. In the Resolver block model, the first secondary winding is oriented such that peak coupling occurs when the rotor is at zero degrees, and therefore the second secondary winding has minimum coupling when the rotor is at zero degrees.
Without loss of generality, it is assumed that the transformer between primary and rotor circuit is ideal with a ratio of 1:1. This results in the rotor current and voltage being equivalent to the primary current and voltage.
You have two options for defining the block equations:
Omit the dynamics by neglecting the transformer inductive terms. This model is only valid if the sensor is driven by a sine wave because any DC component on the primary side will pass to the output side.
Include the inductive terms, thereby capturing voltage amplitude loss and phase differences. This model is valid for any input waveform. Within this option, you can either specify the inductances and the peak coupling coefficient directly, or specify the transformation ratio and measured impedances, in which case the block uses these values to determine the inductive terms.
The equations are based on the superposition of two ideal transformers, both with coupling coefficients that depend on rotor angle. The two ideal transformers have a common primary winding. See the Simscape™ Ideal Transformer block reference page for more information on modeling ideal transformers. The equations are:
Kx = R cos( N Θ )
Ky = R sin( N Θ )
vx = Kx v p
vy = Ky v p
ip = – Kx ix – Ky iy
vp and ip are the rotor (or equivalently primary) voltage and current, respectively.
vx and ix are the first secondary voltage and current, respectively.
vy and iy are the second secondary voltage and current, respectively.
Kx is the coupling coefficient for the first secondary winding.
Ky is the coupling coefficient for the second secondary winding.
R is the transformation ratio.
Θ is the rotor angle.
The equations are based on the superposition of two mutual inductors, both with coupling coefficients that depend on rotor angle. The two mutual inductors have a common primary winding. See the Simscape Mutual Inductor block reference page for more information on modeling mutual inductors. The equations are:
{v}_{p}={R}_{p}{i}_{p}+{L}_{p}\frac{d{i}_{p}}{dt}+\sqrt{{L}_{p}{L}_{s}}k\left(\mathrm{cos}\left(N\mathrm{θ}\right)\frac{d{i}_{x}}{dt}+\mathrm{sin}\left(N\mathrm{θ}\right)\frac{d{i}_{y}}{dt}\right)
{v}_{x}={R}_{s}{i}_{x}+{L}_{s}\frac{d{i}_{x}}{dt}+\sqrt{{L}_{p}{L}_{s}}k\mathrm{cos}\left(N\mathrm{θ}\right)\frac{d{i}_{p}}{dt}
{v}_{y}={R}_{s}{i}_{y}+{L}_{s}\frac{d{i}_{y}}{dt}+\sqrt{{L}_{p}{L}_{s}}k\mathrm{sin}\left(N\mathrm{θ}\right)\frac{d{i}_{p}}{dt}
Rp is the rotor (or primary) resistance.
Lp is the rotor (or primary) inductance.
Rs is the stator (or secondary) resistance.
Ls is the stator (or secondary) inductance.
k is the coefficient of coupling.
It is assumed that coupling between the two secondary windings is zero.
Datasheets typically do not quote the coefficient of coupling and inductance parameters, but instead give the transformation ratio R and measured impedances. If you select Specify transformation ratio and measured impedances for the Parameterization parameter, then the values you provide are used to determine values for the equation coefficients, as defined above.
The resolver draws no torque between the mechanical rotational ports R and C.
The transformer between primary and rotor circuit is ideal with a ratio of 1:1.
The coupling between the two secondary windings is zero.
p1 — Primary winding positive terminal
Electrical conserving port associated with the positive terminal of the primary winding.
p2 — Primary winding negative terminal
Electrical conserving port associated with the negative terminal of the primary winding.
R — Resolver rotor
Mechanical rotational conserving port connected to the rotor.
C — Resolver case
Mechanical rotational conserving port connected to the resolver case.
x1 — Secondary winding x positive terminal
Electrical conserving port associated with the positive terminal of secondary winding x.
x2 — Secondary winding x negative terminal
Electrical conserving port associated with the negative terminal of secondary winding x.
y1 — Secondary winding y positive terminal
Electrical conserving port associated with the positive terminal of secondary winding y.
y2 — Secondary winding y negative terminal
Electrical conserving port associated with the negative terminal of secondary winding y.
Parameterization — Resolver parameterization
Specify transformation ratio and omit dynamics (default) | Specify transformation ratio and measured impedances | Specify equation parameters directly
Specify transformation ratio and omit dynamics — Provide values for transformation ratio, number of pole pairs, and initial rotor angle only. This model neglects the transformer inductive terms, and is only valid if the sensor is driven by a sine wave. The equations are based on the superposition of two ideal transformers, both with coupling coefficients that depend on rotor angle. For more information, see Equations when Omitting Dynamics.
Specify transformation ratio and measured impedances — Provide additional values to determine the transformer inductive terms, to model the voltage amplitude loss and phase differences. This model is valid for any input waveform. The equations are based on the superposition of two mutual inductors, both with coupling coefficients that depend on rotor angle. For more information, see Equations when Including Dynamics.
Specify equation parameters directly — Model the dynamics, but provide values for rotor and stator inductances and the peak coefficient of coupling, instead of transformation ratio and measured impedances. For more information, see Equations when Including Dynamics. This model is valid for any input waveform.
Transformation ratio — Peak output to input voltage ratio
Ratio between the peak output voltage and the peak input voltage assuming negligible secondary voltage drop due to resistance and inductance.
To enable this parameter, set the Parameterization parameter to Specify transformation ratio and omit dynamics or Specify transformation ratio and measured impedances. If you select Specify transformation ratio and measured impedances for the Parameterization parameter, then the transformation ratio takes the voltage drop due to primary winding resistance into account.
Rotor resistance — Primary resistance
70 Ohm (default) | positive number
Rotor ohmic resistance. This resistance is also referred to as the primary resistance.
To enable this parameter, set the Parameterization parameter to Specify transformation ratio and measured impedances or Specify equation parameters directly.
Stator resistance — Secondary resistance
180 Ohm (default) | positive number
Stator ohmic resistance. This resistance is also referred to as the secondary resistance. It is assumed that both secondaries have the same resistance.
