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{\displaystyle f(x,y)={\frac {2xy}{x-y}}.}
{\displaystyle {\frac {\partial }{\partial x}}\left({\frac {f(x)}{g(x)}}\right)={\frac {f'(x)g(x)-g'(x)f(x)}{g(x)^{2}}}}
{\displaystyle {\frac {\partial }{\partial x}}\left({\frac {x^{2}}{x+1}}\right)={\frac {2x(x+1)-x^{2}}{(x+1)^{2}}}}
{\displaystyle {\frac {\partial }{\partial x}}f(x)g(x)=f'(x)g(x)+g'(x)f(x)}
{\displaystyle {\frac {\partial }{\partial x}}x(x+1)=(x+1)+x}
{\displaystyle {\frac {\partial }{\partial y}}xy=x{\frac {\partial }{\partial y}}y=x.}
The word 'marginal' should make you immediately think of a derivative. In this case, the marginal is just the partial derivative with respect to a particular variable.
The teacher has also added the additional restriction that you should not leave your answer with negative exponents.
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Will Trump's pants catch fire on at least five more occasions than Biden's during the month of July? | Metaculus
Will Trump's pants catch fire on at least five more occasions than Biden's during the month of July?
Politifact is an independent fact-checking organisation which focuses primarily, but not entirely, on US politics. Claims are assigned a rating on the “Truth-o-meter” ranging from “True” to “Pants on Fire”. For a statement to qualify as “Pants on Fire”, it should both not be accurate and make what politifact considers to be “a ridiculous claim”.
This question asks whether, in the month of July, the number of claims made by Donald Trump and rated by politifact as “Pants on Fire” will be equal to, or greater than, five plus the number of claims made by Joe Biden and rated by politifact as “Pants on fire” over the same period.
Question resolves positive if, on the date of resolution,
T \geq B + 5
T
is the number of “Pants on Fire” ratings given by politifact to Donald Trump for claims he made in July, and
B
is the number of “Pants on Fire” ratings given to Biden for claims he made in July.
If politifact ceases to operate or changes the labelling on it’s “truth-o-meter”, this question resolves ambiguous, otherwise this question resolves negative.
ETA (2020-06-05) in the case that the question does not resolve positively by July 31, resolution can be delayed by two weeks to see how statements made in late July will be rated.
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Step by step on how to calculate the Luhn check digit
Using the Luhn algorithm calculator – credit card number checker
Our Luhn algorithm calculator can test a number with Luhn validation and find the check digit for a given number. This tool has different applications such as:
Gift card number generator 🎁;
Credit card number checker 💳; and
Keep reading to learn its use cases and find the answer to questions such as 'What is the Luhn algorithm?', 'What is Luhn validation?' or 'How do I calculate check digit with Luhn algorithm?'.
The Luhn algorithm or mod 10 is a method of validating numbers using simple operations on each digit. It can detect common typing errors, and because of that, companies use it as pre-validation with credit card numbers.
It was created by the German computer scientist Hans Peter Luhn.
The Luhn algorithm can only detect single-digit and almost all permutations errors. However, it cannot catch double errors such as 22↔55, 33↔66 and 44↔77.
Now, let's see how a simple algorithm can act as a credit card number checker.
The Luhn algorithm works by taking a number and doing some basic math operations to every digit except the last one. This last digit is called the check digit. According to the Luhn algorithm, if the result of the operations and the check digit are equal, the number is valid.
Credit card, IMEI, and gift card numbers are created so that they pass a Luhn digit check, i.e., after applying the operations to every digit, the result matches the last one.
To begin using the algorithm, we need a number, let's say 23459034.
As we said, the right-most digit is the check digit (2345903[4]), so we need to separate the number and the check digit: 2345903 and 4.
Now, the algorithm operates from right to left, so starting from the last digit of the already cut number (in this case, the last number is 3). We multiply every other digit by two and leave the others intact:
If the multiplied number is equal to or greater than 10, we subtract 9 from it.
Next, we take the modulo 10 of the sum of all the final digits (4 + 3 + 8 + 5 + 9 + 6 = 35), the result is the number that should be our check digit to pass the Luhn validation:
\qquad 35\ \%\ 10 = 5
Finally, we compare the result with our check digit. If it matches, the number is valid. Otherwise, it is not. In this case, we can easily see that 5 ≠ 4, and because of that:
2345903[4] will be an invalid number.
2345903[5] will be a valid number.
This is how a gift card number generator and credit card number checker works; by comparing the check digit with the operations on all other digits.
🙋 Keep in mind that this validation does not guarantee that there actually exists a credit card or gift card number linked with that number. It just means that if it does exist, it will pass a Luhn validation.
The Luhn algorithm calculator uses two methods based on the same algorithm:
Calculate the Luhn check digit for a given number; and
Test a number for Luhn validation.
Calculate the Luhn check digit for a given number
With this method, the Luhn algorithm calculator will basically work like a gift card number generator or Luhn algorithm generator.
Simply input any positive integer, and the calculator will output the digit you should append to it, so it passes a Luhn validation. You can verify that your number is valid using the other method.
Test a number for Luhn validation
This method simply runs the Luhn algorithm in the input number and compares the result with its last digit. A message will automatically appear stating that the number is valid or showing the correct check digit to pass the validation if it's invalid.
