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Wikijunior:Introduction to Mathematics/Negative Numbers - Wikibooks, open books for an open world
Wikijunior:Introduction to Mathematics/Negative Numbers
Adding and SubtractingEdit
~ a positive number
~ a negative number
Subtracting Rules
Adding a negative number is the same as subtracting a positive number.
{\displaystyle 7+-4=7-4}
Subtracting a negative number is the same as adding a positive number.
{\displaystyle 7--4=7+4}
Multiplying and DividingEdit
Multiplying a negative number by a positive number, or a positive number by a negative number makes the answer negative.
{\displaystyle -2\times 3=-6}
{\displaystyle 2\times -3=-6}
Multiplying a negative number by a negative number makes the answer positive.
{\displaystyle -2\times -3=6}
Do the same for dividing.
{\displaystyle -6\div 3=-2}
{\displaystyle 6\div -3=-2}
{\displaystyle -6\div -3=2}
Retrieved from "https://en.wikibooks.org/w/index.php?title=Wikijunior:Introduction_to_Mathematics/Negative_Numbers&oldid=2530159"
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So far, we’ve learned that the equation for a logistic regression model looks like this:
ln(\frac{p}{1-p}) = b_{0} + b_{1}x_{1} + b_{2}x_{2} +\cdots + b_{n}x_{n}
Note that we’ve replaced y with the letter p because we are going to interpret it as a probability (eg., the probability of a student passing the exam). The whole left-hand side of this equation is called log-odds because it is the natural logarithm (ln) of odds (p/(1-p)). The right-hand side of this equation looks exactly like regular linear regression!
In order to understand how this link function works, let’s dig into the interpretation of log-odds a little more. The odds of an event occurring is:
Odds = \frac{p}{1-p} = \frac{P(event\ occurring)}{P(event\ not\ occurring)}
For example, suppose that the probability a student passes an exam is 0.7. That means the probability of failing is 1 - 0.7 = 0.3. Thus, the odds of passing are:
Odds\ of\ passing = \frac{0.7}{0.3} = 2.\overline{33}
This means that students are 2.33 times more likely to pass than to fail.
Odds can only be a positive number. When we take the natural log of odds (the log odds), we transform the odds from a positive value to a number between negative and positive infinity — which is exactly what we need! The logit function (log odds) transforms a probability (which is a number between 0 and 1) into a continuous value that can be positive or negative.
Suppose that there is a 40% probability of rain today (p = 0.4). Calculate the odds of rain and save it as odds_of_rain. Note that the odds are less than 1 because the probability of rain is less than 0.5.
Use the odds that you calculated above to calculate the log odds of rain and save it as log_odds_of_rain. You can calculate the natural log of a value using the numpy.log() function. Note that the log odds are negative because the probability of rain was less than 0.5.
Suppose that there is a 90% probability that my train to work arrives on-time. Calculate the odds of my train being on-time and save it as odds_on_time. Note that the odds are greater than 1 because the probability is greater than 0.5.
Use the odds that you calculated above to calculate the log odds of an on-time train and save it as log_odds_on_time. Note that the log odds are positive because the probability of an on-time train was greater than 0.5.
4. Log-Odds
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Category:MER Tag Riemann sum - UBC Wiki
Category:MER Tag Riemann sum
MER Tag Riemann sum
Typically, the following formulas will be useful
{\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\qquad \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}\qquad \sum _{i=1}^{n}i^{3}=\left({\frac {n(n+1)}{2}}\right)^{2}}
Some pencasts on Riemann sum notation and on converting Riemann sums to definite integrals.
35px The basics of the definite integral and Riemann sums
The basics of the definite integral and Riemann sums
35px The rectangle method
Approximating area under a curve (definite integral) using rectangles.
35px An example computation (Part 1)
An example computation (Part 1)
A two part series on computing definite integrals using Riemann sums.
April 2012 Q01 (g)
April 2005 Q01 (c)MATH 105
April 2010 Q01 (f)
Pages in category "Category:MER Tag Riemann sum"
Retrieved from "https://wiki.ubc.ca/index.php?title=Category:MER_Tag_Riemann_sum&oldid=518252"
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Center of Ellipse Calculator
Where is the center of an ellipse?
How do you find the center of an ellipse in general form?
How to find the center from the vertices or co-vertices
How to use the center of ellipse calculator
The center of ellipse calculator helps you find the center of an ellipse when the equation, vertices, co-vertices, or foci are known. This enables you to determine the center of an ellipse when only a few key points are known or when the equation of the ellipse is known. Read on to learn more about the formulas to find the center based on the given data!
The center of an ellipse is the midpoint of both the major and minor axes. These axes intersect at right angles at the center. So from the center, the distances to the vertices (located along the major axis) are equal, and the distances to the co-vertices (located along the minor axis) are also equal. The center is also equidistant from the foci of the ellipse.
To calculate the center of an ellipse given its equation in general form, we'd need to do the following:
Compare the equation to the general form given below:
\frac{\left(x\ -\ c_{1}\right)^{2}}{a^{2}}+\frac{\left(y\ -\ c_{2}\right)^{2}}{b^{2}}=1
c_{1}
c_{2}
The center will be the point
(c_{1},\ c_{2})
To calculate the center of the ellipse from the vertices or co-vertices, we'd need to do the following:
Get the coordinates of the 2 vertices.
Find the midpoint of the pair of vertices by using the midpoint calculator, which is based on the following formula:
\left(\frac{x_{1}+x_{2}}{2},\ \frac{y_{1}+y_{2}}{2}\right)
\left(x_{1},\ y_{1}\right)
is the first vertex and
\left(x_{2},\ y_{2}\right)
is the second vertex.
The midpoint thus obtained will be the center of the ellipse!
If you know the co-vertices instead of the vertices, you may follow the same steps above using the coordinates of the co-vertices instead!
To use this calculator for finding the center of the ellipse, you'd need to do the following:
Choose which information you have, among the following available options:
Equation of the ellipse;
Vertices of the ellipse;
Co-vertices of the ellipse; or
Foci of the ellipse.
Based on the option you chose, key in the values of variables in the equation or the coordinates of the corresponding points.
Voila! Based on your inputs, the calculator will show you the coordinates of the center of the ellipse!
If you found this tool useful, you may also want to check out our other similar tools for calculations involving ellipse, such as:
Ellipse standard form; and
How do I find the center of an ellipse given the foci?
In order to know how to find the center of an ellipse given the foci, we'd follow these steps:
Write the coordinates of the foci.
Find the midpoint of these two points.
Tada! The center of the ellipse will be this midpoint!
Since the center is equidistant from the foci, the midpoint of the line joining the foci will be the center of the ellipse.
What is the center of an ellipse with vertices at (0, 6) and (0, -6)?
(0, 0) is the center of the ellipse whose vertices are the points (0, 6) and (0, -6). This is because the center is equidistant from the vertices. Thus, we can find the midpoint of the points (0, 6) and (0, -6), which will give us (0, 0) as the center of the ellipse!
the ellipse equation
Check out this tangent calculator if you want to calculate the tangent of any angle.
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The income elasticity of demand of normal and inferior goods
How do you calculate income elasticity of demand?
How to use the income elasticity of demand calculator
How to calculate income elasticity of demand example with steps
Income elasticity of demand example in macroeconomics
The income elasticity of demand calculator (with steps) helps you measure the effect of changes in consumers' incomes on demand for a given good. It is measured as the ratio of the percentage change in quantity demanded to the percentage change in income.
What is income elasticity of demand;
How to find income elasticity of demand - the income elasticity of demand formula; and
The application of income elasticity of demand.
Moreover, we present a practical example to understand the macroeconomic intuition behind the income elasticity of demand.
As you may know, multiple factors can affect the quantity of a good demanded. Its prices, for example, measured by the price elasticity of demand, is a prominent variable that can alter demand.
Another variable that can induce such changes by shifting the demand curve is the income of consumers. More precisely, the income elasticity of demand measures how responsive the demand for a good is to changes in consumers' incomes. We will say more about the formula for calculating the income elasticity of demand in the following paragraphs.
Like the cross-price elasticity of demand between two goods, the income elasticity of demand for a good can also be positive or negative.
The sign of the income elasticity of demand reveals whether a good is normal or inferior. On the one hand, the good is normal when the demand increases when income rises. On the other hand, a good is considered inferior when demand decreases when income rises.
A positive income elasticity of demand coefficient indicates that the good is a normal good: the quantity demanded at any given price increases as income increases.
A negative income elasticity of demand coefficient indicates that the good is inferior good: the quantity demanded at any given price decreases as income increases.
Our income elasticity of demand calculator with steps shows you such a result interpretation after making the computations.
While the income elasticity of demand for a normal good is always positive, its value contains further helpful information. Policymakers are often interested in how a particular industry will grow as consumers' income increases over time to support economic decisions promoting employment or economic growth. By determining whether the income elasticity of demand for a good is larger or less than 1, one can distinguish between income-elastic and income-inelastic goods.
The demand for a good is income-elastic if the income elasticity of demand formula for that good yields more than 1. It means that when income rises, the demand for income-elastic goods rises faster than income.
Luxury goods such as holiday houses, expensive cars and international travel are income-elastic examples.
The demand for a good is income-inelastic if the income elasticity of demand is less than 1. It suggests that when income rises, the demand for income-inelastic goods rises more slowly than income. Necessities such as food and clothes are typically income-inelastic.
The formula for calculating income elasticity of demand is the following:
Find the change in quantity demanded.
Determine the change in income.
Divide the first value by the second:
Income elasticity of demand = Change in quantity demanded / Change in income
You can compute the percentage change in the quantity demanded (
x_1
) and income (
x_2
With the standard way of computation:
\Delta x = (x_{i2} - x_{i1}) / x_{i1}
Calculate income elasticity of demand using midpoint method:
\Delta x = (x_{i2} - x_{i1}) / \lparen (x_{i1} + x_{i2})/2\rparen
\Delta x
- Change in quantity demanded or income;
x_{i1}
- Quantity demanded or income in period 1; and
x_{i2}
- Quantity demanded or income in period 2.
In the default mode of the income elasticity of demand calculator, you need to set the following two parameters to get the result for the income elasticity of demand:
Percent change in income; and
Percentage change in quantity.
You can also input additional numbers for periods 1 and 2 separately, and we also provide the option for choosing between the standard and the midpoint method of estimation:
Method - by default, we use the standard approximation, but you can also calculate income elasticity of demand using the midpoint method;
Income in period 1;
Quantity demanded in period 1; and
Quantity demanded in period 2.
Let's take a simple example to see how income elasticity of demand works. In the demand curve below:
For the first period, while income was 1000, the quantity demanded was 100; and
For the second period, when the income increased to 1200, the quantity demanded increased to 150.
So, the example of how to calculate income elasticity of demand is the following:
Estimate the percentage change in quantity demanded:
Change in quantity demanded = (150 - 100) / 100 = 0.05 = 50%
Compute the percentage change in income:
Change in income = (1200 - 1000) / 1000 = 0.02 = 20%
Calculate income elasticity of demand:
Income elasticity of demand = Change in quantity demanded / Change in income = 0.05 / 0.02 = 2.5
The result suggests that the income elasticity curve represents an income-inelastic normal good, such as foods or clothes.
A practical way to demonstrate the relevance of the income elasticity of demand is to take a real-world example.
You may be able to envisage a picture of an American family living on a farm, but do you know what portion of the population lives on a farm in the U.S. nowadays? The figure is surprisingly low if we make a comparison over history. While in the 18th century, many Americans lived in the countryside and took part in agriculture, only about 1.3 % of the population does so in our times.
Why did this significant change happen? The income elasticity of demand gives the answer. Since food demand is typically income inelastic (its income elasticity of demand is much less than 1), as consumers' income grew, the proportion of food in a general basket of consumers' basket became smaller. Putting it on a macro scale, the falling share of income in the GDP spent on food means a lower income share earned by farmers. Moreover, the considerable technological progress in the agrarian sector triggered high competition between farmers, which depressed prices of agricultural goods, ceteris paribus, further reducing the total revenue of farmers.
To summarize, the combination of the income elasticity of demand of foodstuffs and the fast technological progress in agriculture explains the relative decline of farming in the United States.
What are the factors that affect income elasticity of demand?
The main factors which affect income elasticity of demand are:
The degree of necessity of the good;
The level of income of consumers; and
The rate at which the desire for the good is satisfied as consumption increases.
What is the importance of income elasticity of demand to the government?
By estimating the income elasticity of demand, economists can distinguish industries according to their growth potential when consumer income grows over time. Such information can help policymakers decide on possible interventions in the market.
Can the income elasticity of a good be negative?
Yes. Negative income elasticity of demand coefficient indicates that the good is an inferior good: the quantity demanded at any given price decreases as income increases because people can now afford better quality equivalents.
What is the income elasticity of demand for luxury goods?
Luxury goods typically have a greater than one income elasticity of demand, which means that their demand increases at a greater proportional rate than income.
For example, if the demand for expensive watches increases by 22 percent when aggregate income increases by 20 percent, then watches are considered luxury goods because they have an income elasticity of demand of 1.1.
Quantity demanded in period 1
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Refer to sequences (c) and (i) of problem 5-42.
0-18
0-9
The numbers in the sequence in part (e) of problem 5-42 are called Fibonacci numbers. They are named after an Italian mathematician who discovered the sequence while studying how fast rabbits could breed. What is different about this sequence than the other three you discovered?
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ElementaryGroup - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : ElementaryGroup
ElementaryGroup( p, n )
ElementaryGroup( p, n, form = f )
algebraic; understood to represent a prime number
algebraic; understood to represent a positive integer
An elementary Abelian group of order
{p}^{n}
p
is a prime number, is a direct product (or direct sum) of
n
copies of the cyclic group of order
p
The ElementaryGroup( p, n ) command returns an elementary Abelian group of order
{p}^{n}
, either as a permutation group or as a finitely presented group, according to the value of the form option. By default, a permutation group is returned.
If either of p and n is not an explicit integer, then a symbolic group representing the elementary group of order
{p}^{n}
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔\mathrm{ElementaryGroup}\left(3,4\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{C}}_{\textcolor[rgb]{0,0,1}{3}}^{\textcolor[rgb]{0,0,1}{4}}
\mathrm{GroupOrder}\left(G\right)
\textcolor[rgb]{0,0,1}{81}
\mathrm{IsAbelian}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{Generators}\left(G\right)
[\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\right)\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\right)\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\right)]
G≔\mathrm{ElementaryGroup}\left(3,5,'\mathrm{form}'="fpgroup"\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{C}}_{\textcolor[rgb]{0,0,1}{3}}^{\textcolor[rgb]{0,0,1}{5}}
G≔\mathrm{ElementaryGroup}\left(17,3\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{C}}_{\textcolor[rgb]{0,0,1}{17}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{GroupOrder}\left(G\right)
\textcolor[rgb]{0,0,1}{4913}
\mathrm{GroupOrder}\left(\mathrm{ElementaryGroup}\left(p,n+m\right)\right)
{\textcolor[rgb]{0,0,1}{p}}^{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}}
\mathrm{IsCyclic}\left(\mathrm{ElementaryGroup}\left(p,n\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}1<n
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The GroupTheory[ElementaryGroup] command was introduced in Maple 17.
The GroupTheory[ElementaryGroup] command was updated in Maple 2020.
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Columns description - Input data | CatBoost
A description of the data types contained in the columns of the Dataset description in delimiter-separated values format or the Dataset description in extended libsvm format.
The columns description file is optional. If omitted, it is assumed that the first column in the file with the dataset description defines the label value, and the other columns are the values of numerical features.
The target variable (in other words, the object's label value).
The type of data depends on the machine learning task being solved:
Regression, multiregression and ranking — Numeric values.
Logloss — The value is considered a positive class if it is strictly greater than the value of the parameter of the loss function. Otherwise, it is considered a negative class.
A numerical feature.
A tab-delimited feature ID can be added for this type of column. The specified value replaces the feature ID in the following output files:
A categorical feature.
A text feature.
The value of this column is ignored (the behavior is the same as when this column is omitted in the file with the dataset description).
Alias:DocId
An alphanumeric ID of the object.
The object's weight.
Used as an additional coefficient in the objective functions and metrics. By default, it is set to 1 for all objects.
Do not use this column type if the GroupWeight column is defined in the dataset description.
The group weight.
Used as an additional coefficient in the objective functions and metrics. By default, it is set to 1 for all objects in the group.
The weight must be the same for all objects in one group.
Do not use this column type if the Weight column is defined in the dataset description.
The initial formula values for all input objects.
Used for calculating the final values of trees.
The required number of these columns depends on the machine learning mode:
For classification and regression – one column.
For multiclassification – the same as the number of classes.
Alias:QueryId
All objects in the dataset must be grouped by group identifiers if they are present. I.e., the objects with the same group identifier should follow each other in the dataset.
For example, let's assume that the dataset consists of documents
d_{1}, d_{2}, d_{3}, d_{4}, d_{5}
. The corresponding groups are
g_{1}, g_{2}, g_{3}, g_{2}, g_{2}
, respectively. The feature vectors for the given documents are
f_{1}, f_{2}, f_{3}, f_{4}, f_{5}
respectively. Then the dataset can take the following form:
\begin{pmatrix} d_{2}&g_{2}&f_{2}\\ d_{4}&g_{2}&f_{4}\\ d_{5}&g_{2}&f_{5}\\ d_{3}&g_{3}&f_{3}\\ d_{1}&g_{1}&f_{1} \end{pmatrix}
The grouped blocks of lines can be input in any order. For example, the following order is equivalent to the previous one:
\begin{pmatrix} d_{1}&g_{1}&f_{1}\\ d_{3}&g_{3}&f_{3}\\ d_{2}&g_{2}&f_{2}\\ d_{4}&g_{2}&f_{4}\\ d_{5}&g_{2}&f_{5} \end{pmatrix}
The identifier of the object's subgroup. Used to divide objects within a group. An arbitrary string, possibly representing an integer.
Should be a non-negative integer.
The timestamp of the object.
The ranking position of the object. The value is used to calculate the StochasticFilter metric.
List each column on a new line.
Additional properties are set on the corresponding line.
Use a tab as the delimiter to separate data for a single column.
Columns that contain numerical features don't require descriptions. Any columns that aren't specified in the file are assumed to be Num.
<column ID (numbering starts from zero)><\t><data type><\t><feature id (optional, applicable for Num and Categ column types only)>
The feature indices and the column indices usually differ.
The table below shows the difference between these indices on the columns description example given above.
0 Label —
1 Num 0
3 Categ<\t>wind direction 2
4 Auxiliary —
Multiregression labels are specified in several separate columns.
0<\t>Label
An object contains information about the weather, and the features represent:
wind direction ( south , west , north , east )
The label (target) takes binary values:
0 stands for the absence of precipitation
1 stands for the presence of precipitation
A column with arbitrary data is provided.
The feature representing the wind direction should be renamed to wind direction in the output files with information on the feature strength.
The file with the columns description with tab-separated data looks like this:
3<\t>Categ<\t>wind direction
The following variant is equivalent to the previous but is redundant:
0<\t>Label<\t>
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Rank key features by class separability criteria - MATLAB rankfeatures - MathWorks España
The function uses
Z×\left(1-\alpha ×\rho \right)
to calculate the weight, where ρ is the average of the absolute values of the cross-correlation coefficient between the candidate feature and all previously selected features. α is the CCWeighting value that sets the weighting factor.
Z×\left(1-{e}^{-{\left(\frac{D}{\beta }\right)}^{2}}\right)
to calculate the weight, where D is the distance (in rows) between the candidate feature and previously selected features. β is the NWeighting value that sets the weighting factor. β must be greater than or equal to 0.
"meanvar" —
{X}_{new}=\frac{X-\mu }{\sigma }
"softmax" —
{X}_{new}=\frac{1}{1+{e}^{\left(\frac{\mu -X}{\sigma }\right)}}
"minmax" —
{X}_{new}=\frac{X-{X}_{\mathrm{min}}}{{X}_{\mathrm{max}}-{X}_{\mathrm{min}}}
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Identification of a postmortem redistribution factor (F) for forensic toxicology | Journal of Analytical Science and Technology | Full Text
Identification of a postmortem redistribution factor (F) for forensic toxicology
Iain M McIntyre1
Postmortem redistribution (PMR) refers to the changes that may occur in drug concentrations after death. Consequently, postmortem concentrations in blood may not always replicate the antemortem drug levels. Literature supports the model describing drugs with a liver (L) concentration to peripheral blood (P) concentration ratio less than 5 (L/kg) being prone to little or no PMR. Conversely, drugs with a L/P ratio greater than 20 to 30 (L/kg) have propensity for substantial PMR.
Expanding upon this prior work, the current paper presents the concept of a postmortem redistribution factor (F) for a drug, which characterizes the direct relationship between postmortem peripheral blood and the corresponding antemortem whole blood concentration.
Development of the concept of a "postmortem redistribution factor" will provide a more definitive and authoritative drug ranking, and possibly, numerical interpretation of PMR for forensic toxicologists.
A potentially significant issue complicating interpretation of postmortem drug concentrations results from the phenomenon referred to as postmortem redistribution (PMR). Postmortem drug concentrations in the blood may not always straightforwardly parallel antemortem drug concentrations in the blood due to the movement of the drugs after death. Accordingly, some authors have argued a cautious approach in interpreting postmortem concentrations, and others have taken a far more pessimistic and even cynical perspective. The mechanisms involved in PMR are both complicated and poorly understood. However, postmortem drug concentrations in the blood may follow some commonly accepted trends that aid with interpretation. Generally, the characteristics of the drug itself can be used to predict if a drug is subject to PMR. Substantial changes in blood drug concentrations are predicted for basic, lipophilic drugs with a high volume of distribution (>3 L/kg) (Prouty & Anderson [1990]). When PMR occurs, blood specimens drawn from the central body cavity and heart generally exhibit higher drug concentrations postmortem than specimens drawn from peripheral areas, most commonly the femoral region. Diffusion of drugs from organ tissues into the blood may explain the observed phenomenon.
Previous attempts to assess and account for PMR have utilized postmortem blood specimens collected from at least two areas of the body at autopsy, a peripheral area and a central area (often the heart), so that a comparison could be made. The resulting postmortem blood ratio was considered to reflect a drug's potential for PMR (Prouty & Anderson [1990]; Dalpe-Scott et al. [1995]). Recent work, however, has described ambiguities with this approach (McIntyre et al. [2012]).
The collection, analysis, and comparison of antemortem blood specimens are obviously helpful in assisting with the interpretation of postmortem blood drug concentrations, but relevant specimens are only rarely available. In a set of case studies of six drugs, concentrations in the postmortem femoral blood specimens exceeded the antemortem concentrations in five of the drugs studied, suggesting that even peripheral blood exhibited redistribution (Cook et al. [2000]). The potential for redistribution of other drugs in postmortem peripheral blood has also been documented (Gerostamoulos et al. [2012]).
The liver (L) to peripheral blood (P) ratio has been proposed as a more dependable marker for PMR, with ratios less than 5 (L/kg) indicating little to no propensity towards PMR, and ratios exceeding 20 to 30 (L/kg) indicative of a propensity for substantial PMR (McIntyre et al. [2012]). A number of reports elaborating on, and supporting, this model have now been published (McIntyre & Mallett [2012]; McIntyre & Meyer Escott [2012]; McIntyre & Anderson [2012]; McIntyre et al. [2013a]; McIntyre et al. [2013b]). Furthermore, a direct correlation between the postmortem peripheral blood and corresponding antemortem concentration - by consideration of the L/P ratio - has been expressed (McIntyre et al. [2013c]). The report, describing methamphetamine cases, found that the postmortem peripheral blood concentrations were approximately 1.5 times higher than the corresponding concentrations attained in whole blood specimens collected before death. Given that the L/P ratios for methamphetamine had been confirmed to be approximately 6 (L/kg), it was then projected that drugs exhibiting L/P ratios between 5 and 10 (L/kg) would theoretically yield postmortem peripheral blood concentrations up to twice the corresponding antemortem concentrations - a measure of PMR potential. It was further hypothesized that L/P ratios ranging from 10 to 20 (L/kg) would demonstrate greater potential for PMR with postmortem peripheral blood concentrations 2 to 3 times that of the corresponding antemortem levels and consequently even higher L/P ratios indicative of even greater potential for PMR.
The current document sets out to expound upon this L/P model and its resultant implications by proposing the concept of a postmortem redistribution factor (F) for a drug. The postmortem redistribution factor has been defined as a factor that characterizes the direct relationship between a drug's postmortem peripheral blood and the corresponding antemortem (AM) whole blood concentration.
Equation 1 presents the proposed relationship between the antemortem whole blood concentration of a compound and the corresponding postmortem peripheral blood concentration:
\mathbf{AM}\mathbf{=}\mathbit{P}/\mathbit{F}
where AM = antemortem whole blood concentration, P = postmortem peripheral blood concentration, and F = postmortem redistribution factor.
Rearrangement of Equation 1 gives
\mathbit{F}\mathbf{=}\mathbf{P}/\mathbf{AM}
Thus, an example of an experimental (or actual) F could be determined for a drug where both the postmortem peripheral blood and antemortem whole blood drug concentrations have been determined in the same individual (assuming an insignificant delay between the collection of the antemortem blood and the time of death).
Considering the methamphetamine data (McIntyre et al. [2013c]), an experimental (actual) F for methamphetamine of 1.5 is predicted - postmortem peripheral blood concentrations being 1.5 times (on average) greater than the corresponding antemortem concentrations.
A related approach to assess potential for PMR has also recently been described (Launiainen & Ojanpera [2013]). This study presented data for 129 drugs comparing postmortem femoral blood concentrations to therapeutic plasma concentrations to describe drugs' propensity for PMR. This study analyzed a large number of cases where median postmortem drug concentrations were compared with estimations of the therapeutic concentrations. These authors projected a similar ratio for methamphetamine of 1.8. Although these data represent a practical attempt to describe PMR, it is conceivable that the determination of an F value from analytically determined postmortem data (such as the unique drug L/P ratio) may well produce more consistently accurate estimates.
The principal goal of these endeavors was to attempt to develop a ranking of drugs and indicate their propensity for and, subsequently, their potential extent of PMR. Until now, most efforts in interpretation have simply described PMR by an aphorism, ranging from ‘the drug has not been found to exhibit PMR’ to ‘the drug is subject to PMR.’ Such descriptions have never been particularly useful in the interpretation of postmortem drug concentrations, especially in relation to deducing what the drug concentration may have been at the time of death. The development of the concept of a systematically based postmortem redistribution factor will provide a more definitive and authoritative ranking and possibly numerical interpretation of PMR.
Cook J, Braithwaite RA, Hale KA: Estimating antemortem drug concentrations from postmortem blood samples: the influence of postmortem redistribution. J Clin Path 2000, 53: 282–285. 10.1136/jcp.53.4.282
Dalpe-Scott M, Degouffe M, Garbutt D, Drost M: A comparison of drug concentrations in postmortem cardiac and peripheral blood in 320 cases. Can Soc For Sci J 1995, 28: 113–121. 10.1080/00085030.1995.10757474
Gerostamoulos D, Beyer J, Staikos V, Tayler P, Woodford N, Drummer OH: The effect of the postmortem interval on the redistribution of drugs: a comparison of mortuary admission and autopsy blood specimens. Forensic Sci Med Path 2012, 8: 373–379. 10.1007/s12024-012-9341-2
Launiainen T, Ojanpera I (2013) Drug concentrations in post-mortem femoral blood compared with therapeutic concentrations in plasma. Drug Test Anal. doi:10.1002/dta.1507
McIntyre IM, Anderson DT: Postmortem fentanyl concentrations: a review. J Forensic Res 2012, 3: 157. doi:10.4172/2157–7145.1000157 10.4172/2157-7145.1000e108
McIntyre IM, Mallett P: Sertraline concentrations and postmortem redistribution. Forensic Sci Int 2012, 223: 349–352. 10.1016/j.forsciint.2012.10.020
McIntyre IM, Meyer Escott C: Postmortem drug redistribution. J Forensic Res 2012, 3: e108. doi:10.4172/2157–7145.1000e108 10.4172/2157-7145.1000e108
McIntyre IM, Sherrard J, Lucas J: Postmortem carisoprodol and meprobamate concentrations in blood and liver: lack of significant distribution. J Anal Tox 2012, 36: 177–181. 10.1093/jat/bks011
McIntyre IM, Mallett P, Trochta A, Morhaime J: Hydroxyzine distribution in postmortem cases and potential for redistribution. Forensic Sci Int 2013a, 231: 28–33. 10.1016/j.forsciint.2013.04.013
McIntyre IM, Gary RD, Estrada J, Nelson CL (2013b) Antemortem and postmortem fentanyl concentrations: a case report. Int J Legal Med. , [http://dx.doi.org/10.1007/s00414–013–0897–5]
McIntyre IM, Nelson CL, Schaber B, Hamm CE: Antemortem and postmortem methamphetamine blood concentrations: three case reports. J Anal Tox 2013c, 37(6):386–389. 10.1093/jat/bkt040
Prouty RW, Anderson WH: The forensic science implications of site and temporal influences on postmortem blood-drug concentrations. J Forensic Sci 1990, 35: 243–270.
Forensic Toxicology Laboratory Manager, County of San Diego Medical Examiner’s Office, 5570 Overland Ave., Suite 101, San Diego, 92123, CA, USA
Iain M McIntyre
Correspondence to Iain M McIntyre.
McIntyre, I.M. Identification of a postmortem redistribution factor (F) for forensic toxicology. J Anal Sci Technol 5, 24 (2014). https://doi.org/10.1186/s40543-014-0024-3
Postmortem redistribution factor
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networks(deprecated)/icosahedron - Maple Help
Home : Support : Online Help : networks(deprecated)/icosahedron
creates an icosahedron graph
G:=icosahedron()
is returned as the icosahedron
Important:The networks package has been deprecated. Use the superseding command GraphTheory[SpecialGraphs][IcosahedronGraph]instead.
The simple graph known as a icosahedron is generated and returned as the value of the procedure call.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[icosahedron](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{icosahedron}\left(\right):
\mathrm{edges}\left(G\right)
{\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e10}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e11}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e12}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e13}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e14}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e15}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e16}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e17}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e18}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e19}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e20}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e21}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e22}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e23}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e24}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e25}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e26}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e27}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e28}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e29}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e30}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e5}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e6}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e7}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e8}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e9}}}
\mathrm{vertices}\left(G\right)
{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}}
GraphTheory[SpecialGraphs][IcosahedronGraph]
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Calculating the shorter angle of rotation in 2D — Terresquall Blog
(What looks like) the solution
Here’s an example that illustrates how taking b – a can work:
If we let a = 150° and b = 170°,
\mathrm{b – a = 170° – 150° = 20°}
The case above demonstrates that for the rotations pictured below, the formula correctly gives us a counter-clockwise rotation of 20° as the shorter angle of rotation.
a needs to rotate 20° (i.e. counter-clockwise) to get to b.
If we let a = 150° and b = 140°,
\mathrm{b – a = 140° – 150° = -10°}
The case above demonstrates that for these other rotations pictured below, the formula correctly gives us a clockwise rotation of -10° as the shorter angle of rotation.
a needs to rotate -10° (i.e. clockwise) to get to b.
The problem with angular values
When you begin to dig deeper, however, you will quickly find that things can get surprisingly complex problem for 2 reasons:
Angles loop around beyond 360°, so if a = 380° and b = 40°, we get 340° as the shorter angle of rotation. This is obviously wrong, because the shorter angle of rotation is never larger than 180°.
In most (i.e. 99%) game engines, the lower half of the circle goes from 0° to -180°, like radians do (refer to the picture below). Hence, if we were to consider cases like a = 160° and b = -160°, we will get 320° as the shorter angle of rotation.
If measured in radians, 180° will be π.
The (actual) solution
Well, a – b actually works. Once you convert the result so that it is between -180° and 180° (or -π and π), that is. Here’s how you convert any angle into a value between -180° and 180°:
Firstly, if your result is already between -180° and 180°, then you ignore the following steps.
Next, convert your result into an angle between -360° and 360°. All you have to do is to modulo the result with 360, i.e. result % 360.
Finally, if your result is negative, add 360° to it; otherwise, subtract 360° from it.
Or, in short, expressed mathematically:
\mathrm{f\left(x\right)}=\left\{\begin{array}{ll}\mathrm{sgn x × 360° – |x|}& if |x| > 180\\ x& if 0 \ge |x| \le 180\end{array}
If that is hard to read, here’s a code snippet in C# that expresses the same thing:
float result = b - a; // Shorter angle of rotation.
// If our result's value is more than 180 or less than -180.
if(Mathf.Abs(result) > 180)
// Covert the value to be between -180 and 180.
result = -Math.Sign(result) * (360 - Math.Abs(result));
Did the article help you figure out your angles? Or maybe you spotted some errors with our math? Feel free to leave us a comment below.
game developmentmathsprogramming
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A Model-Based Feed-Forward Controller Scheme for Accurate Chilled Water Temperature Control of Inlet Guide Vane Centrifugal Chillers | J. Sol. Energy Eng. | ASME Digital Collection
Yongzhong Jia,
Yongzhong Jia
Department of Mechanical Engineering and Mechanics, 3141 Chestnut St., Drexel University, Philadelphia, PA 19104
Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received by the ASME Solar Division November 4, 2003; final revision April 7, 2004. Associate Editor: M. Krarti.
Jia , Y., and Reddy, T. A. (February 7, 2005). "A Model-Based Feed-Forward Controller Scheme for Accurate Chilled Water Temperature Control of Inlet Guide Vane Centrifugal Chillers ." ASME. J. Sol. Energy Eng. February 2005; 127(1): 47–52. https://doi.org/10.1115/1.1775225
Capacity control in most commercial centrifugal chillers is achieved by inlet guide vanes which are activated by the leaving chilled water temperature sensor. Due to mechanical reasons, vane control is done discretely, and not continuously. The control module compares the value provided by the temperature sensor to pre-set control band values, and if those bands are exceeded, it sends a signal to the vane control motor to adjust the vane position by one step which could be upwards or downwards. The advantage of this type of discrete control method is its simplicity. Normally, the accuracy in the outlet chilled water temperature is of the order of
0.5°C,
which is acceptable for normal cooling plants such as used in office buildings. However, there are applications such as in pharmaceutical processes, mechanics labs, or instances in chemical processes where more accurate control is required (sometimes as low as
0.05°C).
This paper proposes a simple method to achieve such tight control without any hardware modifications. The basis of this method is a transient physical inverse model of the refrigerant boiling process in the evaporator, in conjunction with a feed-forward control scheme. The model parameters need to be identified from monitored data since they are chiller-specific. This paper describes the model, and applies it to one-minute monitored data from an actual chiller plant of 1580 kW (450 Tons). It is demonstrated that for this specific chiller such a control scheme has the potential to improve control accuracy by about 28% as compared to the traditional control method.
refrigerators, temperature sensors, feedforward, temperature control, cooling, water
Feedforward control, Inlet guide vanes, Transients (Dynamics), Water, Water temperature, Temperature sensors, Temperature, Refrigerants, Signals, Cooling, Control equipment
Multi-bed Regenerative Adsorption Chiller-improving the Utilization of Waste Heat and Reducing the Chilled Water Outlet Temperature Fluctuation
Jeong, S., Kang, B., Lee, C., and Karng, S., 1994, “Computer Simulation on Dynamic Behavior of a Hot Water Driven Absorption Chiller,” Proceedings of the International Absorption Heat Pump Conference, Proc. Int. Absorpt. Heat Pump Conf., Proceedings of the International Absorption Heat Pump Conference, Jan 19–21 1994, New Orleans, LA, USA, pp. 333–338.
Equilibrium Low Pressure Generator Temperatures for Double-effect Series Flow Absorption Refrigeration Systems
Operation and Performance of a 350 kW (100 RT) Single-effect/double-lift Absorption Chiller in a District Heating Network
(Pt IB) 1998, Proceedings of the 1998 ASHRAE Winter Meeting. Part 2 (of 2), Jan. 18–21 1998, San Francisco, SF-98-18-2, pp.
Commercial Absorption Chiller Models for Evaluation of Control Strategies
, 1995, Proceedings of the 1995 ASHRAE Annual Meeting, Jan 29-Feb 1 1995, Chicago, IL, USA, CH-95-18-2, pp.
Differential Pressure, Bypass Chilled Water Systems: Capacity Ratios Between On-off and Modulating Units
Build. Services Eng. Res. Technol.
Sub-optimal On-off Switching Control Strategies for Chilled Water Cooling Systems with Storage
Applied Thermal Eng
Small Chiller System Design
Continuous Capacity Control Type Screw Chiller Unit
(3), Jun, 1990, pp.
Market Trend of Water Chiller Units and Hitachi’s Challenge with Screw Compressors
Cool Thermal Discharge with Time-velocity Variation of Flowing Air In Situ Contact on Water Surface
J. Chin. Inst. Chem. Eng.
Ice Storage System Assures Data Center Cooling
(4), Apr, 1995, pp.
Arima, M., and Hara, E., 1995, “Fuzzy Logic and Rough Sets Controller for HVAC Systems,” IEEE WESCANEX Communications, Power, and Computing, v1, 1995, Proceedings of the 1995 IEEE WESCANEX Communications, Power, and Computing Conference. Part 1 (of 2), Winnipeg, Manit, Can, pp. 133–138.
Ammonia Refrigeration in Supermarkets
Chuntranuluck
Prediction of Chilling Times of Foods in Situations Where Evaporative Cooling is Significant-Part 2. Experimental Testing
Jia, Y., 2002, “Model-Based Generic Approaches for Automated Fault Detection, Diagnosis, Evaluation (FDDE) and for Accurate Control of Field-Operated Centrifugal Chillers,” Ph.D. Thesis, Mechanical Engineering and Mechanics Department, Drexel University, Philadelphia.
Luries, B., and Enright, S., 2000, Classical Feedback Control with MATLAB, ISBN: 0-8247-0370-7.
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Display or compute signal-to-interference-plus-noise (SINR) ratio - MATLAB sinr - MathWorks 日本
SINR Map for Multiple Transmitters
For Plotting SINR
Display or compute signal-to-interference-plus-noise (SINR) ratio
sinr(txs)
sinr(txs,propmodel)
sinr(___,Name,Value)
pd = sinr(txs,___)
r = sinr(rxs,txs,___)
sinr(txs) displays the signal-to-interference-plus-noise ratio (SINR) for transmitter sites txs in the current Site Viewer. The map contours are generated using SINR values computed for receiver site locations on the map. For each location, the signal source is the transmitter site in TXS with the greatest signal strength. The remaining transmitter sites in txs with the same transmitter frequency act as sources of interference. If txs is scalar or there are no sources of interference the resultant map displays signal-to-noise ratio (SNR).
This function only supports plotting for antenna sites with a CoordinateSystem property value of 'geographic'.
sinr(txs,propmodel) displays the SINR map with the propagation model set to the value in propmodel.
sinr(___,Name,Value) sets properties using one or more name-value pairs, in addition to the input arguments in previous syntaxes. For example, sinr(txs,'MaxRange',8000) sets the range from the site location at 8000 meters to include in the SINR map region.
pd = sinr(txs,___) returns computed SINR data in the propagation data object, pd. No plot is displayed and any graphical only name-value pairs are ignored.
r = sinr(rxs,txs,___) returns the sinr in dB computed at the receiver sites due to the transmitter sites.
Define names and location of sites in Boston.
Create a transmitter site array.
txs = txsite("Name", names,...
Display the SINR map, where signal source for each location is selected as the transmitter site with the strongest signal.
rxs — Receiver sites
Receiver site, specified as a rxsite object. Use array inputs to specify multiple sites.
Example: 'MaxRange',8000
SignalSource — Signal source of interest
'strongest' (default) | transmitter site object
Signal source of interest, specified as the comma-separated pair consisting of SignalSource and 'strongest' or as a transmitter site object. When the signal source of interest is 'strongest', the transmitter with the greatest signal strength is chosen as the signal source of interest for that location. When computing sinr, SignalSource can be a txsite array with equal number of elements rxs where each transmitter site element defines the signal source for the corresponding receiver site.
ReceiverNoisePower — Total noise power at receiver
Total noise power at receiver, specified as the comma-separated pair consisting of 'ReceiverNoisePower' and a scalar in dBm. The default value assumes that the receiver bandwidth is 1 MHz and receiver noise figure is 7 dB.
N=â174+10*\mathrm{log}\left(B\right)+F
N = Receiver noise in dBm
F = Noise figure in dB
ReceiverGain — Receiver gain
Mobile receiver gain, specified as the comma-separated pair consisting of 'ReceiverGain' and a scalar in dB. The receiver gain values include the antenna gain and the system loss. If you call the function using an output argument, the default value is computed using rxs.
ReceiverAntennaHeight — Receiver antenna height
Receiver antenna height above the ground, specified as the comma-separated pair consisting of 'ReceiverAntennaHeight' and a scalar in meters. If you call the function using an output argument, the default value is computed using rxs.
Values — Values of SINR for display
[-5:20] (default) | numeric vector
Values of SINR for display, specified as the comma-separated pair consisting of 'Values' and a numeric vector. Each value is displayed as a different colored, filled on the contour map. The contour colors are derived using Colormap and ColorLimits.
Atmospheric or empirical 30 km
Resolution — Resolution of receiver site locations used to compute SINR values
Resolution of receiver site locations used to compute SINR values, specified as the comma-separated pair consisting of 'Resolution' and 'auto' or a numeric scalar in meters. The resolution defines the maximum distance between the locations. If the resolution is 'auto', sinr computes a value scaled to MaxRange. Decreasing the resolution increases the quality of the SINR map and the time required to create it.
Colormap — Colormap for coloring filled contours
'jet' (default) | M-by-3 array of RGB triplets
Colormap for coloring filled contours, specified as the comma-separated pair consisting of 'ColorMap' and an M-by-3 array of RGB triplets, where M is the number of individual colors.
ColorLimits — Color limits for color maps
[-5 20] (default) | two-element vector
Color limits for color maps, specified as the comma-separated pair consisting of 'ColorLimits' and a two-element vector of the form [min max]. The color limits indicate the SINR values that map to the first and last colors in the colormap.
Show signal strength color legend on map, specified as the comma-separated pair consisting of 'ShowLegend' and 'true' or 'false'.
Transparency — Transparency of SINR map
Transparency of SINR map, specified as the comma-separated pair consisting of 'Transparency' and a numeric scalar in the range 0–1. If the value is zero, the map is completely transparent. If the value is one, the map is completely opaque.
r — Signal to interference plus noise ratio at the receiver
Signal to interference plus noise ratio at the receiver due to the transmitter sites, returned as a numeric vector. The vector length is equal to the number of receiver sites.
pd — SINR data
SINR data, returned as a propagationData object consisting of Latitude and Longitude, and a signal strength variable corresponding to the plot type. Name of the propagationData is "SINR Data".
Starting in R2021b, when you use the sinr function and specify the propmodel argument or PropagationModel name-value argument as 'raytracing', the function uses the shooting and bouncing rays (SBR) method and calculates up to two reflections. In previous releases, the sinr function uses the image method and calculates up to one reflection.