Rotor reactance — Primary reactance
Rotor reactance when the secondary windings are open-circuit. This reactance is also referred to as the primary reactance.
To enable this parameter, set the Parameterization parameter to Specify transformation ratio and measured impedances.
Stator reactance — Secondary reactance
Stator reactance when the primary winding is open-circuit. This reactance is also referred to as the secondary reactance.
Frequency at which reactances and transformation ratio are specified — Sinusoidal source frequency
10 kHz (default) | positive number
Frequency of the sinusoidal source used when measuring the reactances.
Rotor inductance — Primary reactance
0.0016 H (default) | positive number
Rotor or primary inductance, Lp.
To enable this parameter, set the Parameterization parameter to Specify equation parameters directly.
Stator inductance — Secondary reactance
Stator or secondary inductance, Ls.
Peak coefficient of coupling — Maximum coupling coefficient
0.35 (default) | number between zero and one, exclusive
Peak coefficient of coupling between the primary and secondary windings.
Number of pole pairs — Rotor pole pairs
Number of pole pairs on the rotor.
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Barry Boehm - Wikipedia
Not to be confused with the Boehm garbage collector created by Hans-Juergen Boehm.
Barry W. Boehm (born 1935) is an American software engineer, distinguished professor[1][2] of computer science, industrial and systems engineering; the TRW Professor of Software Engineering; and founding director of the Center for Systems and Software Engineering at the University of Southern California. He is known for his many contributions to the area of software engineering.
In 1996, Boehm was elected as a member into the National Academy of Engineering for contributions to computer and software architectures and to models of cost, quality, and risk for aerospace systems.
3.1 Software versus hardware costs
3.2 Software economics
3.4 Wideband Delphi
3.5 Incremental Commitment Model
Boehm received a B.A. in mathematics from Harvard University in 1957, and a M.S. in 1961, and Ph.D. from UCLA in 1964, both in mathematics as well. He has also received honorary Sc.D. in Computer Science from the U. of Massachusetts in 2000 and in Software Engineering from the Chinese Academy of Sciences in 2011.[3]
In 1955 he started working as a programmer-analyst at General Dynamics. In 1959 he switched to the RAND Corporation, where he was head of the Information Sciences Department until 1973. From 1973 to 1989 he was chief scientist of the Defense Systems Group at TRW Inc. From 1989 to 1992 he served within the U.S. Department of Defense (DoD) as director of the DARPA Information Science and Technology Office, and as director of the DDR&E Software and Computer Technology Office.[3] Since 1992 he is TRW Professor of Software Engineering, Computer Science Department, and director, USC Center for Systems and Software Engineering, formerly Center for Software Engineering.
He has served on the board of several scientific journals, including the IEEE Transactions on Software Engineering, Computer, IEEE Software, ACM Computing Reviews, Automated Software Engineering, Software Process, and Information and Software Technology.[3]
Recent awards for Barry Boehm include the Office of the Secretary of Defense Award for Excellence in 1992, the ASQC Lifetime Achievement Award in 1994, the ACM Distinguished Research Award in Software Engineering in 1997, and the IEEE International Stevens Award. He is an AIAA Fellow, an ACM Fellow, an IEEE Fellow, and a member of the National Academy of Engineering (1996).[4] He received the Mellon Award for Excellence in Mentoring in 2005[5] and the IEEE Simon Ramo Medal in 2010. He was appointed as a distinguished professor on January 13, 2014[1]
He was awarded the INCOSE Pioneer Award in 2019 by the International Council on Systems Engineering for significant pioneering contributions to the field of systems engineering.[6]
Boehm's research interests include software development process modeling, software requirements engineering, software architectures, software metrics and cost models, software engineering environments, and knowledge-based software engineering.[3]
His contributions to the field, according to Boehm (1997) himself, include "the Constructive Cost Model (COCOMO), the spiral model of the software process, the Theory W (win-win) approach to software management and requirements determination and two advanced software engineering environments: the TRW Software Productivity System and Quantum Leap Environment".[3]
Software versus hardware costs[edit]
In an important 1973 report entitled "Ada - The Project : The DoD High Order Language Working Group" to the Defense Advanced Research Projects Agency (DARPA),[7] Boehm predicted that software costs would overwhelm hardware costs. DARPA had expected him to predict that hardware would remain the biggest problem, encouraging them to invest in even larger computers. The report inspired a change of direction in computing.
Software economics[edit]
Barry Boehm's 1981 book Software Engineering Economics documents his Constructive Cost Model (COCOMO). It relates software development effort for a program, in Person-Months (PM), to Thousand Source Lines of Code (KSLOC).
{\displaystyle PM=A*(KSLOC)^{B}}
Where A is a calibration constant based on project data and B is an exponent for the software diseconomy of scale.
Note: since man-years are not interchangeable with years, Brooks' Law applies:
Thus this formula is best applied to stable software development teams which have completed multiple projects.
Spiral model[edit]
Spiral model (Boehm, 1988).
Boehm also created the spiral model of software development, in which the phases of development are repeatedly revisited. This iterative software development process influenced MBASE and extreme programming.
Wideband Delphi[edit]
Boehm refined the Delphi method of estimation to include more group iteration, making it more suitable for certain classes of problems, such as software development. This variant is called the Wideband Delphi method.
Incremental Commitment Model[edit]
The Incremental Commitment Model (ICM)[8] is a system design, developmental, and evolution process for 21st century systems. The systems' types cover a wide range from COTS based systems to "routine" Information Systems to human intensive and life or safety critical.[9]
It was only in 1998, after the development of the ICM that Barry Boehm along with A Winsor Brown started to focus on reconciling it with the WinWin Spiral Model and its incarnation in MBASE[10] and the follow-on Lean MBASE,[11] and working towards an Incremental Commitment Model for Software (ICMS) by adapting the existing WinWin Spiral Model support tools.[9] In 2008, the evolving ICM for Software with its risk-driven anchor point decisions, proved very useful to several projects which ended up having unusual life cycle phase sequences.[9]
Barry Boehm has published over 170 articles[12] and several books. Books, a selection:
1978. Characteristics of Software Quality. With J.R. Brown, H. Kaspar, M. Lipow, G. McLeod, and M. Merritt, North Holland.