What is Luhn validation?
Luhn validation is the process of comparing a number with its last digit (or check digit) using the Luhn algorithm to test whether a given number is valid. Credit card companies widely use it before testing if the credit card is linked to an account.
What is the check digit for 5435392 using Luhn algorithm?
2. The check digit is the right-most digit. This number is compared with the result of running the Luhn algorithm through all other digits. If the result and the check digit are equal, the number is valid.
How do I generate a gift card number?
To generate a gift card number:
Choose the final length of the number and subtract one from it.
Pick a random number that matches the length of the previous step (after subtracting one).
Multiply by two every other digit starting from the last one.
If any of the new numbers is greater than or equal to 10, subtract 9 from it.
Add all the resulting digits and write down this number.
Take the mod 10 of this final number.
Append this digit to your initial number. This number will pass a Luhn validation and be a valid gift card number. Congrats!
Are credit card numbers random?
No, credit card numbers are created to pass a pre-validation known as Luhn validation. After this initial validation, each company has different methods to assign numbers to a client's ID, account number, etc.
Input any number, and the calculator will find the correct check digit to append to it, so the final number passes the Luhn validation.
Find valid check digit
The power set calculator is here to list and count the number of subsets of a set with up to ten elements.
Estimate properties of your tetrahedron shape using the tetrahedron volume calculator.
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Mathematics/Calculus/Corner cases - Thalesians Wiki
< Mathematics | Calculus
1.1 The derivative of
{\displaystyle {\frac {d}{dx}}x^{x}}
2.1 The integral
{\displaystyle \int x^{x}\,dx}
3.1 The limit of
{\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}}
3.2 The limits of
{\displaystyle \lim _{x\rightarrow +\infty }x\sin {\frac {1}{x}}}
{\displaystyle \lim _{x\rightarrow -\infty }x\sin {\frac {1}{x}}}
{\displaystyle {\frac {d}{dx}}x^{x}}
{\displaystyle {\frac {d}{dx}}x^{x}}
{\displaystyle y=x^{x}}
{\displaystyle \ln }
{\displaystyle \ln y=x\ln x}
Differentiate both sides:
{\displaystyle {\frac {d}{dx}}\ln y={\frac {d}{dx}}x\ln x}
Apply the chain rule on the left-hand side:
{\displaystyle {\frac {d}{dx}}\ln y={\frac {1}{y}}\cdot {\frac {dy}{dx}}}
Apply the product rule on the right-hand side:
{\displaystyle {\frac {d}{dx}}x\ln x=1\cdot \ln x+x\cdot {\frac {1}{x}}=\ln x+1}
Putting it together, we have
{\displaystyle {\frac {1}{y}}\cdot {\frac {dy}{dx}}=\ln x+1}
{\displaystyle {\frac {dy}{dx}}=y(\ln x+1)=x^{x}(\ln x+1)}
{\displaystyle x=e^{\ln x}}
{\displaystyle x^{x}=(e^{\ln x})^{x}=e^{x\ln x}}
Applying the chain rule,
{\displaystyle {\frac {d}{dx}}x^{x}={\frac {d}{dx}}e^{x\ln x}=e^{x\ln x}{\frac {d}{dx}}x\ln x}
Applying the product rule,
{\displaystyle {\frac {d}{dx}}x\ln x=1\cdot \ln x+x\cdot {\frac {1}{x}}=\ln x+1}
{\displaystyle {\frac {d}{dx}}x^{x}=e^{x\ln x}(\ln x+1)=x^{x}(\ln x+1)}
{\displaystyle \int x^{x}\,dx}
{\displaystyle \int x^{x}\,dx}
{\displaystyle x^{x}}
{\displaystyle (e^{\ln x})^{x}=e^{x\ln x}}
Consider the series expansion of
{\displaystyle e^{x\ln x}}
{\displaystyle e^{x\ln x}=1+(x\ln x)+{\frac {(x\ln x)^{2}}{2!}}+{\frac {(x\ln x)^{3}}{3!}}+\ldots +{\frac {(x\ln x)^{i}}{i!}}+\ldots =\sum _{i=0}^{\infty }{\frac {(x\ln x)^{i}}{i!}}}
We can interchange the integration and summation (we can recognize this as a special case of the Fubini/Tonelli theorems) and write
{\displaystyle \int x^{x}\,dx=\int \left(\sum _{i=0}^{\infty }{\frac {(x\ln x)^{i}}{i!}}\right)\,dx=\sum _{i=0}^{\infty }\left(\int {\frac {(x\ln x)^{i}}{i!}}\,dx\right)=\sum _{i=0}^{\infty }\left({\frac {1}{i!}}\int x^{i}(\ln x)^{i}\,dx\right).}
{\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}}
{\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}}
{\displaystyle x=e^{\ln x}}
{\displaystyle x^{x}=(e^{\ln x})^{x}=e^{x\ln x}}
We can further rewrite this as
{\displaystyle x^{x}=e^{x\ln x}=e^{\frac {\ln x}{\frac {1}{x}}}}
{\displaystyle f}
is continuous and the limit of
{\displaystyle g}
exists at the point in question, the limit will commute with composition:
{\displaystyle \lim _{x\rightarrow t}f(g(x))=f(\lim _{x\rightarrow t}g(x)).}
{\displaystyle e(\cdot )}
is continuous, so
{\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}=e^{\lim _{x\rightarrow 0^{+}}{\frac {\ln x}{\frac {1}{x}}}}.}
The question, then, is what is
{\displaystyle \lim _{x\rightarrow 0^{+}}{\frac {\ln x}{\frac {1}{x}}}}
{\displaystyle x\rightarrow 0^{+}}