To display or compute the SINR using the image method instead, create a propagation model by using the propagationModel function. Then, use the sinr function with the propagation model as input. This example shows how to update your code.
sinr(txs,pm)
coverage | propagationModel
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Cursive script 'd' and capital 'D' in the U.S. D'Nealian script style
D (named dee /ˈdiː/[1]) is the fowert letter o maist variants o the basic modern Latin alphabet. Seembol "D" is 500 in Roman numerals.
The Semitic letter Dâlet mey hae developed frae the logogram for a fish or a door. Thare various Egyptian hieroglyphs that micht hae inspired this. In Semitic, Auncient Greek an Latin, the letter representit /d/; in the Etruscan alphabet the letter wis superfluous but still retained (see letter B). The equivalent Greek letter is Delta, 'Δ'.
The minuscule (lawer-case) fairm o 'd' consists o a luip an a taw vertical stroke. It developed bi gradual variations on the majuscule (caipital) form. In haundwritin, it wis common tae stairt the arc tae the left o the vertical stroke, resultin in a serif at the tap o the arc. This serif wis extendit while the rest o the letter wis reduced, resultin in an angled stroke an luip. The angled stroke slawly developed intae a vertical stroke.
The letter D, staundin for "Deutschland" (German for "Germany"), on a boondary stane at the mairch atween Austrick an Germany.
In naur aw leids that uise the Latin alphabet an the Internaitional Phonetic Alphabet 'd' represents the voiced alveolar or voiced dental plosive /d/, but in the Vietnamese alphabet, it represents the soond /z/ (or /j/ in soothren dialects). In Fijian it represents a prenasalized stap /nd/.[2] In some leids whaur voiceless unaspirated staps contrast wi voiceless aspiratit staps, 'd' represents an unaspirated /t/, while 't' represents an aspirated /tʰ/. Examples o sic leids include Icelandic, Scottish Gaelic, Navajo, Estonie an the Pinyin transleeteration o Mandarin.
Đ đ : Latin letter D wi stroke
Ɗ ɗ : Latin letter D wi huik
∂ : the partial derivative seembol,
{\displaystyle \partial }
EBCDIC family 196 0 C4 132 0 84
In Breetish Sign Leid (BSL), the letter 'd' is indicatit bi signin wi the richt haund held wi the index an thumb extendit an slichtly curved, an the tip o the thumb an finger held against the extendit index o the left haund.
↑ "D" Oxford English Dictionary, 2nd edition (1989); Merriam-Webster's Third New International Dictionary of the English Language, Unabridged (1993); "dee", op. cit.
↑ Lynch, John (1998). Pacific languages: an introduction. University of Hawaii Press. p. 97. ISBN 0-8248-1898-9.
Media relatit tae D at Wikimedia Commons
The dictionar defineetion o D at Wiktionary
Taen frae "https://sco.wikipedia.org/w/index.php?title=D&oldid=693896"
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Putting It Together: Elasticity | Macroeconomics | Course Hero
Netflix Pricing Revisited
We began this module discussing a price change that Netflix imposed on its customers. Now that we understand price elasticity, we can better evaluate that case. How did the 60 percent price increase end up for Netflix? It was a very bumpy two-year ride. Before the price increase, there were about 24.6 million U.S. subscribers. After the price increase, 810,000 infuriated customers canceled their Netflix subscriptions, dropping the total number of subscribers to 23.79 million. Fast-forward to June 2013, when there were 36 million streaming Netflix subscribers in the United States. This was an increase of 11.4 million subscribers since the price increase—an average per-quarter growth of about 1.6 million. This growth is less than the 2 million-per-quarter increases Netflix experienced in the fourth quarter of 2010 and the first quarter of 2011.
During the first year after the price increase, the firm's stock price (a measure of future expectations for the firm) fell from about $300 per share to just under $54. By June 2013, the stock price had rebounded to about $200 per share—still off by more than one-third from its high, but definitely improving. What happened? Obviously, Netflix understood the law of demand. Company officials reported, when they announced the price increase, that this could result in the loss of about 600,000 existing subscribers. Using the elasticity of demand formula, it is easy to see that they expected an inelastic response:
\begin{array}{rcl}\text{Percent change in quantity}&=&\large{\frac{-600,000}{(24\text{ million}+24.6\text{ million})\div{2}}} \\ \text{Percent change in price}&=&\large{\frac{\$6}{(\$10+\$16)\div{2}}} \\ {\text{Price Elasticity of Demand}}&=&\large{\frac{-600,000/24.3\text{ million}}{\$6/\$13}}\\&=&\large{\frac{-0.025}{0.46}}\\&=&-0.05 \end{array}
In addition, Netflix officials had expected that the price increase would have little impact on attracting new customers. Netflix anticipated adding up to 1.29 million new subscribers in the third quarter of 2011. It is true that this was slower growth than the firm had experienced over the past year—about 2 million per quarter. Why was the estimate of customers leaving so far off? During the fourteen years after Netflix was founded, there was an increase in the number of close, but not perfect, substitutes. Consumers now had choices ranging from Vudu, Amazon Prime, Hulu, and Redbox to retail stores.
Jaime Weinman reported in Maclean's that Redbox kiosks are "a five-minute drive or less from 68 percent of Americans, and it seems that many people still find a five-minute drive more convenient than loading up a movie online." It seems that, in 2012, many consumers still preferred a physical DVD disk over streaming video. What missteps did the Netflix management make? In addition to misjudging the elasticity of demand, by failing to account for close substitutes, it seems they may have also misjudged customers' preferences and tastes (at the time being). Yet, we now see that as the population has increased, the preference for streaming video has overtaken the desire for physical DVD discs. Netflix, the target of numerous late-night talk-show jabs and laughs in 2011, may have had the last laugh in the end.
Putting It Together: Human Resource Management | Principles of Management.pdf
MGT HUMAN RESO • Strayer University
Closing Case 2- Putting IT Together Answers
MAN 4320 • Florida International University
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Autotransformer Knowpia
An autotransformer is an electrical transformer with only one winding. The "auto" (Greek for "self") prefix refers to the single coil acting alone, not to any kind of automatic mechanism. In an autotransformer, portions of the same winding act as both the primary winding and secondary winding sides of the transformer. In contrast, an ordinary transformer has separate primary and secondary windings which have no metallic conducting path between them.
Single-phase tapped autotransformer with an output voltage range of 40%–115% of input
The autotransformer winding has at least three taps where electrical connections are made. Since part of the winding does "double duty", autotransformers have the advantages of often being smaller, lighter, and cheaper than typical dual-winding transformers, but the disadvantage of not providing electrical isolation between primary and secondary circuits. Other advantages of autotransformers include lower leakage reactance, lower losses, lower excitation current, and increased VA rating for a given size and mass.[1]
An example of an application of an autotransformer is one style of traveler's voltage converter, that allows 230-volt devices to be used on 120-volt supply circuits, or the reverse. An autotransformer with multiple taps may be applied to adjust the voltage at the end of a long distribution circuit to correct for excess voltage drop; when automatically controlled, this is one example of a voltage regulator.
An autotransformer has a single winding with two end terminals and one or more terminals at intermediate tap points. It is a transformer in which the primary and secondary coils have part of their turns in common. The portion of the winding shared by both the primary and secondary is the common section. The portion of the winding not shared by both the primary and secondary is the series section. The primary voltage is applied across two of the terminals. The secondary voltage is taken from two terminals, one terminal of which is usually in common with a primary voltage terminal.[2]
Since the volts-per-turn is the same in both windings, each develops a voltage in proportion to its number of turns. In an autotransformer, part of the output current flows directly from the input to the output (through the series section), and only part is transferred inductively (through the common section), allowing a smaller, lighter, cheaper core to be used as well as requiring only a single winding.[3] However the voltage and current ratio of autotransformers can be formulated the same as other two-winding transformers:[1]
{\displaystyle {\frac {V_{1}}{V_{2}}}={\frac {N_{1}}{N_{2}}}=a}
{\displaystyle (0<V_{2}<V_{1})}
The ampere-turns provided by the series section of the winding:
{\displaystyle F_{S}=(N_{1}-N_{2})I_{1}=\left(1-{\frac {1}{a}}\right)N_{1}I_{1}}
The ampere-turns provided by the common section of the winding:
{\displaystyle F_{C}=N_{2}(I_{2}-I_{1})={\frac {N_{1}}{a}}(I_{2}-I_{1})}
For ampere-turn balance, FS = FC:
{\displaystyle \left(1-{\frac {1}{a}}\right)N_{1}I_{1}={\frac {N_{1}}{a}}(I_{2}-I_{1})}
{\displaystyle {\frac {I_{1}}{I_{2}}}={\frac {1}{a}}={\frac {N_{2}}{N_{1}}}}
One end of the winding is usually connected in common to both the voltage source and the electrical load. The other end of the source and load are connected to taps along the winding. Different taps on the winding correspond to different voltages, measured from the common end. In a step-down transformer the source is usually connected across the entire winding while the load is connected by a tap across only a portion of the winding. In a step-up transformer, conversely, the load is attached across the full winding while the source is connected to a tap across a portion of the winding. For a step-up transformer, the subscripts in the above equations are reversed where, in this situation,
{\displaystyle N_{2}}
{\displaystyle V_{2}}
{\displaystyle N_{1}}
{\displaystyle V_{1}}
As in a two-winding transformer, the ratio of secondary to primary voltages is equal to the ratio of the number of turns of the winding they connect to. For example, connecting the load between the middle of the winding and the common terminal end of the winding of the autotransformer will result in the output load voltage being 50% of the primary voltage. Depending on the application, that portion of the winding used solely in the higher-voltage (lower current) portion may be wound with wire of a smaller gauge, though the entire winding is directly connected.
If one of the center-taps is used for the ground, then the autotransformer can be used as a balun to convert a balanced line (connected to the two end taps) to an unbalanced line (the side with the ground).
An autotransformer does not provide electrical isolation between its windings as an ordinary transformer does; if the neutral side of the input is not at ground voltage, the neutral side of the output will not be either. A failure of the isolation of the windings of an autotransformer can result in full input voltage applied to the output. Also, a break in the part of the winding that is used as both primary and secondary will result in the transformer acting as an inductor in series with the load (which under light load conditions may result in nearly full input voltage being applied to the output). These are important safety considerations when deciding to use an autotransformer in a given application.[4]
Because it requires both fewer windings and a smaller core, an autotransformer for power applications is typically lighter and less costly than a two-winding transformer, up to a voltage ratio of about 3:1; beyond that range, a two-winding transformer is usually more economical.[4]
In three phase power transmission applications, autotransformers have the limitations of not suppressing harmonic currents and as acting as another source of ground fault currents. A large three-phase autotransformer may have a "buried" delta winding, not connected to the outside of the tank, to absorb some harmonic currents.[4]
In practice, losses mean that both standard transformers and autotransformers are not perfectly reversible; one designed for stepping down a voltage will deliver slightly less voltage than required if it is used to step up. The difference is usually slight enough to allow reversal where the actual voltage level is not critical.
Like multiple-winding transformers, autotransformers use time-varying magnetic fields to transfer power. They require alternating currents to operate properly and will not function on direct current. Because the primary and secondary windings are electrically connected, an autotransformer will allow current to flow between windings and therefore does not provide AC or DC isolation.
Power transmission and distributionEdit
Autotransformers are frequently used in power applications to interconnect systems operating at different voltage classes, for example 132 kV to 66 kV for transmission. Another application in industry is to adapt machinery built (for example) for 480 V supplies to operate on a 600 V supply. They are also often used for providing conversions between the two common domestic mains voltage bands in the world (100 V–130 V and 200 V–250 V). The links between the UK 400 kV and 275 kV "Super Grid" networks are normally three phase autotransformers with taps at the common neutral end.
On long rural power distribution lines, special autotransformers with automatic tap-changing equipment are inserted as voltage regulators, so that customers at the far end of the line receive the same average voltage as those closer to the source. The variable ratio of the autotransformer compensates for the voltage drop along the line.
A special form of autotransformer called a zig zag is used to provide grounding on three-phase systems that otherwise have no connection to ground. A zig-zag transformer provides a path for current that is common to all three phases (so-called zero sequence current).
Audio systemEdit
In audio applications, tapped autotransformers are used to adapt speakers to constant-voltage audio distribution systems, and for impedance matching such as between a low-impedance microphone and a high-impedance amplifier input.
In railway applications, it is common to power the trains at 25 kV AC. To increase the distance between electricity Grid feeder points, they can be arranged to supply a split-phase 25-0-25 kV feed with the third wire (opposite phase) out of reach of the train's overhead collector pantograph. The 0 V point of the supply is connected to the rail while one 25 kV point is connected to the overhead contact wire. At frequent (about 10 km) intervals, an autotransformer links the contact wire to rail and to the second (antiphase) supply conductor. This system increases usable transmission distance, reduces induced interference into external equipment and reduces cost. A variant is occasionally seen where the supply conductor is at a different voltage to the contact wire with the autotransformer ratio modified to suit.[5]
Autotransformer starterEdit
Autotransformers can be used as a method of soft starting induction motors. One of the well-known designs of such starters is Korndörfer starter.
Variable autotransformersEdit
A variable autotransformer, with a sliding-brush secondary connection and a toroidal core. Cover has been removed to show copper windings and brush.
Variable Transformer - part of Tektronix 576 Curve Tracer
By exposing part of the winding coils and making the secondary connection through a sliding brush, a continuously variable turns ratio can be obtained, allowing for very smooth control of output voltage. The output voltage is not limited to the discrete voltages represented by actual number of turns. The voltage can be smoothly varied between turns as the brush has a relatively high resistance (compared with a metal contact) and the actual output voltage is a function of the relative area of brush in contact with adjacent windings.[6] The relatively high resistance of the brush also prevents it from acting as a short circuited turn when it contacts two adjacent turns. Typically the primary connection connects to only a part of the winding allowing the output voltage to be varied smoothly from zero to above the input voltage and thus allowing the device to be used for testing electrical equipment at the limits of its specified voltage range.
The output voltage adjustment can be manual or automatic. The manual type is applicable only for relatively low voltage and is known as a variable AC transformer (often referred to by the trademark name Variac). These are often used in repair shops for testing devices under different voltages or to simulate abnormal line voltages.
The type with automatic voltage adjustment can be used as automatic voltage regulator, to maintain a steady voltage at the customers' service during a wide range of line and load conditions. Another application is a lighting dimmer that doesn't produce the EMI typical of most thyristor dimmers.
Variac TrademarkEdit
From 1934 to 2002, Variac was a U.S. trademark of General Radio for a variable autotransformer intended to conveniently vary the output voltage for a steady AC input voltage. In 2004, Instrument Service Equipment applied for and obtained the Variac trademark for the same type of product.[7] The term variac has become a genericised trademark, being used to refer to a variable autotransformer.[citation needed]
^ a b Sen, P. C. (1997). Principles of electric machines and power electronics. John Wiley & Sons. p. 64. ISBN 0471022950.
^ Pansini, Anthony J. (1999). Electrical Transformers and Power Equipment (3rd ed.). Fairmont Press. pp. 89–91. ISBN 9780881733112.
^ "Commercial site explaining why autotransformers are smaller". Archived from the original on 2013-09-20. Retrieved 2013-09-19.
^ a b c Fink, Donald G.; Beaty, H. Wayne (1978). Standard Handbook for Electrical Engineers (Eleventh ed.). New York: McGraw-Hill. pp. 10-44, 10-45, 17-39. ISBN 0-07-020974-X.
^ Fahrleitungen elektrischer Bahnen [Contact Lines for Electric Railways] (in German). Stuttgart: BG Teubner-Verlag. 1997. p. 672. ISBN 9783519061779. An English edition "Contact Lines for Electric Railways" appears to be out of print. This industry standard text describes the various European electrification principles. See the website of the UIC in Paris for the relevant international rail standards, in English. No comparable publications seem to exist for American railways, probably due to the paucity of electrified installations there.
^ Bakshi, M. V.; Bakshi, U. A. Electrical Machines - I. Pune: Technical Publications. p. 330. ISBN 81-8431-009-9.
^ "Trademark Status & Document Retrieval".
Croft, Terrell; Summers, Wilford, eds. (1987). American Electricians' Handbook (Eleventh ed.). New York: McGraw Hill. ISBN 0-07-013932-6.
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Environmental Limits to Population Growth | Boundless Biology | Course Hero
\Delta \text{N} / \Delta \text{T} = \text{B} - \text{D}
\Delta \text{N}
= change in number,
\Delta \text{T}
= change in time,
\text{B}
= birth rate, and
\text{D}
= death rate. The birth rate is usually expressed on a per capita (for each individual) basis. Thus, B (birth rate) = bN (the per capita birth rate "b" multiplied by the number of individuals "N") and D (death rate) = dN (the per capita death rate "d" multiplied by the number of individuals "N"). Additionally, ecologists are interested in the population at a particular point in time: an infinitely small time interval. For this reason, the terminology of differential calculus is used to obtain the "instantaneous" growth rate, replacing the change in number and time with an instant-specific measurement of number and time.
\text{dN} / \text{dT} = \text{BN DN} = (\text{B} \text{D})\text{N}
Notice that the "d" associated with the first term refers to the derivative (as the term is used in calculus) and is different from the death rate, also called "d." The difference between birth and death rates is further simplified by substituting the term "r" (intrinsic rate of increase) for the relationship between birth and death rates:
\text{dN} / \text{dT} = \text{rN}
The value "r" can be positive, meaning the population is increasing in size; negative, meaning the population is decreasing in size; or zero, where the population's size is unchanging, a condition known as zero population growth. A further refinement of the formula recognizes that different species have inherent differences in their intrinsic rate of increase (often thought of as the potential for reproduction), even under ideal conditions. Obviously, a bacterium can reproduce more rapidly and have a higher intrinsic rate of growth than a human. The maximal growth rate for a species is its biotic potential, or rmax, thus changing the equation to:
\text{dN} / \text{dT}=\text{r}_{\text{max}}\text{N}
Logistic growth of a population size occurs when resources are limited, thereby setting a maximum number an environment can support.
Exponential growth is possible only when infinite natural resources are available; this is not the case in the real world. Charles Darwin recognized this fact in his description of the "struggle for existence," which states that individuals will compete (with members of their own or other species ) for limited resources. The successful ones will survive to pass on their own characteristics and traits (which we know now are transferred by genes) to the next generation at a greater rate: a process known as natural selection. To model the reality of limited resources, population ecologists developed the logistic growth model.
In the real world, with its limited resources, exponential growth cannot continue indefinitely. Exponential growth may occur in environments where there are few individuals and plentiful resources, but when the number of individuals becomes large enough, resources will be depleted, slowing the growth rate. Eventually, the growth rate will plateau or level off. This population size, which represents the maximum population size that a particular environment can support, is called the carrying capacity, or K.
The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression "K – N" is indicative of how many individuals may be added to a population at a given stage, and "K – N" divided by "K" is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation:
\text{dN} / \text{dT} = \text{rmax} * (\text{dN} / \text{dT}) = \text{rmax} * \text{N} * ((\text{K N}) / \text{K})
Notice that when N is very small, (K-N)/K becomes close to K/K or 1; the right side of the equation reduces to rmaxN, which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is large, (K-N)/K come close to zero, which means that population growth will be slowed greatly or even stopped. Thus, population growth is greatly slowed in large populations by the carrying capacity K. This model also allows for negative population growth or a population decline. This occurs when the number of individuals in the population exceeds the carrying capacity (because the value of (K-N)/K is negative).
In the real world, the variation of phenotypes among individuals within a population means that some individuals will be better adapted to their environment than others. The resulting competition between population members of the same species for resources is termed intraspecific competition (intra- = "within"; -specific = "species"). Intraspecific competition for resources may not affect populations that are well below their carrying capacity as resources are plentiful and all individuals can obtain what they need. However, as population size increases, this competition intensifies. In addition, the accumulation of waste products can reduce an environment's carrying capacity.
Logistic population growth: (a) Yeast grown in ideal conditions in a test tube show a classical S-shaped logistic growth curve, whereas (b) a natural population of seals shows real-world fluctuation.
Density-Dependent and Density-Independent Population Regulation
Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population.
Differentiate between density-dependent and density-independent population regulation.
The density of a population can be regulated by various factors, including biotic and abiotic factors and population size.
Density-dependent regulation can be affected by factors that affect birth and death rates such as competition and predation.
Density-independent regulation can be affected by factors that affect birth and death rates such as abiotic factors and environmental factors, i.e. severe weather and conditions such as fire.
New models of life history incorporate ecological concepts that are typically included in r- and K-selection theory in combination with population age structures and mortality factors.
interspecific: existing or occurring between different species
In population ecology, density-dependent processes occur when population growth rates are regulated by the density of a population. Most density-dependent factors, which are biological in nature (biotic), include predation, inter- and intraspecific competition, accumulation of waste, and diseases such as those caused by parasites. Usually, the denser a population is, the greater its mortality rate. For example, during intra- and interspecific competition, the reproductive rates of the individuals will usually be lower, reducing their population's rate of growth. In addition, low prey density increases the mortality of its predator because it has more difficulty locating its food source.
An example of density-dependent regulation is shown with results from a study focusing on the giant intestinal roundworm (Ascaris lumbricoides), a parasite of humans and other mammals. The data shows that denser populations of the parasite exhibit lower fecundity: they contained fewer eggs. One possible explanation for this phenomenon was that females would be smaller in more dense populations due to limited resources so they would have fewer eggs. This hypothesis was tested and disproved in a 2009 study which showed that female weight had no influence. The actual cause of the density-dependence of fecundity in this organism is still unclear and awaiting further investigation.
Effect of population density on fecundity: In this population of roundworms, fecundity (number of eggs) decreases with population density.
Many factors, typically physical or chemical in nature (abiotic), influence the mortality of a population regardless of its density. They include weather, natural disasters, and pollution. An individual deer may be killed in a forest fire regardless of how many deer happen to be in that area. Its chances of survival are the same whether the population density is high or low.
In real-life situations, population regulation is very complicated and density-dependent and independent factors can interact. A dense population that is reduced in a density-independent manner by some environmental factor(s) will be able to recover differently than would a sparse population. For example, a population of deer affected by a harsh winter will recover faster if there are more deer remaining to reproduce.
fission. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/fission. License: CC BY-SA: Attribution-ShareAlike
per capita. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/per_capita. License: CC BY-SA: Attribution-ShareAlike
OpenStax College, Environmental Limits to Population Growth. October 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m44872/latest/Figure_45_03_01.jpg. License: CC BY: Attribution
OpenStax College, Environmental Limits to Population Growth. October 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m44872/latest/Figure_45_03_02.png. License: CC BY: Attribution
OpenStax College, Environmental Limits to Population Growth. December 6, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m44872/latest/Figure_45_03_01.jpg. License: CC BY: Attribution
Density dependence. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Density_dependence. License: CC BY-SA: Attribution-ShareAlike
interspecific. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/interspecific. License: CC BY-SA: Attribution-ShareAlike
OpenStax College, Population Dynamics and Regulation. October 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m44882/latest/Figure_45_04_01.jpg. License: CC BY: Attribution
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|
Normal Heptane-Diesel Combustion and Odorous Emissions in Direct Injection Diesel Engines | J. Energy Resour. Technol. | ASME Digital Collection
, Rajshahi 6204, Bangladesh
e-mail: mmroy5767@yahoo.com
Murari Mohon Roy Dr.
Roy, M. M. (January 25, 2008). "Normal Heptane-Diesel Combustion and Odorous Emissions in Direct Injection Diesel Engines." ASME. J. Energy Resour. Technol. March 2008; 130(1): 011101. https://doi.org/10.1115/1.2824295
This study investigated normal heptane (
N
-heptane)-diesel combustion and odorous emissions in a direct injection diesel engine during and after engine warmup at idling. The odor is a little worse with
N
-heptane and blends than that of diesel fuel due to overleaning of the mixture. In addition, formaldehyde (HCHO) and total hydrocarbon (THC) in the exhaust increase with increasing
N
-heptane content. However, 50% and 100%
N
-heptane showed lower eye irritation than neat diesel fuel. Due to low boiling point of
N
-heptane, adhering fuel on the combustion chamber wall is small and as a single-component
C7
fuel, relatively high volatile components present in the exhaust are low. This may cause lower eye irritation. On the contrary, bulk in-cylinder gas temperature is lower and ignition delay significantly increases for 50% and 100%
N
-heptane due to the low boiling point, high latent heat of evaporation, and low bulk modulus of compressibility of
N
-heptane than standard diesel fuel. This longer ignition delay and lower bulk in-cylinder gas temperature of
N
-heptane blends deteriorate exhaust odor and emissions of HCHO and THC.
air pollution, combustion, diesel engines, elastic moduli, exhaust systems, DI diesel engine, odorous emissions, single-component fuel, N-heptane, injection timing, cylinder pressure and temperature, ignition delay
Combustion, Diesel, Diesel engines, Emissions, Heptane, Engines, Fuels, Cylinders, Temperature, Exhaust systems, Ignition delay
Diesel Exhaust Odor of Small, High Speed, Direct Injection Engines
Development of a Diesel Odor Measurement Method and Its Application to Odor Reduction
Effect of Injection Timing and Fuel Properties on Exhaust Odor in DI Diesel Engines
Kubsh
Fuel Effects on Emissions From an Advanced Technology Vehicles
Slodowske
Effect of MTBE and DME on Odorous Emissions in a DI Diesel Engine
Improvements to Diesel Combustion and Emissions by Oxygenated Agent Addition to Diesel Fuels-Influence of Diesel Fuels and Oxygenated Agent Properties
Proceedings of JSAE (in Japanese)
Effect of n-Heptane and n-Decane on Exhaust Odour in Direct Injection Diesel Engines
Owkita
Analysis Method of Low Concentration Gas and Bad Smell
KOUDANNSYA
Investigation of Odorous Components and Improvement in Odor Assessment Procedure in DI Diesel Engines
The Impact of the Bulk Modulus of Diesel Fuels on Fuel Injection Timing
|
Overview - utils | CatBoost
Generate the columns description file with the given structure.
C
C_{i,j}
i
j
Return points of the FNR curve.
Create a pool from a file and quantize it while loading the data. This compresses the size of the initial dataset and provides an opportunity to load huge datasets that can not be loaded to RAM otherwise.
The input data should contain only numerical features (other types are not currently supported).
Return the probability boundary required to achieve the specified false positive or false negative rate.
|
Diameter of a Cone Calculator
Reviewed by Komal Rafay
Diameter of a cone formula
How do I find the diameter of a cone?
Our other cone calculators.
Understanding the properties of a cone can be difficult at the best of times, which is where this diameter of a cone calculator comes into its own! No matter the variables you have - height, surface area, volume of a cone - your diameter won't be far away. We also explain how to calculate the diameter of a cone and the diameter of a cone formula in case you wish to perform the work by hand.
The diameter of a cone equation depends on a number of things, mainly which variables you have to hand. To illustrate this properly, we will go through them all one-by-one. To start with, we have the diameter of a cone from height and slant height:
d = 2 \times \sqrt{l^2-h^2}
l
- Slant height;
h
- Height; and
- Diameter.
This one is quite simple to remember as it relies on using Pythagoreas' theorem to find the radius of the base, then doubling it. Our next formula is to find the diameter of a cone from its volume and height:
d = 2 \times \sqrt{\frac{3 \times V}{\pi \times h}}
V
- Volume.
As you can see, while this formula might not be too difficult to remember, its quite difficult to calculate accurately due to requiring both
\pi
and a square root. That's why we made the diameter of a cone calculator! We now come to a few questions regarding area. First, we use the lateral area and slant height of a cone:
d = 2 \times \frac{A_L}{\pi \times l}
A_L
- Lateral area.
Then we use the base area of a cone to find the diameter:
d = 2 \times \sqrt{\frac{A_B}{\pi}}
A_L
- Base area.
Now, if we have just the whole surface area of a cone and its slant height, we can find its diameter, although we have to solve the following quadratic equation:
0 = \pi r^2 + \pi l r - A
r - Radius; and
A - Surface area.
You must then take the positive root and let
d = 2r
get your answer. That's a lot of work, so just use our diameter of a cone calculator instead!
To use the height and volume of a cone to find diameter:
Multiply the volume of the cone by 3.
Divide the resultant number by pi times the height.
Find the square root of the result of the above division.
Double the result of Step 3 to get the diameter.
While this diameter of a cone calculator answers "How to calculate the diameter of a cone?" in full, you may wish to explore other topics around the topic. In that case, please check out our other relevant tools:
Right circular cone;
Height of a cone;
Radius of a cone;
Lateral area of a cone; and
Slant height of a cone.
How do I find the volume of a cone from its diameter?
To find the volume of a cone from its diameter:
Square the diameter.
Multiply this square by pi and the height of the cone.
Divide the result by 12 to get your cone volume.
Diameter (2r)
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Chameleon_particle Knowpia
The chameleon is a hypothetical scalar particle that couples to matter more weakly than gravity,[1] postulated as a dark energy candidate.[2] Due to a non-linear self-interaction, it has a variable effective mass which is an increasing function of the ambient energy density—as a result, the range of the force mediated by the particle is predicted to be very small in regions of high density (for example on Earth, where it is less than 1mm) but much larger in low-density intergalactic regions: out in the cosmos chameleon models permit a range of up to several thousand parsecs. As a result of this variable mass, the hypothetical fifth force mediated by the chameleon is able to evade current constraints on equivalence principle violation derived from terrestrial experiments even if it couples to matter with a strength equal or greater than that of gravity. Although this property would allow the chameleon to drive the currently observed acceleration of the universe's expansion, it also makes it very difficult to test for experimentally.
Gravity, electroweak
Variable, depending on ambient energy density
In 2021, physicists working at the dark matter detector experiment XENON1T reported the potential incidental discovery of candidate chameleon particles (chameleon-screened dark energy).[3][4][5][further explanation needed]
Hypothetical propertiesEdit
Chameleon particles were proposed in 2003 by Khoury and Weltman.
In most theories, chameleons have a mass that scales as some power of the local energy density:
{\displaystyle m_{\text{eff}}\sim \rho ^{\alpha }}
{\displaystyle \alpha \simeq 1}
Chameleons also couple to photons, allowing photons and chameleons to oscillate between each other in the presence of an external magnetic field.[6]
Chameleons can be confined in hollow containers because their mass increases rapidly as they penetrate the container wall, causing them to reflect. One strategy to search experimentally for chameleons is to direct photons into a cavity, confining the chameleons produced, and then to switch off the light source. Chameleons would be indicated by the presence of an afterglow as they decay back into photons.[7]
A number of experiments have attempted to detect chameleons along with axions.[8]
The GammeV experiment[9] is a search for axions, but has been used to look for chameleons too. It consists of a cylindrical chamber inserted in a 5 T magnetic field. The ends of the chamber are glass windows, allowing light from a laser to enter and afterglow to exit. GammeV set the limited coupling to photons in 2009.[10]
CHASE (CHameleon Afterglow SEarch) results published in November 2010,[11] improve the limits on mass by 2 orders of magnitude and 5 orders for photon coupling.
A 2014 neutron mirror measurement excluded chameleon field for values of the coupling constant
{\displaystyle \beta >5.8\times 10^{8}}
,[12] where the effective potential of the chameleon quanta is written as
{\displaystyle V_{\text{eff}}=V(\Phi )+e^{\beta \Phi /M'_{\text{P}}}\rho }
{\displaystyle \rho }
being the mass density of the environment,
{\displaystyle V(\Phi )}
the chameleon potential and
{\displaystyle M'_{\text{P}}}
the reduced Planck mass.
The CERN Axion Solar Telescope has been suggested as a tool for detecting chameleons.[13]
^ Cho, Adrian (2015). "Tiny fountain of atoms sparks big insights into dark energy". Science. doi:10.1126/science.aad1653.
^ Khoury, Justin; Weltman, Amanda (2004). "Chameleon cosmology". Physical Review D. 69 (4): 044026. arXiv:astro-ph/0309411. Bibcode:2004PhRvD..69d4026K. doi:10.1103/PhysRevD.69.044026. S2CID 119478819.
^ "Have we detected dark energy? Scientists say it's a possibility". University of Cambridge. Retrieved 18 October 2021.
^ Fernandez, Elizabeth. "Signal From The XENON1T Experiment May Be A Hallmark Of Dark Energy". Forbes. Retrieved 18 October 2021.
^ Vagnozzi, Sunny; Visinelli, Luca; Brax, Philippe; Davis, Anne-Christine; Sakstein, Jeremy (15 September 2021). "Direct detection of dark energy: The XENON1T excess and future prospects". Physical Review D. 104 (6): 063023. arXiv:2103.15834. doi:10.1103/PhysRevD.104.063023. S2CID 232417159.
^ Erickcek, A. L.; Barnaby, N; Burrage, C; Huang, Z (2013). "Catastrophic consequences of kicking the chameleon". Physical Review Letters. 110 (17): 171101. arXiv:1204.1488. Bibcode:2013PhRvL.110b1101S. doi:10.1103/PhysRevLett.110.021101. PMID 23679701. S2CID 118730981.
^ Steffen, Jason H.; Gammev Collaboration (2008). "Constraints on chameleons and axion-like particles from the GammeV experiment". Proceedings of "Identification of Dark Matter 2008". August 18-22, 2008, Stockholm, Sweden. Vol. 2008. p. 064. arXiv:0810.5070. Bibcode:2008idm..confE..64S.
^ Rybka, G; Hotz, M; Rosenberg, L. J.; Asztalos, S. J.; Carosi, G; Hagmann, C; Kinion, D; Van Bibber, K; Hoskins, J; Martin, C; Sikivie, P; Tanner, D. B.; Bradley, R; Clarke, J (2010). "Search for chameleon scalar fields with the axion dark matter experiment". Physical Review Letters. 105 (5): 051801. arXiv:1004.5160. Bibcode:2010PhRvL.105a1801B. doi:10.1103/PhysRevLett.105.051801. PMID 20867906. S2CID 55204188.
^ GammeV experiment at Fermilab
^ Chou, A. S.; Wester, W.; Baumbaugh, A.; Gustafson, H. R.; Irizarry-Valle, Y.; Mazur, P. O.; Steffen, J. H.; Tomlin, R.; Upadhye, A.; Weltman, A.; Yang, X.; Yoo, J. (22 Jan 2009). "Search for Chameleon Particles Using a Photon-Regeneration Technique". Physical Review Letters. 102 (3): 030402. arXiv:0806.2438. Bibcode:2009PhRvL.102c0402C. doi:10.1103/PhysRevLett.102.030402. PMID 19257328. S2CID 12327360.
^ Steffen, Jason H. (2010). "The CHASE laboratory search for chameleon dark energy". Proceedings of the 35th International Conference of High Energy Physics (ICHEP 2010). July 22-28, 2010. Paris, France. Vol. 2010. p. 446. arXiv:1011.3802. Bibcode:2010iche.confE.446S.
^ Jenke, T.; Cronenberg, G.; Burgdörfer, J.; Chizhova, L. A.; Geltenbort, P.; Ivanov, A. N.; Lauer, T.; Lins, T.; Rotter, S.; Saul, H.; Schmidt, U.; Abele, H. (Apr 16, 2014). "Gravity Resonance Spectroscopy Constrains Dark Energy and Dark Matter Scenarios". Physical Review Letters. 112 (15): 151105. arXiv:1404.4099. Bibcode:2014PhRvL.112o1105J. doi:10.1103/PhysRevLett.112.151105. PMID 24785025. S2CID 38389662.
^ V. Anastassopoulos; M. Arik; S. Aune; K. Barth; A. Belov; H. Bräuninger; . . . K. Zioutas (March 16, 2015). "Search for chameleons with CAST". Physics Letters B. 749: 172–180. arXiv:1503.04561. Bibcode:2015PhLB..749..172A. doi:10.1016/j.physletb.2015.07.049. S2CID 59375694.
Khoury, J.; Weltman, A. (2004). "Chameleon fields: awaiting surprises for tests of gravity in space". Physical Review Letters. 93 (17): 171104. arXiv:astro-ph/0309300. Bibcode:2004PhRvL..93q1104K. doi:10.1103/PhysRevLett.93.171104. PMID 15525066. S2CID 17570608.
Khoury, J.; Weltman, A. (2004). "Chameleon cosmology". Physical Review D. 69 (4): 044026. arXiv:astro-ph/0309411. Bibcode:2004PhRvD..69d4026K. doi:10.1103/PhysRevD.69.044026. S2CID 119478819.
Brax, P.; van de Bruck, C.; Davis, A.-C.; Khoury, J.; Weltman, A. (2004). "Detecting dark energy in orbit: The cosmological chameleon". Physical Review D. 70 (12): 123518. arXiv:astro-ph/0408415. Bibcode:2004PhRvD..70l3518B. doi:10.1103/PhysRevD.70.123518. S2CID 119100816.
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Reflections on Math, Tensors and Programming - Nextjournal
Erik Engheim / Apr 25 2020
This is not an explanation of Tensors if you are interested in that but rather a reflection upon the process of learning about mathematical concepts such as Tensors as a programmer with limited mathematical background.
Over the last months I have on and off been learning about Deep Learning, which quickly leads one down in the rabbit hole to the strange world of abstract mathematics.
"Alice in Wonderland," written by a mathematician, Lewis Carroll, gives a hint at some of the problems of grasping abstract math. This is about a song, "Haddocks Eyes" sung by the white knight:
You are sad,’ the Knight said in an anxious tone: ‘let me sing you a song to comfort you.’
‘It’s long,’ said the Knight, ‘but very, VERY beautiful. Everybody that hears me sing it--either it brings the TEARS into their eyes, or else--’
We can summarize the confusion about the song in this way:
This is the kind of problems you stumble on as a poor programmer wanting to familiarize them with machine learning gets into. They discover say TensorFlow and immediately ask or Google "What is a Tensor?"
This question will easily cause you a lot of pain. It is the equivalent to getting into functional programming and asking what a Monad is.
So a Tensor is a Matrix?
The typical early mistake you will make on your quest to understand Tensors is that you ask somebody to show you a Tensor and you see something that to you looks like a Matrix.
And so you go "ah... so a tensor is just a matrix?" Beep!! Wrong!
The answer you get after this point typically just make you more confuse than ever.
a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space.
This is a common human habit to focus on what things are, what they look like, we naturally latch on to how something is represented.
So before we get back to discussing Tensors I want to detour and talk more about how to think about objects both in programming and mathematics.
What Something is and Its Representation are not the Same
III, 3 and three are all different representations of the number 3. We should not mistake either representation for actually "being" that number.
Likewise one and the same representation could represent entirely different things. A could mean the letter 'A' or it could mean the hexadecimal digit 0xA corresponding to 10 in the decimal system. Likewise 65 could represent the number 65 or the letter 'A' in ASCII or UTF-8 encoding.
We could look at more complex representations. A list of two numbers
(2, 3)
could in principle refer to a point, a vector, a tuple or a set.
To not mix the two up we often use slightly different notation. A set of two points would often be written
\{2, 3\}
. We usually write a vector as
\left [\begin{matrix}2\\ 3 \end{matrix}\right]
. However because that is often cumbersome in the middle of a text it is also possible to write it as
(2, 3)
. Both are representation of a column-vector. This should not be confused with
[2 \enskip 3]
which usually denotes a 1
\times
2 row matrix.
The point is that how something is written does not uniquely define it
(2, 3)
could mean a tuple, a point or a column vector. They are all written the same way.
You cannot tell by merely looking at tuple
(2, 3)
(2, 3)
what the difference is.
This can be a source of confusion for a programmer, because as programmers we are not used to this kind of ambiguity. In code the vector may have been written Vector(2, 3) and the tuple Tuple(2, 3). Of course many languages have specific literal syntax for very common objects. E.g. in Julia a column-vector would be written [2, 3] and a tuple would be written (2, 3). The point however is that there is a unique syntax for each type of object.
In mathematics in contrast you get to know the type through context. The author will in the text point out what sort of objects we are working with.
Type Hierarchies and Interfaces
In programming each object is of a particular type. This type may be a subtype of another concrete or abstract super-type. The type may also conform to several different interfaces, protocols or abstract types. The terminology used will vary depending on the language used.
We can consider the Julia programming language. An Array is a concrete subtype of AbstractArray. However AbstractArray types also implement an iterator interface, so that its elements can be iterated over. That something is iteratable is separate from it being an array. Things which are not arrays can also be iterated over.
In Julia every type is a subtype of Any. And types such as Array are also actual objects. The type of the Array type object is DataType. DataType is also a subtype of Any. This kind of arrangement is not unusual. In Java or Smalltalk e.g. the class Object is at the top of the type hierarchy.
We find very similar kinds of arrangement within mathematics. Everything is a mathematical object. Instead of speaking of types we speak of mathematical structures. Mathematical structures exist in a hierarchy similar to a type-hierarchy in programming. E.g. under mathematical structures we have algebraic structures. This is subdivided further into other structures. Further down the hierarchy we have vector spaces.
Relating Mathematical Structures to Programming Types
To make sense of this we can compare to the Vector type in the Julia programming language. Vector is a parameterized type, meaning we must specify the type of its elements (components in math-speak).
v = Vector{Float64}()
push!(v, 3, 4)
In this case I have create the vectors v and u in different but equivalent ways. v as first created empty and then I added two elements 3 and 4 to it.
Formally we can say that v is an object the type Vector with type parameter Float64.
A mathematician could express something similar by saying
v = (3, 4)
is a mathematical object belonging to the vector space of vectors
V
F
F =\mathbb{R}
Let us try to unpack this. In programming-speak the type of
v
is a vector space. However in mathematics you cannot simply say it is a vector space anymore than you can say something is a Vector in Julia. You need to specify the type parameter. In mathematics we do that by saying it is a vector space over the field
\mathbb{R}
e.g. In mathematics
\mathbb{R}
is the set of all real numbers. Float64 is a concrete subset of Julia's abstract Real type, so there is some similarity in our definitions.
In Julia the type parameter of a Vector does not need to be a number. Nor does it have to be for a vector space. That is why we are not saying:
vector space over number
F
That is why we are saying field instead. So what is a field? Math is heavy on abstractions. A field is a type of mathematical object that allows addition, subtraction, multiplication and division according to specific rules.
In Julia notation it would be some type T which has the following operations defined:
+(x::T, y::T) where T <: Field
-(x::T, y::T) where T <: Field
*(x::T, y::T) where T <: Field
/(x::T, y::T) where T <: Field
Of course in mathematics everything is far more pedantic. In programming we make quite a lot of assumptions which must be spelled out in mathematics. Let us pick a couple of examples from wikipedia to show how accurate these descriptions must be:
and
b
are in the field then:
a + (b + c) = (a + b) + c
a \cdot (b \cdot c) = (a \cdot b) \cdot c
Cummulativity:
a + b = b +a
a \cdot b = b \cdot a
Because everything in the field is of the same type, we don't have to specify it, but if we had. The we could write that addition is defined as:
This basically says that addition takes two values
and
b
and produce a new value
c
. It says that
+
takes two arguments which each are in the field
F
and produce a value in the field
F
F \times F
means an operation takes two arguments of the same type
F
F \times F \times F
would mean three arguments and so on. We can shorten that to
F^3
. In cases where we want to be specific we can say that we are taking say 3 real values and producing a real value with
\mathbb{R}^3 \rightarrow \mathbb{R}
Formal Specification of a Vector Space
In programming we may define a Vector interface and list method supported or something similar. We have a similar but more pedantic way of specifying the same in mathematics. A Vector space is defined as the set:
V
is a set of vectors,
F
a set of scalars and
+
\cdot
the operations which must be defined/supported for the elements of the vector space. If we where to attempt to frame this in programming language syntax it may look something like this in Julia pseudo-code syntax:
abstract type VectorSpace{V, F} where V <: Array{F}, F <: Field
+(u::V, v::V)
*(k::F, v::V)
This is not really valid Julia code as it is hard to really express in code what is expressed with mathematical notation. It is just trying to get across that
V
F
are similar to type parameters in programming, with some constraints.