1981. Software Engineering Economics. Englewood Cliffs, NJ : Prentice-Hall, 1981 ISBN 0-13-822122-7.
— (1989). "Software Risk Management". In Ghezzi, C.; McDermid, J. A. (eds.). Proceedings of 2nd European Software Engineering Conference. ESEC'89. LNCS. Vol. 387. pp. 1–19. doi:10.1007/3-540-51635-2_29. ISBN 3-540-51635-2. ISSN 0302-9743.
1996. Ada and Beyond: Software Policies for the Department of Defense. National Academy Press.
2007. Software engineering: Barry Boehm's lifetime contributions to software development, management and research. Ed. by Richard Selby. Wiley/IEEE press, 2007. ISBN 0-470-14873-X.
2004. Balancing Agility and Discipline: A Guide for the Perplexed. With Richard Turner. Person Education, Inc 2004 ISBN 0-321-18612-5.
2014. The Incremental Commitment Spiral Model: Principles and Practices for Successful Systems and Software. B. Boehm, J. Lane, S. Koolmanojwong, R. Turner. Addison-Wesley Professional, 2014. ISBN 0-321-80822-3.
1996. "Anchoring the Software Process",. In: IEEE Software, July 1996.
1997. "Developing Multimedia Applications with the WinWin Spiral Model," with A. Egyed, J. Kwan, and R. Madachy. In: Proceedings, ESEC/FSE 97 and ACM Software Engineering Notes, November 1997.
^ "Dr. Barry W. Boehm named USC Distinguished Professor – CSSE". Csse.usc.edu. 2014-01-27. Retrieved 2016-10-23.
^ a b c d e "Biography". csse.usc.edu. Retrieved 2017-05-14.
^ "NAE Directory, 1996".
^ "Pioneer Awards". INCOSE. Retrieved 7 March 2020.
^ William A. Whitaker (1993). Ada - The Project : The DoD High Order Language Working Group Archived 2008-08-12 at the Wayback Machine. Accessdate 2008-08-06.
^ "CSE Website". Sunset.usc.edu. Retrieved 2016-10-23.
^ a b c Boehm, B., Brown, A. W., and Koolmanojwong, S. Demonstration Proposal: Incremental Commitment Model for Software. University of Southern California, Los Angeles, CA. 90089.
^ Boehm, B., Abts, C., Brown, A. W., Chulani, S., Clark, B. K., Horowitz, K., Madachy, R., Reifer, D., and Steece, B. 2000. Software Cost Estimation with COCOMO II. ISBN 0-13-026692-2. Prentice Hall PTR Upper Saddle River, NJ.
^ "DBLP: Barry W. Boehm". Dblp.uni-trier.de. Retrieved 2016-10-23.
Wikimedia Commons has media related to Barry W. Boehm.
Wikiquote has quotations related to Barry Boehm.
Barry Boehm home page
"A View of 20th and 21st Century Software Engineering" — talk by Barry Boehm
Retrieved from "https://en.wikipedia.org/w/index.php?title=Barry_Boehm&oldid=1086424191"
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Perform predictor variable selection for Bayesian linear regression models - MATLAB estimate
Select Variables Using Bayesian Lasso Regression
Select Variables Using SSVS
Access Estimates in Estimation Summary Display
Sigma2Start
Perform predictor variable selection for Bayesian linear regression models
To estimate the posterior distribution of a standard Bayesian linear regression model, see estimate.
PosteriorMdl = estimate(PriorMdl,X,y) returns the model that characterizes the joint posterior distributions of β and σ2 of a Bayesian linear regression model. estimate also performs predictor variable selection.
PriorMdl specifies the joint prior distribution of the parameters, the structure of the linear regression model, and the variable selection algorithm. X is the predictor data and y is the response data. PriorMdl and PosteriorMdl are not the same object type.
To produce PosteriorMdl, estimate updates the prior distribution with information about the parameters that it obtains from the data.
NaNs in the data indicate missing values, which estimate removes using list-wise deletion.
PosteriorMdl = estimate(PriorMdl,X,y,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, 'Lambda',0.5 specifies that the shrinkage parameter value for Bayesian lasso regression is 0.5 for all coefficients except the intercept.
If you specify Beta or Sigma2, then PosteriorMdl and PriorMdl are equal.
[PosteriorMdl,Summary] = estimate(___) uses any of the input argument combinations in the previous syntaxes and also returns a table that includes the following for each parameter: posterior estimates, standard errors, 95% credible intervals, and posterior probability that the parameter is greater than 0.
Consider the multiple linear regression model that predicts US real gross national product (GNPR) using a linear combination of industrial production index (IPI), total employment (E), and real wages (WR).
{\text{GNPR}}_{t}={\beta }_{0}+{\beta }_{1}{\text{IPI}}_{t}+{\beta }_{2}{\text{E}}_{t}+{\beta }_{3}{\text{WR}}_{t}+{\epsilon }_{t}.
t
{\epsilon }_{t}
{\sigma }^{2}
Assume the prior distributions are:
{\beta }_{k}|{\sigma }^{2}
{\sigma }^{2}/\lambda
\lambda
{\sigma }^{2}\sim IG\left(A,B\right)
A
B
Create a prior model for Bayesian lasso regression. Specify the number of predictors, the prior model type, and variable names. Specify these shrinkages:
0.01 for the intercept
10 for IPI and WR
1e5 for E because it has a scale that is several orders of magnitude larger than the other variables
The order of the shrinkages follows the order of the specified variable names, but the first element is the shrinkage of the intercept.
PriorMdl = bayeslm(p,'ModelType','lasso','Lambda',[0.01; 10; 1e5; 10],...
PriorMdl is a lassoblm Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance.
\beta
{\sigma }^{2}
\beta
{\sigma }^{2}
given the data. estimate displays a summary of the marginal posterior distributions in the MATLAB® command line. Rows of the summary correspond to regression coefficients and the disturbance variance, and columns correspond to characteristics of the posterior distribution. The characteristics include:
CI95, which contains the 95% Bayesian equitailed credible intervals for the parameters. For example, the posterior probability that the regression coefficient of IPI is in [4.157, 4.799] is 0.95.