{\displaystyle \ln x\rightarrow -\infty }
{\displaystyle {\frac {1}{x}}\rightarrow +\infty }
. In this situation we can apply l'Hôpital's rule:
{\displaystyle \lim _{x\rightarrow 0^{+}}{\frac {\ln x}{\frac {1}{x}}}=\lim _{x\rightarrow 0^{+}}{\frac {{\frac {d}{dx}}\ln x}{{\frac {d}{dx}}{\frac {1}{x}}}}={\frac {\frac {1}{x}}{-{\frac {1}{x^{2}}}}}={\frac {{\frac {1}{x}}\cdot x^{2}}{-{\frac {1}{x^{2}}}\cdot x^{2}}}=-x.}
{\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}=e^{0}=1}
The limits of
{\displaystyle \lim _{x\rightarrow +\infty }x\sin {\frac {1}{x}}}
{\displaystyle \lim _{x\rightarrow -\infty }x\sin {\frac {1}{x}}}
{\displaystyle \lim _{x\rightarrow +\infty }x\sin {\frac {1}{x}}}
{\displaystyle \lim _{x\rightarrow -\infty }x\sin {\frac {1}{x}}}
Let us rewrite
{\displaystyle x\sin {\frac {1}{x}}}
{\displaystyle {\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}}
{\displaystyle x\rightarrow +\infty }
{\displaystyle {\frac {1}{x}}\rightarrow 0}
{\displaystyle x\sin {\frac {1}{x}}\rightarrow 0}
We have "
{\displaystyle {\frac {0}{0}}}
", so we can apply l'Hôpital's rule.
Differentiating the numerator in
{\displaystyle {\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}}
{\displaystyle \left(\cos {\frac {1}{x}}\right)\left(-{\frac {1}{x^{2}}}\right)}
Differentiating the denominator in
{\displaystyle {\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}}
{\displaystyle -{\frac {1}{x^{2}}}}
{\displaystyle \lim _{x\rightarrow +\infty }x\sin {\frac {1}{x}}=\lim _{x\rightarrow +\infty }{\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}=\lim _{x\rightarrow +\infty }{\frac {\left(\cos {\frac {1}{x}}\right)\left(-{\frac {1}{x^{2}}}\right)}{-{\frac {1}{x^{2}}}}}=\lim _{x\rightarrow +\infty }\cos {\frac {1}{x}}=1.}
Similarly we can find that
{\displaystyle \lim _{x\rightarrow -\infty }x\sin {\frac {1}{x}}=1}
Retrieved from "https://wiki.thalesians.com/index.php?title=Mathematics/Calculus/Corner_cases&oldid=41"
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AreDistinct - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : AreDistinct
test if given objects are distinct
AreDistinct({A, B, C, ...})
points, lines, planes, ...
The routine returns true if the given objects are distinct; false if they are not; and FAIL if it is unable to reach a conclusion.
The command with(geom3d,AreDistinct) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
\mathrm{sphere}\left(\mathrm{s1},[\mathrm{point}\left(\mathrm{o1},[0,0,0]\right),1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{s1}}
\mathrm{line}\left(l,[\mathrm{o1},\mathrm{point}\left(\mathrm{o2},0,0,1\right)]\right)
\textcolor[rgb]{0,0,1}{l}
\mathrm{rotation}\left(\mathrm{s2},\mathrm{s1},\frac{\mathrm{\pi }}{4},l\right)
\textcolor[rgb]{0,0,1}{\mathrm{s2}}
\mathrm{rotation}\left(\mathrm{s3},\mathrm{s2},\frac{7\mathrm{\pi }}{4},l\right)
\textcolor[rgb]{0,0,1}{\mathrm{s3}}
\mathrm{AreDistinct}\left(\mathrm{s1},\mathrm{s3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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LocatePoint - Maple Help
Home : Support : Online Help : Mathematics : Geometry : Polyhedral Sets : Properties of a Set : LocatePoint
find maximal dimensional face containing a point
LocatePoint(point, polyset)
list of rationals, a point in the coordinate system of polyset
This command finds the highest dimensional face of polyset that contains point. A point in a set's interior will return polyset, while a point not in the set will return the empty set.
\mathrm{with}\left(\mathrm{PolyhedralSets}\right):
A point inside the cube belongs to the trivial face that is the cube itself.
c≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right);
\mathrm{origin_location}≔\mathrm{LocatePoint}\left([0,0,0],c\right)
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{{}\begin{array}{lll}\textcolor[rgb]{0,0,1}{\mathrm{Coordinates}}& \textcolor[rgb]{0,0,1}{:}& [{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}]\\ \textcolor[rgb]{0,0,1}{\mathrm{Relations}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\le }\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}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}]\end{array}
\textcolor[rgb]{0,0,1}{\mathrm{origin_location}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{{}\begin{array}{lll}\textcolor[rgb]{0,0,1}{\mathrm{Coordinates}}& \textcolor[rgb]{0,0,1}{:}& [{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}]\\ \textcolor[rgb]{0,0,1}{\mathrm{Relations}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\le }\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}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}]\end{array}
A point on the side of the cube returns the face with dimension 2 that includes the point.