V
has to be some kind of container containing components of type
F
F
has to be some kind of field. A real number is a field while an integer e.g. isn't.
We also try to get across that a vector space is something quite abstract. You don't see a definition of member variables stored. It is all about its behavior. What kind of operations are supported. And it is not like object-oriented programming where these operations are attached to one particular type. It is more like perhaps a module or namespace. It is a collection of functionality and relationships which together forms the vector space.
A vector space must as a minimum it must support addition and scalar multiplication. If addition in a vector space looks like
\mathbf{w} = \mathbf{u} + \mathbf{v}
We need to actually say something about types involved with:
With vector spaces it gets a bit more interesting than with fields, when we get to multiplication. This is defined for a scalar and a vector.
\mathbf{v} = c\mathbf{u}
c
is our scalar. This if formally expressed as:
F \times V
tells us that the multiplication operator
\cdot
, takes two arguments: one which is our field
F
with values such as
c
V
which is a set representing our vectors, such as
\mathbf{u}
\mathbf{v}
\mathbf{w}
You can keep adding functionality to define new subspaces. It must be kept in mind that a vector space is not like a concrete type in programming, but far more like an interface. Lots of things you would not think of as vectors can form vector spaces such as functions, because you can add two functions and multiply a function with a scalar.
Matrix and Tensors
You can form vector spaces out of matrices as well. However not every matrix combination forms a vector space. E.g. You cannot add a
2 \times 3
matrix with a
2 \times 2
matrix. In programming speak you could say these two matrices together don't adhere to the vector space interface, because they don't support addition.
However the set of
m \times n
matrices form a vector space because it supports addition and multiplication with a scalar. Here is an example of proof of this.
In fact tensors are also vector spaces. However pointing that out is not all that interesting.
Nor is it interesting to say both a vectors and matrices can form vector spaces, since so little is required to be a vector space. Matrices support a bunch of other operations which distinguishes them from mere vectors.
E.g. you can multiply matrices, which you cannot do with vectors.
With tensors comes more abilities or axioms if you will that don't exit for regular matrices. And again let me hammer in yet again. A representation of something is not the object.
You can represent a tensor as a matrix but that does not really make it a matrix.
Linear and Multilinear Transformations
You can think of a matrix as something which happens to be useful in defining a linear transformation. A linear transformation is just a fancy way of saying function which takes a vector as input and spits out a vector as output. It is just a word to distinguish it from a regular function which takes a scalar and spits out a scalar.
T
is our transformation it is something like this
It takes a vector
\mathbf{v}
and spits out another vector
\mathbf{u}
. It so happens that we figured out that you can accomplish such linear transformations with matrix multiplication. Thus there exists some matrix
\mathbf{A}
which accomplishes the same as
T
. So we can write:
Tensors is an analogy to this for multilinear transformations. Again this is just a fancy way of saying a function which takes multiple vectors as inputs and spits out a vector as output. So something like this would qualify:
A tensor is just a multidimensional array of numbers which allow us to perform this kind of transformation. Frequently this will look like a matrix to you. For tensors I believe
\mathbf{w}
will usually be a scalar such as a real number. This is possible because reals can also be vector spaces. A real can be a vector space onto itself. The vectors in
V
just all have single components which are fields
F
These where just reflections upon how mathematicians define things in relation to how us software developers think about things. I have not actually shown any practical use of a tensor. My motivation was to write a piece for somebody who wants to watch a video of read about tensors but who find it hard to deal with the math heavy language.
I have tried here to relate that language to concepts from programming.
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networks(deprecated)/petersen - Maple Help
Home : Support : Online Help : networks(deprecated)/petersen
creates the petersen graph
G:=petersen()
is returned as the petersen
Important: The networks package has been deprecated.Use the superseding command GraphTheory[SpecialGraphs][PetersenGraph] instead.
The petersen graph is constructed. This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[petersen]().
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{petersen}\left(\right):
\mathrm{degreeseq}\left(G\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
\mathrm{nops}\left(\mathrm{vertices}\left(G\right)\right)
\textcolor[rgb]{0,0,1}{10}
\mathrm{nops}\left(\mathrm{edges}\left(G\right)\right)
\textcolor[rgb]{0,0,1}{15}
GraphTheory[SpecialGraphs][PetersenGraph]
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IsSplitGraph - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : IsSplitGraph
test if graph is a split graph
IsSplitGraph(G,opts)
(optional) equation of the form decomposition=true or decomposition=false
decomposition : keyword option of the form decomposition=true or decomposition=false. This specifies whether the decomposition into a maximum clique and an independent set should be returned when the graph is a split graph. The default is false.
The IsSplitGraph(G) command returns true if G is a split graph and false otherwise.
An undirected graph G is a split graph if its vertices can be partitioned into a clique and an independent set. The partition is in general not unique. Split graphs are closed under complement.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
K≔\mathrm{Graph}\left(5,{{1,2},{1,3},{2,3},{2,4},{3,4},{4,5}}\right)
\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 5 vertices and 6 edge\left(s\right)}}
\mathrm{IsSplitGraph}\left(K\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsSplitGraph}\left(K,\mathrm{decomposition}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]]
P≔\mathrm{PathGraph}\left(4\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 4 vertices and 3 edge\left(s\right)}}
\mathrm{IsSplitGraph}\left(P\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsSplitGraph}\left(P,\mathrm{decomposition}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]
G≔\mathrm{SpecialGraphs}:-\mathrm{PetersenGraph}\left(\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: an undirected unweighted graph with 10 vertices and 15 edge\left(s\right)}}
\mathrm{IsSplitGraph}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The GraphTheory[IsSplitGraph] command was introduced in Maple 2020.
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Specify normalization type for sensitivity analysis - MATLAB - MathWorks 日本
Specify normalization type for sensitivity analysis
Normalization is a property of the SensitivityAnalysisOptions object. SensitivityAnalysisOptions is a property of the configuration set object. Use Normalization to specify the normalization for the computed sensitivities.
The following values let you specify the type of normalization. The examples show you how sensitivities of a species x with respect to a parameter k are calculated for each normalization type:
'None' specifies no normalization.
\frac{âx\left(t\right)}{âk}
'Half' specifies normalization relative to the numerator (species quantity) only.
\left(\frac{1}{x\left(t\right)}\right)\left(\frac{âx\left(t\right)}{âk}\right)
'Full' specifies that the data should be made dimensionless.
\left(\frac{k}{x\left(t\right)}\right)\left(\frac{âx\left(t\right)}{âk}\right)
Data values 'None', 'Half', 'Full'. Default is 'None'.
Inputs, Outputs, SensitivityAnalysis, SensitivityAnalysisOptions
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A general restricted estimator in binary logistic regression in the presence of multicollinearity
June 2022 A general restricted estimator in binary logistic regression in the presence of multicollinearity
Gargi Tyagi, Shalini Chandra
Gargi Tyagi,1 Shalini Chandra1
1Department of Mathematics & Statistics, Banasthali Vidyapith, Banasthali-304022 Rajasthan, India
The presence of multicollinearity adversely affects the inferential properties of the maximum likelihood (ML) estimator in logistic regression model. It is a well-established fact that the use of restrictions lowers the effect of multicollinearity. In this article, an alternative to the ML estimator has been introduced by combining the exact prior information into the logistic
\mathit{r}-\mathit{k}
class (Lrk) estimator. The estimator is named a logistic restricted
\mathit{r}-\mathit{k}
class estimator. Another estimator, logistic restricted PCR estimator, is also developed as a special case of the LRrk estimator. The asymptotic mean squared error (MSE) matrix properties of the estimators are studied and necessary and sufficient conditions are derived. Further, a Monte Carlo simulation study is performed to compare the performance of the estimators in terms of the scalar MSE and the prediction MSE. It is found that the proposed estimators perform better than the existing estimators in most of the cases considered. Moreover, a numerical example has also been presented for comparing the performance of the estimators.
The authors are grateful to the editor and referees for their valuable comments and suggestions, which certainly improved the quality and presentation of the paper.
Gargi Tyagi. Shalini Chandra. "A general restricted estimator in binary logistic regression in the presence of multicollinearity." Braz. J. Probab. Stat. 36 (2) 287 - 314, June 2022. https://doi.org/10.1214/21-BJPS527
Received: 1 March 2021; Accepted: 1 November 2021; Published: June 2022
Keywords: logistic regression , mean squared error , multicollinearity , predicted mean squared error , restrictions
Gargi Tyagi, Shalini Chandra "A general restricted estimator in binary logistic regression in the presence of multicollinearity," Brazilian Journal of Probability and Statistics, Braz. J. Probab. Stat. 36(2), 287-314, (June 2022)
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Simple Linear Regression - Nextjournal
Intelligence Refinery / Jan 14 2020
Remix of Simple Linear Regression by IRIntelligence Refinery
Simple Linear Regression in R (2018) [STHDA]
Formula and basics
Simple linear regression predicts a quantitative outcome variable y, on the basis of a single predictor variable x.
An important assumption of the linear regression model is that the relationship between the predictor variables and the outcome is linear and additive.
The sum of the squares of the residual errors
b_0
b_1
are determined so that the RSS is minimized
The average variation of points around the fitted regression line
Lower the RSE, the better the fitted regression model
Load & preview the data
library(tidyverse) ## For data manipulation and visualization
library(ggpubr) ## For publication-ready plots
linear_reg_R (R)
The marketing data set describes the impact of Youtube, Facebook and newspaper advertising on sales.
#devtools::install_github('kassambara/datarium')
library(datarium)
head(marketing)
Create a scatterplot displaying the sales in thousands of dollars vs. Youtube advertising budget:
ggplot(marketing, aes(x = youtube, y = sales)) +
Measures the level of association between X and Y
-1: perfect negative correlation
+1: perfect positive correlation
~0: weak relationship between the variables
cor(marketing$sales, marketing$youtube)
The jtools package is very useful for summarizing and visualizing regression results. Read more about it here.
model <- lm(sales ~ youtube, data = marketing)
summ(model)
Intercept=8.43 means that when Youtube advertising budget is $0, we can expect $8,430 in sales
Fit the linear regression line
The confidence band reflect the uncertainty about the line
ggplot(marketing, aes(youtube, sales)) +
Call: the function used to compute the regression model
Residuals: distribution of the residuals (median should be ~0 and min ~ max)
Coefficients: the regression beta coefficients and their statistical significance. Statistical significance indicated by asterisks.
R^2
and F-statistic: metrics of how well the model fits to the data
Measures the variability/accuracy of the beta coefficients. Can be used to compute the confidence intervals of the coefficients.
A t-test is performed to check whether or not these coefficients are significantly different from zero. High t-statistics (which go with low p-values near 0) indicate that a predictor should be retained in a model.
RSE (Closer to zero, the better)
Whether the RSE is acceptable depends on the problem
R^2
(Higher the better)
Represents the proportion of information (i.e. variation) in the data that can be explained by the model, or how well the model fits the data
For a simple linear regression,
R^2
is the square of the Pearson correlation coefficient
R^2
is adjusted for the degrees of freedom. As
R^2
tends to increase when there are more predictors, should consider this metric for multiple linear regression model
F statistic (Higher the better)
Give overall significance of the model, assessing whether at least one predictor variable has a non-zero coefficient
For a simple linear regression, this just duplicates the information of the t-test in the coefficient table
More important in multiple linear regression
A large F statistic corresponds to a statistically significant P value
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ELECTRON SPECTROSCOPY (42113)
SPECTROSCOPY (42112)
PHOTOELECTRON SPECTROSCOPY (33987)
Differential photoelectron holography: A new approach for three-dimensional atomic imaging
Omori, S.; Nihei, Y.; Rotenberg, E.; Denlinger, J.D.; Kevan, S.D.; Tonner, B.P.; Van Hove, M.A.; Fadley, C.S.
Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA (United States). Funding organisation: USDOE Director, Office of Science. Office of Basic Energy Studies. Division of Materials Sciences (United States)
LBNL--47048; AC03-76SF00098; Available from Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA (US); Journal Publication Date: February 4, 2002
Physical Review Letters; ISSN 0031-9007; ; CODEN PRLTAO; v. 88(5); [10 p.]
ATOMS, HOLOGRAPHY, IMAGES, PHOTOELECTRON SPECTROSCOPY
Applications of X-ray photoelectron spectroscopy in fundamental research
Demanet, C.M.
[en] The applications of X-ray photoelectron spectroscopy in biochemistry, organic and inorganic chemistry, elemental and trace analysis are indicated, as well as in polymer chemistry and quantitative analysis, after a short introduction to the basics of the technique has been given
South African Journal of Science; ISSN 0038-2353; ; v. 77(6); p. 249-252
CHEMICAL ANALYSIS, CHEMICAL SHIFT, CHEMISTRY, PHOTOELECTRON SPECTROSCOPY, X-RAY SPECTROSCOPY
Chattarji, D.
[en] Beginning with a chapter on symmetry and invariance, the theory of the Auger process is developed by considering the progressive breaking of symmetry as one goes from the simple hydrogenic model to a more realistic description of the many-electron atom. The calculation of Auger intensities and energies using different coupling schemes is also discussed from this angle, the results of important calculations are compared with experiment, and recent work on Auger satellites and correlation effects is reviewed. With a conscious departure from the emphasis on theory, the last chapter deals with surface Auger phenomena: after a brief survey of analytical methods, it described the features of Auger spectra in solids and gives a critical account of the principal aspects of surface Auger spectroscopy
1976; 274 p; Academic Press Inc; New York; ISBN 0-12-169850-5;
ATOMS, AUGER EFFECT, AUGER ELECTRON SPECTROSCOPY, ELECTRONIC STRUCTURE, SOLIDS, SURFACES, SYMMETRY BREAKING
Calculated variation trends across the actinide dioxide series: electronic structure and charge transfer transitions
[en] Variation trends across the AnO2 series of An5f-electron covalent bonding and An5f→O3s, O2p→An5f charge transfer energies, are studied using DV-Xα relativistic spin-polarised computation of electronic structures applied to 11 'AnO8' clusters, from ThO8 to FmO8. It is found that the binding energies of 5f orbitals, therefore the An5f-O2p hybridisation, are increasing though the 5f orbitals are localising across the An series. In our calculations, the 3s and 3p ionic-like orbitals of O2- ions are included for the first time as LCAO-MO bases. Then, the conduction band is a mixing of O3s and An6d and its lower edge corresponds to an O3s-dominated state. Moreover, the calculated charge transfer (CT) energies of An5f→O3s and O2p→An5f transitions show the so-called tetrad effect when CT energies, respectively increasing and decreasing across the AnO2 series. It is pointed out that the tetrad effect here comes mainly from the special spin-polarised pattern of 5f levels and the increasing general trend of 5f binding energies. (orig.)
Journal of Alloys and Compounds; ISSN 0925-8388; ; CODEN JALCEU; v. 307(1-2); p. 61-69
ACTINIDE COMPOUNDS, CHARGE TRANSPORT, CHEMICAL BONDS, ELECTRONIC STRUCTURE, LATTICE PARAMETERS, PHOTOELECTRON SPECTROSCOPY, VALENCE
Molecular spectroscopy of surfaces - determination of binding states and molecules within monolayers
Holm, R.; Storp, S.
[en] The composition of the surface of a solid body is generally different from that of the bulk. There are many problems that cannot be solved simply by an analysis of the elements, information on the binding states of the detected elements, or rather on the molecules, being required also. The results obtainable depend on the nature of the problem. Where general surface analysis is concerned, e.g. in the areas of corrosion, catalysis, and high polymer technology, ESCA permits substantial characterization of the binding states. SIMS is opening up new opportunities in the use of surfaces and surface analysis methods to identify and investigate the structures of small quantities of substances. In model experiments under idealized conditions (performed by UPS and ELS, for example) it is possible to arrive at fundamental conclusions regarding the adsorbate/substrate interaction. (orig.)
Ein Festkoerper ist an seiner Oberflaeche in der Regel anders zusammengesetzt als im Volumen. Bei zahlreichen Fragestellungen genuegt die Elementanalyse nicht; es werden Angaben zum Bindungszustand der nachgewiesenen Elemente, besser noch Aussagen ueber Molekuele gefordert. Die erzielbaren Ergebnisse haengen von der Problemstellung ab. In der allgemeinen Oberflaechenanalytik, z.B. in den Bereichen Korrosion, Katalyse, Hochpolymertechnologie gestattet ESCA eine weitreichende Charakterisierung der Bindungsverhaeltnisse. Beim Gebrauch der Oberflaeche und der Oberflaechenanalysenmethoden zur Identifizierung und Strukturaufklaerung geringer Substanzmengen weist SIMS neue Wege. In Modellversuchen unter idealisierten Voraussetzungen koennen grundlegende Aussagen zur Wechselwirkung Adsorbat/Substrat z.B. mit UPS und ELS gewonnen werden. (orig.)
Molekuelspektroskopie an Oberflaechen - Nachweis von Bindungsverhaeltnissen und Molekuelen im Monolagenbereich
Work-meeting on applied molecular spectroscopy; Dortmund, Germany, F.R; 24 - 27 Nov 1981
G-I-T (Glas- Instrum.-Tech.) Fachz. Lab; ISSN 0016-3538; ; v. 26(1); p. 13-27
CHEMICAL ANALYSIS, CHEMICAL BONDS, ELECTRON LOSS, LAYERS, MOLECULES, PHOTOELECTRON SPECTROSCOPY, QUANTITATIVE CHEMICAL ANALYSIS, SURFACES, USES
Quantum control of dressed state population for Li2 molecules by intense femtosecond laser pulses
Han, Xiao; Zhan, Wei-Shen; Wang, Shuo; Zai, Jing-Bo; Dang, Hai-Ping, E-mail: zhanwsh@dlut.edu.cn
[en] We investigated the Autler–Townes splitting in photoelectron spectra of
\text{L}{\text{i}}_{2}
molecules steered by ultrashort laser pulses using the time-dependent wave-packet method. Structure of the Autler–Townes splitting was presented to analyze the information of a selective population of the dressed states. It was found that population transfer process, structure of photoelectron spectrum and pattern of Autler–Townes splitting can be controlled by adjusting the intensity, wavelength and delay time of laser pulses. (paper)
Laser Physics (Online); ISSN 1555-6611; ; v. 26(2); [6 p.]
CONTROL, INFORMATION, LASERS, MOLECULES, PHOTOELECTRON SPECTROSCOPY, POPULATIONS, PULSES, TIME DEPENDENCE, WAVE PACKETS, WAVELENGTHS
Valence-Shell-Photoelectron Imaging of Controlled Biomolecules
Wiese, Joss; Trippel, Sebastian; Küpper, Jochen, E-mail: joss.wiese@desy.de
ICPEAC2015: 29. international conference on photonic, electronic, and atomic collisions; Toledo (Spain); 22-28 Jul 2015; Available from http://dx.doi.org/10.1088/1742-6596/635/11/112139; Abstract only; Country of input: International Atomic Energy Agency (IAEA)
Journal of Physics. Conference Series (Online); ISSN 1742-6596; ; v. 635(11); [1 p.]
ELECTRONIC STRUCTURE, IMAGES, MOLECULES, PHOTOELECTRON SPECTROSCOPY, VALENCE
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Tree alignment – Violet-Wall INFO
In computational phylogenetics, tree alignment is a computational problem concerned with producing multiple sequence alignments, or alignments of three or more sequences of DNA, RNA, or protein. Sequences are arranged into a phylogenetic tree, modeling the evolutionary relationships between species or taxa. The edit distances between sequences are calculated for each of the tree’s internal vertices, such that the sum of all edit distances within the tree is minimized. Tree alignment can be accomplished using one of several algorithms with various trade-offs between manageable tree size and computational effort.
This article needs attention from an expert in Computational Biology. (November 2018)
. . . Tree alignment . . .
Input: A set
{displaystyle S}
of sequences, a phylogenetic tree
{displaystyle T}
leaf-labeled by
{displaystyle S}
and an edit distance function
{displaystyle d}
between sequences.
Output: A labeling of the internal vertices of
{displaystyle T}
{displaystyle Sigma _{ein T}d(e)}
{displaystyle d(e)}
is the edit distance between the endpoints of
{displaystyle e}
The task is NP-hard.[1]
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Arithmetic Mean vs. Geometric Mean With Formula
What Is the Difference Between the Arithmetic Mean and the Geometric Mean?
There are many ways to measure financial portfolio performance and determine if an investment strategy is successful. Investment professionals often use the geometric average, more commonly called the geometric mean.
The geometric mean is most appropriate for series that exhibit serial correlation. This is especially true for investment portfolios.
Most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums. The longer the time horizon, the more critical compounding becomes, and the more appropriate the use of the geometric mean.
For volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding.
\begin{aligned} &A = \frac{1}{n} \sum_{i =1}^n a_i = \frac{a_1 + a_2 + \dotso + a_n}{n} \\ &\textbf{where:} \\ &a_1, a_2, \dotso, a_n=\text{Portfolio returns for period } n \\ &n=\text{Number of periods} \\ \end{aligned}
A=n1i=1∑nai=na1+a2+…+anwhere:a1,a2,…,an=Portfolio returns for period nn=Number of periods
An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers.
If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students and then divide that sum by the number of students. For example, if five students took an exam and their scores were 60%, 70%, 80%, 90%, and 100%, the arithmetic class average would be 80%.
This would be calculated as:
\begin{aligned} &\frac {60\% + 70\% + 80\% + 90\% + 100\% }{ 5 } = 80\% \\ \end{aligned}
560%+70%+80%+90%+100%=80%
The reason we use an arithmetic average for test scores is that each score is an independent event. If one student happens to perform poorly on the exam, the next student's chances of performing poorly (or well) on the exam is not affected.
In the world of finance, the arithmetic mean is not usually an appropriate method for calculating an average. Consider investment returns, for example. Suppose you have invested your savings in the financial markets for five years. If your portfolio returns each year were 90%, 10%, 20%, 30%, and -90%, what would your average return be during this period?
With the arithmetic average, the average return would be 12%, which appears at first glance to be impressive—but it's not entirely accurate. That's because when it comes to annual investment returns, the numbers are not independent of each other. If you lose a substantial amount of money in a particular year, you have that much less capital to invest and generate returns in the following years.
We need to calculate the geometric average of your investment returns to arrive at an accurate measurement of what your actual average annual return over the five-year period would be.
\begin{aligned} &\left( \prod_{i = 1}^n x_i \right)^{\frac{1}{n}} = \sqrt[n]{x_1 x_2 \dots x_n} \\ &\textbf{where:} \\ &x_1, x_2, \dots = \text{Portfolio returns for each period} \\ &n = \text{Number of periods} \\ \end{aligned}
(i=1∏nxi)n1=nx1x2…xnwhere:x1,x2,⋯=Portfolio returns for each periodn=Number of periods
The geometric mean for a series of numbers is calculated by taking the product of these numbers and raising it to the inverse of the length of the series.
To do this, we add one to each number (to avoid any problems with negative percentages). Then, multiply all the numbers together and raise their product to the power of one divided by the count of the numbers in the series. Then, we subtract one from the result.
The formula, written in decimals, looks like this:
\begin{aligned} &[ ( 1 + \text{R}_1) \times (1 + \text{R}_2) \times (1 + \text{R}_3) \dotso \times (1 + \text{R}_n) ]^{\frac {1}{n} } - 1 \\ &\textbf{where:} \\ &\text{R} = \text{Return} \\ &n = \text{Count of the numbers in the series} \\ \end{aligned}
[(1+R1)×(1+R2)×(1+R3)…×(1+Rn)]n1−1where:R=Returnn=Count of the numbers in the series
The formula appears complex, but on paper, it's not so difficult. Returning to our example, we calculate the geometric average: Our returns were 90%, 10%, 20%, 30%, and -90%, so we plug them into the formula as:
\begin{aligned} &(1.9 \times 1.1 \times 1.2 \times 1.3 \times 0.1)^{\frac{1}{5}} -1 \\ \end{aligned}
(1.9×1.1×1.2×1.3×0.1)51−1
The result gives a geometric average annual return of -20.08%. The result using the geometric average is a lot worse than the 12% arithmetic average we calculated earlier, and unfortunately, it is also the number that represents reality in this case.
Formula for Calculating Compound Annual Growth Rate (CAGR) in Excel?
Understanding the Harmonic Mean
The harmonic mean is an average which is used in finance to average multiples like the price-earnings ratio.
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Transform IIR lowpass filter to IIR complex N-point filter - MATLAB iirlp2xc - MathWorks í•œêµ
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
b=\left[\begin{array}{ccccc}{b}_{01}& {b}_{11}& {b}_{21}& ...& {b}_{Q1}\\ {b}_{02}& {b}_{12}& {b}_{22}& ...& {b}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {b}_{0P}& {b}_{1P}& {b}_{2P}& \cdots & {b}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
a=\left[\begin{array}{ccccc}{a}_{01}& {a}_{11}& {a}_{21}& \cdots & {a}_{Q1}\\ {a}_{02}& {a}_{12}& {a}_{22}& \cdots & {a}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {a}_{0P}& {a}_{1P}& {a}_{2P}& \cdots & {a}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
IIR lowpass filter to IIR complex N-point filter transformation effectively places N features of the original filter, located at frequencies wo1, … ,woN, at the required target frequency locations, wt1, … ,wtM. The function iirlp2xc requires that N and M are equal.
[1] Krukowski, A., and I. Kale, “High-order complex frequency transformations,†Internal report No. 27/2001, Applied DSP and VLSI Research Group, University of Westminster.
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Directivity of uniform linear array - MATLAB - MathWorks Australia
D = directivity(H,FREQ,ANGLE) computes the Directivity (dBi) of a uniform linear array (ULA) of antenna or microphone elements, H, at frequencies specified by FREQ and in angles of direction specified by ANGLE.
D = directivity(H,FREQ,ANGLE,Name,Value) returns the directivity with additional options specified by one or more Name,Value pair arguments.
H — Uniform linear array
Uniform linear array specified as a phased.ULA System object.
Example: H = phased.ULA;
Compute the directivities of two different uniform linear arrays (ULA). One array consists of isotropic antenna elements and the second array consists of cosine antenna elements. In addition, compute the directivity when the first array is steered in a specified direction. For each case, calculate the directivities for a set of seven different azimuth directions all at zero degrees elevation. Set the frequency to 800 MHz.
Array of isotropic antenna elements
First, create a 10-element ULA of isotropic antenna elements spaced 1/2-wavelength apart.
myAnt1 = phased.IsotropicAntennaElement;
myArray1 = phased.ULA(10,lambda/2,'Element',myAnt1);
d = directivity(myArray1,fc,ang,'PropagationSpeed',c)
Array of cosine antenna elements
Next, create a 10-element ULA of cosine antenna elements spaced 1/2-wavelength apart.
myAnt2 = phased.CosineAntennaElement('CosinePower',[1.8,1.8]);
The directivity of the cosine ULA is greater than the directivity of the isotropic ULA because of the larger directivity of the cosine antenna element.
Steered array of isotropic antenna elements
Finally, steer the isotropic antenna array to 30 degrees in azimuth and compute the directivity.
w = steervec(getElementPosition(myArray1)/lambda,[30;0]);
d = directivity(myArray1,fc,ang,'PropagationSpeed',c,...
The directivity is greatest in the steered direction.
D=4\pi \frac{{U}_{\text{rad}}\left(\theta ,\phi \right)}{{P}_{\text{total}}}
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ParzenWindow - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Windowing Functions : ParzenWindow
multiply an array of samples by a Parzen windowing function
ParzenWindow(A)
The ParzenWindow(A) command multiplies the Array A by the Parzen windowing function and returns the result in an Array having the same length.
The Parzen windowing function
w\left(k\right)
N
w\left(k\right)={\begin{array}{cc}1-6{\left(\frac{2k}{N}-1\right)}^{2}\left(1-|-\frac{2k}{N}+1|\right)& |\frac{N}{2}-k|\le \frac{N}{4}\\ 2{\left(1-|-\frac{2k}{N}+1|\right)}^{3}& \mathrm{otherwise}\end{array}
The SignalProcessing[ParzenWindow] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
N≔1024:
a≔\mathrm{GenerateUniform}\left(N,-1,1\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627829133664612}}
\mathrm{ParzenWindow}\left(a\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627828985630716}}
c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right):
\mathrm{ParzenWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right),'\mathrm{container}'=c\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627828985606620}}
u≔\mathrm{`~`}[\mathrm{log}]\left(\mathrm{FFT}\left(c\right)\right):
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
The SignalProcessing[ParzenWindow] command was introduced in Maple 18.
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ScrewDisplacement - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : Transformations : ScrewDisplacement
find the screw-displacement of a geometric object.
ScrewDisplacement(Q, P, theta, l, AB)
A screw-displacement is the product of a rotation and a translation along the axis of rotation.
For a detailed description of Q (the object created), use the routine detail (i.e., detail(Q);)
The command with(geom3d,ScrewDisplacement) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
\mathrm{point}\left(o,0,0,0\right),\mathrm{point}\left(z,0,0,2\right),\mathrm{dsegment}\left(\mathrm{seg},[o,z]\right),\mathrm{line}\left(l,[o,z]\right):
\mathrm{triangle}\left(T,[\mathrm{point}\left(A,0,0,1\right),\mathrm{point}\left(B,2,1,0\right),\mathrm{point}\left(C,4,3,0\right)]\right):
Define the screw-displacement of triangle T
\mathrm{ScrewDisplacement}\left(\mathrm{T1},T,\mathrm{\pi },l,\mathrm{seg}\right)
\textcolor[rgb]{0,0,1}{\mathrm{T1}}
\mathrm{map}\left(\mathrm{coordinates},\mathrm{DefinedAs}\left(\mathrm{T1}\right)\right)
[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{-2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\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}{-3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
\mathrm{draw}\left([T\left(\mathrm{color}=\mathrm{blue}\right),\mathrm{T1}\left(\mathrm{color}=\mathrm{red}\right),l\left(\mathrm{thickness}=3\right)],\mathrm{orientation}=[122,73],\mathrm{title}=\mathrm{`Screw-displacement of a triangle`},\mathrm{style}=\mathrm{patch}\right)
geom3d[objects]
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§ Fuzzing book
Statement coverage is different from branch coverage, since an if (cond) { s1; } s2 will say that s1 and s2 were executed when cond=True, so we have full statement coverage. On the other hand, this does not guarantee full branch coverage, since we have not exectuted the branch where cond=False. We can't tell that we haven't covered this branch since there is no statement to record that we have taken the else branch!
Branch distances: for conditions a == b, a != b, a < b, a <= b, define the "distance true/distance false" to be the number that is to be added/subtracted to a to make the condition true/false (for a fixed b). So, for example, the "distance true" for a == b is abs(b - a), while "distance false" is 1 - int(a == b).
What are we missing in coverage? The problem here is that coverage is unable to evaluate the quality of our assertions. Indeed, coverage does not care about assertions at all. However, as we saw above, assertions are an extremely important part of test suite effectiveness. Hence, what we need is a way to evaluate the quality of assertions.
Competent Programmer Hypothesis / Finite Nbhd Hypothesis : Mutation Analysis provides an alternative to a curated set of faults. The key insight is that, if one assumes that the programmer understands the program in question, the majority of errors made are very likely small transcription errors (a small number of tokens). A compiler will likely catch most of these errors. Hence, the majority of residual faults in a program is likely to be due to small (single token) variations at certain points in the structure of the program from the correct program (This particular assumption is called the Competent Programmer Hypothesis or the Finite Neighborhood Hypothesis).
Equivalent mutants : However, if the number of mutants are sufficiently large (say > 1000), one may choose a smaller number of mutants from the alive mutants randomly and manually evaluate them to see whether they represent faults. We choose the sample size by sampling theory of binomial distributions .
Chao's estimator : way to estimate the number of true mutants (and hence the number of equivalent mutants) is by means of Chao's estimator:
hat M \equiv \begin{cases} M(n) + k_1^2 / (2 k_2) & \text{if } k_2 > 0 \\ M(n) + k_1(k_1 - 1)/2 & \text{otherwise} \\ \end{cases}
k_1
is the number of mutants that were killed exactly once,
k_2
is the number of mutants that were killed exactly twice.
\hat M
estimates the the true numbe of mutants.
T
is the total mutants generated, then
T - M(n)
represents immortal mutants.
\hat M
is the is the mutants that the testset can detect given an infinite amount of time.
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Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors | EMS Press
Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors
A perfect polynomial
A\in\mathbb{F}_4[x]
is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials
A\in\mathbb{F}_4[x]
3
prime divisors, i.e., of the form
A=P^aQ^bR^c
P,Q,R\in\mathbb{F}_4[x]
are irreducible and
a,b,c
are positive integers. We characterize the perfect polynomials
A
4
prime divisors such that one of them has degree
1
A
has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.
Luis H. Gallardo, Olivier Rahavandrainy, Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors. Port. Math. 64 (2007), no. 1, pp. 21–38
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This workshop, organised by Hakan Eliasson (Paris), Helmut Hofer (New York), and Jean-Christophe Yoccoz (Paris) continued the biannual series at Oberwolfach on Dynamical Systems that started as the ``Moser \& Zehnder meeting'' in 1981. The workshop was attended by more than 50 participants from 12 countries. The main themes of the workshop were the new results and developments in the area of classical dynamical systems, in particular in celestial mechanics and Hamiltonian systems. The workshop covers a large area of dynamical systems and the following samples give an idea about the scope. The topic of Arnold Diffusion was treated in great detail by talks of M. Levi and J. Mather. In the classical field of celestial mechanics new insight has been gained about two interesting families of relative periodic solutions of the spatial n-body problem (the
P_{12}
-family and the hip-hop-family) as they share the property of being global continuations of Lyapunov families which bifurcate from a relative equilibrium solution in the direction orthogonal to the plane of motion (A. Chenciner). K. Kuperberg reported on the construction of flows on three-manifolds where every nonconstant trajectory is wild in a sense related to the Artin-Fox example of an exotic arc in Euclidean three-space. Other results were concerned with one of the main problems in Hamiltonian dynamic, namely the stability of motions in nearly-integrable systems (L. Niedermann). Y. Pesin outlined the construction of hyperbolic volume-preserving flows on manifolds of dimension at least three and V. Ginzburg described the recent developments concerning the Conley Conjecture for periodic points of Hamiltonian symplectic maps. John Franks described his recent results about group actions on surfaces. In addition several talks covered new Floer-theoretic methods in the study of Hamiltonian systems and it will be interesting to see in the future how these symplectic methods can be merged with the more classical dynamical systems methods.
, Håkan Eliasson, Helmut W. Hofer, Jean-Christophe Yoccoz, Dynamische Systeme. Oberwolfach Rep. 4 (2007), no. 3, pp. 1913–1956
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Rotation matrix for rotations around z-axis - MATLAB rotz - MathWorks Nordic
{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]
{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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10^9 electrons pass from A to B in 10^-3 second Calculate the strength of current and give its direction - Physics - Current Electricity - 9100371 | Meritnation.com
10^9 electrons pass from A to B in 10^-3 second.Calculate the strength of current and give its direction.
ans is 1.6*10^-7A.
Current,i=\frac{ne}{t}\phantom{\rule{0ex}{0ex}} i=\frac{1.6×{10}^{-19}×{10}^{9}}{{10}^{-3}}=1.6×{10}^{-7}A\phantom{\rule{0ex}{0ex}}Current will be in opposite direction of flow of electron i.e. from B to A.
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Fabrication and Characterization of Anode-Supported Planar Solid Oxide Fuel Cell Manufactured by a Tape Casting Process | J. Electrochem. En. Conv. Stor | ASME Digital Collection
Connecticut Global Fuel Cell Center (CGFCC),
Sun-Il Park,
Sun-Il Park
Gwangju Research Center,
, Gwangju, 506-824, Republic of Korea
Seongjae Boo,
Seongjae Boo
Hwan Moon,
School of Advanced Materials Science and Engineering,
, Seoul 120-749, Republic of Korea
J. Fuel Cell Sci. Technol. May 2008, 5(2): 021003 (5 pages)
Song, J., Sammes, N. M., Park, S., Boo, S., Kim, H., Moon, H., and Hyun, S. (April 10, 2008). "Fabrication and Characterization of Anode-Supported Planar Solid Oxide Fuel Cell Manufactured by a Tape Casting Process." ASME. J. Fuel Cell Sci. Technol. May 2008; 5(2): 021003. https://doi.org/10.1115/1.2885401
A planar anode-supported electrolyte was fabricated using a tape casting method that involved a single step cofiring process. A standard NiO∕8YSZ cermet anode,
8mol%
YSZ electrolyte, and a lanthanum strontium manganite cathode were used for the solid oxide fuel cell unit cell. A pressurized cofiring technique allows the creation of a thin layer of dense electrolyte about
10μm
without warpage. The open circuit voltage of the unit cell indicated negligible fuel leakage through the electrolyte film due to the dense and crack-free electrolyte layer. An electrochemical test of the unit cell showed a maximum power density up to
0.173W∕cm2
900°C
. Approximated electrochemical properties, e.g., activation energy, Ohmic resistance, and exchange current density, indicated that the cell performance was significantly influenced by the electrode properties of the unit cell.
cermets, electrolytes, solid oxide fuel cells, tape casting, solid oxide fuel cell (SOFC), tape casting, cofiring, thin film electrolyte, anode-supported electrolyte
Anodes, Co-firing, Electrolytes, Manufacturing, Solid oxide fuel cells, Tape casting, Warping, Temperature
SOFC System and Technology
Lashtabeg
Solid Oxide Fuel Cells—A Challenge for Materials Chemists
U. V. T.
Ceramic Materials for Advanced Solid Oxide Fuel Cells
Solid Oxide Fuel Cells for Stationary, Mobile, and Military Applications
Electrochemical Study of IT-SOFC Unit Cell Prepared by Tape Casting Process and Co-Sintering
Electrolyte in High Temperature Solid Oxide Fuel Cells
Fabrication and Characterization of Anode Supported Electrolyte Thin Films for Intermediate Temperature Solid Oxide Fuel Cells
Preparation of Dense Electrolyte Layer Using Dissociated Oxygen Electrochemical Vapor Deposition Technique
Metal-Organic Vapor Deposition of YSZ Electrolyte Layers for Solid Oxide Fuel Cell Applications, Thin Solid Films
Preparation of LaCoO3 Catalytic Thin Film by the Sol-gel Process and its NO Decomposition Characteristics
Morphology and Sintering Behaviour of Yttria Stabilised Zirconia (8-YSZ) Powders Synthesised by Spray Pyrolysis
Development of a Planar SOFC Device Using Screen-Printing Technology
Screen-Printed Thin YSZ Films Used as Electrolytes for Solid Oxide Fuel Cells
Snijker
de Mullens
Aqueous Tape Casting of Yttria Stabilized Zirconia Using Natural Product Binder
Characterization of YSZ-YST Composites for SOFC Anodes
Single Step Co-Firing Technique for SOFC Fabrication
Effect of YSZ Particle Size and Sintering Temperature on the Microstructure and Impedance Property of Ni-YSZ Anode for Solid Oxide Fuel Cell
Characterization of Electrical Properties of GDC Doped A-Site Deficient LSCF Based Composite Cathode Using Impedance Spectroscopy
The Electrical Properties of Solid Oxide Electrolytes
Electrochemical Performance of Solid Oxide Fuel Cells Manufactured by Single Step Co-Firing Process
Complete Polarization Model of a Solid Oxide Fuel Cell and Its Sensitivity to the Change of Cell Component Thickness
SOFCRoll Development at St. Andrews Fuel Cells Ltd.
A Novel Direct Carbon Fuel Cell Concept
Thermal Cycling Evaluation of Rolled Tubular Solid Oxide Fuel Cells
Prepare Electrolyte and Electrolyte/Anode Bi-Layers for Solid Oxide Fuel Cells With Novel Gel-Casting Method
Modeling the Manufacturing Cost of Solid Oxide Fuel Cells
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Enthalpy of fusion - Wikipedia
(Redirected from Latent heat of fusion)
Enthalpy change when a substance melts
Enthalpies of melting and boiling for pure elements versus temperatures of transition, demonstrating Trouton's rule
The enthalpy of fusion of a substance, also known as (latent) heat of fusion, is the change in its enthalpy resulting from providing energy, typically heat, to a specific quantity of the substance to change its state from a solid to a liquid, at constant pressure. For example, when melting 1 kg of ice (at 0 °C under a wide range of pressures), 333.55 kJ of energy is absorbed with no temperature change. The heat of solidification (when a substance changes from liquid to solid) is equal and opposite.
This energy includes the contribution required to make room for any associated change in volume by displacing its environment against ambient pressure. The temperature at which the phase transition occurs is the melting point or the freezing point, according to context. By convention, the pressure is assumed to be 1 atm (101.325 kPa) unless otherwise specified.
3 Solubility prediction
The 'enthalpy' of fusion is a latent heat, because, while melting, the heat energy needed to change the substance from solid to liquid at atmospheric pressure is latent heat of fusion, as the temperature remains constant during the process. The latent heat of fusion is the enthalpy change of any amount of substance when it melts. When the heat of fusion is referenced to a unit of mass, it is usually called the specific heat of fusion, while the molar heat of fusion refers to the enthalpy change per amount of substance in moles.
When liquid water is cooled, its temperature falls steadily until it drops just below the line of freezing point at 0 °C. The temperature then remains constant at the freezing point while the water crystallizes. Once the water is completely frozen, its temperature continues to fall.
The enthalpy of fusion is almost always a positive quantity; helium is the only known exception.[1] Helium-3 has a negative enthalpy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative enthalpy of fusion below 0.77 K (−272.380 °C). This means that, at appropriate constant pressures, these substances freeze with the addition of heat.[2] In the case of 4He, this pressure range is between 24.992 and 25.00 atm (2,533 kPa).[3]
Standard enthalpy change of fusion of period three
methane 13.96 58.99
propane 19.11 79.96
glycerol 47.95 200.62
formic acid 66.05 276.35
acetic acid 45.90 192.09
benzene 30.45 127.40
myristic acid 47.49 198.70
palmitic acid 39.18 163.93
sodium acetate 63–69 264–289[4]
stearic acid 47.54 198.91
gallium 19.2 80.4
paraffin wax (C25H52) 47.8–52.6 200–220
These values are mostly from the CRC Handbook of Chemistry and Physics, 62nd edition. The conversion between cal/g and J/g in the above table uses the thermochemical calorie (calth) = 4.184 joules rather than the International Steam Table calorie (calINT) = 4.1868 joules.