Given the shrinkages, the distribution of E is fairly dense around 0. Therefore, E might not be an important predictor.
Consider the regression model in Select Variables Using Bayesian Lasso Regression.
Create a prior model for performing stochastic search variable selection (SSVS). Assume that
\beta
{\sigma }^{2}\text{\hspace{0.17em}}
\beta
{\sigma }^{2}
. Because SSVS uses Markov chain Monte Carlo for estimation, set a random number seed to reproduce the results.
\beta
{\sigma }^{2}
given the data. estimate displays a summary of the marginal posterior distributions in the command line. Rows of the summary correspond to regression coefficients and the disturbance variance, and columns correspond to characteristics of the posterior distribution. The characteristics include:
CI95, which contains the 95% Bayesian equitailed credible intervals for the parameters. For example, the posterior probability that the regression coefficient of E (standardized) is in [0.000, 0.0.002] is 0.95.
\gamma =1
for a variable). For example, the posterior probability E that should be included in the model is 0.0925.
Consider the regression model and prior distribution in Select Variables Using Bayesian Lasso Regression.
Create a Bayesian lasso regression prior model for 3 predictors and specify variable names. Specify the shrinkage values 0.01, 10, 1e5, and 10 for the intercept, and the coefficients of IPI, E, and WR.
PriorMdl = bayeslm(p,'ModelType','lasso','VarNames',["IPI" "E" "WR"],...
'Lambda',[0.01; 10; 1e5; 10]);
\beta
{\sigma }^{2}=10
[Mdl,SummaryBeta] = estimate(PriorMdl,X,y,'Sigma2',10);
Conditional variable: Sigma2 fixed at 10
| Mean Std CI95 Positive Distribution
Intercept | -8.0643 4.1992 [-16.384, 0.018] 0.025 Empirical
IPI | 4.4454 0.0679 [ 4.312, 4.578] 1.000 Empirical
E | 0.0004 0.0002 [ 0.000, 0.001] 0.999 Empirical
WR | 2.9792 0.1672 [ 2.651, 3.305] 1.000 Empirical
Sigma2 | 10 0 [10.000, 10.000] 1.000 Empirical
\beta
{\sigma }^{2}
is fixed at 10 during estimation, inferences on it are trivial.
IPI | 0 0.1000 [-0.200, 0.200] 0.500 Scale mixture
E | 0 0.0000 [-0.000, 0.000] 0.500 Scale mixture
WR | 0 0.1000 [-0.200, 0.200] 0.500 Scale mixture
Because estimate computes the conditional posterior distribution, it returns the model input PriorMdl, not the conditional posterior, in the first position of the output argument list.
Display the estimation summary table.
SummaryBeta
SummaryBeta=5×6 table
Mean Std CI95 Positive Distribution Covariances
__________ __________ ________________________ ________ _____________ _______________________________________________________________________
Intercept -8.0643 4.1992 -16.384 0.01837 0.0254 {'Empirical'} 17.633 0.17621 -0.00053724 0.11705 0
IPI 4.4454 0.067949 4.312 4.5783 1 {'Empirical'} 0.17621 0.0046171 -1.4103e-06 -0.0068855 0
E 0.00039896 0.00015673 9.4925e-05 0.00070697 0.9987 {'Empirical'} -0.00053724 -1.4103e-06 2.4564e-08 -1.8168e-05 0
WR 2.9792 0.16716 2.6506 3.3046 1 {'Empirical'} 0.11705 -0.0068855 -1.8168e-05 0.027943 0
Sigma2 10 0 10 10 1 {'Empirical'} 0 0 0 0 0
SummaryBeta contains the conditional posterior estimates.
{\sigma }^{2}
\beta
is the conditional posterior mean of
\beta |{\sigma }^{2},X,y
(stored in SummaryBeta.Mean(1:(end – 1))). Return the estimation summary table.
condPostMeanBeta = SummaryBeta.Mean(1:(end - 1));
Conditional variable: Beta fixed at -8.0643 4.4454 0.00039896 2.9792
Intercept | -8.0643 0.0000 [-8.064, -8.064] 0.000 Empirical
IPI | 4.4454 0.0000 [ 4.445, 4.445] 1.000 Empirical
E | 0.0004 0.0000 [ 0.000, 0.000] 1.000 Empirical
WR | 2.9792 0.0000 [ 2.979, 2.979] 1.000 Empirical
Sigma2 | 56.8314 10.2921 [39.947, 79.731] 1.000 Empirical
estimate displays an estimation summary of the conditional posterior distribution of
{\sigma }^{2}
\beta
is condPostMeanBeta. In the display, inferences on
\beta
are trivial.
\beta
{\sigma }^{2}\text{\hspace{0.17em}}
\beta
{\sigma }^{2}
. Because SSVS uses Markov chain Monte Carlo for estimation, set a random number seed to reproduce the results. Suppress the estimation display, but return the estimation summary table.
[PosteriorMdl,Summary] = estimate(PriorMdl,X,y,'Display',false);
\beta
{\sigma }^{2}
given the data. Summary is a table with columns corresponding to posterior characteristics and rows corresponding to the coefficients (PosteriorMdl.VarNames) and disturbance variance (Sigma2).
Display the estimated parameter covariance matrix (Covariances) and proportion of times the algorithm includes each predictor (Regime).
Covariances = Summary(:,"Covariances")
Covariances=5×1 table
Intercept 103.74 1.0486 -0.0031629 0.6791 7.3916
IPI 1.0486 0.023815 -1.3637e-05 -0.030387 0.06611
E -0.0031629 -1.3637e-05 1.3481e-07 -8.8792e-05 -0.00025044
WR 0.6791 -0.030387 -8.8792e-05 0.13066 0.089039
Sigma2 7.3916 0.06611 -0.00025044 0.089039 74.911
Regime = Summary(:,"Regime")
Regime=5×1 table
Intercept 0.8806
IPI 0.4545
E 0.0925
WR 0.1734
Sigma2 NaN
Regime contains the marginal posterior probability of variable inclusion (
\gamma =1
PriorMdl — Bayesian linear regression model for predictor variable selection
mixconjugateblm model object | mixsemiconjugateblm model object | lassoblm model object
Bayesian linear regression model for predictor variable selection, specified as a model object in this table.