\mathrm{cube_face}≔\mathrm{LocatePoint}\left([0,0,1],c\right)
\textcolor[rgb]{0,0,1}{\mathrm{cube_face}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{{}\begin{array}{lll}\textcolor[rgb]{0,0,1}{\mathrm{Coordinates}}& \textcolor[rgb]{0,0,1}{:}& [{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}]\\ \textcolor[rgb]{0,0,1}{\mathrm{Relations}}& \textcolor[rgb]{0,0,1}{:}& [{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\le }\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}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}]\end{array}
\mathrm{Dimension}\left(\mathrm{cube_face}\right)
\textcolor[rgb]{0,0,1}{2}
A point outside of the cube returns the empty set.
\mathrm{outside_cube}≔\mathrm{LocatePoint}\left([2,1,1],c\right)
\textcolor[rgb]{0,0,1}{\mathrm{outside_cube}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{{}\begin{array}{lll}\textcolor[rgb]{0,0,1}{\mathrm{Coordinates}}& \textcolor[rgb]{0,0,1}{:}& [{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{3}}]\\ \textcolor[rgb]{0,0,1}{\mathrm{Relations}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{-1}]\end{array}
\mathrm{IsEmpty}\left(\mathrm{outside_cube}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The PolyhedralSets[LocatePoint] command was introduced in Maple 2015.
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How do I calculate price elasticity of supply?
How to use the price elastisity of supply formula. Price elasticity of supply examples
How to use the price elasticity of supply calculator
The importance of price elasticity of supply to businesses
The price elasticity of supply calculator measures how much the quantity supplied changes after changes in the price of a given good. Our tool not only helps you compute this ratio, but we also explain the price elasticity of supply formula background and show you some practical examples.
Read further and learn the following:
How to calculate price elasticity of demand and supply;
What the importance of price elasticity of supply to businesses is;
What the price elasticity of supply short-run and long-run is; and
What the determinants of the price elasticity of supply are.
You may also try other elasticity related tools, such as:
Income elasticity of demand calculator;
Cross price elasticity calculator.
The price elasticity of supply measures how responsive the quantity supplied is to the price of a good. It is the ratio of the percent change in the quantity supplied to the percent change in the price as we move along the supply curve.
We define the price elasticity of supply in the same way as the price elasticity of demand, with the only difference being that we consider movements along the supply curve instead of the demand curve.
We can formulate the price elasticity of supply equation in the following way:
Find the change in quantity supplied.
Determine change in price.
Divide the first value by the second value:
Price elasticity of supply = Change in quantity supplied / Change in price
You can compute the percentage change in the quantity supplied (
x_1
) and price (
x_2
) in two different ways:
In case of the standard way of computation:
\Delta x = (x_{i2} - x_{i1}) / x_{i1}
For the price elasticity of supply midpoint formula:
\Delta x = (x_{i2} - x_{i1}) / \lparen (x_{i1} + x_{i2})/2\rparen
\Delta x
- Change in quantity supplied or price;
x_{i1}
- Quantity supplied or price in Period 1; and
x_{i2}
- Quantity supplied or price in Period 2.
So far, we have learned that the price of elasticity of supply measures how much the quantity supplied changes in response to changes in the price. Let's suppose that the price of potatoes rises by 10 percent. In general, depending on the response of the quantity of potatoes supplied, the price elasticity of supply (PES) will likely fall into three categories:
If the quantity of potatoes supplied also rises by 10 percent in response, the price elasticity of supply for potatoes is 1 (10% / 10%) and supply is unit-elastic;
If the quantity supplied increases by 5 percent, the price elasticity of supply is 0.5 (PES = 5% / 10%) and supply is inelastic; and
If the quantity increases by 20 percent, the price elasticity of supply is 2 (PES = 20% / 10%) and supply is elastic.
There are other possibilities, however, that constitute two extreme cases of price elasticity of supply:
One example is the supply of cell phone frequencies, i.e., the portion of the radio spectrum which can transmit cell phone signals. Since, for technical reasons, it is a fixed quantity, the supply curve is vertical, implying that the quantity supplied doesn't respond to the price (PES = 0). This is a case of perfectly inelastic supply.
Now, let's consider the supply curve for bread. Suppose that the cost of producing one loaf is 5 dollars, including all opportunity costs. It means that if the price of bread goes under 5 dollars, all bakeries will start losing money and sooner or later go bankrupt. On the other hand, many producers would be happy to produce bread if its price rose above 5 dollars, increasing the supply considerably. The implied supply curve is a horizontal line at 5 dollars in this hypothetical case. Since even a slight rise in the price would increase the quantity supplied, the price elasticity of supply would be close to infinite (PES = ∞). This is a case of perfectly elastic supply.