To heat 1 kg of liquid water from 0 °C to 20 °C requires 83.6 kJ (see below). However, heating 0 °C ice to 20 °C requires additional energy to melt the ice. We can treat these two processes independently; thus, to heat 1 kg of ice from 273.15 K to water at 293.15 K (0 °C to 20 °C) requires:
(1) 333.55 J/g (heat of fusion of ice) = 333.55 kJ/kg = 333.55 kJ for 1 kg of ice to melt, plus
(2) 4.18 J/(g⋅K) × 20 K = 4.18 kJ/(kg⋅K) × 20 K = 83.6 kJ for 1 kg of water to increase in temperature by 20 K
(1 + 2) 333.55 kJ + 83.6 kJ = 417.15 kJ for 1 kg of ice to increase in temperature by 20 K
From these figures it can be seen that one part ice at 0 °C will cool almost exactly 4 parts water from 20 °C to 0 °C.
Silicon has a heat of fusion of 50.21 kJ/mol. 50 kW of power can supply the energy required to melt about 100 kg of silicon in one hour:
50 kW = 50kJ/s = 180000kJ/h
180000kJ/h × (1 mol Si)/50.21kJ × 28gSi/(mol Si) × 1kgSi/1000gSi = 100.4kg/h
Solubility prediction[edit]
The heat of fusion can also be used to predict solubility for solids in liquids. Provided an ideal solution is obtained the mole fraction
{\displaystyle (x_{2})}
of solute at saturation is a function of the heat of fusion, the melting point of the solid
{\displaystyle (T_{\text{fus}})}
{\displaystyle (T)}
of the solution:
{\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}
{\displaystyle R}
is the gas constant. For example, the solubility of paracetamol in water at 298 K is predicted to be:
{\displaystyle x_{2}=\exp {\left[-{\frac {28100~{\text{J mol}}^{-1}}{8.314~{\text{J K}}^{-1}~{\text{mol}}^{-1}}}\left({\frac {1}{298~{\text{K}}}}-{\frac {1}{442~{\text{K}}}}\right)\right]}=0.0248}
Since the molar mass of water and paracetamol are 18.0153gmol−1 and 151.17gmol−1 and the density of the solution is 1000gL−1, an estimate of the solubility in grams per liter is:
{\displaystyle {\frac {0.0248\times {\frac {1000~{\text{g L}}^{-1}}{18.0153~{\text{g mol}}^{-1}}}}{1-0.0248}}\times 151.17~{\text{g mol}}^{-1}=213.4~{\text{g L}}^{-1}}
which is a deviation from the real solubility (240 g/L) of 11%. This error can be reduced when an additional heat capacity parameter is taken into account.[5]
At equilibrium the chemical potentials for the pure solvent and pure solid are identical:
{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{solution}}^{\circ }\,}
{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{liquid}}^{\circ }+RT\ln X_{2}\,}
{\displaystyle R\,}
the gas constant and
{\displaystyle T\,}
{\displaystyle RT\ln X_{2}=-\left(\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\right)\,}
{\displaystyle \Delta G_{\text{fus}}^{\circ }=\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\,}
the heat of fusion being the difference in chemical potential between the pure liquid and the pure solid, it follows that
{\displaystyle RT\ln X_{2}=-\left(\Delta G_{\text{fus}}^{\circ }\right)\,}
Application of the Gibbs–Helmholtz equation:
{\displaystyle \left({\frac {\partial \left({\frac {\Delta G_{\text{fus}}^{\circ }}{T}}\right)}{\partial T}}\right)_{p\,}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{T^{2}}}}
ultimately gives:
{\displaystyle \left({\frac {\partial \left(\ln X_{2}\right)}{\partial T}}\right)={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}}
{\displaystyle \partial \ln X_{2}={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \delta T}
and with integration:
{\displaystyle \int _{X_{2}=1}^{X_{2}=x_{2}}\delta \ln X_{2}=\ln x_{2}=\int _{T_{\text{fus}}}^{T}{\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \Delta T}
the end result is obtained:
{\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}
Joback method (Estimation of the heat of fusion from molecular structure)
^ Atkins & Jones 2008, p. 236.
^ Ott & Boerio-Goates 2000, pp. 92–93.
^ Ibrahim Dincer and Marc A. Rosen. Thermal Energy Storage: Systems and Applications, page 155
Atkins, Peter; Jones, Loretta (2008), Chemical Principles: The Quest for Insight (4th ed.), W. H. Freeman and Company, p. 236, ISBN 978-0-7167-7355-9
Ott, BJ. Bevan; Boerio-Goates, Juliana (2000), Chemical Thermodynamics: Advanced Applications, Academic Press, ISBN 0-12-530985-6
Retrieved from "https://en.wikipedia.org/w/index.php?title=Enthalpy_of_fusion&oldid=1083054638"
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Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps
1 October 2014 Attracting cycles in
p
-adic dynamics and height bounds for postcritically finite maps
Robert Benedetto, Patrick Ingram, Rafe Jones, Alon Levy
A rational function of degree at least
2
with coefficients in an algebraically closed field is postcritically finite (PCF) if and only if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattès maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattès PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a nonarchimedean version of Fatou’s classical result that every attracting cycle of a rational function over
\mathbb{C}
attracts a critical point.
Robert Benedetto. Patrick Ingram. Rafe Jones. Alon Levy. "Attracting cycles in
p
-adic dynamics and height bounds for postcritically finite maps." Duke Math. J. 163 (13) 2325 - 2356, 1 October 2014. https://doi.org/10.1215/00127094-2804674
Keywords: $p$-adic dynamics , arithmetic dynamics , heights , postcritically finite
Robert Benedetto, Patrick Ingram, Rafe Jones, Alon Levy "Attracting cycles in
p
-adic dynamics and height bounds for postcritically finite maps," Duke Mathematical Journal, Duke Math. J. 163(13), 2325-2356, (1 October 2014)
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Sensitivities of Medial Meniscal Motion and Deformation to Material Properties of Articular Cartilage, Meniscus and Meniscal Attachments Using Design of Experiments Methods | J. Biomech Eng. | ASME Digital Collection
Paul D. Funkenbusch,
, Rochester, NY 14627-0168
e-mail: amy.lerner@rochester.edu
Yao, J., Funkenbusch, P. D., Snibbe, J., Maloney, M., and Lerner, A. L. (December 27, 2005). "Sensitivities of Medial Meniscal Motion and Deformation to Material Properties of Articular Cartilage, Meniscus and Meniscal Attachments Using Design of Experiments Methods." ASME. J Biomech Eng. June 2006; 128(3): 399–408. https://doi.org/10.1115/1.2191077
This study investigated the role of the material properties assumed for articular cartilage, meniscus and meniscal attachments on the fit of a finite element model (FEM) to experimental data for meniscal motion and deformation due to an anterior tibial loading of
45N
in the anterior cruciate ligament-deficient knee. Taguchi style L18 orthogonal arrays were used to identify the most significant factors for further examination. A central composite design was then employed to develop a mathematical model for predicting the fit of the FEM to the experimental data as a function of the material properties and to identify the material property selections that optimize the fit. The cartilage was modeled as isotropic elastic material, the meniscus was modeled as transversely isotropic elastic material, and meniscal horn and the peripheral attachments were modeled as noncompressive and nonlinear in tension spring elements. The ability of the FEM to reproduce the experimentally measured meniscal motion and deformation was most strongly dependent on the initial strain of the meniscal horn attachments
(ε1H)
, the linear modulus of the meniscal peripheral attachments
(EP)
and the ratio of meniscal moduli in the circumferential and transverse directions
(Eθ∕ER)
. Our study also successfully identified values for these critical material properties (
ε1H=−5%
EP=5.6MPa
Eθ∕ER=20
) to minimize the error in the FEM analysis of experimental results. This study illustrates the most important material properties for future experimental studies, and suggests that modeling work of meniscus, while retaining transverse isotropy, should also focus on the potential influence of nonlinear properties and inhomogeneity.
biomechanics, biological tissues, deformation, materials properties, finite element analysis, physiological models, design of experiments, Taguchi methods, meniscus, finite element analysis, design of experiments, orthogonal array design, central composite design, meniscal attachments
Cartilage, Deformation, Design, Errors, Experimental design, Finite element model, Materials properties, Composite materials, Finite element analysis, Knee
Structure and Function Relationships of the Menisci of the Knee
Is the Circumferential Tensile Modulus within a Human Medial Meniscus Affected by the Test Sample Location and Cross-Sectional Area?
Motion of the Meniscus During Knee Flexion
Sanchez-Ballester
R. de W. M.
The Posteromedial Corner Revisted: An Anatomical Description of the Passive Restraining Structures of the Medial Aspect of the Human Knee
G. J. M. A.
A Numerical Model of the Load Transmission in the Tibio-Femoral Contact Area
A Biphasic Finite Element Model of the Meniscus for Stress-Strain Analysis
How the Stiffness of Meniscal Attachments and Meniscal Material Properties Affects Tibio-Femoral Contact Pressure Computed Using a Validated Finite Element Model of the Human Knee Joint
Statistical Methods in Finite Element Analysis
Practical Guide to Designed Experiments: A Unified Modular Approach
Response Surfaces: Design and Analyses
Optimizing the Configuration of Cement Keyholes for Acetabular Fixation in Total Hip Replacement Using Taguchi Experimental Design
Design of Monolimb using Finite Element Modeling and Statistics-Based Taguchi Method
MR Assessment of Meniscal Movement During Knee Flexion: Correlation With the Severity of Cartilage Abnormality in the Femorotibial Joint
Meniscal Subluxation: Association with Osteoarthritis and Joint Space Narrowing
Radial Displacement of the Medial Meniscus and Fairbank’s Signs
Mechanical Response of Bovine Articular Cartilage under Dynamic Unconfined Compression Loading at Physiological Stress Levels
ABAQUS Version 6.3, Standard User Manual
Hibbitt, Karlsson & Sorensen
Viscoelastic and Mechanical Properties of Bovine Meniscal Horn Attachments
In 50th Annual Meeting of the Orthopaedic Research Society
The Incidence of Meniscal Tears Associated with Acute Anterior Cruciate Ligament Disruption Secondary to Snow Skiing Accidents
The Posteromedial Corner of the Knee: Medial-Sided Injury Patterns Revisited
Healing Potential of Experimental Meniscal Tears in the Rabbit: Preliminary Results
Repairable Posterior Menisco-Capsular Disruption in Anterior Cruciate Ligament Injuries
Transverse Ligament and its Effect on Meniscal Motion: Correlation of Kinematic MR imaging and Anatomic Section
Tensile Strength of the Tibial Meniscal Attachments in the Rabbit
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A Transmission Quality Index for a Class of Four-Limb Parallel Schönflies Motion Generators | J. Mechanisms Robotics | ASME Digital Collection
Guanglei Wu,
High-tech Park District, Dalian 116024, China
e-mail: gwu@dlut.edu.cn
Shaoping Bai,
Department of Materials and Production,
Aalborg DK-9220, Denmark
e-mail: shb@mp.aau.dk
CNRS, Laboratoire des Sciences du Numérique
Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 17, 2017; final manuscript received May 9, 2018; published online July 30, 2018. Assoc. Editor: Marc Gouttefarde.
Wu, G., Bai, S., and Caro, S. (July 30, 2018). "A Transmission Quality Index for a Class of Four-Limb Parallel Schönflies Motion Generators." ASME. J. Mechanisms Robotics. October 2018; 10(5): 051014. https://doi.org/10.1115/1.4040353
This paper presents a uniform method of evaluating both transmission quality and singularity applicable for a class of parallel Schönflies-motion generators (SMGs) with four
RRΠRR
limbs. It turns out that the determinant of the forward Jacobian matrices for this class of parallel robots can be expressed as the scalar product of two vectors, the first vector being the cross product of the four unit vectors along the parallelograms, and the second one being related to the rotation of the mobile platform (MP). The pressure angles, derived from the determinants of forward and inverse Jacobians, respectively, are used for the evaluation of the transmission quality and the detection of robot singularities. Four robots are compared based on the proposed indices as illustrative examples.
Mechanism design, Parallel platforms, Robot design, Theoretical kinematics
Robots, Pressure, Jacobian matrices
Optimal Design of a 4-DOF Parallel Manipulator: From Academia to Industry
Kinematic Analysis and Optimal Design of a 3T1R Type Parallel Mechanism
A Symmetric Parallel Schönflies-Motion Manipulator for Pick-and-Place Operations
Optimal Design of a 4-DOF SCARA Type Parallel Robot Using Dynamic Performance Indices and Angular Constraints
Hjønet
On the Stiffness of Three/Four Degree-of-Freedom Parallel Pick-and-Place Robots With Four Identical Limbs
Internal Singularity Analysis of a Class of Lower Mobility Parallel Manipulators With Articulated Traveling Plate
I4: A New Parallel Mechanism for SCARA Motions
de la O Rodriguez
IEEE/RSJ International Conference on Intelligent Robotic System
), Edmonton, AB, Canada, Aug. 2–6, pp.
Adpet Technology, Inc.
Adept Quattro Parallel Robots
,” Adpet Technology, Livermore, CA, accessed June 1, 2018, http://www1.adept.com/main/ke/data/Archived/Quattro/sQuattro_UG.pdf
Veloce. Robot
,” Penta Robotics, Delft, The Netherlands, accessed June 1, 2018, http://pentarobotics.com/products/#brochure
Synthesis and Design of a Novel 3T1R Fully-Parallel Manipulator
Design and Development of a High-Speed and High-Rotation Robot With Four Identical Arms and a Single Platform
Architecture Optimization of a Parallel Schönfliesmotion Robot for Pick-and-Place Applications in a Predefined Workspace
Type Synthesis of 3T1R 4-DOF Parallel Manipulators Based on Screw Theory
Structural Synthesis of Fully-Isotropic Parallel Robots With Schönflies Motions Via Theory of Linear Transformations and Evolutionary Morphology
Kinematic Analysis and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipulator
PAMINSA: A New Family of Partially Decoupled Parallel Manipulators
Delta: A Simple and Efficient Parallel Robot
Kinetostatic Design of an Innovative Schönflies-Motion Generator
Proc. Ins. Mech. Eng. Part C: J. Mech. Eng. Sci.
Singularity and Workspace Analysis of Three Isoconstrained Parallel Manipulators With Schoenflies Motion
Kinematics and Singularity Analysis of a CRRHHRRC Parallel Schönflies Motion Generator
CSME Trans.
Development of a Novel Two-Limbed Parallel Mechanism Having Schönflies Motion
Kinematic Analysis and Optimal Design of a Wall-Mounted Four-Limb Parallel Sch¨Nflies-Motion Robot for Pick-and-Place Operations
New Dimensionally Homogeneous Jacobian Matrix Formulation by Three End-Effector Points for Optimal Design of Parallel Manipulators
Gonzalez-Palacios
The Mechanical Design of a Seven-Axes Manipulator With Kinematic Isotropy
Multiobjective Optimum Design of a Symmetric Parallel Schönflies-Motion Generator
Design of Planar Parallel Robots With Preloaded Flexures for Guaranteed Backlash Prevention
Is There a Characteristic Length of a Rigid-Body Displacement?
Definition of Pressure and Transmission Angles Applicable to Multi-Input Mechanisms
Rectified Synthesis of Six-Bar Mechanisms With Well-Defined Transmission Angles for Four-Position Motion Generation
Transmission Angle in Mechanisms (Triangle in Mech)
A Transmission Index for in-Parallel Wire-Driven Mechanisms
JSME Inter. J. Ser. C Mech. Syst., Mach. Elem. Manuf.
Performance Evaluation of Parallel Manipulators: Motion/Force Transmissibility and Its Index
A Dual Space Approach for Force/Motion Transmissibility Analysis of Lower Mobility Parallel Manipulators
Constant Motion/Force Transmission Analysis and Synthesis of a Class of Translational Parallel Mechanisms
Kinematic Performance Evaluation of Highspeed Delta Parallel Robots Based on Motion/Force Transmission Indices
A Generalized Approach for Computing the Transmission Index of Parallel Mechanisms
IEEE Trans. Robot. Autom
Int. J. Rot. Res
Mokhiamar
Transmission Quality Evaluation for a Class of Four-Limb Parallel Schönflies-Motion Generators With Articulated Platforms
Mech. Mach. Sci.: Computational Kinematics
Dynamic Modeling and Design Optimization of a 3-DOF Spherical Parallel Manipulator
Rob. Autom. Syst.
Operation Modes Comparison of a Reconfigurable 3-PRS Parallel Manipulator Based on Kinematic Performance
Brogårdh
A Comparison of the Yaw Constraining Performance of SCARA-Tau Parallel Manipulator Variants Via Screw Theory
One Novel Isoconstrained Parallel Robot With Schoenflies-Motion
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Itô isometry - Wikipedia
In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals.
{\displaystyle W:[0,T]\times \Omega \to \mathbb {R} }
denote the canonical real-valued Wiener process defined up to time
{\displaystyle T>0}
{\displaystyle X:[0,T]\times \Omega \to \mathbb {R} }
be a stochastic process that is adapted to the natural filtration
{\displaystyle {\mathcal {F}}_{*}^{W}}
of the Wiener process. Then
{\displaystyle \operatorname {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)^{2}\right]=\operatorname {E} \left[\int _{0}^{T}X_{t}^{2}\,\mathrm {d} t\right],}
{\displaystyle \operatorname {E} }
denotes expectation with respect to classical Wiener measure.
In other words, the Itô integral, as a function from the space
{\displaystyle L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}
of square-integrable adapted processes to the space
{\displaystyle L^{2}(\Omega )}
of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products
{\displaystyle {\begin{aligned}(X,Y)_{L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}&:=\operatorname {E} \left(\int _{0}^{T}X_{t}\,Y_{t}\,\mathrm {d} t\right)\end{aligned}}}
{\displaystyle (A,B)_{L^{2}(\Omega )}:=\operatorname {E} (AB).}
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
{\displaystyle \operatorname {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)\left(\int _{0}^{T}Y_{t}\,\mathrm {d} W_{t}\right)\right]=\operatorname {E} \left[\int _{0}^{T}X_{t}Y_{t}\,\mathrm {d} t\right]}
{\displaystyle X,Y\in L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Itô_isometry&oldid=920241915"
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Feistel cipher - Wikipedia
Cryptography construction
In cryptography, a Feistel cipher (also known as Luby–Rackoff block cipher) is a symmetric structure used in the construction of block ciphers, named after the German-born physicist and cryptographer Horst Feistel, who did pioneering research while working for IBM (USA); it is also commonly known as a Feistel network. A large proportion of block ciphers use the scheme, including the US Data Encryption Standard, the Soviet/Russian GOST and the more recent Blowfish and Twofish ciphers. In a Feistel cipher, encryption and decryption are very similar operations, and both consist of iteratively running a function called a "round function" a fixed number of times.
4.1 Unbalanced Feistel cipher
4.3 Feistel networks as a design component
5 List of Feistel ciphers
Many modern symmetric block ciphers are based on Feistel networks. Feistel networks were first seen commercially in IBM's Lucifer cipher, designed by Horst Feistel and Don Coppersmith in 1973. Feistel networks gained respectability when the U.S. Federal Government adopted the DES (a cipher based on Lucifer, with changes made by the NSA) in 1976. Like other components of the DES, the iterative nature of the Feistel construction makes implementing the cryptosystem in hardware easier (particularly on the hardware available at the time of DES's design).
A Feistel network uses a round function, a function which takes two inputs – a data block and a subkey – and returns one output of the same size as the data block.[1] In each round, the round function is run on half of the data to be encrypted, and its output is XORed with the other half of the data. This is repeated a fixed number of times, and the final output is the encrypted data. An important advantage of Feistel networks compared to other cipher designs such as substitution–permutation networks is that the entire operation is guaranteed to be invertible (that is, encrypted data can be decrypted), even if the round function is not itself invertible. The round function can be made arbitrarily complicated, since it does not need to be designed to be invertible.[2]: 465 [3]: 347 Furthermore, the encryption and decryption operations are very similar, even identical in some cases, requiring only a reversal of the key schedule. Therefore, the size of the code or circuitry required to implement such a cipher is nearly halved.
The structure and properties of Feistel ciphers have been extensively analyzed by cryptographers.
Michael Luby and Charles Rackoff analyzed the Feistel cipher construction and proved that if the round function is a cryptographically secure pseudorandom function, with Ki used as the seed, then 3 rounds are sufficient to make the block cipher a pseudorandom permutation, while 4 rounds are sufficient to make it a "strong" pseudorandom permutation (which means that it remains pseudorandom even to an adversary who gets oracle access to its inverse permutation).[4] Because of this very important result of Luby and Rackoff, Feistel ciphers are sometimes called Luby–Rackoff block ciphers.
Further theoretical work has generalized the construction somewhat and given more precise bounds for security.[5][6]
{\displaystyle \mathrm {F} }
{\displaystyle K_{0},K_{1},\ldots ,K_{n}}
{\displaystyle 0,1,\ldots ,n}
Split the plaintext block into two equal pieces: (
{\displaystyle L_{0}}
{\displaystyle R_{0}}
{\displaystyle i=0,1,\dots ,n}
{\displaystyle L_{i+1}=R_{i},}
{\displaystyle R_{i+1}=L_{i}\oplus \mathrm {F} (R_{i},K_{i}),}
{\displaystyle \oplus }
means XOR. Then the ciphertext is
{\displaystyle (R_{n+1},L_{n+1})}
{\displaystyle (R_{n+1},L_{n+1})}
{\displaystyle i=n,n-1,\ldots ,0}
{\displaystyle R_{i}=L_{i+1},}
{\displaystyle L_{i}=R_{i+1}\oplus \operatorname {F} (L_{i+1},K_{i}).}
{\displaystyle (L_{0},R_{0})}
The diagram illustrates both encryption and decryption. Note the reversal of the subkey order for decryption; this is the only difference between encryption and decryption.
Unbalanced Feistel cipher[edit]
Unbalanced Feistel ciphers use a modified structure where
{\displaystyle L_{0}}
{\displaystyle R_{0}}
are not of equal lengths.[7] The Skipjack cipher is an example of such a cipher. The Texas Instruments digital signature transponder uses a proprietary unbalanced Feistel cipher to perform challenge–response authentication.[8]
The Thorp shuffle is an extreme case of an unbalanced Feistel cipher in which one side is a single bit. This has better provable security than a balanced Feistel cipher but requires more rounds.[9]
The Feistel construction is also used in cryptographic algorithms other than block ciphers. For example, the optimal asymmetric encryption padding (OAEP) scheme uses a simple Feistel network to randomize ciphertexts in certain asymmetric-key encryption schemes.
A generalized Feistel algorithm can be used to create strong permutations on small domains of size not a power of two (see format-preserving encryption).[9]
Feistel networks as a design component[edit]
Whether the entire cipher is a Feistel cipher or not, Feistel-like networks can be used as a component of a cipher's design. For example, MISTY1 is a Feistel cipher using a three-round Feistel network in its round function, Skipjack is a modified Feistel cipher using a Feistel network in its G permutation, and Threefish (part of Skein) is a non-Feistel block cipher that uses a Feistel-like MIX function.
List of Feistel ciphers[edit]
Feistel or modified Feistel:
Generalised Feistel:
Lifting scheme for discrete wavelet transform has pretty much the same structure
^ Menezes, Alfred J.; Oorschot, Paul C. van; Vanstone, Scott A. (2001). Handbook of Applied Cryptography (Fifth ed.). p. 251. ISBN 978-0849385230.
^ Schneier, Bruce (1996). Applied Cryptography. New York: John Wiley & Sons. ISBN 0-471-12845-7.
^ Stinson, Douglas R. (1995). Cryptography: Theory and Practice. Boca Raton: CRC Press. ISBN 0-8493-8521-0.
^ Luby, Michael; Rackoff, Charles (April 1988), "How to Construct Pseudorandom Permutations from Pseudorandom Functions", SIAM Journal on Computing, 17 (2): 373–386, doi:10.1137/0217022, ISSN 0097-5397 .
^ Patarin, Jacques (October 2003), Boneh, Dan (ed.), "Luby–Rackoff: 7 Rounds Are Enough for 2n(1−ε) Security" (PDF), Advances in Cryptology—CRYPTO 2003, Lecture Notes in Computer Science, 2729: 513–529, doi:10.1007/b11817, ISBN 978-3-540-40674-7, S2CID 20273458, retrieved 2009-07-27
^ Zheng, Yuliang; Matsumoto, Tsutomu; Imai, Hideki (1989-08-20). On the Construction of Block Ciphers Provably Secure and Not Relying on Any Unproved Hypotheses. Advances in Cryptology — CRYPTO' 89 Proceedings. Lecture Notes in Computer Science. Vol. 435. pp. 461–480. doi:10.1007/0-387-34805-0_42. ISBN 978-0-387-97317-3.
^ Schneier, Bruce; Kelsey, John (1996-02-21). Unbalanced Feistel networks and block cipher design. Fast Software Encryption. Lecture Notes in Computer Science. Vol. 1039. pp. 121–144. doi:10.1007/3-540-60865-6_49. ISBN 978-3-540-60865-3. Retrieved 2017-11-21.
^ Bono, Stephen; Green, Matthew; Stubblefield, Adam; Juels, Ari; Rubin, Aviel; Szydlo, Michael (2005-08-05). "Security Analysis of a Cryptographically-Enabled RFID Device" (PDF). Proceedings of the USENIX Security Symposium. Retrieved 2017-11-21.
^ a b Morris, Ben; Rogaway, Phillip; Stegers, Till (2009). How to Encipher Messages on a Small Domain (PDF). Advances in Cryptology – CRYPTO 2009. Lecture Notes in Computer Science. Vol. 5677. pp. 286–302. doi:10.1007/978-3-642-03356-8_17. ISBN 978-3-642-03355-1. Retrieved 2017-11-21.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Feistel_cipher&oldid=1060573741"
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Evaluate an Expression at a Point - Maple Help
Home : Support : Online Help : Tasks : Evaluating : Evaluate an Expression at a Point
Evaluate an Expression at a Point
Evaluate an expression at a point.
\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{\mathrm{cos}}\left(\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{\mathrm{π}} \textcolor[rgb]{0.372549019607843,0,0.850980392156863}{x}\right)\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{-}{\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{x}}^{\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{2}}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{+}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{4} \textcolor[rgb]{0.372549019607843,0,0.850980392156863}{x}
\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{π}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
Specify a point, and then evaluate the expression.
\mathrm{eval}\left(, \textcolor[rgb]{0.372549019607843,0,0.850980392156863}{x}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{=}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{2}\right)
\textcolor[rgb]{0,0,1}{5}
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§ Fenwick trees and orbits
I learnt of a nice, formal way to prove the correctness of Fenwick trees in terms of orbits that I wish to reproduce here. One can use a Fenwick tree to perform cumulative sums
Sum(n) \equiv \sum_{i=0}^n A[i]
, and updates
Upd(i, v) \equiv A[i] += v
. Naively, cumulative sums can take
O(n)
time and updates take
O(1)
time. A Fenwick tree can perform both in
\log(n)
. In general, we can perform any monoid-based catenation and update in
\log(n)
, as long as our queries start from the 0th index till some index
n
. So we can only handle monoidal fold queries of the form
\sum_{i=0}^n
and point updates.
We allow indexes
[1, 2, \dots n]
. The node with factorization
i \equiv 2^k \times l
2 \not \vert l
k
is the highest power of
2
i
) is responsible for the interval
[i-2^k+1, i] = (i-2^k, i]
. I'm going to state all the manipulations in terms of prime factorizations, since I find it far more intuitive than bit-fiddling. In general, I want to find a new framework to discover and analyze bit-fiddling heavy algorithms. Some examples of the range of responsibility of an index are:
1 = 2^0 \times 1 = (0, 1]
(Subtract
2^0 = 1
2 = 2\times 1 = (0, 2]
2^1 = 2
3 = 3 = (2, 3]
4 = 2^2 = (0, 4]
5 = 5 = (4, 5]
6 = 2\times 3 = (4, 6]
7 = 7 = (6,7]
8 = 2^3 = (0,8]
9 = 9 = (8, 9]
10 = 2\times 5 = (8, 10]
11 = 11 = (10, 11]
12 = 2^2\times 3 = (8, 12]
13 = 13 = (12, 13]
14 = 2\times 7 = (12, 14]
15 = 15 = (14, 15]
16 = 2^4 = (0, 16]
To perform a cumulative sum, we need to read from the correct overlap regions that cover the full array. For example, to read from
15
, we would want to read:
a[15] = (14, 15], a[14] = (12, 14], a[12] = (8, 12], a[8] = (0, 8]
So we need to read the indices:
15=2^0 \cdot 15 \xrightarrow{-2^0} 14=2^1 \cdot 7 \xrightarrow{-2^1} 12=2^2\cdot3 \xrightarrow{-2^2} 8=2^3\cdot1 \xrightarrow{-2^3} 0
At each location, we strip off the value
2^r
. We can discover this value with bit-fiddling: We claim that
a \& (-a) = 2^r
a \equiv \langle x 1 0^r \rangle
. We wish to extract the output as
2^r = \langle 1 0^r \rangle
, which is the power of 2 that we need to subtract from
a
to strip the rightmost
1
\begin{aligned} &-a = \lnot a + 1 = x01^r + 1 = \overline{x}10^r \\ &a \& (-a) = a \& (\lnot a + 1) = (x 10^r) \& (\overline{x}10^r) = 0^{|\alpha|}10^r = 2^r \end{aligned}
a - (a \& (-a)) = \langle x 1 0^r \rangle - \langle 1 0^r \rangle = \langle x 0 0^r \rangle
That is, we successfully strip off the trailing
1
. Armed with the theory, our implemtation becomes:
#define LSB(x) x&(-x)
int q(int i) {
i -= LSB(i); // strip off trailing 1.
To perform an update at
i
, we need to update all locations which on querying overlap with
i
. For example, to update the location
9
, we would want to update:
a[9] = (8, 9], a[10] = (8, 10], a[12] = (8, 12], a[16] = (0, 16]
So we need to update the indices:
9=2^0 \cdot 9 \xrightarrow{+2^0} 10=2^1 \cdot 5 \xrightarrow{+2^1} 12=2^2\cdot3 \xrightarrow{+2^2} 16=2^4\cdot1 \xrightarrow{+2^4} \dots
We use the same bit-fiddling technique as above to strip off the value
2^r
int u(int i, int v) {
while (i < N) { tree[i] += v; i += LSB(i); }
§ correctness
We wish to analyze the operations
Query(q) \equiv \sum_{i=1}^q a[i]
Update(i, val) \equiv a[i]~\texttt{+=}~val
. To do this, we are allowed to maintain an auxiliary array
d
which we will manipuate. We will stipulate the conditions of operations on
d
such that they will reflect the values of
Query
Update
, albeit much faster. We will analyze the algorithm in terms of orbits. We have two operators, one for update called
U
, and one for query called
Q
. Given an index
i
, repeatedly applying the query operator gives us the indeces we need to read and accumulate from the underlying array
a
to get the total sum
a[0..i]
Query(i) = \sum_i d[Q^i(q)]
Given an index
u
, repeatedly applying the update operator
U
gives us all the indeces we need to add the change to update:
Update(i, val) = \forall j~, d[U^j(i)]~\texttt{+=}~ val
For query and update to work, we need the condition that:
q \geq u \iff \left\vert \{ Q^i(q)~:~ i \in \mathbb{N} \} \cap \{ U^i(u)~:~i \in \mathbb{N} \} \right\vert = 1
That is, if and only if the query index
q
includes the update location
u
, will the orbits intersect. The intuition is that we want updates at an index
u
to only affect queries that occur at indeces
q \geq u
. Hence, we axiomatise that for an update to be legal, it must the orbits of queries that are at indeces greater than it. We will show that our operators:
Q(i=2^r\cdot a) = i - 2^r = 2^r(a-1)
U(j=2^s\cdot b) = j + 2^{s} = 2^{s}(b+1)
do satisfy the conditions above. For a quick numerical check, we can use the code blow to ensure that the orbits are indeed disjoint:
# calculate orbits of query and update in fenwick tree
def lsb(i): return i & (-i)
def U(i): return i + lsb(i)
def Q(i): return i - lsb(i)
def orbit(f, i):
while i not in s and i > 0 and i < 64:
s.add(i); i = f(i)
for q in range(1, 16):
for u in range(1, 16):
qo = orbit(Q, q); uo = orbit(U, u)
c = qo.intersection(uo)
print("q:%4s | u:%4s | qo: %20s | uo: %20s | qu: %4s" %
(q, u, qo, uo, c))
§ Case 1:
q = u
Q
always decreases the value of
q
u
always increases it. Hence, if
q = u
, they meet at this point, and
\forall i, j \geq 1, \quad Q^i (q) \neq U^j(u)
. Hence, they meet exactly once as required.
q < u
q
always decreases and
u
always increases, hence in this case they will never meet as required.
q > u
Let the entire array have size
2^N
q = \texttt{e1 f\_q}, u = \texttt{e0 f\_u}
\texttt{e, f\_q, f\_u}
may be empty strings. Notice that
Q
will always strip away rightmost ones in
f_q
q = \texttt{e10...0}
at some point. Similarly,
U
will keep on adding new rightmost ones, causing the state to be
u = \texttt{e01...10...0} \xrightarrow{U} \texttt{e100...}
. Hence, at some point
q = u
Fenwick trees on PolyMath
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2014 A Suzuki Type Coupled Fixed Point Theorem for Generalized Multivalued Mapping
Pushpendra Semwal, Ramesh Chandra Dimri
We obtain a new Suzuki type coupled fixed point theorem for a multivalued mapping
T
X×X
\text{C}\text{B}\left(X\right)
, satisfying a generalized contraction condition in a complete metric space. Our result unifies and generalizes various known comparable results in the literature. We also give an application to certain functional equations arising in dynamic programming.
Pushpendra Semwal. Ramesh Chandra Dimri. "A Suzuki Type Coupled Fixed Point Theorem for Generalized Multivalued Mapping." Abstr. Appl. Anal. 2014 (SI56) 1 - 8, 2014. https://doi.org/10.1155/2014/820482
Pushpendra Semwal, Ramesh Chandra Dimri "A Suzuki Type Coupled Fixed Point Theorem for Generalized Multivalued Mapping," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI56), 1-8, (2014)
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Positive Solution for a Class of Boundary Value Problems with Finite Delay
2012 Positive Solution for a Class of Boundary Value Problems with Finite Delay
We study a class of boundary value problems with equation of the form
{x}^{″}\left(t\right)+f\left(t,x\left(t\right),{x}^{\prime }\left(t-\tau \right)\right)=0
. Some sufficient conditions for existence of positive solution are obtained by using the Krasnoselskii fixed point theorem in cones.
Hongzhou Wang. "Positive Solution for a Class of Boundary Value Problems with Finite Delay." J. Appl. Math. 2012 1 - 7, 2012. https://doi.org/10.1155/2012/382392
Hongzhou Wang "Positive Solution for a Class of Boundary Value Problems with Finite Delay," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-7, (2012)
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On the Role of Diffusion Behaviors in Stability Criterion for p-Laplace Dynamical Equations with Infinite Delay and Partial Fuzzy Parameters under Dirichlet Boundary Value
2013 On the Role of Diffusion Behaviors in Stability Criterion for p-Laplace Dynamical Equations with Infinite Delay and Partial Fuzzy Parameters under Dirichlet Boundary Value
Ruofeng Rao, Zhilin Pu, Shouming Zhong, Jialin Huang
By the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space
{W}_{0}^{1,p}\left(Ω\right)
, a global asymptotical stability criterion for p-Laplace partial differential equations with partial fuzzy parameters is derived under Dirichlet boundary condition, which gives a positive answer to an open problem proposed in some related literatures. Different from many previous related literatures, the nonlinear p-Laplace diffusion item plays its role in the new criterion though the nonlinear p-Laplace presents great difficulties. Moreover, numerical examples illustrate that our new stability criterion can judge what the previous criteria cannot do.
Ruofeng Rao. Zhilin Pu. Shouming Zhong. Jialin Huang. "On the Role of Diffusion Behaviors in Stability Criterion for p-Laplace Dynamical Equations with Infinite Delay and Partial Fuzzy Parameters under Dirichlet Boundary Value." J. Appl. Math. 2013 1 - 8, 2013. https://doi.org/10.1155/2013/940845
Ruofeng Rao, Zhilin Pu, Shouming Zhong, Jialin Huang "On the Role of Diffusion Behaviors in Stability Criterion for p-Laplace Dynamical Equations with Infinite Delay and Partial Fuzzy Parameters under Dirichlet Boundary Value," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-8, (2013)
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\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\mathrm{f3}{!}^{k}\dots
i,j,k
\mathrm{f1}
\mathrm{f2}
\mathrm{f3}
\sqrt{\mathrm{\pi }}=\mathrm{\Gamma }\left(\frac{1}{2}\right)
0<i
j,k<0
\mathrm{f1}-\mathrm{f2}-\mathrm{f3}=n
\frac{\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)c\mathrm{f2}!\mathrm{f3}!}{\mathrm{f1}!}
c
is a correction factor depending on
\mathrm{f3}
i,0<j
k<0
\mathrm{f3}-\mathrm{f1}-\mathrm{f2}=n
\frac{c\mathrm{f3}!}{\mathrm{f1}!\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)}
\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\dots
i,j
\frac{\mathrm{f1}}{\mathrm{f2}}
r
1<|r|
\frac{\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)\left(\mathrm{f1}-\mathrm{f2}\right)!}{\mathrm{f1}!}
|r|<1
\frac{\mathrm{f2}!}{\mathrm{f1}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)\left(\mathrm{f2}-\mathrm{f1}\right)!}
a≔\frac{n!}{k!\left(n-k\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{n}}{\textcolor[rgb]{0,0,1}{k}}\right)
a≔\frac{n\left({n}^{2}+m-k+2\right)\left({n}^{2}+m\right)!}{k!\left({n}^{2}+m-k+2\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{k}}\right)}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
a≔\frac{{m!}^{3}}{\left(3m\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{!}}^{\textcolor[rgb]{0,0,1}{3}}}{\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)}
a≔\frac{\mathrm{\Gamma }\left(m+\frac{3}{2}\right)}{\mathrm{sqrt}\left(\mathrm{\pi }\right)\mathrm{\Gamma }\left(m\right)}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\right)}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}\right)
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§ Stable homotopy theory
We like stable homotopy groups because of the Freudenthal suspension theorem which tells us that homotopy groups stabilise after many suspensions. The basic idea seems to be something like a tensor-hom adjunction. We have the loop spaces which are like
S^1 \rightarrow X
and the suspension which is like
S^1 \wedge X
. The theory begins by considering the tensor-hom-adjunction between these objects as fundamental. So curry stuff around to write things as (S^1, A) -> B and A -> (S^1 -> B), which is Suspension(A) -> B and A -> Loop(B). This gives us the adjunction between suspension and looping.
We then try to ask: how can one invert the suspension formally? One tries to do some sort of formal nonsense, by declaring that maps between
\Sigma^{-n}X
\Sigma^{-m} Y
, but this doesn't work due to some sort of grading issue.
Instead, one repaces a single object
X
with a family of objects
\{ X_i \}
called as the spectrum. Then, we can invert the suspension by trying to invert maps between objects of the same index.
Homotopy Extension
Stable homotopy theory 1
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Most people believe that topology is about some notion of "nearness" or "closeness", which has been abstracted out from our usual notion of continuity that we have from a metric space. Here, I make the claim that topology is really about computation , and more specifically, decidability . These are not new ideas. I learnt of this from a monograph Synthetic topology of data types and classical spaces. Martın Escardo . This does not seem very well known, so I decided to write about it. The idea is this: We have turing machines which can compute things. We then also have a set
S
. Now, a topology
\tau \subset 2^S
precisely encodes which of the subsets of
S
can be separated from the rest of the space by a turing machine. Thus, a discrete space is a very nice space, where every point can be separated from every other point. An indescrete space is one where no point can be separated. Something like the reals is somewhere in between, where we can separate stuff inside an open interval from stuff clearly outside, but there's some funny behaviour that goes on at the boundary due to things like (0.999... = 1), which we'll see in detail in a moment.
§ Semidecidability
Q\subseteq S
is semidecidable , if there exists a turing machine
\hat Q: Q \rightarrow \{ \bot, \top \}
\begin{aligned} \hat Q(q) = \top \iff q \in Q \\ \hat Q(q) = \bot \iff q \notin Q \\ \end{aligned}
\top
signifies stopping at a state and returning TRUE, and
\bot
signifies never halting at all !. So, the subset
Q
is semidedicable , in that, we will halt and say TRUE if the element belongs in the set. But if an element does not belong in the set, we are supposed to never terminate.
§ Deep dive: semidecidability of the interval
(1, 2)
Let's start with an example. We consider the interval
I = (1, 2)
\mathbb{R}
.Let the turing machine recieve the real number as a function
f: \mathbb N \rightarrow \{0, 1, \dots 9\}
, such that given a real number
{(a_0 \cdot a_1 \cdot a_2 \dots)}
, this is encoded as a function
{f_a(i) = a_i}
. We now build a turing machine
\hat I
which when given the input the function
f_a
, semi-decides whether
{a \in I}
. Let's consider the numbers in
I
\begin{aligned} &0 \rightarrow \texttt{NO} \\ &0.\overline{9} \rightarrow \texttt{NO} \\ &1.00\dots \rightarrow \texttt{NO} \\ &1.a_1 a_2 \dots \rightarrow \texttt{YES} \\ &1.\overline{9} \rightarrow \texttt{NO} \\ &2.0 \rightarrow \texttt{NO} \\ &2.a_1 a_2 \rightarrow \texttt{NO} \end{aligned}
So, we can write a turing machine (ie, some code) that tries to decide whether a real number
a
's encoding
f_a
I = (1, 2)
def decide_number_in_open_1_2(f):
# if the number is (1.abcd)
if f(0) == 1:
# (1.0...0<NOT0>) | look for the <NOT0>
# If the number is 1.00..., do not terminate.
if f(1) == 0: while f(i) == 0: i += 1
# (1.99...9<NOT9>) | look for the <NOT9>
if f(1) == 9: while f(i) == 9: i += 1
# if the number is not 1.abcd, do not terminate
Hence, we say that the interval
I = (1, 2)
is semi-decidable , since we have a function
\hat I \equiv \texttt{decide-number-in-open-1-2}
\hat I (f_a) \text{ terminates } \iff a \in I
. We don't make any claim about what happens if
a \notin I
. This is the essence of semidecidability: We can precisely state when elements in the set belong to the set, but not when they don't.
§ Semi decidability in general
To put this on more solid ground, we define a topology on a set
S
by considering programs which recieve as input elements of
S
, suitably encoded. For example, the way in which we encoded real numbers as functions from the index to the digit. Similarly, we encode other mathematical objects in some suitable way. Now, we define:
For every program
P
which takes as inputs elements in
S
{halts(P) \equiv \\{ s \in S \vert P(s) \text{halts} \\}}
is called as a semidecidable set .
Alternatively, we can say for a subset
{T \subset S}
, if there exists a program
{\hat T}
{s \in T \iff \hat T(s) \text{ halts}}
T
is semi-dedecidable.
These are just two viewpoints on the same object. In one, we define the set based on the program. In the other, we define the program based on the set.