Display — Flag to display Bayesian estimator summary to command line
Flag to display Bayesian estimator summary to the command line, specified as the comma-separated pair consisting of 'Display' and a value in this table.
The estimation information includes the estimation method, fixed parameters, the number of observations, and the number of predictors. The summary table contains estimated posterior means, standard deviations (square root of the posterior variance), 95% equitailed credible intervals, the posterior probability that the parameter is greater than 0, and a description of the posterior distribution (if known). For models that perform SSVS, the display table includes a column for variable-inclusion probabilities.
If you specify either Beta or Sigma2, then estimate includes your specification in the display. Corresponding posterior estimates are trivial.
Monte Carlo simulation adjusted sample size, specified as the comma-separated pair consisting of 'NumDraws' and a positive integer. estimate actually draws BurnIn – NumDraws*Thin samples. Therefore, estimate bases the estimates off NumDraws samples. For details on how estimate reduces the full Monte Carlo sample, see Algorithms.
mixconjugateblm model object | mixsemiconjugateblm model object | lassoblm model object | empiricalblm model object
Bayesian linear regression model storing distribution characteristics, returned as a mixconjugateblm, mixsemiconjugateblm, lassoblm, or empiricalblm model object.
If you do not specify either Beta or Sigma2 (their values are []), then estimate updates the prior model using the data likelihood to form the posterior distribution. PosteriorMdl characterizes the posterior distribution and is an empiricalblm model object. Information PosteriorMdl stores or displays helps you decide whether predictor variables are important.
If you specify either Beta or Sigma2, then PosteriorMdl equals PriorMdl (the two models are the same object storing the same property values). estimate does not update the prior model to form the posterior model. However, Summary stores conditional posterior estimates.
Summary of Bayesian estimators, returned as a table. Summary contains the same information as the display of the estimation summary (Display). Rows correspond to parameters, and columns correspond to these posterior characteristics:
Positive – Posterior probability that the parameter is greater than 0
Regime – Variable-inclusion probabilities for models that perform SSVS; low probabilities indicate that the variable should be excluded from the model
\ell \left(\beta ,{\sigma }^{2}|y,x\right)=\prod _{t=1}^{T}\varphi \left({y}_{t};{x}_{t}\beta ,{\sigma }^{2}\right).
Monte Carlo simulation is subject to variation. If estimate uses Monte Carlo simulation, then estimates and inferences might vary when you call estimate multiple times under seemingly equivalent conditions. To reproduce estimation results, before calling estimate, set a random number seed by using rng.
This figure shows how estimate reduces the Monte Carlo sample using the values of NumDraws, Thin, and BurnIn.
Rectangles represent successive draws from the distribution. estimate removes the white rectangles from the Monte Carlo sample. The remaining NumDraws black rectangles compose the Monte Carlo sample.
mixconjugateblm | mixsemiconjugateblm | lassoblm
summarize | forecast | simulate | plot
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Rotation matrix for rotations around y-axis - MATLAB roty - MathWorks Switzerland
{R}_{y}\left(\beta \right)=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]
{v}^{\prime }=Av={R}_{z}\left(\gamma \right){R}_{y}\left(\beta \right){R}_{x}\left(\alpha \right)v
{R}_{x}\left(\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]
{R}_{y}\left(\beta \right)=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]
{R}_{z}\left(\gamma \right)=\left[\begin{array}{ccc}\mathrm{cos}\gamma & -\mathrm{sin}\gamma & 0\\ \mathrm{sin}\gamma & \mathrm{cos}\gamma & 0\\ 0& 0& 1\end{array}\right]
{A}^{-1}A=1
{R}_{x}^{-1}\left(\alpha \right)={R}_{x}\left(-\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & \mathrm{sin}\alpha \\ 0& -\mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]={R}_{x}^{\prime }\left(\alpha \right)
i,j,k
{i}^{\prime },j{,}^{\prime }{k}^{\prime }
\begin{array}{ll}{i}^{\prime }\hfill & =Ai\hfill \\ {j}^{\prime }\hfill & =Aj\hfill \\ {k}^{\prime }\hfill & =Ak\hfill \end{array}
\left[\begin{array}{c}{i}^{\prime }\\ {j}^{\prime }\\ {k}^{\prime }\end{array}\right]={A}^{\prime }\left[\begin{array}{c}i\\ j\\ k\end{array}\right]
v={v}_{x}i+{v}_{y}j+{v}_{z}k={{v}^{\prime }}_{x}{i}^{\prime }+{{v}^{\prime }}_{y}{j}^{\prime }+{{v}^{\prime }}_{z}{k}^{\prime }
\left[\begin{array}{c}{{v}^{\prime }}_{x}\\ {{v}^{\prime }}_{y}\\ {{v}^{\prime }}_{z}\end{array}\right]={A}^{-1}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]={A}^{\prime }\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]
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Fourt–Woodlock equation - Wikipedia
Fourt–Woodlock equation
The Fourt–Woodlock equation (sometimes misspelled Fort-Woodlock equation) is a market research tool to describe the total volume of consumer product purchases per year based on households which initially make trial purchases of the product and those households which make a repeat purchase within the first year. Since it includes the effects of initial trial and repeat rates, the equation is useful in new product development.
The Fourt–Woodlock equation itself is
{\displaystyle V=(HH\cdot TR\cdot TU)+(HH\cdot TR\cdot MR\cdot RR\cdot RU)}
The left-hand-side of the equation is the volume of purchases per unit time (usually taken to be one year). On the right-hand-side, the first parentheses describes trial volume, and the second describes repeat volume.