In the default mode of the price elasticity of supply calculator, you need to set the following two parameters to get the result:
Percent change in price; and
Percentage change in quantity supplied.
You can also input additional numbers for Periods 1 and 2 separately, and we also provide the option for choosing between the standard and midpoint methods of estimation:
Method - The standard approximation is selected by default, but you can calculate price elasticity of supply using the midpoint method;
Price in Period 1;
Quantity supplied in Period 1; and
Quantity supplied in Period 2.
"Time is money" – says the well-known aphorism. It is particularly true in the case of the price elasticity of supply. The business' quick reaction to changing market conditions is crucial: firms aim to make supply more elastic to respond to increased demand and thereby obtain a greater profit.
There are multiple ways to get at this end:
In case of volatile prices, they may invest in a spare and flexible capacity that can adapt to changes in demand;
Paying employees overtime in case of increased production;
Outsource production to other agents; and
Introduce time management techniques, such as just in time, to increase supply and efficiency.
Flexibility doesn't come without a cost: firms need to find the optimal way to manage flexible supply and costs to reach long-term profitability.
The price elasticity of supply measures the responsiveness of the quantity supplied to changes in the price of a given good. If the price elasticity of supply is less than 1, the supply is inelastic; if it is larger than 1, the supply is elastic.
Can the price elasticity of supply be zero?
Yes, if there is no change in the quantity of a supplied good in response to a change in price in that good, we say that the supply is perfectly inelastic and its value is zero.
There are two main determinants of the price elasticity of supply:
The availability of inputs - Typically, the price elasticity of supply is high when it is easy to obtain its input used for production, and producers can enter and leave the market at a relatively low cost.
Time - The price elasticity of supply tends to increase if producers have a longer time to respond to a price change. It means that the long-run price elasticity of supply is typically higher than the short-run elasticity.
What is the price elasticity of supply short-run and long-run?
Besides the availability of inputs, time is a crucial factor in the price elasticity of supply. While it is often difficult to alter the production of a given good in response to a price change, it can be adjusted according to the market condition in the longer run.
A real-life example is the agricultural sector. When farmers receive much higher prices for wheat, they are likely to increase the proportion of their land for planting wheat in the following season. Therefore the price elasticity of supply is typically higher in the long-run than in the short-run.
Quantity supplied in Period 1
Hourly to salary calculator tells you how your hourly wage would look like as an annual salary.
Hourly to Salary - Wage Calculator
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Will Metaculus's predictions significantly improve in the next 6 months? | Metaculus
Will Metaculus's predictions significantly improve in the next 6 months?
One of the goals of Metaculus is to produce well-calibrated probabilistic forecasts. According to the Metaculus FAQ: "Like many mental capabilities, prediction is a talent that persists over time and is a skill that can be developed. By giving steady quantitative feedback and assessment, predictors can improve their skill and accuracy, as well as develop a quantified track record." So, as the size and experience of the Metaculus community increases so should the quality of the predictions generated by it.
Recently, Metaculus assembled an interactive display of its track record so far. Several metric are given, including the Brier score, which is essentially the mean square difference between the prediction and the actual outcome:
p_i
are the predicted probabilities and
o_i
are the observed outcomes, equal to zero if the question resolved negatively and equal to one if it resolved positively. Brier scores range from zero to one and smaller scores are better, with the ideal score being zero and the worst possible being one.
As of question launch, the mean Brier score for the "Metaculus prediction" is 0.150. The Metaculus prediction was the same as the community prediction until June 2017; thereafter it began to use a more sophisticated aggregation and recalibration scheme that promises more accurate predictions.
But how much more accurate? The "Metaculus post-diction" applies the current aggregation method to old data (excluding each question from the data used for "training"), and should give an estimate of future performance if future questions and predictors are comparable to past ones. As of question launch the "postdiction" Brier score is at 0.10.
Over the next 6 months prior to Jan 23, 2018, of order 25-75 binary questions are likely to resolve, about 1/3 to 1/2 as many as have previously resolved. If the community (and the aggregation) is steadily improving, we'd expect the mean Brier score to steadily drop. We'll ask:
Will the mean Brier score of the Metaculus prediction on Jan 23, 2018 be lower than 0.13?
Resolution will be determined via the Metaculus track record page.
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3 Simple Ways to Identify Dependent and Independent Variables
How to Identify Dependent and Independent Variables
1 Understanding Independent and Dependent Variables
2 Identifying Variables in Equations
3 Graphing Independent and Dependent Variables
Whether you’re conducting an experiment or learning algebra, understanding the relationship between independent and dependent variables is a valuable skill. Learning the difference between them can be tricky at first, but you’ll get the hang of it in no time.
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Think of an independent variable as a cause that produces an effect. A variable is a category or characteristic that’s measured in an equation or experiment. An independent variable stands alone and isn’t affected by other variables. In a scientific experiment, a researcher changes an independent variable to see how it affects other variables.[1] X Expert Source Michael Simpson, PhD
For example, if a researcher wants to see how well different doses of a medication work, the dose is the independent variable.
Suppose you want to see if studying more improves your test scores. The amount of time you spend studying is the independent variable.