§ Semi decidability of the empty set and the universe set.
The empty set is semi-decidable, due to the existence of the program:
def semidecide_empty(x):
while True: continue
The universe set is semi-decidable, due to the existence of the program:
def semidecide_univ(x): return
§ Semi decidability of the union of sets
infinite unions of sets are semi decidable, since we can "diagonalize" on the steps of all programs. That way, if any program halts, we will reach the state where it halts in our diagonalized enumeration. Let A00, A01... A0n be the initial states of the machines. We are trying to semidecide whether any of them halt. We lay out the steps of the machines in an imaginary grid:
Am0 Am1 Am2 ... Amn
For example, machine A0 has states:
A00 -> A10 -> .. -> Am0
We can walk through the combined state-space of the machines as:
Where on the k'th line, we collect all states
A_{ij}
(i + j = k)
. Now, if any of the machines have a state that is HALT, we will reach the state as we enumerate the diagonals, and the machine that explores the combined state space can also return HALT.
§ Semi decidability of the intersection of sets
infinite intersections of sets are not semi decidable, since by running these programs in parallel, we cannot know if an infinite number of programs halt in finite time. We can tell if one of them halts, but of if all of them halt. For example, consider the sequence of machines produced by machine_creator:
# creates a machine that stops after n steps
def machine_creator(n):
# f terminates after n steps
We wish to check if the intersection of all machine_creator(n) halt, for all
n \geq 0, n \in \mathbb N
. Clearly, the answer is an infinite number of steps, even though every single machine created by machine_creator halts in a finite number of steps.
§ An even deeper dive: re-examining the topology of the reals
We generally think of the topology of reals as being generated from the base of intervals
(a, b)
. But really, this is a perverse perspective from the point of view of computation. Structurally speaking, the only comparison operator we have on the reals is a
<
operator. So we should ideally start by taking as a base the sets
(-\infty, r)
. This is elegant, because:
it relates to the order theoretic notion of downward closed set
By yoneda , the downward closed set
(-\infty, r)
contains the exact same order theoretic information as
r
It makes our "code" easier to write:
z = 10 # arbitrary fixed integer
def is_in_downward_closed_set(x_int, x_frac):
return true if x_int.x_frac < z
if x_int < z:
elif x_frac[i] == 9: i += 1 # examine next number.
else: return True # x_frac[i] < r_frac[i]
def loop(): while True: pass
Synthetic topology of data types and classical spaces. Martın Escardo
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The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: [math] What is the optimal velocity which minimizes the fuel consumption of the vehicle in gallons PER MILE?
To solve this problem, we need to minimize the following function of [math]
v
g(v) =
Hint for the above: Assume the vehicle is moving at constant velocity [math]
v
. How long will it take to travel 1 mile? How much gas will it use during that time?
We find that this function has one critical number at [math]
v=
To verify that [math]
g(v)
has a minimum at this critical number we compute the second derivative [math]
g''(x)
and find that its value at the critical number is , a positive number.
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Friction in contact between moving bodies - MATLAB - MathWorks Deutschland
Breakaway friction force
Friction in contact between moving bodies
The Translational Friction block represents friction in contact between moving bodies. The friction force is simulated as a function of relative velocity and is assumed to be the sum of Stribeck, Coulomb, and viscous components, as shown in the following figure.
The Stribeck friction, FS, is the negatively sloped characteristics taking place at low velocities [1]. The Coulomb friction, FC, results in a constant force at any velocity. The viscous friction, FV, opposes motion with the force directly proportional to the relative velocity. The sum of the Coulomb and Stribeck frictions at the vicinity of zero velocity is often referred to as the breakaway friction, Fbrk. The friction is approximated with the following equations:
F=\sqrt{2e}\left({F}_{brk}-{F}_{C}\right)\cdot \mathrm{exp}\left(-{\left(\frac{v}{{v}_{St}}\right)}^{2}\right)\cdot \frac{v}{{v}_{St}}+{F}_{C}\cdot \mathrm{tanh}\left(\frac{v}{{v}_{Coul}}\right)+fv
{v}_{St}={v}_{brk}\sqrt{2}
{v}_{Coul}={v}_{brk}/10
v={v}_{R}-{v}_{C}
F is friction force.
FC is Coulomb friction.
Fbrk is breakaway friction.
vbrk is breakaway friction velocity.
vSt is Stribeck velocity threshold.
vCoul is Coulomb velocity threshold.
vR and vC are absolute velocities of ports R and C, respectively.
v is relative velocity.
The hyperbolic tangent function used in the Coulomb portion of the force equation ensures that the equation is smooth and continuous through v = 0, but quickly reaches its full value at nonzero velocities.
The block positive direction is from port R to port C. This means that if the port R velocity is greater than that of port C, the block transmits force from R to C.
Mechanical translational conserving port associated with the rod, that is, the moving body.
Mechanical translational conserving port associated with the case, that is, the stationary body.
Breakaway friction force — Sum of Coulomb and the static frictions
The breakaway friction force, which is the sum of the Coulomb and the static frictions. It must be greater than or equal to the Coulomb friction force value.
Breakaway friction velocity — Peak velocity for Stribeck friction
The velocity at which the Stribeck friction is at its peak. At this point, the sum of the Stribeck and Coulomb friction is the Breakaway friction force. This parameter specifies the velocity threshold, which affects the tradeoff between the simulation accuracy and speed.
Coulomb friction force — Friction that opposes motion
The Coulomb friction force, which is the friction that opposes motion with a constant force at any velocity.
100 N/(m/s) (default) | nonnegative scalar
Proportionality coefficient between the friction force and the relative velocity. The parameter value must be greater than or equal to zero.
Translational Damper | Translational Hard Stop | Translational Spring
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For an AM signal modulated by an audio signal, maximum frequency is <math>F_{Max}=F_0+F_{Max Audio}</math>. Direct sampling of such signal is not possible with conventionnal hardware such as low cost SDR dongle. If the carrier frequency is close to 1 GHz, the sampling rate should be 2 GHz. This is obviously too much then computer cans handle (higher then most computer clock).
{\displaystyle z=a+jb}
{\displaystyle {\text{Re}}\{z\}=a}
{\displaystyle {\text{Im}}\{z\}=b}
{\displaystyle r=|z|={\sqrt {a^{2}+b^{2}}}}
{\displaystyle \phi =\arg(z)=\arctan(b/a)}
{\displaystyle z=r\left(cos(\phi )+jsin(\phi )\right)=re^{j\phi }}
{\displaystyle z_{1}=r_{1}e^{j\phi _{1}}}
{\displaystyle z_{1}=r_{2}e^{j\phi _{2}}}
{\displaystyle z=z_{1}z_{2}=r_{1}r_{2}e^{j(\phi _{1}+\phi _{2})}}
{\displaystyle +1=e^{j0}}
{\displaystyle +j=e^{j\pi /2}}
{\displaystyle -1=e^{j\pi }}
{\displaystyle -j=e^{j3\pi /2}}
{\displaystyle c(t)=i(t)+jq(t)=a(t)e^{j\phi (t)}}
{\displaystyle m(t)=a(t)c(t)=a(t)\cos(2\pi f_{0}t)}
{\displaystyle M(f)={\frac {1}{2}}{\big (}A(f-f_{0})+A(f+f_{0}){\big )}}
{\displaystyle f_{s}>F_{Max}}
{\displaystyle F_{MaxAudio}}
{\displaystyle F_{Max}=F_{0}+F_{MaxAudio}}
|
{\displaystyle F=P\cdot A}
{\displaystyle P={\frac {F_{p}}{A_{p}-A_{r}}}}
Cylinder blockPneumatic cylinderTelescopic cylinderPneumaticsGas cylinderControl valveMechanical engineeringHydraulic headBrake (sheet metal bending)Oil filterHydraulic machineryFluid powerAngle grinderLubricationAutomatic lubricationTribologyCrankcase
This article uses material from the Wikipedia article "Hydraulic cylinder", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
|
Classification of flat slant surfaces in complex Euclidean plane
October, 2002 Classification of flat slant surfaces in complex Euclidean plane
It is well-known that the classification of flat surfaces in Euclidean 3space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane
{C}^{2}
endowed with almost complex structure
J
, flat surfaces are the simplest ones from intrinsic point of views. On the other hand, from
J
-action point of views, the most natural surfaces in
{C}^{2}
are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classifies flat slant surfaces in
{C}^{2}
. The main result states that, beside the totally geodesic ones, there are five large classes of flat slant surfaces in
{C}^{2}
. Conversely, every non-totally geodesic flat slant surfaces in
{C}^{2}
is locally a surface given by these five classes.
Bang-Yen CHEN. "Classification of flat slant surfaces in complex Euclidean plane." J. Math. Soc. Japan 54 (4) 719 - 746, October, 2002. https://doi.org/10.2969/jmsj/1191591991
Keywords: complex Euclidean space , flat surface , slant submanifold , slant surface , wave equation
Bang-Yen CHEN "Classification of flat slant surfaces in complex Euclidean plane," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 54(4), 719-746, (October, 2002)
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Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit
2022 Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit
Mouad Ramil
1Cermics (Ecole des Ponts ParisTech), France
Consider the Langevin process, described by a vector (positions and momenta) in
{\mathbb{R}}^{d}×{\mathbb{R}}^{d}
\mathcal{O}
{\mathcal{C}}^{2}
open bounded and connected set of
{\mathbb{R}}^{d}
. Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain
D:=\mathcal{O}×{\mathbb{R}}^{d}
. In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain
\mathcal{O}
Supported by the Région Ile-de- France through a PhD fellowship of the Domaine d’Intérêt Majeur (DIM) Math Innov.
This work also benefited from the support of the project ANR QuAMProcs (ANR-19-CE40-0010) from the French National Research Agency. The author would also like to thank Tony Lelièvre and Julien Reygner for fruitfull discussions throughout this work.
Mouad Ramil. "Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit." Electron. J. Probab. 27 1 - 18, 2022. https://doi.org/10.1214/22-EJP789
Primary: 35B25 , 47B07 , 60H10 , 82C31
Keywords: Langevin process , overdamped Langevin process , overdamped limit , quasi-stationary distribution
Mouad Ramil "Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit," Electronic Journal of Probability, Electron. J. Probab. 27(none), 1-18, (2022)
|
Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators
2012 Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators
Lupas-type operators and Szász-Mirakyan-type operators are the modifications of Bernstein polynomials to infinite intervals. In this paper, we investigate the convergence of Lupas-type operators and Szász-Mirakyan-type operators on
\left[0,\infty \right)
Hee Sun Jung. Ryozi Sakai. "Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators." J. Appl. Math. 2012 1 - 28, 2012. https://doi.org/10.1155/2012/546784
Hee Sun Jung, Ryozi Sakai "Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-28, (2012)
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A Novel Iterative Method for Solving Systems of Fractional Differential Equations
2013 A Novel Iterative Method for Solving Systems of Fractional Differential Equations
E. Hesameddini, A. Rahimi
The Reconstruction of Variational Iteration Method (RVIM) technique has been successfully applied to obtain solutions for systems of nonlinear fractional differential equations:
{D}_{\ast }^{\alpha }{x}_{i}\left(t\right)={N}_{i}\left(t,{x}_{1},\dots ,{x}_{n}\right)
{x}_{i}^{\left(k\right)}={c}_{k}^{i}
0\le k\le \left[{\alpha }_{i}\right]
1\le i\le n
{D}_{\ast }^{\alpha }
denote Caputo fractional derivative. The RVIM, for differential equations of integer order is extended to derive approximate analytical solutions for systems of fractional differential equations. Advantage of the RVIM, is simplicity of the computations and convergent successive approximations without any restrictive assumptions or transform functions. Some illustrative examples are given to show the validity of this method for solving linear and nonlinear systems of fractional differential equations.
E. Hesameddini. A. Rahimi. "A Novel Iterative Method for Solving Systems of Fractional Differential Equations." J. Appl. Math. 2013 (SI11) 1 - 7, 2013. https://doi.org/10.1155/2013/428090
E. Hesameddini, A. Rahimi "A Novel Iterative Method for Solving Systems of Fractional Differential Equations," Journal of Applied Mathematics, J. Appl. Math. 2013(SI11), 1-7, (2013)
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BabyMonster - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : BabyMonster
BabyMonster()
The Baby Monster
𝔹
group is the second largest among the sporadic finite simple groups. The Baby Monster was constructed in 1977 by Jeffrey Leon and Charles Sims as a permutation group of degree 13571955000, but the existence of the Baby Monster had been predicted earlier in the 1970s by Bernd Fischer.
The BabyMonster() command returns a symbolic group that represents the Baby Monster.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔\mathrm{BabyMonster}\left(\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{𝔹}
\mathrm{GroupOrder}\left(G\right)
\textcolor[rgb]{0,0,1}{4154781481226426191177580544000000}
\mathrm{IsSimple}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The GroupTheory[BabyMonster] command was introduced in Maple 17.
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Engine Design and Operational Impacts on Particulate Matter Precursor Emissions | J. Eng. Gas Turbines Power | ASME Digital Collection
Stephen P. Lukachko,
Stephen P. Lukachko
Gas Turbine Laboratory, Department of Aeronautics and Astronautics,
e-mail: sluka@mit.edu
e-mail: iaw@mit.edu
Richard C. Miake-Lye,
Center for Aerothermodynamics,
, Billerica, MA 01821
e-mail: rick@aerodyne.com
Lukachko, S. P., Waitz, I. A., Miake-Lye, R. C., and Brown, R. C. (February 29, 2008). "Engine Design and Operational Impacts on Particulate Matter Precursor Emissions." ASME. J. Eng. Gas Turbines Power. March 2008; 130(2): 021505. https://doi.org/10.1115/1.2795758
Aircraft emissions of trace sulfur and nitrogen oxides contribute to the generation of fine volatile particulate matter (PM). Resultant changes to ambient PM concentrations and radiative properties of the atmosphere may be important sources of aviation-related environmental impacts. This paper addresses engine design and operational impacts on aerosol precursor emissions of
SOx
NOy
species. Volatile PM formed from these species in the environment surrounding an aircraft is dependent on intraengine oxidation processes occurring both within and downstream of the combustor. This study examines the complex response of trace chemistry to the temporal and spatial evolution of temperature and pressure along this entire intraengine path after combustion through the aft combustor, turbine, and exhaust nozzle. Low-order and higher-fidelity tools are applied to model the interaction of chemical and fluid mechanical processes, identify important parameters, and assess uncertainties. The analysis suggests that intraengine processing is inefficient. For in-service engine types in the large commercial aviation fleet, mean conversion efficiency
(ε)
is estimated to be 2.8–6.5% for sulfate precursors and 0.3–5.7% for nitrate precursors at the engine exit plane. These ranges reflect technological differences within the fleet, a variation in oxidative activity with operating mode, and modeling uncertainty stemming from variance in rate parameters and initial conditions. Assuming that sulfur-derived volatile PM is most likely, these results suggest emission indices of
0.06–0.13g∕kg
fuel, assuming particles nucleated as
2H2SO4∙H2O
for a fuel sulfur content of
500ppm
aerosols, aerospace engines, air pollution, design engineering, particulate matter, sulfate, nitrate, emissions, inventory, environmental impact
Aircraft, Chemistry, Combustion chambers, Emissions, Engines, Exhaust systems, Flow (Dynamics), Nozzles, Oxidation, Particulate matter, Pressure, Temperature, Turbines, Uncertainty, Aerosols, Cycles, Sulfur, Fuels, Modeling
J. Geophys. Res., [Atmos.]
Air Quality Criteria for Particulate Matter, Volumes 1 and 2
,” National Center for Environmental Assessment-RTP Office, Office of Research and Development, Report No. EPA∕600∕P-99∕002bF.
1999, Intergovernmental Panel on Climate Change, 1999,
Aviation and the Global Atmosphere: A Special Report of the Intergovernmental Panel on Climate Change
Ovarlez
Pollution From Aircraft Emissions in the North Atlantic Flight Corridor: Overview on the POLINAT Projects
Airborne Observations of Aircraft Aerosol Emissions. I: Total Nonvolatile Particle Emission Indices
SOx Oxidation and Volatile Aerosol in Aircraft Exhaust Plumes Depend on Fuel Sulfur Content
Airborne Observations of Aircraft Aerosol Emissions. II: Factors Controlling Volatile Particle Production
Quantities, Characteristics and Reduction Potentials of Aircraft Engine Emissions
World Aviation Congress and Exposition
The Possible Role of Organics in the Formation and Evolution of Ultrafine Aircraft Particles
Model Simulations of Fuel Sulfur Conversion Efficiencies in an Aircraft Engine: Dependence on Reaction Rate Constants and Initial Species Mixing Ratios
Modeling of Sulfur Gases and Chemiions in Aircraft Engines
Laboratory Flow Reactor Measurements of the Reaction SO3+H2O+M−H2SO4+M-Implications for Gaseous H2SO4 and Aerosol Formation in the Plumes of Jet Aircraft
Ultrafine Aerosol Particles in Aircraft Plumes: In Situ Observations
In Situ Observations of Particles in Jet Aircraft Exhausts and Contrails for Different Sulfur-Containing Fuels
Gas-Phase Reaction of Sulfur-Trioxide With Water-Vapor
Contrail Formation—Homogeneous Nucleation of H2SO4∕H2O Droplets
The Role of Ions in the Formation and Evolution of Particles in Aircraft Plumes
Aerosol Dynamics in Near-Field Aircraft Plumes
Physicochemistry of Aircraft-Generated Liquid Aerosols, Soot, and Ice Particles. 1. Model Description
Surv. Geophys.
Calculations of Condensation and Chemistry in an Aircraft Contrial
Impact of Emissions From Aircraft and Spacecraft Upon the Atmosphere
Proceedings of an International Scientific Colloquium
, Cologne, Germany, April 18–20, DLR-Mitteilung 94-06, Deutsches Zentrum für Luft- und Raumfahrt, Oberpfaffenhofen and Cologne, Germany.
The Aerosol Dynamics of H2O–H2SO4–HNO3 Mixtures in Aircraft Wakes. a Modeling Study
Climatology: Contrails Reduce Daily Temperature Range—A Brief Interval When the Skies Were Clear of Jets Unmasked an Effect on Climate
Regional Radiative Forcing by Line-Shaped Contrails Derived From Satellite Data
Future Development of Contrail Cover, Optical Depth, and Radiative Forcing: Impacts of Increasing Air Traffic and Climate Change
Aviation, Atmosphere, and Climate—What Has Been Learned
European Conference on Aviation, Atmosphere and Climate (AAC), Proceedings of an International Conference
Jet Engine Exhaust Chemiion Measurements: Implications for Gaseous SO3 and H2SO4
Sulfuric Acid Measurements in the Exhaust Plume of a Jet Aircraft in Flight: Implications for the Sulfuric Acid Formation Efficiency
First Gaseous Sulfur (VI) Measurements in the Simulated Internal Flow of an Aircraft Gas Turbine Engine During Project Partemis
A Unified Model for Ultrafine Aircraft Particle Emissions
Conversion of Sulfur-Dioxide to Sulfur-Trioxide in Gas-Turbine Exhaust
Measurements of Jet Aircraft Emissions at Cruise Altitude. I. The Odd-Nitrogen Gases NO, NO2, HNO2 and HNO3
Evolution of Carbonaceous Aerosol and Aerosol Precursor Emissions Through a Jet Engine
Observation of NO and NO2 in the Young Plume of an Aircraft Jet Engine
Plume and Wake Dynamics, Mixing, and Chemistry Behind a High-Speed Civil Transport Aircraft
Formation of SO3 in Gas-Turbines
Chemical Kinetic Modeling of the Evolution of Gaseous Aerosol Precursors Within a Gas Turbine Engine
Kinetic Modeling of the CO∕H2O∕O2∕NO∕SO2 System: Implications for High Pressure Fall-Off in the SO2+O(+M)=SO3(+M) Reaction
DamJohansen
Heterogeneous Reactions in Aircraft Gas Turbine Engines
The Simulation of 3-Dimensional Viscous-Flow in Turbomachinery Geometries Using a Solution-Adaptive Unstructured Mesh Methodology
Vode—A Variable-Coefficient Ode Solver
CHEMKIN-II: A FORTAN Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics, SAND89–8009
,” Sandia National Laboratories, Report No. SAND89-8009.
Gas Turbine Engine Durability Impacts of High Fuel-Air Ratio Combustors, Part I: Potential for Secondary Combustion of Partially Reacted Fuel
Performance of the Clean Exhaust Engine Concept
A CFD Model for Reacting Flows in an Aero-Engine Hot End Simulator
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Does completing the square always work for quadratic equations?
What is completing the square and when do you use it?
Does completing the square always work?
What is the benefit of completing the square?
Is there ever an advantage to using the quadratic formula?
Completing the square is a method to solve quadratic equations that always works! We will explain the logic behind this method and we'll show you an example of how it works. Also, we discuss the advantages of this method over the quadratic formula and vice versa.
Completing the square is an algebraic technique that lets you solve quadratic equations (or equivalently, factor quadratic trinomials). With this method, you apply basic algebraic operations to transform your problem so that there's a perfect square trinomial on the left side of the equation and a constant term on the right side. Formally, for the equation
x^2 +bx + c = 0
we perform the following transformations:
\frac{b^2}{4} - c
x^2 + bx + c + (\frac{b^2}{4} - c) = \frac{b^2}{4} - c
\therefore x^2 + bx + \frac{b^2}{4} = \frac{b^2}{4} - c
Use the short multiplication formula in reverse to transform the left side:
x^2 + bx + \frac{b^2}{4} = (x^2 + \frac{b}{2})^2
\therefore (x^2 + \frac{b}{2})^2 = \frac{b^2}{4} - c
\frac{b^2}{4} - c
on the right side.
If it's positive, we take the square root of both sides. The equation then has two solutions.
If it's equal to zero, we immediately know that the solution is
x = -\frac{b}{2}
If it's negative, then our equation has no real solutions and only complex solutions.
The method of completing the square will work for every quadratic equation you can imagine. Above, we assumed that the coefficient for
x^2
1
. If your equation starts with
ax^2
a
1
(for example, if you need to solve
2x^2+3x-7=0
), then divide both sides of the equation by
a
(in this example it's
2
). If done correctly, the left side will now start with
x^2
, and you can then apply the steps we've discussed above.
Yes, you can solve any quadratic equation by completing the square, even if the equation has no real solutions! However, in some cases, quite a bit of computation will be required. Here's an example of an equation without real roots solved by completing the square:
"Solve
x^2-4x+13 = 0
9
x^2-4x+13-9 = -9
x^2-4x+4 = -9
Recognize the perfect square on the left-hand side:
(x-2)^2 = -9 = (3i)^2
x-2 = \sqrt{-9} = \pm 3i
x = 2 \pm 3i
The method of solving quadratic equations by completing the square always works because of the short multiplications formula:
(x+b)^2 = x^2+bx+b^2
Consequently, for every quadratic expression of the form
x^2+bx+c
you can add (or subtract) a constant term on both sides of the equation so that we obtain the perfect square trinomial
(x+b)^2
It is certainly a good idea to learn the completing the square method. First, it is much more intuitive than simply plugging the coefficients into the quadratic formula. It gives you a feeling of understanding what happens to your equation and why a particular number is a solution. In fact, it's almost impossible to forget how this method works once you've seen it in action!
Also, the method of completing the square can be used to derive the quadratic formula, should you ever forget it — and it's easy to forget a formula that you simply learned by heart and never fully understood.
Finally, if you ever face integration problems, you'll be thankful you've mastered completing the square, because it pops up repeatedly there.
The advantage of using the quadratic formula method for solving quadratic equations is that it usually requires fewer steps and takes less time than completing the square.
Moreover, it can be easily implemented if you need to solve quadratic equations inside a computer program.
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Borel set - Wikipedia
(Redirected from Borel algebra)
1 Generating the Borel algebra
2 Standard Borel spaces and Kuratowski theorems
3 Non-Borel sets
4 Alternative non-equivalent definitions
Generating the Borel algebraEdit
{\displaystyle T_{\sigma }}
be all countable unions of elements of T
{\displaystyle T_{\delta }}
be all countable intersections of elements of T
{\displaystyle T_{\delta \sigma }=(T_{\delta })_{\sigma }.}
For the base case of the definition, let
{\displaystyle G^{0}}
be the collection of open subsets of X.
{\displaystyle G^{i}=[G^{i-1}]_{\delta \sigma }.}
{\displaystyle G^{i}=\bigcup _{j<i}G^{j}.}
{\displaystyle G\mapsto G_{\delta \sigma }.}
{\displaystyle \aleph _{1}\cdot 2^{\aleph _{0}}\,=2^{\aleph _{0}}.}
In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of Lebesgue measurable sets that exist, which is strictly larger and equal to
{\displaystyle 2^{2^{\aleph _{0}}}}
Standard Borel spaces and Kuratowski theoremsEdit
Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function
{\displaystyle f:X\rightarrow Y}
is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, the set
{\displaystyle f^{-1}(B)}
is measurable in X.
Non-Borel setsEdit
An example of a subset of the reals that is non-Borel, due to Lusin,[4] is described below. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.
{\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}}
{\displaystyle a_{0}}
is some integer and all the other numbers
{\displaystyle a_{k}}
{\displaystyle A}
be the set of all irrational numbers that correspond to sequences
{\displaystyle (a_{0},a_{1},\dots )}
with the following property: there exists an infinite subsequence
{\displaystyle (a_{k_{0}},a_{k_{1}},\dots )}
such that each element is a divisor of the next element. This set
{\displaystyle A}
is not Borel. In fact, it is analytic, and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris, especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.
It's important to note, that while
{\displaystyle A}
can be constructed in ZF, it cannot be proven to be non-Borel in ZF alone. In fact, it is consistent with ZF that
{\displaystyle \mathbb {R} }
is a countable union of countable sets,[5] so that any subset of
{\displaystyle \mathbb {R} }
is a Borel set.
Another non-Borel set is an inverse image
{\displaystyle f^{-1}[0]}
of an infinite parity function
{\displaystyle f\colon \{0,1\}^{\omega }\to \{0,1\}}
. However, this is a proof of existence (via the axiom of choice), not an explicit example.
Alternative non-equivalent definitionsEdit
Norberg and Vervaat[7] redefine the Borel algebra of a topological space
{\displaystyle X}
{\displaystyle \sigma }
–algebra generated by its open subsets and its compact saturated subsets. This definition is well-suited for applications in the case where
{\displaystyle X}
is not Hausdorff. It coincides with the usual definition if
{\displaystyle X}
is second countable or if every compact saturated subset is closed (which is the case in particular if
{\displaystyle X}
is Hausdorff).
Borel isomorphism
Descriptive set theory – Subfield of mathematical logic
Polish space – Concept in topology
^ Mackey, G.W. (1966), "Ergodic Theory and Virtual Groups", Math. Ann., 166 (3): 187–207, doi:10.1007/BF01361167, ISSN 0025-5831, S2CID 119738592
^ Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology?
^ Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 978-0-387-98412-4
^ Lusin, Nicolas (1927), "Sur les ensembles analytiques", Fundamenta Mathematicae (in French), 10: Sect. 62, pages 76–78, doi:10.4064/fm-10-1-1-95
^ Jech, Thomas (2008). The Axiom of Choice. Courier Corporation. p. 142.
^ (Halmos 1950, page 219)
^ Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150
"Borel set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Formal definition of Borel Sets in the Mizar system, and the list of theorems Archived 2020-06-01 at the Wayback Machine that have been formally proved about it.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Borel_set&oldid=1086945090"
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Control and Implementation of Single-Inverter Microgrid
Control and Implementation of Single-Inverter Microgrid ()
Mehdi Moradian, Milad Moradian
Roham Sanat Engineering Co., Tehran, Iran.
In this paper, simulation and implementation way for practical control of Single Inverter Microgrid (SIMG) is presented. This system is equipped by solar system, wind energy conversion system (WECS), and microturbine system. Each DG’s has controlled independently. This is a kind of decentralize control because each DG’s has difference controller. Control of Microgrid (MG) during both grid tie and islanding modes is presented. Solar system and WECS are modeled based on santerno products. This system is compared with three inverter MGs with Centralize control strategy. Controlled signals show that SIMG is more reliable and economical. THD is improved and strategy is simplified for SIMG.
Single Inverter Microgrid, Solar System, Wind Energy Conversion System, Microturbine, Buck and Boost Chopper
Moradian, M. and Moradian, M. (2019) Control and Implementation of Single-Inverter Microgrid. Smart Grid and Renewable Energy, 10, 29-41. doi: 10.4236/sgre.2019.102003.
Renewable energy emerges as an alternative way of generating clean energy. As a result, increasing the use of “green” energy benefits the global environment, making it global concern. This topic relies on a variety of manufacturing and installation industries for its development. As a solution, continuously small and smart grid energy systems appear including renewable energy resources (RES), micro-generators, small energy storage systems, critical and noncritical loads, forming among them a special type of distributed generation system called the Microgrid.
Microgrids knowledge presents a wide range of solution for all of traditional grids problems such as: energy management, system stability, voltage quality, active and reactive power flow control, islanding detection, grid synchronization, and system recovery, all together and provides optimal use of small scale energy generation interaction, increases the penetration of RES, and operates in both grid connected or in autonomous modes. These small but smart grids present a new paradigm for low voltage distribution systems, in which a multilevel control system must be performed in order to ensure the proper operation of the MG [1] .
Recent advances using smart grid technology to maximize operations efficiency, monitoring and supervisory control, power management, and utility grid supplying, make this kind of system a suitable solution for decentralizing the electricity production. Thus, the study of these small grids is imperative because they are helpful to fulfill the maximization of the following issues:
・ Efficiency and demand trends involving technological changes.
・ Advanced energy storage systems.
・ Reduced system restoration time due to transition between the utility grid and the smart grid, improving network reliability.
・ Increased integration of distributed generation resources.
・ Increased security and tolerance to faults.
・ Power quality and system reliability.
・ Decentralized power management: how to generate an amount of power in a lot of places, rather than a lot of power in one place [2] .
The main aim of this paper is to solve problems related to the modeling, control, and power management of distributed generation systems based on the MG’s operating modes. Case study system is equipped by a solar system, WECS, and a microturbine system that jointed to system by dc link and choppers to a common link. All of DG’s connected to the grid by an inverter. Some loads are connected to MG. Control process explained in islanding and grid tie modes.
System modeling is developed based on the variables for an optimal control of active power flow. Voltage source inverters (VSI) are considered as electronic interface in the MG, and need for new analysis tools. Each DG’s and its control strategies are discussed in this article.
2. Photovoltaic System Description
Basic implementation method for PV systems by an auxiliary system is shown in Figure 1 [3] .
The flexibility of the santerno components allows optimal configuration of the various plants, responding to every kind of need [3] . Reliability, considerable reduction of construction times, ease of maintenance, diagnostics and efficient repairs: these are the essential aspects of the modern photovoltaic plants aimed at constantly guaranteeing of maximizing of energy production. A typical implementation of a MV plant is shown in Figure 2.
Options: 1) fiscal meters; 2) a70-terna compliant interface protection relay; 3) general safety device on the MV board; 4) self- powering system; 5) UPS; 6) premium remote control; 7) Rs485 optical-fiber converter for the connection to a string box; 8) optical-fiber switch LAN for the connection between multiple substations; 9) Anti-theft device; 10) fire device; 11) panic bar on inner side of doors.
Figure 1. Typical model of photovoltaic system implementation.
Figure 2. Multiple cabinets with chain or ring configuration [3] .
Voltage and current waveform of simulated solar panel (Based on Table 1 parameters-Appendix) is shown in Figure 3.
3. Wind Power Energy Conversion Description
With a long track record in the industrial and photovoltaic field, santerno designs and manufactures wind turbine inverters of various sizes. All of products are designed for global use and therefore certified according to standards. A wide range of accessories are also available.
Displacement of pitch gearbox, yaw gearbox and generation inverter of WECS is shown in Figure 4.
Technical features of WECS are: high quality design, extremely high reliability, output power factor, top efficiency, maximum operating voltage range, true sin-wave, output grid current, galvanic isolation from grid by means of integrated transformer, freely configurable power curve (32 points) related to turbine PM generator speed. PMSG parameters of WECS have mentioned in Table 2 (Appendix).
Case study control cabinet is shown in Figure 5. Rectifier, inverter, filter and its control blocks are considered.
As it can be seen, output of WECS is shown in Figure 6. Torque-speed, Cp-λ curve, produced power and voltage of DC link of simulated wind power system are displayed. After changing the voltage to a same value by choppers, DC link tied to a mutual inverter for connecting the MGS to grid.
4. Micriturbine System
Recently Microturbine (MT) systems have been much applicable because of their small size, relative low cost, repair and cheap maintenance and relatively simple control. Basic model of microturbine system with its output is discussed in [4] . Different dynamic models have been discussed for microturbines by Rowen, Hannet, Saha and Nern for combustion gas turbine [5] [6] [7] . In 1993 mathematical method of gas turbine by Rowen was developed [5] . While in 1993, Prime Mover Working Group by considering the control of speed, acceleration, fuel and
Figure 3. Non-linear characteristics of V-I and P-I.
Figure 4. Wind power displacement.
Figure 5. Block diagram of WECS.
temperature made this model wider [6] . MT in this article is a small combustion turbine with an installed capacity 25 to 500 KW and a high rotation speed (between 50,000 to 120,000 rpm). This model includes the speed governor, acceleration control block, fuel system control and temperature control. Single-shaft turbine model is considered. Power producer with a Permanent Magnet Synchronous Generator (PMSG) has two poles and smooth poles rotor. Because of high speed
Figure 6. Output of WECS. (a) Torque-Speed characteristics of WT; (b) Cp-λ curve of WT; (c) Produced mechanical power; (d) DC link Voltage in battery bank terminal.
shaft, generators of an AC voltage source will be a high frequency (frequency angular higher than 100,000 rad/sec) [7] . Since turbines moves at high speed, so AC generator is a high-frequency generator which cannot be directly coupled the AC grid [8] . One way to model a system of distributed generation MT, based on all classification system are three following separate parts [9] [10] :
Module 1: mechanical system of turbine and fuel.
Module 2: PMSG and AC/DC rectifier and energy storage devices.
Module 3: AC/DC voltage source inverter, PWM controller.
Mechanical Model and MT Control Functions:
Analysed MT is based on Rowen and Hannet model and dynamic equations of MTS in [5] are investigated.
A typical model of MT system and its electrical connection process is shown in Figure 7.
According to the principle of energy conversion and ignore the inverter losses, total of instantaneous powers in output of AC terminal must be equal to the instant powers in dc terminal like.
{V}_{dc}{I}_{dc}={v}_{a}{i}_{a}+{v}_{b}{i}_{b}+{v}_{c}{i}_{c}
{I}_{DC}
{V}_{DC}
are dc link voltage and current.
In Figure 8 SIMG is evaluated. Benefits of SIMG compare with three-IMG [4] are: low implementation cost (using one inverter is more inexpensive), simple control (control of single grid-side inverter is simpler rather than three inverters), and maintenance. It’s considerable that in DC-side all DG’s should have same voltages by choppers (buck and boost). Sensitive loads are parallel in inverter output and joint to the non-sensitive load by a circuit breaker considered in point of common coupling (PCC) and then all of the system is connected to the main grid. Parameters amount are listed in Appendix.
6. Buck and Boost Design for SIMG
DG’s output DC voltages level are different, hereby choppers should design for assimilated DC voltage surfaces in common bus. By considering the modulation factor (0.8), DC voltage should regulate on 275 volt [11]. By this means, boost chopper for solar system and WECS also buck chopper for microturbine designed.
Design of boost chopper for solar system [11] with consideration of 20 KHz switching frequency, 0.1 ampere current variation and 0.5 volt for voltage variation (input voltage is 35 volt):
Figure 7. Simplified electrical model of MTS.
Figure 8. Case study system by single inverter.
L=\frac{{V}_{s}\left({V}_{a}-{V}_{s}\right)}{f{V}_{a}\Delta I}=\frac{35\left(275-35\right)}{20\times {10}^{3}\times 275\times 0.1}=15\text{\hspace{0.17em}}\text{mH}
{I}_{a}=\frac{P}{{V}_{a}}=\frac{103}{275}=0.374\left(\text{A}\right)
C=\frac{{I}_{a}\left({V}_{a}-{V}_{s}\right)}{f{V}_{a}\Delta {V}_{c}}=\frac{0.374\left(275-35\right)}{2\times {10}^{4}\times 275\times 0.5}=32\text{\hspace{0.17em}}\mu \text{F}
Boost chopper design for wind power system by considering Equations (1)-(3) is as below (input voltage is 242 volt):
L=\frac{{V}_{s}\left({V}_{a}-{V}_{s}\right)}{f{V}_{a}\Delta I}=\frac{242\left(275-242\right)}{20\times {10}^{3}\times 275\times 0.1}=14.5\text{\hspace{0.17em}}\text{mH}
{I}_{a}=\frac{P}{{V}_{a}}=\frac{7100}{275}=25.81\left(\text{A}\right)
C=\frac{{I}_{a}\left({V}_{a}-{V}_{s}\right)}{f{V}_{a}\Delta {V}_{c}}=\frac{25.81\left(275-242\right)}{2\times {10}^{4}\times 275\times 0.5}=309\text{\hspace{0.17em}}\mu \text{F}
Buck chopper design for microturbine system by considering 850 volt for input voltage, 20 kHz switching frequency, 0.5 (V) voltage variation, and 0.5 (A) ampere variation:
L=\frac{{V}_{a}\left({V}_{s}-{V}_{a}\right)}{f{V}_{s}\Delta I}=\frac{275\left(850-275\right)}{20\times {10}^{3}\times 850\times 0.5}=18.6\text{\hspace{0.17em}}\text{mH}
C=\frac{\Delta I}{8f\Delta {V}_{c}}=\frac{0.5}{8\times 2\times {10}^{4}\times 0.5}=6.25\text{\hspace{0.17em}}\mu \text{F}
In 0.25 sec MG cuts by CB on PCC because of fault occur in grid, but in 0.42 sec reclosing process befall on PCC after fault abating. MG synchronization process with the grid is accomplished by inverter controller. Transferred power between MG and grid 1 Kw is considered (Figure 9).
In Figure 10, adaption of voltages of MG and grid for one phase is showed. This operation is feasible by grid side applied controller and existent of Phase Loucked Loop (PLL) on it.
Rotor Angle variations of microturbine and wind power generation are shown in Figure 11 and Figure 12 respectively. Microturbine speed variation also illustrated in Figure 13.
THD of SIMG in grid tie and islanding modes are shown in Figure 14 and Figure 15. Achieved values from simulation are in standard range. By a simple comparison between single inverter and three inverters MG concluded that THD of SIMG is less than THD of three-IMG. This is a benefit of SIMG than three-IMG.
Figure 9. Line three phase current (transferred current between MG and grid).
Figure 10. Current adaption between grid and MG.
Figure 11. Rotor angle variation in microturbine generators by radian.
Figure 12. Rotor angle variation in wind power generators by radian.
Figure 13. Microturbine speed on time unit.
Figure 14. Total harmonic distortion of SIMG for grid tie mode.
Figure 15. Total harmonic distortion of SIMG for islanding mode.
In this paper, simulation and analysis of a typical MG system has been investigated. For this system, decentralize control strategy with one inverter in grid side discussed and simulated. Practical implementation method for the case study system by santerno production has been reviewed. Sensitive loads are for SIMG and non-sensitive loads have been considered for the grid. In disconnecting time of MG from grid by PCC (exerted fault detector), SIMG operated in autonomous mode. Each DG’s produced power applied to sensitive loads. After fault clearance, grid and MG reconnect again and micro smart grid continues to operation. Buck and boost converters are designed to have same voltages at inverters input. In DC link for each DG, battery bank applied for storing the excess energy of DG’s. Reliability of grid has increased by storage devices.
Carrier frequency in VMPPT PWM generator, 3000 Hz and in grid-side controller, 5000 Hz, boost converter parameters:
L=0.0034\text{\hspace{0.17em}}\text{H}
C=0.00561\text{\hspace{0.17em}}\text{F}
. PI coefficients in grid-side controller:
{K}_{p{V}_{dc}}=0.05
{K}_{i{V}_{dc}}=3
{K}_{p{I}_{d}}=2.5
{K}_{i{I}_{d}}=700
{K}_{pIq}=2.5
{K}_{iIq}=700
Table 1. Values and coefficients used in PV cell.
Table 2. Synchronous generator parameters amounts.
[1] (2006) European Smart Grids Technology Platform. Printed on White Chlorine, ISBN 92-79-01414-5.
[2] Quintero, J.C.V. (2009) Decentralized Control Techniques Applied to Electric Power Distributed Generation in Microgrid. Doctorate Thesis, Universitat Plitecnica De Catalunya.
[3] (2012) Solar and Wind Energy Handbook. Santerno-Carraro Group, Italy.
[4] Moradian, M., Tabatabaei, F. and Moradian, S. (2013) Modeling, Control & Fault Management of Microgrids. Smart Grid and Renewable Energy, 4, 99-112.
[5] Rowen, W.I. (1983) Simplified Mathematical Representations of Heavy Duty Gas Turbines. Journal of Engineering for Power, 105, 865-869.
[6] Hannet, L.N. and Khan, A. (1993) Combustion Turbine Dynamic Model Validation from Tests. IEEE Transactions on Power Systems, 8, No. 1.
[7] Saha, A.K., Chowdhury, S., Chowdhury, S.P. and Crossley, P.A. (2009) Modeling and Performance Analysis of a Microturbine as a Distributed Energy Resource. IEEE Transactions on Energy Conversion, 24, No. 2.
[8] Working Group on Prime Mover and Energy Supply Models for System Dynamic Performance Studies. (1994) Dynamic Models for Combined Cycle Plants in Power System Studies. IEEE Transactions on Power Systems, 9, 1698-1708.
[9] Zamora, I., San Martin, J., Mazon, A., San Martin, J. and Aperribay, V. (2005) Emergent Technologies in Electrical Micro-generation. International Journal of Emerging Electric Power Systems, 3, No. 2.
[10] Ong, C.-M. (1998) Dynamic Simulation of Electric Machinery Using Matlab/Simulink. Prentice Hall, Upper Saddle River.
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Unicorn - Uncyclopedia, the content-free encyclopedia
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The unicorn in its larval state.
Fodder; Fantasy Fodder
LegendsRmadeOfUs
Greek Tragi-Comic References
1 Horn Power (HP)
{\displaystyle {\sqrt {\pi }}}
5000 metric tonnes of Bible reprints on your stupid head
All your intelligence are belong to us
The common unicorn (Equus monoclonius) is a quadruped terrestrial horse-like animal. They are acknowledged in the magical world as "completely dead." If you cut a wild ox in half, you get a Unicorn. If you cut a unicorn in half, you get a bloody mess. All unicorns must live! (Also note, do not anger a unicorn, they may try to butthead you). Unicorns are known for their love of juicy couture. Many an attempt as been made to lure unicorns out from their hiding places using said clothing. So far, none of these attempts have been successful. They are believed to be the most manly thing in the world even more than the color Periwinkle.