HH is the total number of households in the geographic area of projection, and TR ("trial rate") is the percentage of those households which will purchase the product for the first time in a given time period. TU ("trial units") is the number of units purchased on this first purchase occasion. MR is "measured repeat," or the percentage of those who tried the product who will purchase it at least one more time within the first year of the product's launch. RR is the repeats per repeater: the number of repeat purchases within that same year. RU is the number of repeat units purchased on each repeat event.
The applied science of product forecasting is used to estimate each term on the right-hand-side of this equation. Estimating the trial rate is complex and typically requires sophisticated models to predict, while the number of households is usually well known (except in some unusually complicated markets such as China).
Fourt L.A., Woodlock J.W., 1960. 'Early prediction of market success for new grocery products.' Journal of Marketing 25: 31–38.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fourt–Woodlock_equation&oldid=612357065"
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When trying to evaluate the accuracy of our multiple linear regression model, one technique we can use is Residual Analysis.
The difference between the actual value y, and the predicted value ŷ is the residual e. The equation is:
e = y - \hat{y}
In the StreetEasy dataset, y is the actual rent and the ŷ is the predicted rent. The real y values should be pretty close to these predicted y values.
sklearn‘s linear_model.LinearRegression comes with a .score() method that returns the coefficient of determination R² of the prediction.
The coefficient R² is defined as:
1 - \frac{u}{v}
where u is the residual sum of squares:
((y - y_predict) ** 2).sum()
and v is the total sum of squares (TSS):
((y - y.mean()) ** 2).sum()
The TSS tells you how much variation there is in the y variable.
R² is the percentage variation in y explained by all the x variables together.
For example, say we are trying to predict rent based on the size_sqft and the bedrooms in the apartment and the R² for our model is 0.72 — that means that all the x variables (square feet and number of bedrooms) together explain 72% variation in y (rent).
Now let’s say we add another x variable, building’s age, to our model. By adding this third relevant x variable, the R² is expected to go up. Let say the new R² is 0.95. This means that square feet, number of bedrooms and age of the building together explain 95% of the variation in the rent.
The best possible R² is 1.00 (and it can be negative because the model can be arbitrarily worse). Usually, a R² of 0.70 is considered good.
Use the .score() method from LinearRegression to find the mean squared error regression loss for the training set.
Use the .score() method from LinearRegression to find the mean squared error regression loss for the testing set.
8. Evaluating the Model's Accuracy
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Pseudocode Knowpia
No broad standard for pseudocode syntax exists, as a program in pseudocode is not an executable program; however, certain limited standards exist (such as for academic assessment). Pseudocode resembles skeleton programs, which can be compiled without errors. Flowcharts, drakon-charts and Unified Modelling Language (UML) charts can be thought of as a graphical alternative to pseudocode, but need more space on paper. Languages such as HAGGIS bridge the gap between pseudocode and code written in programming languages.
procedure fizzbuzz;
print_number := true;
if i is divisible by 3 then begin
print_number := false;
print_number = true;
if (i is divisible by 3) {
print_number = false;
if (print_number) print i;
Mathematical style pseudocodeEdit
{\displaystyle \sum _{k\in S}x_{k}}
Common mathematical symbolsEdit
Machine compilation of pseudocode style languagesEdit
Natural language grammar in programming languagesEdit
Mathematical programming languagesEdit
^ Invitation to Computer Science, 8th Edition by Schneider/Gersting, "Keep statements language independent" as quoted in this stackexchange question
Look up pseudocode in Wiktionary, the free dictionary.
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A Trie in Swift - Donnacha Oisín Kidney
A Trie in Swift
Tags: Swift, Data Structures
If you google “cool data structures” you’ll get this as your first result. It’s a stackoverflow question: “What are the lesser known but useful data structures?”. And the top answer is a Trie. I read up on them, and found out a lot of cool things about their use (as well as finding out that I’m now the kind of person who googles “cool data structures”). So I rocked on up to my playground, and got writing.
A Trie is a prefix tree. It’s another recursive data structure: each Trie contains other children Tries, identifiable by their prefixes.
It’s a bit of a hipster data structure, not very widely used, but it’s got some useful applications. It’s got set-like operations, with insertion and searching each at
O(n)
n
is the length of the sequence being searched for. A Set is the only way to go for hashable, unordered elements. But, if you’ve got sequences of hashable elements, a Trie might be for you. (one thing to note is that Sets are hashable themselves, so if the sequences you want to store are unordered, a Set of Sets is more applicable)
A trie for keys
In Swift, we can do this by having every Trie contain a dictionary of prefixes and Tries. Something like this:
public struct Trie<Element : Hashable> {
private var children: [Element:Trie<Element>]
We don’t run into the problem of structs not being allowed to be recursive here, because we don’t directly store a Trie within a Trie - we store a dictionary, and therefore a reference to the child Tries. In this dictionary, the keys correspond to the prefixes. So how do we fill it up? Like lists, we can use the decomposition properties of generators:
private init<G : GeneratorType where G.Element == Element>(var gen: G) {
if let head = gen.next() {
children = [head:Trie(gen:gen)]
<S : SequenceType where S.Generator.Element == Element>
self.init(gen: seq.generate())
That’s not really enough. That can store one sequence, but we need an insert function. Here ya go:
private mutating func insert
<G : GeneratorType where G.Element == Element>
(var gen: G) {
children[head]?.insert(gen) ?? {children[head] = Trie(gen: gen)}()
public mutating func insert
insert(seq.generate())
There’s a line in there that some may find offensive:
And, to be honest, I’m not a huge fan of it myself. It’s making use of the fact that you can call mutating methods on optionals with chaining. When you do it in this example, the optional is returned by the dictionary lookup: we then want to mutate that value, if it’s there, with an insertion.
If it’s not there, though, we want to add it in, so we’ve got to have some way of understanding and dealing with that. We could try and extract the child Trie, like this:
if var child = children[head] {
child.insert(gen)
children[head] = Trie(gen: gen)
But the child there is just a copy of the actual child in the Trie we want to mutate. We could then set it back to the dictionary entry - but at this stage it feels like a lot of extra, inefficient work.