Treat the dependent variable as an outcome. A dependent variable is an effect or result, and it always depends on another factor. The goal of an experiment or study is to explain or predict the dependent variables caused by the independent variable.[3] X Expert Source Michael Simpson, PhD
Say a researcher is testing an allergy medication. Allergy relief after taking the dose is the dependent variable, or the outcome caused by taking the medicine.
Remember that a dependent variable can’t change an independent variable. When distinguishing between variables, ask yourself if it makes sense to say one leads to the other. Since a dependent variable is an outcome, it can’t cause or change the independent variable. For instance, "Studying longer leads to a higher test score" makes sense, but "A higher test score leads to studying longer" is nonsense.[5] X Research source
Tip: When you encounter variables, plug them into this sentence: "Independent variable causes Dependent Variable, but it isn't possible that Dependent Variable could cause Independent Variable.
For example: "A 5 mg dose of medication causes allergy relief, but it isn’t possible that allergy relief could cause a 5 mg dose of medication."
Identifying Variables in Equations
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Use letters to represent variables in word problems. Turning statements with variables into math equations makes it easy to see which variable is which. For example, suppose your parents give you $3 for every chore you complete. You want to figure out how much you'll earn if you do a certain number of chores.[6] X Research source
The $3 per chore is a constant. Your parents set that in stone, and that number isn't going to change. On the other hand, the number of chores you do and the total amount of money you earn aren't constant. They're variables that you want to measure.
To set up an equation, use letters to represent the chores you do and the money you'll earn. Let t represent the total amount of money you earn and n stand for the number of chores you do.
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Set up an equation with the variables. If you get $3 for every chore you complete, say out loud, "The total amount of money I'll earn (or t) equals $3 times the number of chores I do (or n)." That gives you the equation
{\displaystyle t=3n}
Notice that the amount of money you'll earn depends on the number of chores to do. Since it depends on other variables, it's the dependent variable.
Practice solving equations to see how variables are connected. If, in the chores example,
{\displaystyle t=3n}
, and you do 5 chores, then
{\displaystyle t=(3)(5)=15}
. Doing 5 chores causes t to equal $15, so t depends on n.[8] X Research source
Say an episode of your favorite TV show is 30 minutes. The total time in minutes (m) you'll spend watching TV equals 30 times the number of episodes (e) you watch. That gives you the equation
{\displaystyle m=30e}
. If you watch 3 episodes,
{\displaystyle m=(30)(3)=90}
Plug different values into the independent variable. Remember that, in an experiment, a researcher changes the independent variable to see how it affects other variables.[9] X Expert Source Michael Simpson, PhD
Registered Professional Biologist Expert Interview. 25 June 2021. Equations work the same way! Try solving your practice equations using different numbers for the independent variables.[10] X Research source
Say you want to know how much you'll earn if you do 8 chores instead of 5. Plug 8 into n:
{\displaystyle t=(3)(8)=24}
. It's the same principle as a researcher changing the dose of a medication from 2 mg to 4 mg to test its effects.
Create a graph with x and y-axes. Draw a vertical line, which is the y-axis. Then make the x-axis, or a horizontal line that goes from the bottom of the y-axis to the right. The y-axis represents a dependent variable, while the x-axis represents an independent variable.
Say you sell apples and want to see how advertising affects your sales. The amount of money you spent in a month on advertising is the independent variable, or the factor that causes the effect you’re trying to understand. The number of apples you sold that month is the dependent variable.
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Label the x-axis with units to measure your independent variable. Next, make dashes in even increments along the horizontal line. The line should now look a bit like a ruler. These dashes will stand for units, which you’ll use to measure your independent variables.
Suppose you’re trying to see if advertising more increases the number of apples you sold. Divide the x-axis into units to measure your monthly advertising budget.
If you’ve spent between $0 and $500 a month in the last year on advertising, draw 10 dashes along the x-axis. Label the left end of the line "$0." Then label each dash with a dollar amount in $50 increments ($50, $100, $150, and so on) until you’ve reached the last dash, or "$500."
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Draw dashes along the y-axis to measure the dependent variable. As with the x-axis, make dashes along the y-axis to divide it into units. If you're studying the effects of advertising on your apple sales, the y-axis measures how many apples you sold per month.
Suppose your monthly apple sales have ranged between 60 and 250 over the last year. Draw 10 dashes across the y-axis, label the first "50," and label the rest of the dashes in increments of 25 (50, 75, 100, and so on), until you’ve written 275 next to the last dash.
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Enter your variables' coordinates onto the graph. Use your variables’ number values as coordinates, and place a dot on the corresponding point on your graph. The coordinate is where invisible lines running from the x and y-axes cross each other.
For instance, if you spent $350 on advertising last month, find the dash labeled "350" on the x-axis. If last month’s apple sales totaled 225, find the dash labeled "225" on the y-axis. Draw a dot at the point at the graph coordinate ($350, 225), then continue graphing points for the rest of your monthly numbers.
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Look for patterns in the points you’ve graphed. If the dots form a recognizable pattern, such as a roughly organized line, there’s a relationship between the independent and dependent variables. The independent variable probably doesn’t affect the dependent variable if the dots are randomly scattered across the graph without any recognizable order.[11] X Research source
For example, say you’ve graphed your advertising expenses and monthly apple sales, and the dots are arranged in an upward sloped line. This means that your monthly sales were higher when you spent more on advertising.