1 Origin of the name, leading to a particularly weak pun
2 Creation of the Unicorn
7 Unicorn World
8 Attraction to virgins
10 Unicorn droppings
12 Forgotten by Noah
13 Unicorn Holocaust
14 The True Story: Unicorns and Pegasi
Origin of the name, leading to a particularly weak pun
This function was first observed among the Italian children's choirs, especially the castrati, who respected the adage that form follows function. They named it the "Eunuch-horn." When this animal was first discovered it was widely used in such manner giving birth to a unique kind of porn. From this fact the animal was named "unic(que) (p)orn" (hehe, get it?!?!?). When Pamela Anderson was fisting herself one day, the animal walked past and she replaced her fist with it's horn and as she orgasmed she screamed "Unicorn! Unicorn!" She then walked down the road a way and let a large boy with dreadlocks fuck her against a tree.
Its name is sometimes believed to have come from the fact that all unicorns are born with one corn on their hoof, and they tend to be eunuches.
Some people have differing opinions about Unicorns.
For those without comedic tastes, the "questionable parody" of this website called Wikipedia have an article about Unicorn.
The unicorn was one of the first living things on the planet(not). As you can see from the name uni-porn, it evolved around a single piece of porn. The single piece of porn then reproduced with its self creating more, but this was not not a "uniporn" but a multiporn, soon there were hundreds of these bits of porn, and they formed together to creat the UNIPORN! The unicorn was created on Thursday afternoon by God as a follow-up on the failed and woefully-inadequate Leopleurodon 4004BCE model. The standard unicorn features genuine-leather ergonomic bucket seats, hair conditioning, moon-roof, all-weather hoofs that provide superior traction on grassy surfaces, turbo concorde jet engines and affordably low monthly payments; whereas the pegasus, in comparison, just stands there stupidly and dumps huge piles of smelly horse shit. Gorgeous chicks just love guys who are savvy enough to own a Unicorn; so why not "C'mon down!" and trade in your worthless broken-down pegasus for a brand new Unicorn today at your friendly neighborhood Unicorn World? Long story short: Unicorns are far superior to Pegasi.
Unicorns are widely distributed on every continent, including Antarctica and Euthanasia. They are often seen by the millions in vast thundering herds, viciously trampling and skewering anything and everything that have the misfortune of getting in their way. Sometimes also seen at Creationism parties. The unicorn is the official national symbol of the hidden realm of Euthanasia, which has the largest population of unicorns in the world. The Euthanasian banner even has a unicorn somewhere on it. Unicorns are also found on Planet Unicorn, whom an 8 year old gay boy named Shannon wished into existence.
Your average uni- whoops. My mistake.
Individual unicorns are quite docile in temperament, and they make great pets for young children and the elderly, provided that they are continuously fed with large quantities of their favourite foods, such as bratwurst, anchovies, and live kittens. Also, it is a well-known fact that unicorns love to impale naughty children with their magical horns and suck out their intestines like so much spaghetti. They then trample upon the bodies until they are nothing but bloody corpses, thus making a prime example of them. Unicorns who have not been domesticated are glossy-eyed rape machines who will rape you with their vile horn. Stay the fuck away from them.
in the 1950's rumors spread that russia was making an army of unicorns to attack america with, the president ordered that the millitary start an army of unicorns as well, this project, called Project wonderland, failed, and was discontinued after the unicorns rebelled and killed all the trainers. later it was found that a bastard commie told this as a lie.
Uniporns LOVE to eat wheat thins! If they see a box of wheat thins, they will headbutt it, tear it apart, and eat all of its wheaty deliciousness!!! They also eat babies, especially evil babies, and are referred to as the "agressive baby-eaters", unlike the new-and-improved Leonard Cohen. These "gentle" creatures also love cheerios, Irn Bru and chocolate milkshakes (mixed together with shark caviar and windows, for strength).
In 1903, Al Gore invented Unicorn World, a state where everything is truly possible -i.e., the one place in the universe where we dare to dream to the fullest extent. He later renamed it the Uninet. For example, In Unicorn World, many people are rich, famous and married to John Gotti (the little one, not the dead one). This is also where the great tennis players are born. It has been prophesied by the unicorns that Uninet and Euthanasia shall one day merge into one, Unitanasia, and that it shall become the one true superpower of the 4th millennium.
Attraction to virgins
Virgina like rocking horses, as it moves things they like moved. Here is the best of both worlds - a unicorn rocking horse, with a little communism thrown in!
It has often been said that unicorns are attracted to Max Donnelly. I mean, who isn't? In 1978, the Rand Corporation decided to finally determine the truth of that statement. Unfortunately, no virgins could be found for the study.
It is theorized by some that unicorns aren't actually attracted to virgins so much as they are particularly violent towards non-virgins. This is said to account for their low numbers, as this quality was also expressed towards members of their own species, resulting in many a "unibortion". This theory is said to tie in with the Unicorn Holocaust (see below).
Unicorn's are actually attracted to 8 year old gay boys named Shannon. The most popular names for unicorns are Feathers, Cadillac, and Tom Cruise.
Unicorn horn, commonly referred to as Uyi Nee Hoo Ne in Chinese medications, has been used in treating many disorders, but is notably used in the treatment and prevention of cases of incompetence. It also refers to a state of excitement not usually seen this side of Alpha Centauri. On an interesting note, if you polish a unicorn's horn, it squirts out magical "unicorn mayonnaise".
This "unicorn mayonnaise" is a byproduct of the unicorn's sperm. Uni's can reproduce in two different ways, the regular ordinary way with the humping, but their real favourite is ramming that sharped pointed horn up each others arses. Unicorns unleashing their unholy mayo into a receptive human female will result in the creation of Centaur.
On another interesting note, Unicorn mayonnaisse is considered quite a delicacy in Eastern Russia, where restuarant owners send teams of highly skilled "sauce collectors" out into the night to "farm" the substance from unsuspecting sleeping unicorns. A popular serving suggestion is to drizzle a little on hot dog made using bavarian cat meat. Side effects include nausea, loss of interest in life, and general self-loathing.
For thousands of years, people believed that the single horn of the unicorn offered a number of positive and medicinal effects; but what is less known is the magical power of the animal’s scat. Those brave few that own actual unicorn wastes have found themselves in the best of all possible worlds, good fortune, and dreams made real, with a rapid increase in dial tone and personal charm, all the result of a strict diet of rainbows and pure love! A rare collection of unicorn turds are keep at The Center for Fantasticalogical Studies at the Robert Joseph Bell Institute for the Advancement of the Future, in Houston, Texas.
In 1876, this fully-formed fossilized unicorn was discovered in pre-flood Cambrian strata, thereby disproving the Theory of Evolutionism once and for all.
You may also be interested to know that the modern day rhinoceros is said to be a descendant of the unicorn that actually existed a boring amount of time ago. Then again, you might not be interested to know that at all.
The original pair of unicorns was left out in the rain by Noah in 2525 BCE. Most modern historians believe Noah was jealous at how much his wife seemed to like the unicorn Ark. However, the unicorns were able to survive by eating dead flounders. History States that half of the time they floated on a dead cows carcasses. Another theroy is that the unicorns were however brought onto the ship and eaten by lions.
Unicorn Holocaust
In the 1930s, while attending a Nazi Camping Jamboree, a young Adolf Hitler tried to impress his comrades with stories of his amazing sexual prowess. Awed, they lauded Hitler, even declaring him to be The Uberrist Menschen In The Hizzy. This revelry was interrupted, however, by a renegade band of unicorns that came across the young campers. Enticed by their saucy virginal loins, the unicorns brutally sodomized all of the campers, with the sole exception of Hitler, who had never made it past second base anyway. Years later, when experimenting with a Ouija board, Hitler's former camping buddies taunted him mercilessly, making various lewd comments about his weinershnitzel. Enraged and embarrassed, Hitler ordered that all unicorn-kind be exterminated and their bones made into glockenspiels. Fortunately for unicorns everywhere, the monstrous plan was indefinitely shelved in 1944, when Hitler finally found true love with his pet chihuahua, Eva Braun.
The True Story: Unicorns and Pegasi
Unicorn enthusiasts are becoming increasingly disgruntled as of late due to the growing opinion of the masses that unicorns are actually not the same thing as Pegasi. Like seriously, they are. Take for instance the coloring of said majestic beasts. Unicorns are primarily white and various shades of pink depending on rank (darker pink represents higher ranking unicorns in their delightful socitey). The pegasus is also a white creature. Clearly they are the same. Of course many skeptics will mention the solitary horn atop every unicorn's head, and the fact that it is lacking on pegasi. This doesn't matter. Also, some say if unicorns and pegaz0rz were the same then unicorns would have wings like their pega-brothers. NO. Not true. And as I have now proved beyond all reasonable doubt that both the unicorn and the pegasus are very different from marauding centaurs I will leave. Peace.
Pegasus-Unicorn War
Retrieved from "https://uncyclopedia.com/w/index.php?title=Unicorn&oldid=6093308"
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Yvonne_Choquet-Bruhat Knowpia
Yvonne Choquet-Bruhat (French: [ivɔn ʃɔkɛ bʁy.a] ( listen); born 29 December 1923) is a French mathematician and physicist. She has made seminal contributions to the study of Einstein's general theory of relativity, by showing that the Einstein equations can be put into the form of an initial value problem which is well-posed. In 2015, her breakthrough paper was listed by the journal Classical and Quantum Gravity as one of thirteen 'milestone' results in the study of general relativity, across the hundred years in which it had been studied.[1]
Yvonne Choquet-Bruhat in 2006
Well-posedness of the vacuum Einstein Equations
Grand Officier of the Légion d'honneur
Elected to the French Academy of Sciences
Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires (1951)
André Lichnérowicz
She was the first woman to be elected to the French Academy of Sciences and is a Grand Officier of the Légion d'honneur.[2]
Yvonne Bruhat was born in Lille in 1923.[3] Her mother was the philosophy professor Berthe Hubert and her father was the physicist Georges Bruhat, who died in 1945 in the concentration camp Oranienburg-Sachsenhausen. Her brother François Bruhat also became a mathematician, making notable contributions to the study of algebraic groups.
Bruhat undertook her secondary school education in Paris. In 1941 she entered the prestigious Concours Général national competition, winning the silver medal for physics. From 1943 to 1946 she studied at the École Normale Supérieure in Paris, and from 1946 was a teaching assistant there and undertook research advised by André Lichnerowicz.
From 1949 to 1951 she was a research assistant at the French National Centre for Scientific Research, as a result of which she received her doctorate.[4]
In 1951, she became a postdoctoral researcher at the Institute for Advanced Study in Princeton, New Jersey. Her supervisor, Jean Leray, suggested that she study the dynamics of the Einstein field equations. He also introduced her to Albert Einstein, whom she consulted with a few times further during her time at the Institute.
In 1952, Bruhat and her husband were both offered jobs at Marseilles, precipitating her early departure from the Institute. In the same year, she published the local existence and uniqueness of solutions to the vacuum Einstein Equations, her most renowned achievement. Her work proves the well-posedness of the Einstein equations, and started the study of dynamics in General Relativity.
Choquet-Bruhat at the University of California, Berkeley, in 1974.
In 1947, she married fellow mathematician Léonce Fourès. Their daughter Michelle is now (as of 2016) an ecologist. Her doctoral work and early research is under the name Yvonne Fourès-Bruhat. In 1960, Bruhat and Fourès divorced, with her later marrying the mathematician Gustave Choquet and changing her last name to Choquet-Bruhat. She and Choquet had two children; her son, Daniel Choquet, is a neuroscientist and her daughter, Geneviève, is a doctor.
In 1958, she was awarded the CNRS Silver Medal.[5] From 1958 to 1959 she taught at the University of Reims. In 1960 she became a professor at the Université Pierre-et-Marie-Curie (UPMC) in Paris, and has remained professor or professor emeritus until her retirement in 1992.
At the Universite Pierre et Marie Curie she continued to make significant contributions to mathematical physics, notably in general relativity, supergravity, and the non-Abelian gauge theories of the standard model. Her work in 1981 with Demetrios Christodoulou showed the existence of global solutions of the Yang-Mills, Higgs, and Spinor Field Equations in 3+1 Dimensions.[6] Additionally in 1984 she made perhaps the first study by a mathematician of supergravity with results that can be extended to the currently important model in D=11 dimensions.[7]
In 1978 Yvonne Choquet-Bruhat was elected a correspondent to the Academy of Sciences and on 14 May 1979 became the first woman to be elected a full member. From 1980 to 1983 she was President of the Comité international de relativité générale et gravitation ("International committee on general relativity and gravitation"). In 1985 she was elected to the American Academy of Arts and Sciences. In 1986 she was chosen to deliver the prestigious Noether Lecture by the Association for Women in Mathematics.
Technical research contributionsEdit
Choquet-Bruhat's best-known research deals with the mathematical nature of the initial data formulation of general relativity. A summary of results can be phrased purely in terms of standard differential geometric objects.
An initial data set is a triplet (M, g, k) in which M is a three-dimensional smooth manifold, g is a smooth Riemannian metric on M, and k is a smooth (0,2)-tensor field on M.
Given an initial data set (M, g, k), a development of (M, g, k) is a four-dimensional Lorentzian manifold (M, g) together with a smooth embedding f : M → M and a smooth unit normal vector field along f such that f *g = g and such that the second fundamental form of f, relative to the given normal vector field, is k.
In this sense, an initial data set can be viewed as the prescription of the submanifold geometry of an embedded spacelike hypersurface in a Lorentzian manifold.
An initial data set (M, g, k) satisfies the vacuum constraint equations, or is said to be a vacuum initial data set, if the following two equations are satisfied:
{\displaystyle {\begin{aligned}R^{g}-|k|_{g}^{2}+(\operatorname {tr} ^{g}k)^{2}&=0\\\operatorname {div} ^{g}k-d(\operatorname {tr} ^{g}k)&=0.\end{aligned}}}
Here Rg denotes the scalar curvature of g.
One of Choquet-Bruhat's seminal 1952 results states the following:
Every vacuum initial data set (M, g, k) has a development f : M → (M, g) such that g has zero Ricci curvature, and such that every inextendible timelike curve in the Lorentzian manifold (M, g) intersects f(M) exactly once.
Briefly, this can be summarized as saying that (M, g) is a vacuum spacetime for which f(M) is a Cauchy surface. Such a development is called a globally hyperbolic vacuum development. Choquet-Bruhat also proved a uniqueness theorem:
Given any two globally hyperbolic vacuum developments f1 : M → (M1, g1) and f2 : M → (M2, g2) of the same vacuum initial data set, there is an open subset U1 of M1 containing f1(M) and an open subset U2 of M2 containing f1(M), together with an isometry i : (U1, g1) → (U2, g2) such that i(f1(p)) = f2(p) for all p in M.
In a slightly imprecise form, this says: given any embedded spacelike hypersurface M of a Ricci-flat Lorentzian manifold M, the geometry of M near M is fully determined by the submanifold geometry of M.
In an article written with Robert Geroch in 1969, Choquet-Bruhat fully clarified the nature of uniqueness. With a two-page argument in point-set topology using Zorn's lemma, they showed that Choquet-Bruhat's above existence and uniqueness theorems automatically imply a global uniqueness theorem:
Any vacuum initial data set (M, g, k) has a maximal globally hyperbolic vacuum development, meaning a globally hyperbolic vacuum development f : M → (M, g) such that, for any other globally hyperbolic vacuum development f1 : M → (M1, g1), there is an open subset U of M containing f(M) and an isometry i : M1 → U such that i(f1(p)) = f(p) for all p in M.
Any two maximal globally hyperbolic vacuum developments of the same vacuum initial data are isometric to one another.
It is now common to study such developments. For instance, the well-known theorem of Demetrios Christodoulou and Sergiu Klainerman on stability of Minkowski space asserts that if (ℝ3, g, k) is a vacuum initial data set with g and k sufficiently close to zero (in a certain precise form), then its maximal globally hyperbolic vacuum development is geodesically complete and geometrically close to Minkowski space.
Choquet-Bruhat's proof makes use of a clever choice of coordinates, the wave coordinates (which are the Lorentzian equivalent to the harmonic coordinates), in which the Einstein equations become a system of hyperbolic partial differential equations, for which well-posedness results can be applied.
Fourès-Bruhat, Y. Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires. Acta Math. 88 (1952), 141–225. doi:10.1007/bf02392131 Bibcode:1952AcM....88..141F Zbl 0049.19201 MR53338
Choquet-Bruhat, Yvonne; Geroch, Robert. Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14 (1969), 329–335. doi:10.1007/BF01645389 MR0250640
Bruhat, Yvonne. The Cauchy problem. Gravitation: An introduction to current research, pp. 130–168, Wiley, New York, 1962.
Choquet-Bruhat, Yvonne; York, James W., Jr. The Cauchy problem. General relativity and gravitation, Vol. 1, pp. 99–172, Plenum, New York-London, 1980.
Choquet-Bruhat, Yvonne. Positive-energy theorems. Relativity, groups and topology, II (Les Houches, 1983), 739–785, North-Holland, Amsterdam, 1984.
Choquet-Bruhat, Yvonne. Results and open problems in mathematical general relativity. Milan J. Math. 75 (2007), 273–289.
Choquet-Bruhat, Yvonne. Beginnings of the Cauchy problem for Einstein's field equations. Surveys in differential geometry 2015. One hundred years of general relativity, 1–16, Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.
Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile; Dillard-Bleick, Margaret. Analysis, manifolds and physics. Second edition. North-Holland Publishing Co., Amsterdam-New York, 1982. xx+630 pp. ISBN 0-444-86017-7
Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile. Analysis, manifolds and physics. Part II. North-Holland Publishing Co., Amsterdam, 1989. xii+449 pp. ISBN 0-444-87071-7
Choquet-Bruhat, Y. Distributions. (French) Théorie et problèmes. Masson et Cie, Éditeurs, Paris, 1973. x+232 pp.
Choquet-Bruhat, Yvonne. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. ISBN 978-0-19-923072-3
Choquet-Bruhat, Y. Géométrie différentielle et systèmes extérieurs. Préface de A. Lichnerowicz. Monographies Universitaires de Mathématiques, No. 28 Dunod, Paris 1968 xvii+328 pp.
Choquet-Bruhat, Yvonne. Graded bundles and supermanifolds. Monographs and Textbooks in Physical Science. Lecture Notes, 12. Bibliopolis, Naples, 1989. xii+94 pp. ISBN 88-7088-223-3
Choquet-Bruhat, Yvonne. Introduction to general relativity, black holes, and cosmology. With a foreword by Thibault Damour. Oxford University Press, Oxford, 2015. xx+279 pp. ISBN 978-0-19-966645-4, 978-0-19-966646-1
Choquet-Bruhat, Y. Problems and solutions in mathematical physics. Translated from the French by C. Peltzer. Translation editor, J.J. Brandstatter Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1967 x+315 pp.
Choquet-Bruhat, Yvonne. A lady mathematician in this strange universe: memoirs. Translated from the 2016 French original. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. x+351 pp. ISBN 978-981-3231-62-7
Médaille d'Argent du Centre National de la Recherche Scientifique, 1958
Prix Henri de Parville of the Académie des Sciences, 1963
Member (since 1965), Comite International de Relativite Generale et Gravitation (President 1980-1983) [8]
Member, Académie des Sciences, Paris (elected 1979)
Association for Women in Mathematics Noether Lecturer, 1986
She was elevated to the 'Grand Officier' and 'Grand Croix' dignities in the Légion d'Honneur in 2008.[9]
^ Focus issue: Milestones of general relativity. Classical and Quantum Gravity (2015).
^ (in French) Décret of 11 July 2008, published in the JO of 13 July 2008
^ (in French)Notice biographique sur le site de l'Institut des hautes études scientifiques
^ Yvonne Choquet-Bruhat at the Mathematics Genealogy Project
^ Yvonne Choquet-Bruhat page Archived February 19, 2012, at the Wayback Machine at Contribution of 20th Century Women to Physics pages Archived October 29, 2014, at the Wayback Machine of UCLA
^ "Existence of Global Solutions of the Yang-Mills, Higgs, and Spinor Field Equations in 3+1 Dimensions," (with D. Christodoulou) MR654209 Zbl 0499.35076 doi:10.24033/asens.1417
^ Causalite des Theories de Supergravite," Societe Mathematique de France, Asterisque 79-93
^ Presentation on the site for the Association for Women in Mathematics
^ O'Connor, John J.; Robertson, Edmund F., "Yvonne Suzanne Marie-Louise Choquet-Bruhat", MacTutor History of Mathematics archive, University of St Andrews
Wikimedia Commons has media related to Yvonne Choquet-Bruhat.
Contributions of 20th Century Women to Physics'
"Yvonne Choquet-Bruhat", Biographies of Women Mathematicians, Agnes Scott College
Videos of Yvonne Choquet-Bruhat in the AV-Portal of the German National Library of Science and Technology
Christina Sormani, C. Denson Hill, Paweł Nurowski, Lydia Bieri, David Garfinkle, and Nicolás Yunes (August 2017). "A two-part feature: The Mathematics of Gravitational waves". Notices of the American Mathematical Society. American Mathematical Society. 64 (7): 684–707. doi:10.1090/noti1551. ISSN 1088-9477. {{cite journal}}: CS1 maint: uses authors parameter (link)
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Generalization of $p$-adic cohomology ; bounded Witt vectors. A canonical lifting of a variety in characteristic $p \ne 0$ back to characteristic zero
p
-adic cohomology ; bounded Witt vectors. A canonical lifting of a variety in characteristic
p\ne 0
back to characteristic zero
author = {Lubkin, Saul},
title = {Generalization of $p$-adic cohomology ; bounded {Witt} vectors. {A} canonical lifting of a variety in characteristic $p \ne 0$ back to characteristic zero},
AU - Lubkin, Saul
TI - Generalization of $p$-adic cohomology ; bounded Witt vectors. A canonical lifting of a variety in characteristic $p \ne 0$ back to characteristic zero
Lubkin, Saul. Generalization of $p$-adic cohomology ; bounded Witt vectors. A canonical lifting of a variety in characteristic $p \ne 0$ back to characteristic zero. Compositio Mathematica, Tome 34 (1977) no. 3, pp. 225-277. http://archive.numdam.org/item/CM_1977__34_3_225_0/
[1] Saul Lubkin: A p-Adic Proof of Weil's Conjectures. Annals of Mathematics, 87, Nos. 1-2, Jan-March, 1968, 105-255. | MR 224616 | Zbl 0188.53004
[2] Saul Lubkin: Generalization of p-adic Cohomology (to appear).
[3] Ernst Witt: Zyklische Körper und Algebren der Charackteristik p von Grade pn. J. Reine angew. Math., 176 (1936) 126-140. | Zbl 0016.05101
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Loma Prieta Earthquake Analysis - MATLAB & Simulink Example - MathWorks Korea
Create a variable, Time, containing the timestamps sampled at 200Hz with the same length as the other vectors. Represent the correct units with the seconds function and multiplication to achieve the Hz (
{\mathit{s}}^{-1}
) sampling rate. This results in a duration variable which is useful for representing elapsed time.
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Mechanisms - Aloe Protocol
Aloe Blend
How Blend can operate as a hyperstructure
The information provided by Aloe Labs, Inc. (“we,” “us” or “our”) on docs.aloe.capital (the “Site”) is for general informational purposes only. All information on the Site is provided in good faith, however we make no representation or warranty of any kind, express or implied, regarding the accuracy, adequacy, validity, reliability, availability or completeness of any information on the Site.
Rebalance Incentives
Most products require consistent maintenance and upkeep in order to operate properly. Lending markets like Compound and AAVE rely on liquidators to eliminate bad debt, and AMMs like Uniswap rely on arbitrageurs to keep prices up-to-date with centralized exchanges. In general, people who perform these upkeep tasks are called "keepers."
With Aloe Blend, the keepers' job is to rebalance pools every day. Unfortunately this isn't free -- each rebalance costs around $100 -- so Blend has to give keepers some sort of incentive. There are a few off-the-shelf solutions for this (keep3r.network, Gelato, etc) but they tend to inject awkward tokenomics and constrain the incentive design space. So we built our own.
Since each pool needs to be rebalanced, each pool manages its own incentives. The payout is proportional to "urgency" which scales linearly with time. It resets to 0 whenever the pool's primary Uniswap position is re-centered around the current price.
urgency = 100000 * (block.timestamp - _recenterTimestamp) / 24 hours;
reward = gasPrice * gasUsed * urgency / 100000;
As you can see, the payout is dependent on gasPrice. Blend estimates this value by assuming that keepers break-even when they call rebalance -- i.e. reward is approximately equal to transaction cost. Therefore,
gasPrice_{new}=\frac{reward}{gasUsed}
This gas price estimate is performed at the end of every rebalance and appended to a circular buffer so that the pool can track the moving average. There's a separate circular buffer for each ERC20 that's been used for rewards.
Just because a pool has been rebalanced doesn't mean it has been re-centered. Re-centering only takes place when the pool's inventory ratio is 50% ± 1%. If that condition isn't met, limit orders (range orders that are tickSpacing ticks wide) will be placed to get back to 50/50. What this means is that the rebalance incentive may remain high even after multiple rebalance calls. It only goes to 0 after re-centering. You can see which case is playing out by checking rebalance event logs or getRebalanceUrgency().
The previous section showed how Blend determines how much it wants to pay keepers. But how much can it pay keepers, and where does that money come from?
To fund rebalances, Blend streams up to 10% of earnings to its maintenance budgets. There's one for each token: maintenanceBudget0 and maintenanceBudget1. With sufficient TVL, the contract will reach equilibrium and each maintenance budget will be clipped to maxBudget.
maxBudget = K * gasPrice * maxRebalanceGas;
K is set to 20. This creates a buffer so that the pool can incentivize rebalances despite fluctuating earnings and gas prices.
maxRebalanceGas is the maximum amount of gas that has ever been used during a rebalance call.
Once maxBudget is reached, the maintenance budgets can be replenished at-cost. So instead of paying 10% of earnings to the maintenance budgets, the pool only has to pay around $100/day (or however much rebalance transactions cost).
Blend maintains a flag called maintenanceIsSustainable. This flag is true if either maintenanceBudget0 or maintenanceBudget1 is greater than maxBudget, and false if either of them drops below maxBudget / L (L=4).
When maintenance is sustainable, the width of Blend's Uniswap position is based on implied volatility -- usually a few thousand ticks wide. But if maintenance isn't sustainable (i.e., the pool doesn't have enough TVL to incentivize daily rebalances), Blend will expand its Uniswap position considerably -- tens of thousands of ticks wide. In this mode, the pool could go for months without rebalancing, albeit with slightly lower capital efficiency.
If some teams/protocols want to ensure maximum capital efficiency despite low TVL, they can subsidize the cost of rebalancing or simply do it themselves (at a loss).
For token0 and token1, payouts to keepers are constrained to be less than the maintenance budgets. If the contract happens to hold other tokens, keepers can request those and no constraint will be applied (just the usual reward computation described here).
New entries into the gas price oracle's circular buffer are clipped such that they're not less than currentAverage - currentAverage / D where D=10. This helps protect against a certain form of attack, described and simulated here.
After each rebalance, 5% of pool funds will be left sitting in the contract (not deployed to Uniswap or silos). While this slightly reduces capital efficiency, it drastically reduces gas costs for depositing, withdrawing, and placing small limit orders.
To save gas, maintenance budgets are only updated during rebalances. This causes deposit and withdraw to give you slightly (very slightly) less than what you're owed. So if you're depositing or withdrawing a significant fraction of the pool, it's best to do so immediately after a rebalance. You can trigger one yourself or wait for keepers to do it.
Aloe Blend - Previous
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Healthcare is fragmented, not equitable, expensive, and very hard to scale. That said, governments and other organizations are working very hard to change the face of healthcare and its delivery, even with the massive setbacks from the COVID-19 pandemic. Among the many wins over the past few decades, there has been a global increase in life expectancy [Ref], decrease in malaria moralities[Ref] and under 5 mortality[Ref], as well as the eradication of diseases [Ref].
Despite those leaps of progress, we still have a long way to go as the challenges seem to be evolving faster than the solutions can [AMR, super bugs, covid-19, climate change + spillovers]. Luckily for us, technology can grow and develop just as fast, empower novel solutions, and fast track the trial and error processes to filter out sub-par solutions. In this article we will look at one dimension of the multidimensional problems in healthcare - Health Equity - and explore how technology can be leveraged for global health improvement.
We achieve health equity when everyone, everywhere, is able to attain their full health potential and no one is “disadvantaged from achieving this potential because of social position or other socially determined circumstances.”[Ref].
“Health equity is defined as the absence of unfair and avoidable or remediable differences in health among population groups defined socially, economically, demographically or geographically”. - WHO
A key ingredient of Health Equity is ensuring that everyone has access to high quality care services and decisions through qualified and experienced physicians and specialists. In overly simplistic terms, this means increasing the number of doctors and specialists to a suitable physician-patient ratio (PPR) so that there are enough highly-trained care providers to meet demand. According to the WHO, this means achieving a recommended minimum ratio of 4.5 physicians for every 1,000 population (4.5:1,000). Currently, the African continent averages 1.3 physicians per 1,000 population [Ref] - well below the minimum ratio for quality care.
Its easy to see that closing this gap is not easy to do. Humans are notoriously difficult to scale. Training a physician could take close to a decade and training a specialist takes even longer, not accounting for the (important) years needed to get field experience treating patients and encountering new situations.
Diseases do not read books, always trust your clinical acumen. [Ref]
The narrative gets even worse when considering specialists such as dermatologists, pediatricians, pulmonologists, cardiologists, and radiologists. These specialists can be found in developed cities and towns where the compensation rightfully matches their skills. Unfortunately, this leaves a majority of the continent (rural), severely underserved.
The unequal distribution of health specialists, in addition to the low numbers of them to begin with, results in the inequity of access to quality health decisions. Patients in rural areas are often misdiagnosed[Ref], poorly treated, and often given treatments like antibiotics without regard for downstream effects[Ref].
It is important to note that this is (at least) partially responsible for the high mortality and morbidity rates in Africa. Another side effect of this unequal distribution of knowledge and skills is how it affects the quality of prevalence data collected when the data collectors have high misdiagnoses rates.
Technology for Improved Health Decisions
For a few decades now, technology has been affecting every aspect of our lives and healthcare is no different - from the use of EHRs for patient history management to the use of telemedicine to reach specialists thousands of kilometres away.
The technologies of interest in this article are Artificial Intelligence, Machine Learning and Federated/Collaborative Learning.
Artificial Intelligence: Intelligence exhibited by computers and machines. This is a fast growing field of research at the intersection of many domains with computer science and mathematics.
Machine Learning: Similar to how humans learn through examples, extrapolation, analogies, and educated guesses with trial & error, machine learning refers to the process of computers learning to solve a given problem just by looking at the examples provided (historical data) and taking the learned skills to be applied to new situations.
Federated Learning: Federated learning is a machine learning technique that trains an algorithm across multiple decentralized edge devices or servers holding local data samples, without exchanging them [Ref].
Simple Scenario Let's consider a community healthcare worker/clinical officer in a rural area of East Africa. We will call her Rose. Despite Rose's constant hard work and passion, she has received minimal training and her diagnostic abilities are poor compared to the specialists she one day hopes to become.
Within a 100 kilometre radius of Rose's jurisdiction, there are 4 hospitals with medical testing equipment, and of those 4, only one has a very experienced pediatrician.
Now that the scene has been set, let's explore how we can leverage advances in technology to increase the availability of quality health decisions.
Symptom Assessment, the healthcare provider: There is an abundance of engineers and health professionals worldwide working on Artificial Intelligence for symptom assessment, including the WHO/ITU Focus group on the subject[Ref].
These algorithms can be loaded onto a low cost smartphone of the care provider (availability of smartphones in rural areas is another problem entirely), and can be used offline to help assess the symptoms of the patient.
For simplicity, we will assume the disease identification algorithm is simply the product of likelihood of a given condition and the prevalence of that condition:
\begin{align} p(A \, | \, B) &= \frac{p(A, B)}{p(B)} \\ &= \frac{p(B \, | \, A) p(A)}{p(B)} \end{align}
which, for Malaria, roughly translates to:
\begin{align} P(Mal \, | \, Syms) &= \frac{p(Mal, Syms)}{p(Syms)} \\ &= \frac{p(Syms \, | \, Mal) p(Mal)}{p(Syms)} \end{align}
The likelihood function
P(B \, | \, A)
of a given condition is the main algorithm for identifying the underlying illness, whether its using Machine Learning or not.
Collaborative Care, standing on the shoulders of giants: The epidemiology and prevalence of disease presents itself in complex and very convoluted patterns. However, the prevalence of a condition can make an almost impossible diagnosis in one area, the best differential in another area.
Our community healthcare provider's technology, connected to the same technology used by the other 3 hospitals with testing equipment and the one with a specialist, can have access to both their improved decision support models (P(B | A)) and disease prevalence information (P(A)).
This is powerful for the poorly trained community healthcare worker, and this can scale almost infinitely.
The flow of knowledge, through the prevalence and likelihood models, is not limited to from the top (specialists) to the bottom (community care providers) - it can also be from the bottom to the top. Take for example the case of the skin condition scabies: most scabies cases are easily identified at the community level and do not make it to the higher levels of care, often for years.
The proposed solutions are not without their challenges, mainly a lack of reliable infrastructure, but the benefits and opportunities created far outweigh the costs.
Using Federated/Collaborative Learning approaches, we can leverage patient data within connected facilities in a privacy preserving ways that require no data to be transmitted from the host facility. Only all or some aspects of the models are shared with other members of the network.
A healthcare provider with 6 months of basic training can now assess patients with specialist algorithms that are supported by prevalence information from all other connected providers. This is a pretty good start.
Simplified Demo
Adjust the distances (Kms) between the healthcare provider and each facility as well as the condition prevalences to see how her decisions can be affected just by leveraging the skills of others.
Hosp. A
Hosp. B
Hosp. C
Leveraging the results of surrounding providers
Malaria: P(θ) = 0.15
Typhoid: P(θ) = 0.16
Malaria : P(θ) = 0.15
Typhoid : P(θ) = 0.15
What we are doing at Elsa Health
At Elsa Health, we build AI technologies for decision support in rural and low resource areas of Africa. In our work, our algorithms benefit heavily from the shared and distributed architecture for sharing models and priors.
We use these solutions in:
HIV: Medication adherence prediction, symptom assessment for HIV + patients, and tracking lost to follow up patients (LTFU).
Pediatrics: Symptom assessment for children under 14 years.
Anti-microbial Stewardship: Working with stakeholders to keep track of resistant pathogens (bacterial, enteric and fungal).
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What is the velocity of money? — Velocity of money equation
How to calculate velocity of money
Example: Using the velocity of money calculator
The velocity of money calculator determines how many times the money moves between the population or a group of people. It is a concept of economics that affects the money supply, demand and inflation. The velocity of money is a function of the gross national product and the money supply. The money supply refers to the amount of money currently in circulation in the economy. As an individual spends money by purchasing services or goods from another individual, the money moves from person to person.
As the second individual spends that money on something else, this loop continues, and money keeps circulating. You can apply the concept of velocity of money to determine how many times the money has changed hands in this process of transferring money and considering it over a given time and group of individuals.
The velocity of money is connected to an equation of exchange that also factors in the expenditure index and is also a part of the quantity theory of money. You can begin by inputting some numbers in the calculator or reading on to understand what the velocity of money is and how to calculate the velocity of money.
The velocity of money, by definition, is the number of times all the money in the economy has been circulated. Consider a small group of 4 people. A factory owner gives $1000 as salary to his worker.
The worker uses that money to shop at a local supermarket and pays $1000 to the supermarket owner.
The supermarket owner uses that money to restock his shelves and pays $1000 to the wholesaler or supplier.
The supplier who uses the warehouse owned by the factory owner pays the rent for the warehouse of $1000.
In the above group of 4 people, the total transactions made were of value worth $4000. However, the total money in circulation or supply is only $1000. Therefore, money changed hands 4 times per unit time, say, an year.
Movement of money from person to person.
Mathematically, the velocity of money formula is:
\quad V_t = \frac{T}{M}
T
– Total sum of all the transactions;
M
– Money supplied or circulated; and
V_t
– Velocity of money.
The sum of all transactions is the product of the volume of transactions,
N
, and the price index,
P
\quad T = NP
In a broader perspective, the velocity of money is the ratio of a country's gross national product and the money supply in the country. If the velocity of money is higher, one can conclude that the money is being used to purchase goods and services at a faster rate.
To calculate the velocity of money:
Enter the price index,
P
Fill in the volume or the number of transactions,
N
The calculator will return the sum of all transactions,
T
Insert the amount of money in circulation,
M
The calculator returns the velocity of money,
V_t
Calculate the velocity of money if the price index is $15 with 6 transactions in a year. Take the money in circulation as $30.
Enter the price index, P = $15.
Fill in the volume or number of transactions, N = 6.
The calculator will return the sum of all transactions, T = $90 .
Insert the amount of money in circulation, M = $30.
Using the velocity of money formula:
\qquad V_t = \frac{15\times6}{30} = 3/\text{year}
What do you mean by velocity of money?
The velocity of money is the number of times the total money supplied into the economy is circulated or has changed hands. It is the ratio of the gross national product or the sum of all transactions to the amount of money in circulation per unit period of time.
How do I calculate velocity of money?
Multiply the volume of transactions with the price index to obtain the sum of all transactions.
Divide the sum of all transactions by the money supply in the economy to obtain the velocity of money per unit time, say, an year.
What are the factors affecting velocity of money?
The factors affecting the velocity of money are:
Number or frequency of transactions;
Demand of goods and services;
Value of money; and
The expanding type of economy where the demand of goods is higher results in a higher velocity of money compared to a contracting one.
What is quantity theory of money?
It is a concept in macroeconomics that relates the money and goods value. The quantity theory of money states that the price index of goods and services depends on the amount of money in circulation or being supplied.
For instance, if the money in the economy is increased, the prices for goods will also go up proportionally, which would result in inflation. In other words, the increase in the supply of money will reduce its value.
Price index (P)
Volume of transactions (N)
Sum of all transaction (T)
Amount of money in circulation (M)
Velocity of money (Vt)
IRR calculator (Internal Rate of Return) finds the IRR metric for all potential investments.
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Bézout's identity - Wikipedia
Relation between two numbers and their greatest common divisor
Here the greatest common divisor of 0 and 0 is taken to be 0. The integers or polynomials x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that
{\displaystyle |x|\leq |b/d|}
{\displaystyle |y|\leq |a/d|;}
equality occurs only if one of a and b is a multiple of the other. In the polynomial case, the extended Euclidean algorithm produces the unique pair such that
{\displaystyle \deg x<\deg b}
{\displaystyle \deg y<\deg a}
(both inequalities are verified except one of a and b is a multiple of the other).
1 Structure of solutions
3.1 For three or more integers
3.2 For polynomials
3.3 For principal ideal domains
Structure of solutions[edit]
{\displaystyle \left(x-k{\frac {b}{d}},\ y+k{\frac {a}{d}}\right),}
{\displaystyle |x|\leq \left|{\frac {b}{d}}\right|\quad {\text{and}}\quad |y|\leq \left|{\frac {a}{d}}\right|,}
and equality may occur only if one of a and b divides the other.
The two pairs of small Bézout's coefficients are obtained from the given one (x, y) by choosing for k in the above formula either of the two integers next to
{\displaystyle {\frac {x}{b/d}}}
{\displaystyle {\begin{aligned}\vdots \\12&\times ({\color {blue}{-10}})&+\;\;42&\times \color {blue}{3}&=6\\12&\times ({\color {red}{-3}})&+\;\;42&\times \color {red}{1}&=6\\12&\times \color {red}{4}&+\;\;42&\times ({\color {red}{-1}})&=6\\12&\times \color {blue}{11}&+\;\;42&\times ({\color {blue}{-3}})&=6\\12&\times \color {blue}{18}&+\;\;42&\times ({\color {blue}{-5}})&=6\\\vdots \end{aligned}}}
If (x, y) = (18, −5) is the original pair of Bézout coefficients, then
{\displaystyle {\frac {18}{42/6}}\in [2,3]}
yields the minimal pairs via k = 2, respectively k = 3; that is, (18 − 2 ⋅ 7, −5 + 2 ⋅ 2) = (4, −1), and (18 − 3 ⋅ 7, −5 + 3 ⋅ 2) = (−3, 1).
Given any nonzero integers a and b, let
{\displaystyle S=\{ax+by\mid x,y\in \mathbb {Z} {\text{ and }}ax+by>0\}.}
The set S is nonempty since it contains either a or –a (with x = ±1 and y = 0). Since S is a nonempty set of positive integers, it has a minimum element
{\displaystyle d=as+bt}
. To prove that d is the greatest common divisor of a and b, it must be proven that d is a common divisor of a and b, and that for any other common divisor c, one has c ≤ d.
The Euclidean division of a by d may be written
{\displaystyle a=dq+r\quad {\text{with}}\quad 0\leq r<d.}
The remainder r is in
{\displaystyle S\cup \{0\}}
{\displaystyle {\begin{aligned}r&=a-qd\\&=a-q(as+bt)\\&=a(1-qs)-bqt.\end{aligned}}}
Thus r is of the form
{\displaystyle ax+by}
{\displaystyle r\in S\cup \{0\}}
. However, 0 ≤ r < d, and d is the smallest positive integer in S: the remainder r can therefore not be in S, making r necessarily 0. This implies that d is a divisor of a. Similarly d is also a divisor of b, and d is a common divisor of a and b.
{\displaystyle {\begin{aligned}d&=as+bt\\&=cus+cvt\\&=c(us+vt).\end{aligned}}}
That is, c is a divisor of d. Since d > 0, this implies c ≤ d.
For three or more integers[edit]
Bézout's identity can be extended to more than two integers: if
{\displaystyle \gcd(a_{1},a_{2},\ldots ,a_{n})=d}
then there are integers
{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
{\displaystyle d=a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}}
d is the smallest positive integer of this form
every number of this form is a multiple of d
For polynomials[edit]
Main article: Polynomial greatest common divisor § Bézout's identity and extended GCD algorithm
For univariate polynomials f and g with coefficients in a field, there exist polynomials a and b such that af + bg = 1 if and only if f and g have no common root in any algebraically closed field (commonly the field of complex numbers).
For principal ideal domains[edit]
AF+BG theorem, an analogue of Bézout's identity for homogeneous polynomials in three indeterminates
^ Bézout, É. (1779). Théorie générale des équations algébriques. Paris, France: Ph.-D. Pierres.
^ Tignol, Jean-Pierre (2001). Galois' Theory of Algebraic Equations. Singapore: World Scientific. ISBN 981-02-4541-6.
^ Claude Gaspard Bachet (sieur de Méziriac) (1624). Problèmes plaisants & délectables qui se font par les nombres (2nd ed.). Lyons, France: Pierre Rigaud & Associates. pp. 18–33. On these pages, Bachet proves (without equations) "Proposition XVIII. Deux nombres premiers entre eux estant donnez, treuver le moindre multiple de chascun d’iceux, surpassant de l’unité un multiple de l’autre." (Given two numbers [which are] relatively prime, find the lowest multiple of each of them [such that] one multiple exceeds the other by unity (1).) This problem (namely, ax - by = 1) is a special case of Bézout's equation and was used by Bachet to solve the problems appearing on pages 199 ff.