So, you can make use of the fact the functions which don’t return anything actually do return something: a special value called Void, or (). Except that, in this case, it’s ()? (or Optional<Void>). We’re not interested in the void itself, obviously, just whether or not it’s nil. So, one way you could use it would be like this:
if let _ = children[head]?.insert(gen) { return }
Or, to use guard:
guard let _ = children[head]?.insert(gen) else { children[head] = Trie(gen: gen) }
But I think the nil coalescing operator is a little clearer, without the distraction of let or _.
This data structure, as you can see, has a very different feel to the list. For a start, it’s much more mutable, with in-place mutating methods being a little easier than methods that return a new Trie. Also, laziness is pretty much out of the question: almost every imaginable useful method would involve evaluation of the entire Trie. (if anyone does have a useful way of thinking about Tries lazily, I’d love to hear it)
The contains function, the most important of them all, is here:
private func contains
(var gen: G) -> Bool {
return gen.next().map{self.children[$0]?.contains(gen) ?? false} ?? true
public func contains
(seq: S) -> Bool {
return contains(seq.generate())
So this uses more generators. If the generator is empty (gen.next() returns nil), then the Trie contains that sequence, as we have not yet found a dictionary without that element. Within the map() we search for the next element from the generator. If that returns nil, then the Trie doesn’t contain that sequence. Finally, if none of that works, return whether or not the child Trie contains the rest of the generator. Let’s try it out:
var jo = Trie([1, 2, 3])
jo.insert([4, 5, 6])
jo.contains([4, 5, 6]) // true
jo.contains([2, 1, 3]) // false
There’s a catch. The contains method doesn’t work as we’d like it to:
jo.contains([1, 2]) // true
Because we return true whenever the generator runs out, our Trie “contains” every prefix of the sequences that have been inserted. This is not what we want. One way to solve this may be to return true only if the last Trie found has no children. Something like this:
return gen.next().map{self.children[$0]?.contains(gen) ?? false} ?? children.isEmpty
But this doesn’t really work either. what if we did jo.insert([1, 2])? Now, if we check if the Trie contains [1, 2], we’ll get back false.
It’s time for flags. We need to add an extra variable to our Trie: a Boolean, which describes whether or not that Trie represents the end of a sequence.
private var endHere : Bool
We’ll also need to change our insert and init functions, so that when the generator returns nil, endHere gets initialised to true.
(children, endHere) = ([head:Trie(gen:gen)], false)
(children, endHere) = ([:], true)
endHere = true
And the contains function now returns endHere, instead of true:
public extension Trie {
return gen.next().map{self.children[$0]?.contains(gen) ?? false} ?? endHere
While we’re improving the contains function, we could use guard to make it much more readable:
private func contains<
G : GeneratorType where G.Element == Element
>(var gen: G) -> Bool {
guard let head = gen.next() else { return endHere }
return children[head]?.contains(gen) ?? false
Chris Eidhof gave me this idea. (Apparently there’s a Trie implementation in Functional Programming in Swift, his book. I’ve not read it, but it’s on my list. If Advanced Swiftis anything to go by, it should be fantastic.)
The objective of this Trie is to replicate all of the Set methods: Union, Intersect, etc. Most of those are manageable to build from just insert, init, and contains, but there’s one other function that comes in handy: remove.
Remove is deceptively difficult. You could just walk to the end of your given sequence to remove, and switch endHere from true to false, but that’s kind of cheating. I mean, you’ll be storing the same amount of information that way after a removal. No, what you need is something that deletes branches of a tree that aren’t being used any more.
Again, this is a little complicated. You can’t just find the head of the sequence you want to remove, and then delete all children: you may be deleting other entries along with that. You also can’t just delete when a given Trie only contains one child: that child may branch off subsequently, or it may contain prefixes for the sequence you want to remove.
Crucially, all of the information telling you whether or not you can delete a given entry in a given Trie will come from the children of that Trie. What I decided to go with was this: I’ll have some mutating method that does the work recursively. However, this method also returns a value, representing some important information for whatever called it. In this case, the remove method would remove, as you’d imagine, but it will also return a Boolean, signifying whether the Trie it was called on can be removed. Since I used the normal structure of having a private method take a generator, and then a public wrapper method take a sequence, I could have the public method just discard the Boolean.
Let’s go through it. Here’s the signature:
private mutating func remove<
>(var g: G) -> Bool {
No surprises there. Similar to the other methods. Then, get the head from the generator:
if let head = g.next() {
Within that if block is the meat of the logic, so I might skip to what happens if g.next() returns nil for the start:
if let head = g.next() {...}
endHere = false
return children.isEmpty
So the sequence being removed has ended. That means that whatever Trie you’re on should have its endHere set to false. To the user of the Trie, that’s all that matters: from now on, if the contains method on that Trie is used with that sequence, it will return false.
However, to find out if you can delete the data itself, it returns children.isEmpty. If it has no children, it does not hold any other sequences or information, so it can be deleted.
Now for inside the if block:
guard children[head]?.remove(g) == true else { return false }
children.removeValueForKey(head)
return !endHere && children.isEmpty
So it calls remove on the child Trie corresponding to head. That guard statement will fail for two distinct reasons: if children doesn’t contain head, then the sequence being removed wasn’t in the Trie in the first place. The method will then return false, so that no removal or mutation is done.
If it does contain head, but the Bool returned from the remove method is false, that means that its child is not removable, so it is also not removable, so it should return false.
Otherwise, it will remove that member (children.removeValueForKey(head)). Then, the Trie can decide whether or not it itself is removable: return !endHere && children.isEmpty. If the endHere is set to true, then it is the end of some sequence: it is not removable. Otherwise, it’s removable if it has no children. Here’s the whole thing, with its public version:
>(var g: G) -> Bool { // Return value signifies whether or not it can be removed
public mutating func remove<
S : SequenceType where S.Generator.Element == Element
>(seq: S) {
remove(seq.generate())
That was a little heavy. And kind of ugly. Let’s lighten things up for a second, with one of the loveliest count properties I’ve seen:
return children.values.reduce(endHere ? 1 : 0) { $0 + $1.count }
All it’s really doing is counting the instances of a true endHere. If the current Trie is an end, then it knows that it adds one to the count (endHere ? 1 : 0), and it adds that to the sum of the counts of its children.