What are dependent and independent variables in a study?
In a study design, the dependent variables are the responses that you measure on, in or around the subjects you are studying. The results you get will be based on the independent variable.
What is the main difference between dependent and independent variables?
The difference between independent and dependent variables is essentially the difference between cause and effect. For example, if you are studying the growth of a plant species under different fertilizer concentrations, you might choose height as your variable if your hypothesis is that fertilizer concentration affects plant height. Height is your dependent variable because, according to your hypothesis, it is dependent on fertilizer concentration.
What is the independent and dependent variable for y+5 = x^2 / 3+1?
The variable that is expressed in the first degree (having an exponent of 1) is the dependent variable. In this case, it's y.
↑ https://researchbasics.education.uconn.edu/variables/
↑ https://libguides.usc.edu/writingguide/variables
↑ https://www.khanacademy.org/math/algebra/introduction-to-algebra/alg1-dependent-independent/e/dependent-and-independent-variables
↑ https://www.ixl.com/math/algebra-1/identify-independent-and-dependent-variables
↑ https://www.khanacademy.org/math/pre-algebra/pre-algebra-equations-expressions/pre-algebra-dependent-independent/a/dependent-and-independent-variables-review
↑ http://www.stat.yale.edu/Courses/1997-98/101/linreg.htm
This article was co-authored by Michael Simpson, PhD. Dr. Michael Simpson (Mike) is a Registered Professional Biologist in British Columbia, Canada. He has over 20 years of experience in ecology research and professional practice in Britain and North America, with an emphasis on plants and biological diversity. Mike also specializes in science communication and providing education and technical support for ecology projects. Mike received a BSc with honors in Ecology and an MA in Society, Science, and Nature from The University of Lancaster in England as well as a Ph.D. from the University of Alberta. He has worked in British, North American, and South American ecosystems, and with First Nations communities, non-profits, government, academia, and industry. This article has been viewed 73,261 times.
Español:identificar las variables dependientes e independientes
Deutsch:Abhängige und unabhängige Variablen feststellen
Français:distinguer les variables dépendantes et indépendantes
Nederlands:Afhankelijke en onafhankelijke variabelen onderscheiden
Bahasa Indonesia:Mengenali Variabel Dependen dan Independen
Português:Identificar Variáveis Dependentes e Independentes
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Section 59.58 (0DVG): Tate's continuous cohomology—The Stacks project
Section 59.58: Tate's continuous cohomology (cite)
59.58 Tate's continuous cohomology
Tate's continuous cohomology ([Tate]) is defined by the complex of continuous inhomogeneous cochains. We can define this when $M$ is an arbitrary topological abelian group endowed with a continuous $G$-action. Namely, we consider the complex
\[ C^\bullet _{cont}(G, M) : M \to \text{Maps}_{cont}(G, M) \to \text{Maps}_{cont}(G \times G, M) \to \ldots \]
where the boundary map is defined for $n \geq 1$ by the rule
\begin{align*} \text{d}(f)(g_1, \ldots , g_{n + 1}) & = g_1(f(g_2, \ldots , g_{n + 1})) \\ & + \sum \nolimits _{j = 1, \ldots , n} (-1)^ jf(g_1, \ldots , g_ jg_{j + 1}, \ldots , g_{n + 1}) \\ & + (-1)^{n + 1}f(g_1, \ldots , g_ n) \end{align*}
and for $n = 0$ sends $m \in M$ to the map $g \mapsto g(m) - m$. We define
\[ H^ i_{cont}(G, M) = H^ i(C^\bullet _{cont}(G, M)) \]
Since the terms of the complex involve continuous maps from $G$ and self products of $G$ into the topological module $M$, it is not clear that this turns a short exact sequence of topological modules into a long exact cohomology sequence. Another difficulty is that the category of topological abelian groups isn't an abelian category!
However, a short exact sequence of discrete $G$-modules does give rise to a short exact sequence of complexes of continuous cochains and hence a long exact cohomology sequence of continuous cohomology groups $H^ i_{cont}(G, -)$. Therefore, on the category $\text{Mod}_ G$ of Definition 59.57.1 the functors $H^ i_{cont}(G, M)$ form a cohomological $\delta $-functor as defined in Homology, Section 12.12. Since the cohomology $H^ i(G, M)$ of Definition 59.57.2 is a universal $\delta $-functor (Derived Categories, Lemma 13.16.6) we obtain canonical maps
\[ H^ i(G, M) \longrightarrow H^ i_{cont}(G, M) \]
for $M \in \text{Mod}_ G$. It is known that these maps are isomorphisms when $G$ is an abstract group (i.e., $G$ has the discrete topology) or when $G$ is a profinite group (insert future reference here). If you know an example showing this map is not an isomorphism for a topological group $G$ and $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}_ G)$ please email stacks.project@gmail.com.
Comment #7013 by Joshua Ruiter on February 02, 2022 at 14:12
I believe the assertion regarding isomorphisms when
G
is profinite is false. A counterexample is given in Gille & Szamuely's Central Simple Algebras and Galois Cohomology remark 4.2.4.