^ See also: Maarten Bullynck (February 2009). "Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany" (PDF). Historia Mathematica. 36 (1): 48–72. doi:10.1016/j.hm.2008.08.009.
Weisstein, Eric W. "Bézout's Identity". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bézout%27s_identity&oldid=1075412739"
Lemmas in number theory
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Psywave (move) - Bulbapedia, the community-driven Pokémon encyclopedia
サイコウェーブ Psychowave
Psywave (Japanese: サイコウェーブ Psychowave) is a damage-dealing Psychic-type move introduced in Generation I. It was TM46 in Generation I. In Generation II, it was the signature move of Misdreavus.
Psywave inflicts a random amount of damage, varying between 1 damage and 1.5× the user's level.
In a link battle, the damage dealt varies between 1 damage and 1.5× the user's level on the user's side, and between 0 damage and 1.5× the user's level on its opponent's side in a link battle. As a result, if a 0 is generated in a link battle, it causes the games to desynchronize as the target will receive 0 damage on the opponent's game, while the user's side will keep generating pseudorandom numbers until it produces a number between 1 and 1.5× the user's level. Additionally, the game will freeze if a Level 0, 1, or 171 Pokémon uses the move, though Pokémon cannot normally be obtained at these levels without the use of glitches.
In Pokémon Stadium (English), Psywave always deals at least 1 damage.
Psywave inflicts a random amount of damage, varying between 1 HP and 1.5× the user's level. The damage is always rounded down; however, Psywave will always deal at least 1 HP of damage.
Psywave does not take weaknesses or resistances into account; however, type immunity is not ignored.
Psywave inflicts damage equal to
{\displaystyle \left\lfloor {\frac {Level_{user}\times (10r+50)}{100}}\right\rfloor }
, where r is a random number from 0 to 10. However, if this calculation produces a value of 0, Psywave deals 1 HP of damage instead.
Psywave can be used as part of a Pokémon Contest combination, with the user gaining extra two appeal points if Calm Mind was used in the prior turn.
{\displaystyle \left\lfloor {\frac {Level_{user}\times (r+50)}{100}}\right\rfloor }
, where r is a random number from 0 to 100. However, if this calculation produces a value of 0, Psywave deals 1 HP of damage instead.
Psywave's accuracy is now 100% instead of 80%.
Psywave cannot be selected in a battle.
StadStad2 A Psychic-type attack of varying intensity. It occasionally inflicts heavy damage.
GSC An attack with variable power.
RSEColo.XD Attacks with a psychic wave of varying intensity.
FRLG The foe is attacked with an odd, hot energy wave that varies in intensity.
DPPtHGSSPBR The foe is attacked with an odd, hot energy wave. The attack varies in intensity.
SMUSUMPE The target is attacked with an odd psychic wave. The attack varies in intensity.
Reason: Pokken
In Red Rescue Team and Blue Rescue Team, Psywave has 19PP and 100% accuracy. The attack travels up to 10 tiles away and hits the first Pokémon it encounters, ally or enemy, dealing damage between 1/2 and 383/256 (~1.49x) of the user's level. It is affected by type matchup and STAB unlike in main series. Before accounting for type matchup and STAB, the move is guaranteed to deal at least 1HP of damage, but no more than 199 (although that number is impossible to achieve under normal gameplay).
MDRB Inflicts damage on the target, even at a distance. The amount of damage depends on the user's level.
MDTDS Inflicts damage on the target, even at a distance. The higher the user's level, the greater the damage.
BSL はなれたポケモンに ダメージをあたえる じぶんの レベルがたかいほど ダメージが おおきくなりやすい
MDGtI Its damage varies. The higher your level, the greater the possible damage.*
It damages even a faraway Pokémon. The higher your level, the greater the damage.*
SMD Its damage varies. The higher your level, the greater the possible damage.*
It damages an enemy. The higher your level, the greater the damage.*
Mismagius Duosion Solrock Exeggutor
The foe is attacked with an odd, hot energy wave.
Jynx Jynx's eyes glow and she raises her hands in the air. The opponent then lifts up into the air, and by waving her arms, Jynx can control wherever the opponent goes.
Exeggutor Exeggutor's eyes glow green and it bends down, firing a white ring at the opponent from the middle of its leaves.
Mandi's Exeggutor Round One - Begin! None
Gastly Gastly shoots the fog around its body at the opponent and the fog morphs into an arrow-like shape that hits the opponent.
Officer Jenny's Gastly Pikachu Re-Volts None
Mr. Mime Mr. Mime raises its hands then brings them down in the shape of a circle and a white circle appears in front of it. It then fires the ring at the opponent.
Toku's Mr. Mime The Psychic Sidekicks! None
Solrock Solrock's body becomes outlined in light blue, and its eyes also glow light blue. Solrock then fires light blue rings at the opponent from its body.
Mismagius Mismagius's outlines glow light blue and it releases a wave of energy out of its body. When it hits the opponent, its body glows multicolored and Mismagius can control the opponent, or Mismagius's body becomes outlined in light blue and it fires invisible or light blue shockwaves at the opponent from its body, or the gems on Mismagius's body glow pink and its body becomes outlined in light blue. It then fires multiple light blue tentacles from the outline at the opponent, or Mismagius's eyes become surrounded in multicolored energy and it fires a beam of multicolored energy from its eyes at the opponent.
Cocoa's Mismagius Arriving in Style! None
Zoey's Mismagius Coming Full - Festival Circle! None
Grings Kodai's Mismagius Zoroark: Master of Illusions None
Chimecho Chimecho's body becomes outlined in light blue and it releases multiple light blue rings from its mouth at the opponent.
Maya's Chimecho Mastering Current Events! None
Reuniclus Reuniclus puts its hands together in front of it and fires a multicolored beam of energy from its hands at the opponent.
Duosion Duosion puts its arms out in front of it and multicolored energy appears and gathers in front of its body. A wavy multicolored beam of energy then fires from the energy at the opponent.
Mewtwo Mega Mewtwo Y creates a white force field that expands rapidly from around it and explodes on contact with the opponent.
Mewtwo (M16) Genesect and the Legend Awakened Mewtwo cannot legally know Psywave in Generation V
Grumpig Grumpig fires a beam of multicolored energy from between its hands at the opponent.
Spoink Spoink fires a beam of multicolored energy from between its hands at the opponent.
Sigilyph Sigilyph's two eyes on its head start to glow and then it fires a beam of multicolored energy from its eyes at the opponent.
Carrie's Sigilyph Cloudy Fate, Bright Future! None
Mewtwo (Tornado)
Mewtwo (Spoon)
Mewtwo (Fork)
Jynx Jynx releases a powerful wave of psychic energy from her body that distorts the air around the opponent and creates a horrible sound in the target's head.
Mewtwo Mewtwo releases a powerful tornado from its body that traps the opponent inside, or Mewtwo creates a spoon out of psychic energy and hits the opponent with it. When in the spoon form, Mewtwo can fire the energy from the spoon by transforming it into a fork and extending it.
Blaine's Mewtwo And Mewtwo Too?! None
Misdreavus Misdreavus releases a beam of energy from its face at the opponent.
Morty's Misdreavus Misdreavus Misgivings None
Girafarig Girafarig releases a powerful wave of psychic energy from its antennae that distorts the air around and causes the opponent's head to hurt.
Harry's Girafarig Great Girafarig None
Solrock Solrock glares at the opponent and releases a powerful wave of psychic energy at the opponent from its body.
Tate's Solrock You Can Fight Day or Night With Lunatone & Solrock None
Mewtwo Mega Mewtwo X creates a spoon out of psychic energy and hits the opponent with it.
Blaine's Mewtwo PS591 None
The user releases a powerful blast of energy at the opponent.
Mismagius Mismagius glows and shoots a blast of energy at the opponent.
Mime Jr. Mime Jr. glows and shoots a blast of energy at the opponent.
Hiori's Mime Jr. Lily Regains Her Memories! Used via Mimic
The user fires a beam of psychic energy at the opponent.
Spoink Spoink fires a beam of psychic energy at the opponent.
Soro's Spoink TA30 Debut
Chinese Cantonese 精神波 Jīngsàhn Bō *
幻象波 Waahnjeuhng Bō *
幻像波浪 Waahnjeuhng Bōlohng *
Mandarin 精神波 Jīngshén Bō *
幻象波 Huànxiàng Bō *
Czech Psychická vlna
Dutch Psygolf
Finnish Psykoaalto
French Vague Psy
German Psywelle
Greek Ψυχοκύμα Psychokýma
Hindi मानसिक लहर Maansik leher
Indonesian Psywave
Italian Psiconda*
Psico-onda*
Korean 사이코웨이브 Saikoweibeu
Polish Psychofala
Brazilian Portuguese Onda Psíquica
Romanian Psywave
Serbian Psihički Talas
Spanish Psicoonda*
Swedish Psykvåg
Hypnos*
Vietnamese Sóng Tâm Linh
Retrieved from "https://bulbapedia.bulbagarden.net/w/index.php?title=Psywave_(move)&oldid=3509936"
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§ Weird free group construction from adjoint functor theorem
We wish to construct the free group on a set
S
. Call the free group
\Gamma S
Call the forgetful functor from groups to sets as
U
The defining property of the free group is that if we are given a mapping
\phi: S \to UG
, a map which tells us where the generators go, there is a unique map
\Gamma \phi: \Gamma S \to G
which maps the generators of the free group via a group homomorphism into
G
. Further, there is a bijection between
\phi
\Gamma \phi
Written differently, there is a bijection
\hom_\texttt{Set}(S, UG) \simeq \hom_\texttt{Group}(\Gamma S, G)
. This is the condition for an adjunction.
The idea to construct
\Gamma S
is roughly, to take all possible maps
f_i: S \to UG
for all groups
G
, take the product of all such maps, and define
\Gamma S \equiv im(\pi_i f_i)
. The details follow.
First off, we can't take all groups, that's too large. So we need to cut down the size somehow. We do this by considering groups with at most
|S|
generators, since that's all the image of the maps
f_i
can be anyway. We're only interested in the image at the end, so we can cut down the groups we consider to be set-sized.
Next, we need to somehow control for isomorphisms. So we first take isomorphism classes of groups with at most
|S|
generators. Call this set of groups
\mathcal G
We then construct all possible maps
f_i: S \to UG
for all possible maps
f
G \in \mathcal G
This lets us construct the product map
f : S \to \prod_{G \in \mathcal G} UG
f(s) \equiv \prod_{G \in \mathcal G} f_i(s)
Now we define the free group
\gamma S \equiv im(f)
. Why does this work?
Well, we check the universal property. Suppose we have some map
h: S \to UH
. This must induce a map
\Gamma h: \Gamma S \to H
We can cut down the map, by writing the map as
h_{im}: S \to im(h)
. This maps into some subset of
UH
, from which we can generate a group
H_{im} \subseteq H
First off, there must be some index
k
f_k = h_{im}
, since the set of maps
\{ f_i \}
covers all possible maps from
S
into groups with those many generators.
This implies we can project the group
\Gamma S
k
th index to get a map from
\Gamma S
H_{im}
We can then inject
H_{im}
H
, giving us the desired map!
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Classification loss for linear classification models - MATLAB - MathWorks Switzerland
Estimate Test-Sample Classification Loss
Find Good Lasso Penalty Using Classification Loss
Classification loss for linear classification models
L = loss(Mdl,X,Y) returns the classification losses for the binary, linear classification model Mdl using predictor data in X and corresponding class labels in Y. L contains classification error rates for each regularization strength in Mdl.
L = loss(Mdl,Tbl,ResponseVarName) returns the classification losses for the predictor data in Tbl and the true class labels in Tbl.ResponseVarName.
L = loss(Mdl,Tbl,Y) returns the classification losses for the predictor data in table Tbl and the true class labels in Y.
L = loss(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can specify that columns in the predictor data correspond to observations or specify the classification loss function.
If you supply weights, then for each regularization strength, loss computes the weighted classification loss and normalizes weights to sum up to the value of the prior probability in the respective class.
L — Classification losses
Classification losses, returned as a numeric scalar or row vector. The interpretation of L depends on Weights and LossFun.
L is the same size as Mdl.Lambda. L(j) is the classification loss of the linear classification model trained using the regularization strength Mdl.Lambda(j).
Estimate the training- and test-sample classification error.
ceTrain = loss(CMdl,X(trainIdx,:),Ystats(trainIdx))
ceTrain = 1.3572e-04
ceTest = loss(CMdl,X(testIdx,:),Ystats(testIdx))
ceTest = 5.2804e-04
Because there is one regularization strength in CMdl, ceTrain and ceTest are numeric scalars.
Load the NLP data set. Preprocess the data as in Estimate Test-Sample Classification Loss, and transpose the predictor data.
Train a binary, linear classification model. Specify to hold out 30% of the observations. Optimize the objective function using SpaRSA. Specify that the predictor observations correspond to columns.
L=\frac{\sum _{j}-{w}_{j}{y}_{j}{f}_{j}}{\sum _{j}{w}_{j}}.
{w}_{j}
{y}_{j}
{f}_{j}
is the raw classification score of observation j. Custom loss functions must be written in a particular form. For rules on writing a custom loss function, see the LossFun name-value pair argument.
linearloss = @(C,S,W,Cost)sum(-W.*sum(S.*C,2))/sum(W);
Estimate the training- and test-sample classification loss using the linear loss function.
ceTrain = loss(CMdl,X(:,trainIdx),Ystats(trainIdx),'LossFun',linearloss,...
'ObservationsIn','columns')
ceTrain = -7.8330
ceTest = loss(CMdl,X(:,testIdx),Ystats(testIdx),'LossFun',linearloss,...
ceTest = -7.7383
1{0}^{-6}
1{0}^{-0.5}
Estimate the test-sample classification error.
ce = loss(Mdl,X(:,testIdx),Ystats(testIdx),'ObservationsIn','columns');
log10(Lambda),log10(numNZCoeff + 1));
1{0}^{-4}
1{0}^{-1}
\sum _{j=1}^{n}{w}_{j}=1.
L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.
L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},
{\stackrel{^}{y}}_{j}
{c}_{{y}_{j}{\stackrel{^}{y}}_{j}}
{\stackrel{^}{y}}_{j}
L=\sum _{j=1}^{n}{w}_{j}I\left\{{\stackrel{^}{y}}_{j}\ne {y}_{j}\right\},
L=-\sum _{j=1}^{n}\frac{{\stackrel{˜}{w}}_{j}\mathrm{log}\left({m}_{j}\right)}{Kn},
{\stackrel{˜}{w}}_{j}
L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).
L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.
L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).
{\gamma }_{jk}={\left(f{\left({X}_{j}\right)}^{\prime }C\right)}_{k}.
{\stackrel{^}{y}}_{j}=\underset{k=1,...,K}{\text{argmin}}{\gamma }_{jk}.
L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.
L=\sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}.
By default, observation weights are prior class probabilities. If you supply weights using Weights, then the software normalizes them to sum to the prior probabilities in the respective classes. The software uses the renormalized weights to estimate the weighted classification loss.
ClassificationLinear | predict | fitclinear
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(Redirected from Double precision floating-point format)
{\displaystyle e}
{\displaystyle (-1)^{\text{sign}}(1.b_{51}b_{50}...b_{0})_{2}\times 2^{e-1023}}
{\displaystyle (-1)^{\text{sign}}\left(1+\sum _{i=1}^{52}b_{52-i}2^{-i}\right)\times 2^{e-1023}}
{\displaystyle 2^{1-1023}=2^{-1022}}
{\displaystyle 2^{1023-1023}=2^{0}}
{\displaystyle 2^{1029-1023}=2^{6}}
{\displaystyle 2^{2046-1023}=2^{1023}}
{\displaystyle (-1)^{\text{sign}}\times 2^{e-1023}\times 1.{\text{fraction}}}
{\displaystyle (-1)^{\text{sign}}\times 2^{1-1023}\times 0.{\text{fraction}}=(-1)^{\text{sign}}\times 2^{-1022}\times 0.{\text{fraction}}}
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Zeno's Paradox - Uncyclopedia, the content-free encyclopedia
For those without comedic tastes, the so-called experts at Wikipedia have an article about Zeno's Paradox.
“Now I understand the importance of not being Zeno.”
~ Oscar Wilde on Zeno
Zeno's niece, Helen of Troy
Zeno claimed to be a great philosopher and mathemagician, but everyone knows that's not true. He was formally known as Zeno of Elea, but he was the only Zeno living from 490 to 430 B.C., so everyone just called him Zeno. Except his niece, Helen of Troy. She just called him "Elle".
1 Of Noble Birth
1.2 A Great Day Of Morning
2.1 Zeno, A Common Thief?
3 Objections Zeno's Theory
3.1 The Denial Objection
3.2 Irrelevance of Points Objection
3.3 The Zeno's Paradox Paradox Objection
Of Noble Birth[edit]
Zeno was born on November 31st, 2012 AD. to proud parents Jesus Christ and Mary Magdalene. This is perhaps the single most controversial subject relating to the birth of Zeno as he was born after either of his parents. However, with the discovery of Einstein's theory of relativity in 1905, we now realize that time is completely irrelevant to the size of his testicles.
Zeno's father, Jesus Christ
Having been born before either of his parents, Zeno had a disturbing childhood. He was raised by his great grandfather Socrates. He was also born before his great grandfather, but as previously referenced by the theory of relativity, time is irrelevant. It is vitally important to the remainder of the article that one remembers this, so I will say it once more: time is irrelevant! He was raised by Socrates, regardless of the birthdates involved.
Zeno's great grandfather, Socrates
A Great Day Of Morning[edit]
When Zeno awoke on Friday, October 13, 478 B.C., he awoke in a pool of blood. Socrates had performed a Pagan ritual the night before which only few men have survived (among them Adolf Hitler and Al Gore). Socrates was not one of these men. The ritual involved cutting out the heart of a pig, then cutting out one's own heart.
The idea is then to replace one's own heart with the recently removed pig's heart before bleeding to death. If successful, you must then replace the stolen pig's heart with your heart. There is really no point to this, it is just the type of thing Pagans like to do.
Emancipated from his great grandfather's "training", Zeno took up his surrogate father's profession and began his career in philosophy. That very same morning, to his great joy, he proved that movement is completely impossible.
Zeno's Paradox[edit]
Zeno began his theory with a very basic idea. His idea was this: people want to move from one place to another. They want to go from their homes to the marketplace to buy food. They want to go from the sofa to the toilet. People have a sort of natural, built-in desire to move. To create a generic example, Zeno decided to label the starting point as point A, and the destination as point B. People want to move from point A to point B.
Zeno realized that to move from point A to point B, one must first move halfway from point A to point B. From the midpoint (which we'll call point C), one must then move halfway toward point B (from point C; placing the person in question 3/4 of the way between points A and B). From this point, D, one must again move halfway toward point B (placing him at 7/8 of the way from A to B). At this point most sane and sensible persons would become very frustrated at always moving halfway between where they are and where they want to be without ever getting to where they want to be, thus giving up on reaching point B altogether.
What most people don't realize however is that in order to move toward point C from point A, one must first move halfway from point A to point C. Thus it becomes impossible for anyone to reach point C, which is only halfway to where they actually wanted to go in the first place. Further, to move toward the midpoint between point A and point C from point A, one must first move halfway toward that midpoint between points A and C from the point A, thus making the movement to the midpoint between points A and C impossible. And so on and so on...
It is in this manner that Zeno managed to prove that all movement is impossible. To move from one place to another defied the laws of philosophics, mathematics, physics, and every other word ending in "ics".
Zeno, A Common Thief?[edit]
Recent discoveries have lead to the belief that Zeno may have created his paradox theory with the sole purpose of robbing his friends and family. Reports show that Zeno himself did not believe his theory, which voided its control over his movement. Some recently discovered ancient scrolls state that Zeno would sneak up behind his peers, explain his theory to them in great detail, then while they were paralyzed with the fear of doing something impossible, he would strip them of everything, and run away cackling in the way that so many mad men do.
Objections Zeno's Theory[edit]
Zeno's assertion that no movement is possible has been hotly contested throughout the ages, most notably by people in motion. While no definitive proof has been developed to counter Zeno's theory, there are several prominent objections to Zeno's work:
The Denial Objection[edit]
Denial, against what you may have been told, is more than just a river in Egypt. It is also a common method of disproving everything. All one must do is refuse to accept something, and they have effectively disproved it. This is the least acceptable method of disproving something, and most people deny it any credibility.
Irrelevance of Points Objection[edit]
This graph probably has nothing to do with Zeno's infinite points, but it looks kind of cool.
Anyone who has taken even the most basic algebra class knows that points are really only relevant when drawn on a piece of paper, on some sort of graph.
Because no one actually wants to walk two inches across a piece of paper, Zeno's ideas that people wanted to move from one point to another is completely ludacris. The majority of the "educated" world refuses to except this idea, however, saying that points also exist in three-dimensional planes and therefore hold real-world applications, such as in Zeno's Paradox. Most of these people are just drowning in Denial.
Another way to look at it is that
{\displaystyle \sum _{k=1}^{\infty }2^{-k}=1}
Also, calculus relies on Zeno's Paradox being false. In fact, the integral
{\displaystyle \int _{0}^{\pi }\sin(x){\sqrt {447-{\sqrt {36}}}}}
yields the number 42.
The Zeno's Paradox Paradox Objection[edit]
If we consider Zeno's Paradox to be true, then we must accept that life is impossible. This is a very dangerous thing to consider, so we shall not dwell on the personal implications this holds. Instead we shall look at Zeno's life.
If life is impossible, then it is impossible for Zeno to actually have existed. If it is impossible for Zeno to have existed, then it is impossible for Zeno to have theorized about movement. If it is impossible for Zeno to have theorized about movement, then it is impossible for Zeno to have created his Paradox Theory. If it is impossible for Zeno to have created his Paradox Theory, it is impossible for the Paradox Theory to exist. If it is impossible for Zeno's Paradox Theory to exist, then we must consider the possibility that life is possible. If we consider that life could be possible, we must consider that Zeno's life may have been possible.
Unfortunately, if we must consider that Zeno's life may have been possible, we must consider that he may have created his Paradox Theory. This creates the Zeno's Paradox Paradox. It is both highly possible, and highly impossible for Zeno's Paradox to exist. Therefore, although things such as movement, life, and wars are impossible, they are also extremely possible.
Although during his lifetime, which surprisingly didn't end until 45 years after he proved its impossibility, Zeno never realized the predicament in which he had placed everyone. When the Nobel Prizes were created it was recognized that Zeno had already been awarded four of these prizes:
Nobel Peace Prize: Having proved that movement was completely impossible, Zeno had effectively proved it impossible to engage in wars of any kind. Zeno effectively ended every war which ever had, or would, happen. For this, he was awarded the Nobel Peace Prize for having brought the world to peace.
Nobel War Prize: Seeing as movement was impossible, so too was life itself proved impossible. For having killed every known living thing, by proving that what they were doing was impossible, Zeno was awarded the Nobel War Prize.
Nobel Mathematics Prize: For having applied mathematics to a real-world, real-life situation in order to end all wars, all life, and all movement, Zeno was awarded the Nobel Mathematics Prize.
Nobel Alpha Geek Prize: For having done something so ghastly Geek-ish as ending every war ever, killing all life, and winning both the Nobel Peace and War Prizes for doing so, in addition to the Nobel Mathematics Prize, Zeno was awarded the Nobel Alpha Geek Prize.
This infuriated Alfred Nobel, founder of the Nobel Prize Association of America and Other Lesser Countries. He demanded that Zeno return all of his awards. Much to Zeno's personal benefit in this case, he had already been dead for more than 2000 years, and the location of his burial was not known. He had taken his Nobel Prizes to his grave with him. To this day there is still a $2.5 million reward for information leading to the discovery of Zeno's grave site and the Nobel Prizes within.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Zeno%27s_Paradox&oldid=5960885"
Tedious know-it-alls of Ancient Greece
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Implemented metrics - Key Features | CatBoost
Parameters for trained model
Parameters for trained or applied model
custom_loss
use-best-model
--loss-function
--custom-metric
--use-best-model
--eval-metric
CatBoost provides built-in metrics for various machine learning problems. These functions can be used for model optimization or reference purposes. See the Objectives and metrics section for details on the calculation principles.
The following parameters can be set for the corresponding classes and are used when the model is trained.
The metric to use in training. The specified value also determines the machine learning problem to solve. Some metrics support optional parameters (see the Objectives and metrics section for details on each metric).
<Metric>[:<parameter 1>=<value>;..;<parameter N>=<value>]
LogLinQuantile
MultiRMSE
MultiCrossEntropy
PairLogit
PairLogitPairwise
QueryRMSE
QuerySoftMax
YetiRank
YetiRankPairwise
StochasticRank
A custom python object can also be set as the value of this parameter (see an example).
For example, use the following construction to calculate the value of Quantile with the coefficient
\alpha = 0.1
Quantile:alpha=0.1
Metric values to output during training. These functions are not optimized and are displayed for informational purposes only. Some metrics support optional parameters (see the Objectives and metrics section for details on each metric).
Calculate the value of CrossEntropy
Calculate the value of Quantile with the coefficient
\alpha = 0.1
Quantilealpha=0.1
Calculate the values of Logloss and AUC
['Logloss', 'AUC']
Values of all custom metrics for learn and validation datasets are saved to the Metric output files (learn_error.tsv and test_error.tsv respectively). The directory for these files is specified in the --train-dir (train_dir) parameter.
Use the visualization tools to see a live chart with the dynamics of the specified metrics.
The metric used for overfitting detection (if enabled) and best model selection (if enabled). Some metrics support optional parameters (see the Objectives and metrics section for details on each metric).
A user-defined function can also be set as the value (see an example).
The following parameters can be set for the corresponding methods and are used when the model is trained or applied.
fit (CatBoost)
fit (CatBoostClassifier)
fit (CatBoostRegressor)
Output the measured evaluation metric to stderr.
Method: catboost.train
\alpha = 0.1
c('CrossEntropy')
'CrossEntropy'
c('Logloss', 'AUC')
\alpha = 0.1
c('Quantilealpha=0.1')
The following command keys can be specified for the corresponding commands and are used when the model is trained or applied.
Params for the catboost fit command:
\alpha = 0.1
<Metric 1>[:<parameter 1>=<value>;..;<parameter N>=<value>],<Metric 2>[:<parameter 1>=<value>;..;<parameter N>=<value>],..,<Metric N>[:<parameter 1>=<value>;..;<parameter N>=<value>]
\alpha = 0.1
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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : VFPDO : Builtins
VFPDO Object Overloaded Builtins
overview of overloaded builtins for VFPDO object.
The functionalities of some Maple builtin commands are extended for use on VFPDO object.
Let Delta be a VFPDO object.
(i) The call type(Delta, t) returns true if t is any of the following types: module, object, anything, appliable and VFPDO. See examples below.
(ii) The calls type(Delta, dependent(x)) and type(Delta, freeof(x)) respectively return true if the differential operator or the independent variables of Delta contain (respectively don't contain) x. See example below.
The indets, has, hastype builtin commands accept a VFPDO object and apply their methods onto the differential operator and the independent variables of the object.
These overloaded builtins are associated with the VFPDO object. For more detail, see Overview of the VFPDO object.
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
Construct an VFPDO object from some differential expressions...
X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}[x]+\mathrm{\eta }\left(x,y\right)\mathrm{D}[y],\mathrm{space}=[x,y]\right)
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}
\mathrm{\Delta }≔\mathrm{VFPDO}\left([a[1]\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)-a[2]\mathrm{\xi }\left(x,y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x,x\right)-\left({a[1]}^{2}+{a[2]}^{2}\right)\mathrm{\eta }\left(x,y\right)],X\right)
\textcolor[rgb]{0,0,1}{\mathrm{\Delta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{→}\left[{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{2}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\right]
[\mathrm{type}\left(\mathrm{\Delta },'\mathrm{VFPDO}'\right),\mathrm{type}\left(\mathrm{\Delta },'\mathrm{object}'\right),\mathrm{type}\left(\mathrm{\Delta },'\mathrm{`module`}'\right),\mathrm{type}\left(\mathrm{\Delta },'\mathrm{appliable}'\right)]
[\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}]
The VFPDO object contains x
\mathrm{type}\left(\mathrm{\Delta },\mathrm{dependent}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{\Delta },\mathrm{freeof}\left(a[1]\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(\mathrm{\Delta },\mathrm{dependent}\left([x,y]\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{indets}\left(\mathrm{\Delta }\right)
{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{has}\left(\mathrm{\Delta },a[1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{hastype}\left(\mathrm{\Delta },'\mathrm{name}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{hastype}\left(\mathrm{\Delta },'\mathrm{list}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The VFPDO Object Overloaded Builtins command was introduced in Maple 2020.
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Comparing Nominal and Real GDP | Macroeconomics | Course Hero
Explain and demonstrate the difference between nominal and real GDP
In the last section, we introduced the difference between real measurements and nominal measurements of the same economic statistic. On this page, we explore this challenging, but important, distinction in more depth.
Table 1. U.S. Nominal GDP (1960-2010)
If an unwary analyst compared nominal GDP in 1960 to nominal GDP in 2010, it might appear that national output had risen by a factor of nearly twenty-seven over this time. This conclusion comes from the simple growth rate formula (or percentage change formula):
(Final GDP – Initial GDP) / Initial GDP = Growth of Nominal GDP
($14,958 - $543) / $543 = 2653%
This conclusion, though, would be highly misleading. Recall that nominal GDP is defined as the quantity of every final good or service produced multiplied by the price at which it was sold, summed up for all goods and services. In other words, nominal GDP is the value of output produced:
\text{Nominal Value of Output}=\text{Price}\times\text{Quantity of Output}
We'll call this the Real-to-Nominal formula.
Watch this video to see an example of how inflation can distort our perception of GDP. In this example, we focus on a simplified economy with only one good: apples.
GDP in year one is $1000 and the GDP in year two is $1200. The price for apples in year one was $0.50 per pound, but it rose to $0.55 per pound in year two. We know that the value of apple production increased, but we want to determine the extent to which we are producing more apples (i.e. more quantity of goods and services). Since the value of apples is the price of apples times the quantity produced, we can determine the quantity of apples produced in any year by dividing the value of apples in that year (e.g. $1000 in year one) by the price of apples in that year (e.g. $0.50 in year one):
\frac{1000}{0.50}=2000\text{ lbs of apples in year one}
We can do the same calculation for year two:
\frac{1200}{0.55}=2182\text{ lbs of apples in year two}
The difference in the number of apples produced is 182 lbs. The growth rate (percentage increase) is
\frac{182}{2000}=.091\text{ or }9.1\%
Now compare this with the growth in the value of apples:
\frac{1200-1000}{1000}=\frac{200}{1000}=0.20\text{ or }20\%
So, we appeared to be producing 20% more apples, but in reality we were only producing less than half that or 9.1%.
In sum, nominal GDP was $1000 in year one and $1200 in year two, while real GDP was 2000 lbs of apples in year one and 2182 lbs in year two. To compare these GDPs in dollars, you can look at Year Two's output using Year One's dollar amount. So
2182 lbs x $0.50 = $1091
Year Two's real GDP in dollars is $1091. If prices were held constant, the growth in GDP would have been $91, and not the $200 implied by the nominal GDP.
Nominal output is the value of what's produced, while real output is the quantity of what's produced (in the previous case, pounds of apples). If we produce more apples we can say our real output has increased.
Now suppose our apply economy from above now produces two goods: apples and xylophones.
Year Price of Apples Quantity of Apples Value of Apples Price of Xylophones Quantity of Xylophones Value of Xylophones
Year One $0.50 2000 lbs $1,000 $10 100 $1,000
Year Two $0.55 2182 lbs $1,200 $12 150 $1,800
It’s easy to compute how much nominal GDP has increased. In year one, the value of apples produced was $1000, and the value of xylophones produced was $1000, so nominal GDP (assuming these are the only two goods in the economy) was $2000. Similarly in year 2, the value of apples produced was $1200, and the value of xylophones produced was $1800, so nominal GDP was $3000. Thus, nominal GDP increased by $1000 (the increase)/$2000 (the nominal GDP in year one)= 50%.
But what has happened to real GDP? Real output of apples has increased from 2000 lbs to 2182 lbs. Real output of xylophones has increased from 100 to 150. How much has real output increased? Can we say that Real GDP has increased from 2100 to 2332 items? The answer is no, because it doesn’t make sense to add apples & xylophones together, since they are used for different purposes and have different values. What we need is a common denominator, which would allow us to compare apples and xylophones. The common denominator economists use for this purpose is price, since price reflects the value of what something is worth.
Remember that we’re using price here as a common denominator to enable us to “add” quantities of apples and xylophones together. But which year's prices should we use? The answer is arbitrary. We can choose the prices from any year as long as we use them with each year's quantities. Suppose we choose the set of prices from year one. We use those prices, both in year one and in year two. Thus, real GDP in year one is
\text{real GDP}_\text{year 1}=\text{Price of apples}_\text{year 1}\times\text{Quantity of apples}_\text{year 1}+\text{Price of xylophones}_\text{year 1}\times\text{Quantity of xylophones}_\text{year 1}
But note that real GDP in year two is
\text{real GDP}_\text{year 2}=\text{Price of apples}_\text{year 1}\times\text{Quantity of apples}_\text{year 2}+\text{Price of xylophones}_\text{year 1}\times\text{Quantity of xylophones}_\text{year 2}
In other words, we compute real GDP in every year using the prices that existed in a single year, in this case year 1. That’s why real GDP is often described as being based on “constant dollars” or “year one dollars”.
Plugging in the values from the table above, yields:
\text{real GDP}_\text{year 1}=(\$0.50\times2000)+(\$10\times100)=\$2000
\text{real GDP}_\text{year 2}=(\$0.50\times2182)+(\$10\times150)=\$2591
In other words real GDP increased by $591/$2000 = 29.6%, which is significantly less than the increase in nominal GDP of 50%.
Why does the distinction between real and nominal GDP matter?
We explained earlier that nominal measures are distorted by the effects of inflation. Thus, nominal GDP inflates the actual quantity of goods and services produced (i.e. real GDP) making it look bigger than it really is. Let’s think of this another way. Real GDP is highly correlated with employment and the standard of living. When real GDP increases, we tend to have more jobs and more goods and services to consume. When businesses need to produce more goods and services, they typically need to hire more workers, which means incomes are up. By contrast, when inflation drives nominal GDP up, there may be no effect on jobs and the standard of living. If businesses are producing the same quantity of goods and services, they don’t need to hire more workers. The same quantity of things just cost more.
an economic statistic measured using actual market prices; i.e. nominal values are not adjusted for inflation; contrast with real valuereal-to-nominal formula: the nominal value of some economic variable (e.g. GDP) is the price level times the real value of that economic variable. real value: an economic statistic measured using prices that existed in an earlier year (or period); real values have been adjusted for inflation since that earlier year; contrast with nominal valuesimple growth rate formula: the growth rate (or percentage change) of any variable X over time is (the value of X in the final period - the value of X in the initial period)/(the value of X in the initial period)
Adjusting Nominal Values to Real Values. Authored by: OpenStax College. Provided by: Rice University. Located at: https://cnx.org/contents/[email protected]:[email protected]/Adjusting-Nominal-Values-to-Re. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected]
Real GDP and nominal GDP. Provided by: Khan Academy. Located at: https://www.khanacademy.org/economics-finance-domain/macroeconomics/gdp-topic/real-nominal-gdp-tutorial/v/real-gdp-and-nominal-gdp. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
Tracking Real GDP over Time. Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:[email protected]/Tracking-Real-GDP-over-Time. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected]
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Created by Mehjabin Abdurrazaque
What is a hexagonal based pyramid?
How do I find the surface area of a hexagonal pyramid?
How do I calculate the lateral area of a hexagonal pyramid?
How do I calculate the base area of hexagonal pyramid?
How do I find the face area of a hexagonal pyramid?
How to use the hexagonal pyramid surface area calculator
Other pyramid related calculators
In this hexagonal pyramid surface area calculator, we help you find the surface area, the lateral area, and the base area of your hexagon-based pyramid.
We also guide you on how to find the surface area of a hexagonal pyramid by hand and show you the formula for the surface area of a regular hexagonal pyramid. Keep reading!
The hexagonal pyramid is a hexagon-based pyramid. Its base has 6 edges and hence, six isosceles (in some cases, equilateral) triangular faces.
Altitude or height (h): The distance between the center of the hexagonal base and the common vertex.
Base edge or base length (a): The length of the base's side.
Slant height (l): The altitude or the distance between the midpoint of the base edge and the vertex.
Apothem (ap): The distance between the midpoint of the base edge and the center of the hexagonal base.
🙋 In Greek, hexa denotes the number 6 and gon means corner. Thus, hexagon is a shape having six corners!
The total surface area of a hexagonal pyramid (SA) is the sum of the area of its hexagonal base (BA) and the areas of its six triangular faces (FA):
SA = BA + (6 × FA)
The formula for the surface area of a regular hexagonal pyramid using base edge (a) and height (h) is:
SA = \frac{3\sqrt{3}}{2}a^2+3a\sqrt{h^2+\frac{3a^2}{4}}
The lateral surface area (LSA) of a hexagonal pyramid is the sum of the areas of its 6 triangular faces. We calculate a hexagonal-based pyramid using the formula:
LSA = 3a\sqrt{h^2+\frac{3a^2}{4}}
A hexagonal pyramid has a hexagonal base. Therefore, the base area of the hexagonal pyramid is half of the product of its base perimeter (P) and apothem (ap):
BA = ap×P/2
In terms of the base edge of the pyramid or the length of the hexagon's side (a), the base area is:
BA = \frac{3\sqrt{3}}{2}a^2
💡 The base area of a hexagonal pyramid is 2.598 times the square of its base length (a).
To learn more about the measurements related to hexagons like apothem and radius, you can use our hexagon calculator.
The face area of a hexagonal pyramid is the area of one of its six triangular faces.
In terms of the height (h) and base edge (a) of the pyramid, we calculate the pyramid's face area as:
FA = \frac{a}{2}\sqrt{h^2+\frac{3a^2}{4}}
⚠️ We use base edge or base length and height of the pyramid to estimate the areas in this surface area of a regular hexagonal pyramid calculator.
To find the surface area of a hexagonal pyramid using our tool:
Choose the desired units from the drop-down list for each variable. For example, base length and height of a hexagonal pyramid are 5 cm each. Select cm for lengths and cm2 for areas.
Enter the base length in the input box for Base edge (a) - 5 cm.
Enter the height of the pyramid in the input box for Height (h) - 5 cm.
Our hexagonal pyramid surface area calculator displays the following measurements immediately:
Slant height (l) - 6.61 cm;
Base perimeter (P) - 30 cm;
Total surface area (SA) - 164.17 cm2;
Base area (BA) - 64.95 cm2; and
Lateral surface area (LSA) - 99.22 cm2.
And that's it! Our right hexagonal pyramid calc finds A (surface area) in no time.
How do I calculate the surface area of hexagonal pyramids using slant height?
You can calculate the surface area (SA) of a hexagonal pyramid with slant height (l) using the formula:
SA = 3×ap×a + 3×a×l
where ap is the apothem of the pyramid and a is the base, and l is the slant height of the pyramid.
How do I find the base area of a pyramid with side of length 4cm?
The given pyramid has an area of 41.568 cm2 if you consider the hexagon-based pyramid. The base area (BA) of a hexagonal pyramid is 2.598 times the square of its base length (a):
BA = 2.598 × a2
Now when you know how to find surface area of a hexagonal pyramid, be sure to check out our other pyramid calculators:
Hexagonal pyramid calculator;
Hexagonal pyramid volume calculator;
Square pyramid calculator;
Square pyramid volume calculator; and
Surface area of a rectangular pyramid calculator.
Base edge (a)
Base perimeter (P)
Total surface area (SA)
Base surface area (BA)
The complementary angles calculator determines the complementary angle to the one you input. You can also use this tool to check if two angles are complementary or not.
The polynomial graphing calculator is here to help you with one-variable polynomials up to degree four. It not only draws the graph, but also finds the functions roots and critical points (if they exist).
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Representation Theory of Finite Dimensional Algebras | EMS Press
Representation theory of finite dimensional algebras has always been inspired by interactions with other subjects, and Oberwolfach meetings traditionally serve as a forum for such exchange of ideas. The main source of interactions are the many problems in representation theory and in other parts of mathematics which can be formulated in terms of representations of finite dimensional associative algebras. The study of non-semisimple representations took off in the late 20th century with key advances, such as the link to Lie algebras and quantum groups via quivers and Hall algebras, and the use of tilting theory and derived categories to pass from known algebras to new classes of algebras.
In modern work, instead of studying an algebra through its category of representations, or derived category, one may study similar but more general categories. Thus the classification of some classes of hereditary abelian categories or Calabi-Yau triangulated categories fits into this setup. Another recent development, which had just started at the time of the last Oberwolfach meeting in February 2005, and is still being played out, is the interaction with cluster algebras.
At the workshop, there were 46 participants. Among them, there were experts from neighbouring subjects like commutative algebra, algebraic topology, and combinatorics. Compared to previous meetings, the number of participants was reduced, which made it difficult to include representatives of many other fields with close links to representation theory of finite dimensional algebras. What follows is a quick survey of the main themes of the 23 lectures given at the meeting.
\textbf{Cluster combinatorics and Calabi-Yau categories arising from representations of algebras.} Cluster algebras were invented by Fomin and Zelevinsky in 2000 with motivations coming from the study of canonical bases in quantum groups and total positivity in algebraic groups. The combinatorics of these algebras were soon recognized to be closely related to those of tilting theory for hereditary algebras. A collective effort over the last few years has led to a good understanding of these relations for certain classes of cluster algebras. This was made possible by the use of
2
-Calabi-Yau categories constructed from representations of algebras. The introductory talks by Reiten and Iyama were devoted to these developments as well as to the impact of recent important work by Derksen-Weyman-Zelevinsky. In an informal evening presentation, Keller put Derksen-Weyman-Zelevinsky's work into a beautiful homological framework. The talk by Geiss presented cutting-edge results towards the construction of `dual PBW-bases' in large classes of cluster algebras. The proofs are based on subtle techniques from the study of quasi-hereditary algebras, as demonstrated in Schr\"oer's talk. Marsh analyzed fine points of the correspondence between cluster variables and rigid indecomposables and disproved a recent conjecture by Fomin-Zelevinsky. A powerful representation-theoretic model for `higher cluster combinatorics' was presented in the talk by Bin Zhu.
\textbf{Categorification via representations.} The method of categorification has been developed and studied successfully in representation theory by Chuang and Rouquier. They constructed
sl_2
-categorifications for blocks of symmetric groups and used them to establish Brou\'e's abelian defect group conjecture for the symmetric groups. A similar philosophy led to the categorification of cluster algebras via certain
2
-Calabi-Yau categories, where the multiplication in the cluster algebra is modeled by direct sums. A more recent and very promising approach due to Leclerc was presented by Keller. In this case the multiplication is modeled by the tensor product in certain categories of representations of quantum affine algebras. Categorifications also play an important role in low dimensional topology, thanks to important work of Khovanov. This connection was the motivation for Stroppel's talk on convolution algebras arising from Springer fibres.