Now then. SequenceType. Getting tree-like structures to conform to SequenceType is a bit of a pain, mainly because of their recursiveness. Getting a linear representation is easy enough:
public var contents: [[Element]] {
return children.flatMap {
(head: Element, child: Trie<Element>) -> [[Element]] in
child.contents.map { [head] + $0 } + (child.endHere ? [[head]] : [])
And then you could just return the generate method from that for your Trie’s generate method.
The problem is that it’s not very proper: you’re translating your data structure into another data structure just to iterate through it. What you really want is something that generates each element on demand.
But it gets ugly quick. You’ve got to do a lot of stuff by hand which it isn’t nice to do by hand, and you’ve got to employ some dirty tricks (like using closures as a kind of homemade indirect). At any rate, here it is:
public struct TrieGenerator<Element : Hashable> : GeneratorType {
private var children: DictionaryGenerator<Element, Trie<Element>>
private var curHead : Element?
private var curEnd : Bool = false
private var innerGen: (() -> [Element]?)?
private mutating func update() {
guard let (head, child) = children.next() else { innerGen = nil; return }
curHead = head
var g = child.generate()
innerGen = {g.next()}
curEnd = child.endHere
public mutating func next() -> [Element]? {
for ; innerGen != nil; update() {
if let next = innerGen!() {
return [curHead!] + next
} else if curEnd {
curEnd = false
return [curHead!]
private init(_ from: Trie<Element>) {
children = from.children.generate()
It’s got a similar logic to the lazy flatMap I did from a while ago.
The code is all available here, as a playground, or here, in SwiftSequence, where it’s accompanied by some tests.
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Experimental Study on Homogeneous Charge Compression Ignition Combustion With Fuel of Dimethyl Ether and Natural Gas | J. Eng. Gas Turbines Power | ASME Digital Collection
Experimental Study on Homogeneous Charge Compression Ignition Combustion With Fuel of Dimethyl Ether and Natural Gas
Mingfa Yao,
Mingfa Yao
State Key Laboratory of Engine Combustion,
e-mail: y_mingfa@tju.edu.cn
Zunqing Zheng,
Zunqing Zheng
Yao, M., Zheng, Z., and Qin, J. (September 22, 2005). "Experimental Study on Homogeneous Charge Compression Ignition Combustion With Fuel of Dimethyl Ether and Natural Gas." ASME. J. Eng. Gas Turbines Power. April 2006; 128(2): 414–420. https://doi.org/10.1115/1.2130731
The homogeneous charge compression ignition (HCCI) combustion fueled by dimethyl ether (DME) and compressed natural gas (CNG) was investigated. The experimental work was carried out on a single-cylinder diesel engine. The results show that adjusting the proportions of DME and CNG is an effective technique for controlling HCCI combustion and extending the HCCI operating range. The combustion process of HCCI with dual fuel is characterized by a distinctive two-stage heat release process. As CNG flow rate increases, the magnitude of peak cylinder pressure and the peak heat release rate in the second stage goes up. As DME flow rate increases, the peak cylinder pressure, heat release rate, and
NOx
emissions increase while
THC
CO
emissions decrease.
diesel engines, engine cylinders, combustion, ignition, air pollution
Combustion, Emissions, Fuels, Homogeneous charge compression ignition engines, Ignition, Engines, Nitrogen oxides
Experimental Studies on Controlled Auto-Ignition (CAI) Combustion of Gasoline in a 4-Stroke Engine
,” SAE Paper No. 2001-01-1030, SP-1623.
U.S. Department of Energy Efficiency and Renewable Energy Office of Transportation Technologies, 2001, “Homogeneous Charge Compression Ignition (HCCI) Technology-A Report to the U.S. Congress.”
Research and Development of Controlled Auto-Ignition (CAI) Combustion in a 4-Stroke Multi-Cylinder Gasoline Engine
A Compound Technology for HCCI Combustion in a DI Diesel Engine Based on the Multi-Pluse Injection and the BUMP Combustion Chamber
Influence of Hydrogen and Carbon Monoxide on HCCI Combustion of Dimethyl Ether
Experimental Study of CI Natural-Gas/DME Homogeneous Charge Engine
Numerical Study on the Dimethyl Ether/Compressed Natural Gas HCCI Combustion Using Chemical Kinetics Model
The Effect of Toplland Geometry on Emissions of Unburned Hydrocarbons From a Homogeneous Charge Compression Ignition (HCCI) Engine
A Decoupled Model of Detailed Fluid Mechanics Followed by Detailed Chemical Kinetics for Prediction of Iso-Octane HCCI Combustion
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Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : Gamma
Gamma(b, c)
GammaDistribution(b, c)
The gamma distribution is a continuous probability distribution with probability density function given by:
f\left(t\right)={\begin{array}{cc}0& t<0\\ \frac{{\left(\frac{t}{b}\right)}^{c-1}{ⅇ}^{-\frac{t}{b}}}{b\mathrm{\Gamma }\left(c\right)}& \mathrm{otherwise}\end{array}
0<b,0<c
\mathrm{\Gamma }\left(c\right)
Some sources use other parametrizations for this distribution; they might describe this distribution as
\mathrm{Gamma}\left(c,b\right)
\mathrm{Gamma}\left(c,\frac{1}{b}\right)
Note that the Gamma command is inert and should be used in combination with the RandomVariable command.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{GammaDistribution}\left(b,c\right)\right):
\mathrm{PDF}\left(X,u\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{0}\\ \frac{{\left(\frac{\textcolor[rgb]{0,0,1}{u}}{\textcolor[rgb]{0,0,1}{b}}\right)}^{\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{u}}{\textcolor[rgb]{0,0,1}{b}}}}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\right)}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{PDF}\left(X,0.5\right)
\frac{{\left(\frac{\textcolor[rgb]{0,0,1}{0.5}}{\textcolor[rgb]{0,0,1}{b}}\right)}^{\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1.}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{0.5}}{\textcolor[rgb]{0,0,1}{b}}}}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\right)}
\mathrm{Mean}\left(X\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
\mathrm{Variance}\left(X\right)
{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
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