\textbf{Representation dimension of algebras and complexity of triangulated categories.} The representation dimension of an algebra is a homological invariant which Auslander introduced in 1971 and which remained mysterious for many years thereafter. Some of the modern techniques in representation theory provide now a better understanding. An introductory talk by Ringel discussed the basic ideas and some interesting new phenomena for hereditary algebras. Dimensions of triangulated categories were introduced by Rouquier to obtain lower bounds for representation dimensions and Iyengar presented some new techniques to compute them. The talk of Buchweitz provided a more general perspective for the computation of these dimensions by reviewing the work of Beligiannis and Christensen on projective classes and ghosts in triangulated categories. A description of triangulated structures on additive categories in terms of Hochschild cohomology was presented by Pirashvili.
\textbf{Hereditary categories of geometric origin.} Hereditary categories are in some sense the building blocks for many interesting structures in modern representation theory. Typical examples are categories of coherent sheaves which come equipped with some additional geometric structure. Using this extra structure, Lenzing presented a new description of the stable category of vector bundles on a weighted projective line. The talk of Burban discussed an intriguing connection between vector bundles on elleptic curves and solutions of Yang-Baxter equations. A complete classification of abelian 1-Calabi-Yau categories up to derived equivalence was presented by van Roosmalen.
\textbf{Representations of quivers.} Quivers and their representations have always played a central role in the representation theory of finite dimensional algebras. They provide the link to Lie theory, either through the theorems of Gabriel and Kac, relating possible dimension vectors of indecomposable representations to positive roots, or more directly via Ringel's construction of quantum groups using Ringel-Hall algebras. Progress since the last meeting includes Hausel's announcement of a positive solution of Kac's conjecture that the constant term of the polynomial counting the number of absolutely indecomposable representations over a finite field is the corresponding root multiplicity. Hausel was invited to the meeting, but sadly in the end it was not possible for him to attend. Hausel's result involves hyper-K\"ahler geometry, and in his talk Reineke also used geometry, namely the cohomology of moduli spaces of quiver representations, to prove a formula similar to one conjectured by Kontsevich and Soibelman concerning Donaldson-Thomas type invariants. Chapoton and Hille both gave intriguing talks involving tilting modules for quivers, exceptional sequences and braid group actions. Hubery discussed the connections between Hall algebras and cluster algebras and the existence of Hall polynomials for non-simply laced affine diagrams, using species rather than quivers.
\textbf{Further aspects of algebras and their representations.} Representation theory of finite dimensional algebras has developed immensely since its origin, and it has now, as demonstrated above, profound connections to many other fields. However, the `internal' theory of representation theory is still pushed forward: The talk of Skowro\'nski presented results on algebras with generalized standard almost cyclic coherent Auslander-Reiten components. Representation theory of Lie algebras and algebraic groups is intimately related to finite dimensional algebras which are cellular or quasi-hereditary. These are algebras given by a specific filtration of ideals. K\"onig presented work on how to generalize such a filtration further in order to deal with possibly infinite dimensional building blocks. Benson, Carlson and others developed a theory of support varieties for finitely generated modules over a finite group, and they obtained deep structural information about modular representations of finite groups in terms of the group cohomology ring. These results found their analogous twin results for Lie algebras and Steenrod algebras arising in topology. Similar support varieties have since then been defined for instance for complete intersections, quantum groups and arbitrary finite dimensional algebras. A common denominator for these situations is the presence of a ring of cohomological operations, and in the latter case this is provided by the Hochschild cohomology ring. The talk of Avramov gave an overview over recent results and questions on the Hochschild cohomology ring of an algebra arising in this context. Nakano presented results on the cohomology and support varieties for quantum groups in a quest to find relationships between representations for quantum groups and geometric constructions in complex Lie theory.
The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress. In addition there was plenty of time for informal discussions. Thus the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.
Henning Krause, William Crawley-Boevey, Bernhard Keller, Oeyvind Solberg, Representation Theory of Finite Dimensional Algebras. Oberwolfach Rep. 5 (2008), no. 1, pp. 401–472
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Majorana fermion - Simple English Wikipedia, the free encyclopedia
Majorana fermions are named after Ettore_Majorana. This shows him in the 1930s.
A Majorana fermion(/maɪəˈrɑːnə ˈfɛərmiːɒn/[1]), also referred to as a Majorana particle, is a fermion that has the same properties as its antiparticle. Ettore Majorana, an Italian physicist, thought they would exist, in 1937. Majorana disappeared in 1938, and the particles are named after him. As Majorana fermionns are thought to have the same properties as their antiparticles, they cannot have an electric charge. Today, atomic particles with an electric charge are called Dirac fermions. An example for Dirac fermions are electrons, and positrons; they have the same properties, but their electric charge is different.
Neutrinos do not have an electric charge, and might be Majorana fermions, but their status is unclear
2 Standard model of particle physics
3 Extensions to the standard model
The concept goes back to Majorana's suggestion of 1937.[2] Majorana suggested that neutral spin-1⁄2 particles can be described by a real wave equation (the Majorana equation). The wave functions of particle and antiparticle are related by complex conjugation. For this reason, they would be identical to their antiparticle.
The difference between Majorana fermions and Dirac fermions can be expressed mathematically in terms of the creation and annihilation operators of second quantization: The creation operator
{\displaystyle \gamma _{j}^{\dagger }}
creates a fermion in quantum state
{\displaystyle j}
(described by a real wave function), whereas the annihilation operator
{\displaystyle \gamma _{j}}
annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators
{\displaystyle \gamma _{j}^{\dagger }}
{\displaystyle \gamma _{j}}
are distinct, whereas for a Majorana fermion they are identical. The ordinary fermionic annihilation and creation operators
{\displaystyle f}
{\displaystyle f^{\dagger }}
can be written in terms of two Majorana operators
{\displaystyle \gamma _{1}}
{\displaystyle \gamma _{2}}
{\displaystyle f=(\gamma _{1}+i\gamma _{2})/{\sqrt {2}},}
{\displaystyle f^{\dagger }=(\gamma _{1}-i\gamma _{2})/{\sqrt {2}}.}
In supersymmetry models, neutralinos—superpartners of gauge bosons and Higgs bosons—are Majorana.
Standard model of particle physics[change | change source]
The standard model of physics has no Majorana fermions, all particles are Dirac fermions. There are neutrinos, and anti-neutrinos. In the standard model, neutrinos have no mass, though. The question how to differentiate between neutrinos and anti-neutrinos is not settled yet.
Extensions to the standard model[change | change source]
There are different extensions of the standard model. One of them is called Minimal Supersymmetric Standard Model. It allows for supersymmetry, and can transform one kind of particle, the bosons, into the other, the fermions. In condensed matter physics, bound Majorana fermions can appear as quasiparticle excitations—the collective movement of several individual particles, not a single one, and they are governed by non-abelian statistics.
↑ "Quantum Computation possible with Majorana Fermions" at YouTube, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the physicist's name's pronunciation.
↑ Majorana, Ettore; Maiani, Luciano (2006). "A symmetric theory of electrons and positrons". In Bassani, Giuseppe Franco (ed.). Ettore Majorana Scientific Papers. pp. 201–33. doi:10.1007/978-3-540-48095-2_10. ISBN 978-3-540-48091-4. Translated from: Majorana, Ettore (1937). "Teoria simmetrica dell'elettrone e del positrone". Il Nuovo Cimento (in Italian). 14 (4): 171–84. Bibcode:1937NCim...14..171M. doi:10.1007/bf02961314. S2CID 18973190.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Majorana_fermion&oldid=8070349"
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Ground state - Wikipedia
Lowest energy level of a quantum system
Energy levels for an electron in an atom: ground state and excited states. After absorbing energy, an electron may jump from the ground state to a higher-energy excited state.
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In quantum field theory, the ground state is usually called the vacuum state or the vacuum.
If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator that acts non-trivially on a ground state and commutes with the Hamiltonian of the system.
1 Absence of nodes in one dimension
Absence of nodes in one dimension[edit]
In one dimension, the ground state of the Schrödinger equation can be proven to have no nodes.[1]
Consider the average energy of a state with a node at x = 0; i.e., ψ(0) = 0. The average energy in this state would be
{\displaystyle \langle \psi |H|\psi \rangle =\int dx\,\left(-{\frac {\hbar ^{2}}{2m}}\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}+V(x)|\psi (x)|^{2}\right),}
where V(x) is the potential.
With integration by parts:
{\displaystyle \int _{a}^{b}\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}dx=\left[\psi ^{*}{\frac {d\psi }{dx}}\right]_{a}^{b}-\int _{a}^{b}{\frac {d\psi ^{*}}{dx}}{\frac {d\psi }{dx}}dx=\left[\psi ^{*}{\frac {d\psi }{dx}}\right]_{a}^{b}-\int _{a}^{b}\left|{\frac {d\psi }{dx}}\right|^{2}dx}
Hence in case that
{\displaystyle \left[\psi ^{*}{\frac {d\psi }{dx}}\right]_{-\infty }^{\infty }=\lim _{b\to \infty }\psi ^{*}(b){\frac {d\psi }{dx}}(b)-\lim _{a\to -\infty }\psi ^{*}(a){\frac {d\psi }{dx}}(a)}
is equal to zero, one gets:
{\displaystyle -{\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{\infty }\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}dx={\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{\infty }\left|{\frac {d\psi }{dx}}\right|^{2}dx}
Now, consider a small interval around
{\displaystyle x=0}
{\displaystyle x\in [-\varepsilon ,\varepsilon ]}
. Take a new (deformed) wave function ψ'(x) to be defined as
{\displaystyle \psi '(x)=\psi (x)}
{\displaystyle x<-\varepsilon }
{\displaystyle \psi '(x)=-\psi (x)}
{\displaystyle x>\varepsilon }
; and constant for
{\displaystyle x\in [-\varepsilon ,\varepsilon ]}
{\displaystyle \varepsilon }
is small enough, this is always possible to do, so that ψ'(x) is continuous.
{\displaystyle \psi (x)\approx -cx}
{\displaystyle x=0}
{\displaystyle \psi '(x)=N{\begin{cases}|\psi (x)|,&|x|>\varepsilon ,\\c\varepsilon ,&|x|\leq \varepsilon ,\end{cases}}}
{\displaystyle N={\frac {1}{\sqrt {1+{\frac {4}{3}}|c|^{2}\varepsilon ^{3}}}}}
is the norm.
Note that the kinetic-energy densities hold
{\textstyle {\frac {\hbar ^{2}}{2m}}\left|{\frac {d\psi '}{dx}}\right|^{2}<{\frac {\hbar ^{2}}{2m}}\left|{\frac {d\psi }{dx}}\right|^{2}}
everywhere because of the normalization. More significantly, the average kinetic energy is lowered by
{\displaystyle O(\varepsilon )}
by the deformation to ψ'.
Now, consider the potential energy. For definiteness, let us choose
{\displaystyle V(x)\geq 0}
. Then it is clear that, outside the interval
{\displaystyle x\in [-\varepsilon ,\varepsilon ]}
, the potential energy density is smaller for the ψ' because
{\displaystyle |\psi '|<|\psi |}
On the other hand, in the interval
{\displaystyle x\in [-\varepsilon ,\varepsilon ]}
{\displaystyle {V_{\text{avg}}^{\varepsilon }}'=\int _{-\varepsilon }^{\varepsilon }dx\,V(x)|\psi '|^{2}={\frac {\varepsilon ^{2}|c|^{2}}{1+{\frac {4}{3}}|c|^{2}\varepsilon ^{3}}}\int _{-\varepsilon }^{\varepsilon }dx\,V(x)\simeq 2\varepsilon ^{3}|c|^{2}V(0)+\cdots ,}
which holds to order
{\displaystyle \varepsilon ^{3}}
However, the contribution to the potential energy from this region for the state ψ with a node is
{\displaystyle V_{\text{avg}}^{\varepsilon }=\int _{-\varepsilon }^{\varepsilon }dx\,V(x)|\psi |^{2}=|c|^{2}\int _{-\varepsilon }^{\varepsilon }dx\,x^{2}V(x)\simeq {\frac {2}{3}}\varepsilon ^{3}|c|^{2}V(0)+\cdots ,}
lower, but still of the same lower order
{\displaystyle O(\varepsilon ^{3})}
as for the deformed state ψ', and subdominant to the lowering of the average kinetic energy. Therefore, the potential energy is unchanged up to order
{\displaystyle \varepsilon ^{2}}
, if we deform the state
{\displaystyle \psi }
with a node into a state ψ' without a node, and the change can be ignored.
We can therefore remove all nodes and reduce the energy by
{\displaystyle O(\varepsilon )}
, which implies that ψ' cannot be the ground state. Thus the ground-state wave function cannot have a node. This completes the proof. (The average energy may then be further lowered by eliminating undulations, to the variational absolute minimum.)
As the ground state has no nodes it is spatially non-degenerate, i.e. there are no two stationary quantum states with the energy eigenvalue of the ground state (let's name it
{\displaystyle E_{g}}
) and the same spin state and therefore would only differ in their position-space wave functions.[1]
The reasoning goes by contradiction: For if the ground state would be degenerate then there would be two orthonormal[2] stationary states
{\displaystyle \left|\psi _{1}\right\rangle }
{\displaystyle \left|\psi _{2}\right\rangle }
— later on represented by their complex-valued position-space wave functions
{\displaystyle \psi _{1}(x,t)=\psi _{1}(x,0)\cdot e^{-iE_{g}t/\hbar }}
{\displaystyle \psi _{2}(x,t)=\psi _{2}(x,0)\cdot e^{-iE_{g}t/\hbar }}
— and any superposition
{\displaystyle \left|\psi _{3}\right\rangle :=c_{1}\left|\psi _{1}\right\rangle +c_{2}\left|\psi _{2}\right\rangle }
{\displaystyle c_{1},c_{2}}
fulfilling the condition
{\displaystyle |c_{1}|^{2}+|c_{2}|^{2}=1}
would also be a be such a state, i.e. would have the same energy-eigenvalue
{\displaystyle E_{g}}
and the same spin-state.
{\displaystyle x_{0}}
be some random point (where both wave functions are defined) and set:
{\displaystyle c_{1}={\frac {\psi _{2}(x_{0},0)}{a}}}
{\displaystyle c_{2}={\frac {-\psi _{1}(x_{0},0)}{a}}}
{\displaystyle a={\sqrt {|\psi _{1}(x_{0},0)|^{2}+|\psi _{2}(x_{0},0)|^{2}}}>0}
(according to the premise no nodes).
Therefore the position-space wave function of
{\displaystyle \left|\psi _{3}\right\rangle }
{\displaystyle \psi _{3}(x,t)=c_{1}\psi _{1}(x,t)+c_{2}\psi _{2}(x,t)={\frac {1}{a}}\left(\psi _{2}(x_{0},0)\cdot \psi _{1}(x,0)-\psi _{1}(x_{0},0)\cdot \psi _{2}(x,0)\right)\cdot e^{-iE_{g}t/\hbar }.}
{\displaystyle \psi _{3}(x_{0},t)={\frac {1}{a}}\left(\psi _{2}(x_{0},0)\cdot \psi _{1}(x_{0},0)-\psi _{1}(x_{0},0)\cdot \psi _{2}(x_{0},0)\right)\cdot e^{-iE_{g}t/\hbar }=0}
{\displaystyle t}
{\displaystyle \left\langle \psi _{3}|\psi _{3}\right\rangle =|c_{1}|^{2}+|c_{2}|^{2}=1}
{\displaystyle x_{0}}
is a node of the ground state wave function and that is in contradiction to the premise that this wave function cannot have a node.
Note that the ground state could be degenerate because of different spin states like
{\displaystyle \left|\uparrow \right\rangle }
{\displaystyle \left|\downarrow \right\rangle }
while having the same position-space wave function: Any superposition of these states would create a mixed spin state but leave the spatial part (as a common factor of both) unaltered.
Initial wave functions for the first four states of a one-dimensional particle in a box
The wave function of the ground state of a particle in a one-dimensional box is a half-period sine wave, which goes to zero at the two edges of the well. The energy of the particle is given by
{\textstyle {\frac {h^{2}n^{2}}{8mL^{2}}}}
, where h is the Planck constant, m is the mass of the particle, n is the energy state (n = 1 corresponds to the ground-state energy), and L is the width of the well.
The wave function of the ground state of a hydrogen atom is a spherically symmetric distribution centred on the nucleus, which is largest at the center and reduces exponentially at larger distances. The electron is most likely to be found at a distance from the nucleus equal to the Bohr radius. This function is known as the 1s atomic orbital. For hydrogen (H), an electron in the ground state has energy −13.6 eV, relative to the ionization threshold. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom.
The exact definition of one second of time since 1997 has been the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest at a temperature of 0 K.[3]
^ a b See, for example, Cohen, M. (1956). "Appendix A: Proof of non-degeneracy of the ground state" (PDF). The energy spectrum of the excitations in liquid helium (Ph.D.). California Institute of Technology. Published as Feynman, R. P.; Cohen, Michael (1956). "Energy Spectrum of the Excitations in Liquid Helium" (PDF). Physical Review. 102 (5): 1189. Bibcode:1956PhRv..102.1189F. doi:10.1103/PhysRev.102.1189.
^ i.e.
{\displaystyle \left\langle \psi _{1}|\psi _{2}\right\rangle =\delta _{ij}}
^ "Unit of time (second)". SI Brochure. International Bureau of Weights and Measures. Retrieved 2013-12-22.
Feynman, Richard; Leighton, Robert; Sands, Matthew (1965). "see section 2-5 for energy levels, 19 for the hydrogen atom". The Feynman Lectures on Physics. Vol. 3.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Ground_state&oldid=1078301841"
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Munin: Vertical fluxes of nitrate in the seasonal nitracline of the Atlantic sector of the Arctic Ocean
This study compiles colocated oceanic observations of high-resolution vertical profiles of nitrate concentration and turbulent microstructure around the Svalbard shelf slope, covering both the permanently ice-free Fram Strait and the pack ice north of Svalbard. The authors present an overview over the seasonal evolution of the distribution of nitrate and its relation to upper ocean stratification. The average upward turbulent diffusive nitrate flux across the seasonal nitracline during the Arctic summer season is derived, with average values of 0.3 and 0.7 mmol m−2 d−1 for stations with and without ice cover, respectively. The increase under ice-free conditions is attributed to different patterns of stratification under sea ice versus open water. The nitrate flux obtained from microstructure measurements lacked a seasonal signal. However, bottle incubations indicate that August nitrate uptake was reduced by more than an order of magnitude relative to the May values. It remains inconclusive whether the new production was limited by an unidentified factor other than
N{O}_{3}^{-}
supply in late summer, or the uptake was underestimated by the incubation method.
Source at http://dx.doi.org/10.1002/2016JC011779
Randelhoff A, Fer I, Sundfjord A, Tremblay J, Reigstad M. Vertical fluxes of nitrate in the seasonal nitracline of the Atlantic sector of the Arctic Ocean. Journal of Geophysical Research - Oceans. 2016;121(7):5282-5285
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The \emph{Discrete Geometry} workshop was attended by 53 participants from a wide range of geographic regions, many of them young researchers (some supported by a grant from the European Union). The morning sessions consisted of survey talks providing an overview of recent developments in Discrete Geometry: \begin{itemize} \item Extremal problems concerning convex lattice polygons. (Imre B\'ar\'any) \item Universally optimal configurations of points on spheres. (Henry Cohn) \item Polytopes, Lie algebras, computing. (Jes\'us A. De Loera) \item On incidences in Euclidean spaces. (Gy\"orgy Elekes) \item Few-distance sets in
d
-dimensional normed spaces. (Zolt\'an F\"uredi) \item On norm maximization in geometric clustering. (Peter Gritzmann) \item Abstract regular polytopes: recent developments. (Peter McMullen) \item Counting crossing-free configurations in the plane. (Micha Sharir) \item Geometry in additive combinatorics. (J\'ozsef Solymosi) \item Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) \item Forbidden patterns. (J\'anos Pach) \item Projected polytopes, Gale diagrams, and polyhedral surfaces. (G\"unter M. Ziegler) \item What is known about unit cubes? (Chuanming Zong) \end{itemize} There were 16 shorter talks in the afternoon, an open problem session chaired by Jes\'us De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (J\"urgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants.
Martin Henk, Jiří Matoušek, Emo Welzl, Discrete Geometry. Oberwolfach Rep. 2 (2005), no. 2, pp. 925–994
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§ Von neumann: foundations of QM
I wanted to understand what von neumann actually did when he "made QM rigorous", what was missing, and why we need
C^\star
algebras for quantum mechanics, or even "rigged hilbert spaces".
I decided to read Von Neumann: Mathematical foundations of quantum mechanics.
It seems he provides a rigorous footing for QM, without any dirac deltas. In particular, he proves the Reisez representation theorem, allow for transforming bras to kets and vice versa. On the other hand, it does not allow for dirac delta as bras and kets.
The document The role of rigged hilbert spaces in QM provides a gentle introduction on how to add in dirac deltas.
Rigged hilbert spaces (by Gelfand) combine the theory of distributions (by Schwartz), developed to make dirac deltas formal, and the theory of hilbert spaces (by Von Neumann) developed to make quantum mechanics formal.
To be even more abstract, we can move to
C^\star
algebras, which allow us to make QFT rigorous.
So it seems that in total, to be able to write, say, a "rigorous shankar" textbook, one should follow Chapter 2 of Von Neumann, continuing with the next document which lays out how to rig a hilbert space.
At this point, one has enough mathematical machinery to mathematize all of Shankar.
Von neumann: Mathematical foundations of quantum mechanics.
The role of rigged hilbert spaces in QM
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PolynomialMapImage - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ConstructibleSetTools Subpackage : PolynomialMapImage
compute the image of a variety under a polynomial map
PolynomialMapImage(F, PM, R, S)
PolynomialMapImage(F, H, PM, R, S)
PolynomialMapImage(CS, PM, R, S)
list of polynomials in R
polynomial ring (source)
polynomial ring (target)
The command PolynomialMapImage(F, PM, R, S) returns a constructible set cs which is the image of the variety
V\left(F\right)
under the polynomial map PM.
The command PolynomialMapImage(F, H, PM, R, S) returns a constructible set cs which is the image of the difference of the variety
V\left(F\right)
by the variety
V\left(H\right)
The command PolynomialMapImage(CS, PM, R, S) returns a constructible set cs which is the image of the constructible set CS under the polynomial map PM.
Both rings R and S should be over the same base field.
The variable sets of R and S should be disjoint.
The number of polynomials in PM is equal to the number of variables of ring S.
This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form PolynomialMapImage(..) only after executing the command with(RegularChains[ConstructibleSetTools]). However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][PolynomialMapImage](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):
The following example is related to the Whitney umbrella.
R≔\mathrm{PolynomialRing}\left([u,v]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
S≔\mathrm{PolynomialRing}\left([x,y,z]\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
\mathrm{PM}≔[uv,u,{v}^{2}]
\textcolor[rgb]{0,0,1}{\mathrm{PM}}\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}{u}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}^{\textcolor[rgb]{0,0,1}{2}}]
\mathrm{cs}≔\mathrm{PolynomialMapImage}\left([],\mathrm{PM},R,S\right)
\textcolor[rgb]{0,0,1}{\mathrm{cs}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{cs}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs},S\right)
\textcolor[rgb]{0,0,1}{\mathrm{cs}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{Info}\left(\mathrm{cs},S\right)
[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{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}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{y}]]
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find the area of the figur given in ncert book exercise 122 question number 3 page no 206 please illustrate the calculations - Mathematics - TopperLearning.com | kcyjeeuu
Find the area of the Figur given in ncert book Exercise 12.2 Question Number 3 Page no. 206 Please illustrate the calculations.
Asked by Shivam Saxena | 14th Aug, 2010, 04:41: PM
\mathrm{There}\quad \mathrm{are}\quad 5\quad \mathrm{areas}.
\mathrm{Area}\quad I\quad :\quad \mathrm{It}\quad \mathrm{is}\quad \mathrm{an}\quad \mathrm{isosceles}\quad \mathrm{triangle}\quad \mathrm{with}\quad \mathrm{equal}\quad \mathrm{side}\quad 5\mathrm{cm}\quad \mathrm{and}\quad \mathrm{base}\quad 1\mathrm{cm}.
\mathrm{Using}\quad \mathrm{Heron}\text{'}s\quad \mathrm{formula}\quad ,\quad s=\frac{5+5+1}{2}=\frac{11}{2}
s\left(s-a\right)\left(s-b\right)\left(s-c\right)=\frac{11}{2}\left(\frac{11}{2}-5\right)\left(\frac{11}{2}-5\right)\left(\frac{11}{2}-1\right)
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac{11}{2}\frac{1}{2}\frac{1}{2}\frac{9}{2}=\frac{99}{16}
\mathrm{Therefore}\quad \mathrm{Area}\quad I\quad =\quad \mathrm{square}\quad \mathrm{root}\quad \mathrm{of}\quad \frac{99}{16}=\frac{3\sqrt{11}}{4}=2.5\left(\mathrm{approx}\right)
\mathrm{It}\quad \mathrm{is}\quad a\quad \mathrm{rectangle}\quad \mathrm{with}\quad \mathrm{sides}\quad 6.5\mathrm{cm}\quad \mathrm{and}\quad 1\mathrm{cm}.
\mathrm{Therefore}\quad \mathrm{its}\quad \mathrm{area}\quad =\quad 6.5\mathrm{x1}=\frac{13}{2}\mathrm{x1}=\frac{13}{2}=6.5\mathrm{sq}\quad \mathrm{cm}
\mathrm{Area}\quad \mathrm{III}\quad :\quad
\mathrm{This}\quad \mathrm{is}\quad \mathrm{an}\quad \mathrm{isosceles}\quad \mathrm{trapezium}\quad \mathrm{with}\quad \mathrm{parallel}\quad \mathrm{sides}\quad 1\mathrm{cm}\quad \mathrm{and}\quad 2\quad \mathrm{cm}.\quad
\mathrm{Therefore}\quad \mathrm{its}\quad \mathrm{area}\quad =\quad \frac{1}{2}\mathrm{xheight}\quad (\mathrm{sum}\quad \mathrm{of}\quad \mathrm{parallel}\quad \mathrm{sides})
\mathrm{Find}\quad \mathrm{the}\quad \mathrm{height}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{trapezium}\quad \mathrm{by}\quad \mathrm{drawing}\quad \mathrm{perpendicular}\quad \mathrm{on}\quad \mathrm{the}\quad \mathrm{base}\quad \mathrm{and}\quad \mathrm{using}\quad \mathrm{pythagoras}\quad \mathrm{theorem}.
\mathrm{In}\quad \mathrm{the}\quad \mathrm{right}\quad -\mathrm{angled}\quad \mathrm{triangle},\quad \mathrm{hyotenuse}\quad =\quad 1\quad \mathrm{cm}\quad \mathrm{and}\quad \mathrm{base}\quad =\quad 0.5\quad \mathrm{cm}
\mathrm{therefore}\quad (\mathrm{height}{)}^{2}\quad =\quad {1}^{2}-{\left(\frac{1}{2}\right)}^{2}=\quad \frac{3}{4}.\quad \quad
\mathrm{height}\quad =h\quad =\quad \frac{\mathrm{sq}\quad \mathrm{rt}\quad 3}{2}
\mathrm{Area}\quad \mathrm{III}\quad =\frac{1}{2}x\frac{\mathrm{sq}\quad \mathrm{rt}\quad 3}{2}x(1+2)=1.3\mathrm{sq}\quad \mathrm{cm}\quad \left(\mathrm{aprox}\right)
\mathrm{These}\quad \mathrm{are}\quad \mathrm{two}\quad \mathrm{right}\quad \mathrm{angled}\quad \mathrm{triangles}\quad \mathrm{with}\quad \mathrm{base}\quad 6\quad \mathrm{cm}\quad \mathrm{and}\quad \mathrm{height}\quad 1.5\quad \mathrm{cm}\quad
\mathrm{Therefore}\quad \mathrm{the}\quad \mathrm{area}\quad \mathrm{of}\quad \mathrm{one}\quad \mathrm{triangle}=\frac{1}{2}\mathrm{x6x}\frac{3}{2}=\frac{9}{2}=4.5.
\mathrm{The}\quad \mathrm{total}\quad \mathrm{Area}\quad \mathrm{IV}\quad \mathrm{and}\quad V\quad =\quad 2x\quad 4.5=9\mathrm{sq}\quad \mathrm{cm}
=2.5+6.5+1.3+9\quad =19.3\quad \mathrm{sq}\quad \mathrm{cm}\quad \left(\mathrm{aprox}\right)
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(Redirected from Bid-offer spread)
Financial markets concept
Order book depth chart on a currency exchange. The x-axis is the unit price, the y-axis is cumulative order depth. Bids (buyers) on the left, asks (sellers) on the right, with a bid–ask spread in the middle.
The bid–ask spread (also bid–offer or bid/ask and buy/sell in the case of a market maker) is the difference between the prices quoted (either by a single market maker or in a limit order book) for an immediate sale (ask) and an immediate purchase (bid) for stocks, futures contracts, options, or currency pairs. The size of the bid–ask spread in a security is one measure of the liquidity of the market and of the size of the transaction cost.[1] If the spread is 0 then it is a frictionless asset.
2 Types of spreads
2.1 Quoted spread
2.2 Effective spread
2.3 Realized spread
3 Example: currency spread
The trader initiating the transaction is said to demand liquidity, and the other party (counterparty) to the transaction supplies liquidity. Liquidity demanders place market orders and liquidity suppliers place limit orders. For a round trip (a purchase and sale together) the liquidity demander pays the spread and the liquidity supplier earns the spread. All limit orders outstanding at a given time (i.e. limit orders that have not been executed) are together called the Limit Order Book. In some markets such as NASDAQ, dealers supply liquidity. However, on most exchanges, such as the Australian Securities Exchange, there are no designated liquidity suppliers, and liquidity is supplied by other traders. On these exchanges, and even on NASDAQ, institutions and individuals can supply liquidity by placing limit orders.
The bid–ask spread is an accepted measure of liquidity costs in exchange traded securities and commodities. On any standardized exchange, two elements comprise almost all of the transaction cost—brokerage fees and bid–ask spreads. Under competitive conditions, the bid–ask spread measures the cost of making transactions without delay. The difference in price paid by an urgent buyer and received by an urgent seller is the liquidity cost. Since brokerage commissions do not vary with the time taken to complete a transaction, differences in bid–ask spread indicate differences in the liquidity cost.[2]
Types of spreads[edit]
Quoted spread[edit]
The simplest type of bid-ask spread is the quoted spread. This spread is taken directly from quotes, that is, posted prices. Using quotes, this spread is the difference between the lowest asking price (the lowest price at which someone will sell) and the highest bid price (the highest price at which someone will buy). This spread is often expressed as a percent of the midpoint, that is, the average between the lowest ask and highest bid:
{\displaystyle {\text{Quoted Spread}}={\frac {{\hbox{ask}}-{\hbox{bid}}}{\hbox{midpoint}}}\times 100}
Effective spread[edit]
Quoted spreads often over-state the spreads finally paid by traders, due to "price improvement", that is, a dealer offering a better price than the quotes, also known as "trading inside the spread".[3] Effective spreads account for this issue by using trade prices, and are typically defined as:
{\displaystyle {\text{Effective Spread}}=2\times {\frac {|{\hbox{Trade Price}}-{\hbox{Midpoint}}|}{\hbox{Midpoint}}}\times 100}
. The effective spread is more difficult to measure than the quoted spread, since one needs to match trades with quotes and account for reporting delays (at least pre-electronic trading). Moreover, this definition embeds the assumption that trades above the midpoint are buys and trades below the midpoint are sales.[4]
Realized spread[edit]
Quoted and effective spreads represent costs incurred by traders. This cost includes both a cost of asymmetric information, that is, a loss to traders that are more informed, as well as a cost of immediacy, that is, a cost for having a trade being executed by an intermediary. The realized spread isolates the cost of immediacy, also known as the "real cost".[5] This spread is defined as:
{\displaystyle {\text{Realized Spread}}_{k}=2\times {\frac {|{\hbox{Midpoint}}_{k+1}-{\hbox{Traded Price}}_{k}|}{{\hbox{Midpoint}}_{k}}}\times 100}
where the subscript k represents the kth trade. The intuition for why this spread measures the cost of immediacy is that, after each trade, the dealer adjusts quotes to reflect the information in the trade (and inventory effects).
Inner price moves are moves of the bid-ask price where the spread has been deducted.
Example: currency spread[edit]
^ "Spreads – definition". Riskglossary.com. Archived from the original on 2012-08-15. Retrieved 2019-04-24.
^ Demsetz, H. 1968. "The Cost of Transacting." Quarterly Journal of Economics 82: 33–53 [1] doi:10.2307/1882244 JSTOR 1882244
^ Lee, Charles M. C. (July 1993). "Market Integration and Price Execution for NYSE-Listed Securities". The Journal of Finance. 48 (3): 1009–1038. doi:10.2307/2329024. ISSN 0022-1082. JSTOR 2329024.
^ Lee, Charles M. C.; Ready, Mark J. (June 1991). "Inferring Trade Direction from Intraday Data". The Journal of Finance. 46 (2): 733. doi:10.2307/2328845. ISSN 0022-1082. JSTOR 2328845.
^ Huang, Roger D.; Stoll, Hans R. (July 1996). "Dealer versus auction markets: A paired comparison of execution costs on NASDAQ and the NYSE". Journal of Financial Economics. 41 (3): 313–357. doi:10.1016/0304-405x(95)00867-e. ISSN 0304-405X.
Bartram, Söhnke M.; Fehle, Frank R.; Shrider, David (May 2008). "Does Adverse Selection Affect Bid-Ask Spreads for Options?". Journal of Futures Markets. 28 (5): 417–437. doi:10.1002/fut.20316. S2CID 154229351. SSRN 1089222.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bid–ask_spread&oldid=1070505996"
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Astronauts can see the Earth as a shining ball from outer space though Earth does not give out its own - Science - Light Shadows and Reflections - 9254991 | Meritnation.com
Astronauts can see the Earth as a shining ball from outer space though Earth does not give out its own light . Give reason?
\mathrm{astronauts} \mathrm{can} \mathrm{see} \mathrm{the} \mathrm{earth} \mathrm{as} \mathrm{a} \mathrm{shining} \mathrm{ball} \mathrm{from} \mathrm{outer} \mathrm{space} ,\mathrm{since} \mathrm{as} \mathrm{we} \phantom{\rule{0ex}{0ex}}\mathrm{know} \mathrm{that} \mathrm{earth} \mathrm{has} \frac{2}{3}\mathrm{rd} \mathrm{part} \mathrm{of} \mathrm{water} \mathrm{and} \mathrm{remaining} \mathrm{part} \mathrm{of} \mathrm{earth} \mathrm{so} \mathrm{earth} \mathrm{looks} \mathrm{like} \phantom{\rule{0ex}{0ex}}\mathrm{a} \mathrm{blue} \mathrm{shining} \mathrm{ball} \mathrm{from} \mathrm{outer} \mathrm{space} ,\mathrm{also} \mathrm{the} \mathrm{due} \mathrm{to} \mathrm{radiations} \mathrm{which} \mathrm{comes} \mathrm{from}\phantom{\rule{0ex}{0ex}}\mathrm{sun} \mathrm{which} \mathrm{reflects} \mathrm{back} ,\mathrm{due} \mathrm{to} \mathrm{this} \mathrm{asronauts} \mathrm{see} \mathrm{the} \mathrm{earth} \mathrm{as} \mathrm{shining} \mathrm{ball}.
Due to property of reflection of light, the Earth reflect some part of Sunlight to outer space. so Astronauts can see the Earth as a shining ball from outer space or moon. Same as we see the different phases of moon from the Earth.
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The Rigorous Renormalization Group | EMS Press
The workshop on \emph{The Rigorous Renormalization Group}, was attended by more than 40 participants coming mainly from Western Europe and from America. The official programme consisted in 19 lectures of 60 minutes each (plus discussion). Four of them were devoted to noncommutative field theory, three of them presented methods used for and results on the construction of a non-gaussian fixed point in a statistical mechanics/quantum field theory model, and two lectures concerned, respectively, nonlinear
\sigma
-models, the functional renormalization group, and quantum electrodynamics. The remaining six lectures were on the Brockett-Wegner version of the renormalization group, on random walks, on Fermi liquids, on anomalies in quantum field theory, on renormalization in curved spaces and on functional integrals for many boson systems. The scientific programme, the atmosphere and the Oberwolfach style of the meeting, leaving much room for informal discussions and joint work, were generally highly appreciated. The abstracts of the lectures are presented in chronological order.
Christoph Kopper, Vincent Rivasseau, Manfred Salmhofer, The Rigorous Renormalization Group. Oberwolfach Rep. 3 (2006), no. 2, pp. 1027–1076
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Odd primary Steenrod algebra, additive formal group laws, and modular invariants
April, 2006 Odd primary Steenrod algebra, additive formal group laws, and modular invariants
Masateru INOUE
We give a conceptual clarification of Milnor's theorem, which tells us the Hopf algebra structure of the stable co-operations
{H}_{*}H
in the odd primary ordinary cohomology. Directly connecting
{H}_{*}H
with the quasi-strict automorphism group of some
1
-dimensional additive formal group law and modular invariants, we give a new proof of this theorem of Milnor.
Masateru INOUE. "Odd primary Steenrod algebra, additive formal group laws, and modular invariants." J. Math. Soc. Japan 58 (2) 311 - 332, April, 2006. https://doi.org/10.2969/jmsj/1149166777
Secondary: 55N22 , 55P20
Keywords: Eilenberg-MacLane spectrum , formal group laws , modular invariants , multiplicative operations , reduced power operations , Steenrod algebra
Masateru INOUE "Odd primary Steenrod algebra, additive formal group laws, and modular invariants," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 58(2), 311-332, (April, 2006)
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Dynamic range expander - Simulink - MathWorks Switzerland
The Expander block performs dynamic range expansion independently across each input channel. Dynamic range expansion attenuates the volume of quiet sounds below a given threshold. The block uses specified attack, release, and hold times to achieve a smooth applied gain curve.
The Expander block outputs a signal with the same data type as the input signal. The size of the output depends on the size of the input:
Assuming a hard knee characteristic and a steady-state input such that x[n] dB < Threshold (dB), the expansion ratio is defined as
R=\frac{\left(y\left[n\right]-T\right)}{\left(x\left[n\right]-T\right)}
Knee width (dB) — Transition area in the compression characteristic
0 (default) | scalar in the range 0 to 20
y=x+\frac{\left(1-R\right)×{\left(x-T-\frac{W}{2}\right)}^{2}}{\left(2×W\right)}
\left(2×|x-T|\right)\le W
View static characteristic — Open static characteristic plot of dynamic range expander
The plot is updated automatically when parameters of the Expander block change.
Attack time is the time the expander gain takes to rise from 10% to 90% of its final value when the input goes below the threshold. The Attack time (s) parameter smooths the applied gain curve.
Release time is the time the expander gain takes to drop from 90% to 10% of its final value when the input goes above the threshold. The Release time (s) parameter smooths the applied gain curve.
When you select this parameter, an additional input port SC is added to the block. The SC port enables dynamic range expansion of the input signal x using a separate sidechain signal.
The Expander block processes a signal frame by frame and element by element.
{x}_{\text{dB}}\left[n\right]=20×{\mathrm{log}}_{10}|x\left[n\right]|
{x}_{\text{sc}}\left({x}_{\text{dB}}\right)=\left\{\begin{array}{cc}T+\left({x}_{\text{dB}}-T\right)×R& {x}_{\text{dB}}<\left(T-\frac{W}{2}\right)\\ {x}_{\text{dB}}+\frac{\left(1-R\right){\left({x}_{\text{dB}}-T-\frac{W}{2}\right)}^{2}}{2W}& \left(T-\frac{W}{2}\right)\le {x}_{\text{dB}}\le \left(T+\frac{W}{2}\right)\\ {x}_{\text{dB}}& {x}_{\text{dB}}>\left(T+\frac{W}{2}\right)\end{array}\text{ },
where T is the threshold, R is the expansion ratio, and W is the knee width.
{x}_{\text{sc}}\left({x}_{\text{dB}}\right)=\left\{\begin{array}{cc}T+\left({x}_{\text{dB}}-T\right)×R& {x}_{\text{dB}}<T\\ {x}_{\text{dB}}& {x}_{\text{dB}}\ge T\end{array}
{g}_{\text{c}}\left[n\right]={x}_{\text{sc}}\left[n\right]-{x}_{\text{dB}}\left[n\right].
gc[n] is smoothed using specified attack, release, and hold time parameters:
{g}_{\text{s}}\left[n\right]=\left\{\begin{array}{cc}{\alpha }_{\text{A}}{g}_{\text{s}}\left[n-1\right]+\left(1-{\alpha }_{\text{A}}\right){g}_{\text{c}}\left[n\right]& \text{\hspace{0.17em}}\left({C}_{\text{A}}>{T}_{\text{H}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}&\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({g}_{\text{c}}\left[n\right]\le {g}_{\text{s}}\left[n-1\right]\right)\\ {g}_{\text{s}}\left[n-1\right]\text{\hspace{0.17em}}& {C}_{\text{A}}\le {T}_{\text{H}}\\ {\alpha }_{\text{R}}{g}_{\text{s}}\left[n-1\right]+\left(1-{\alpha }_{\text{R}}\right){g}_{\text{c}}\left[n\right]\text{ }& {g}_{\text{c}}\left[n\right]>{g}_{\text{s}}\left[n-1\right]\end{array}
{\alpha }_{\text{A}}=\mathrm{exp}\left(\frac{-\mathrm{log}\left(9\right)}{Fs×{T}_{\text{A}}}\right)\text{\hspace{0.17em}}.
{\alpha }_{\text{R}}=\mathrm{exp}\left(\frac{-\mathrm{log}\left(9\right)}{Fs×{T}_{\text{R}}}\right)\text{\hspace{0.17em}}.
{g}_{\text{lin}}\left[n\right]={10}^{\left(\frac{{g}_{\text{s}}\left[n\right]}{20}\right)}\text{ }.
y\left[n\right]=x\left[n\right]×{g}_{\text{lin}}\left[n\right].
expander | Limiter | Compressor | Noise Gate
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An electric power station (100MW) transmits power to distant load through long and thin cable which of the - Physics - Current Electricity - 7540139 | Meritnation.com
An electric power station (100MW) transmits power to distant load through long and thin cable .
which of the two modes of transmission would result in lesser power wastage : power transmission of: (i)20000V or (ii)200V ?
Gurpreet K. answered this
Power P = 100 MW
Current in the cable when power is transmitted at 20000 V will be'
I=\frac{100×{10}^{6}}{20,000}=5000 A
Power dissipation in this case:
{I}^{2}R={\left(5000\right)}^{2}R \left(where R is the resistance of cables\right)
Current in the cable when power is transmitted at 200 V will be
I\text{'}=\frac{100×{10}^{6}}{200}=500000 A
I{\text{'}}^{2}R={\left(500000\right)}^{2}R
Comparing the two, we can see that power dissipation is much less in first case. So (i) will be correct choice.
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