Are you sure? Because that reference has a different definition for
H^i_{cont}(G, -)
, see Definition 4.2.2 in your reference. Moreover, the definition of
H^i(G, -)
that is used there is certainly different from what is our
H^i(G, -)
; theirs doesn't use the topology on
G
and ours does.
In fact, I think that the fact that our
H^i_{cont}
is equal to their
H^i_{cont}
is the result that you claim is wrong and that the text above claims is true.
Sorry, but this is just incredibly confusing; so can you please very carefully match the definitions used in both locations and then comment again. Thanks a bunch!
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DVG. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 0DVG, in case you are confused.
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Lemma 10.137.13 (00TC)—The Stacks project
A ring map is smooth if and only if it is smooth at all primes of the target
Lemma 10.137.13. Let $R \to S$ be a ring map. Then $R \to S$ is smooth if and only if $R \to S$ is smooth at every prime $\mathfrak q$ of $S$.
Proof. The direct implication is trivial. Suppose that $R \to S$ is smooth at every prime $\mathfrak q$ of $S$. Since $\mathop{\mathrm{Spec}}(S)$ is quasi-compact, see Lemma 10.17.10, there exists a finite covering $\mathop{\mathrm{Spec}}(S) = \bigcup D(g_ i)$ such that each $S_{g_ i}$ is smooth. By Lemma 10.23.3 this implies that $S$ is of finite presentation over $R$. According to Lemma 10.134.13 we see that $\mathop{N\! L}\nolimits _{S/R} \otimes _ S S_{g_ i}$ is quasi-isomorphic to a finite projective $S_{g_ i}$-module. By Lemma 10.78.2 this implies that $\mathop{N\! L}\nolimits _{S/R}$ is quasi-isomorphic to a finite projective $S$-module. $\square$
Suggested slogan: "A ring map is smooth if and only if it is smooth at all primes of the target."
Comment #4736 by Andy on December 04, 2019 at 19:17
Hi, could you elaborate on why being quasi-isomorphic to finite projective module in degree
0
on cover implies it is true over the entire ring? It's not clear to me that 00NX as stated applies?
The lemma applies to the cohomology module.
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KBC Fall 2020 : anti-induction | Metaculus
KBC Fall 2020 : anti-induction
Everybody loves Keynesian Beauty Contests on Metaculus. Let's have another one.
Which value will Metaculus least predict?
At a secret, randomly chosen time on 2020-10-04, an admin* will retroactively close this question to that time, then use Elicit to input the current community pdf
^†
into the Metaculus prediction UI. Then they will open the current points tab (option "current") and hand pick the value that gives the lowest score to that pdf. This question resolves as this value.
We would like to remind predictors that the use of puppet accounts is prohibited by the Metaculus user agreement.
The close/resolve date is set to 2020-10-04 23:59 for convenience, but the actual time will be chosen randomly using a Random Clock Time Generator and kept secret until resolution.
The admin resolving this question will not predict on it.
The boundaries were randomly generated too. The lower boundary is open, so this question will resolve as <843721001 if this gives the lowest score.
If Elicit is unavailable at close time, the admin will do their best to fit the community prediction with the 5 logistics by hand and resolve as otherwise stated from there.
* Probably the author.
^†
Actually the best fit Elicit found, using Metaculus' 5 logistic distributions.
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networks(deprecated)/draw - Maple Help
Home : Support : Online Help : networks(deprecated)/draw
draws a graph
draw(Concentric(L), G)
draw(Linear(L), G)
sequence of disjoint lists of vertices
Important: The networks package has been deprecated. Use the superseding commands GraphTheory[DrawGraph] and GraphTheory[DrawNetwork] instead.
This routine is used to provide a visual display of the edges and vertices of a graph.
If no vertex partition is specified with the Linear or Concentric options, all vertices are drawn at equal intervals around a circle.
The Linear option draws the vertex groups specified by the lists in L in lines.
The Concentric option draws the vertex groups as specified by lists in L in concentric circles, with the first group in L forming the innermost circle.
Vertices not included in one of the lists in L are formed into a last group.
The method of drawing a specific graph can be specified by assigning a plotting procedure, as in the statement G(_Draw) := proc(G) draw(Concentric([6, 8, 10, 7, 9])) end proc. These procedures can invoke the draw() command with specific arguments, or build an appropriate PLOT data structure directly. An example of such a procedure can be found associated with the Petersen graph.
This routine is normally loaded by using the command with(networks), but it may also be referenced using the full name networks[draw](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{petersen}\left(\right):
\mathrm{draw}\left(G\right)
\mathrm{draw}\left(\mathrm{Concentric}\left([1,2,3,4,5]\right),G\right)
\mathrm{draw}\left(\mathrm{Linear}\left([1,2,3]\right),\mathrm{complete}\left(3,3\right)\right)
\mathrm{draw}\left(\mathrm{Concentric}\left([1,2,3,4,5],[6,8,10,7,9]\right),G\right)
networks(deprecated)[show]
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The Case for a Quantum Theory on a Hilbert Space with an Inner Product of Indefinite Signature
Abstract: We present the theoretical considerations for the case of looking into a generalization of quantum theory corresponding to having an inner product with an indefinite signature on the Hilbert space. The latter is essentially a direct analog of having the Minkowski spacetime with an indefinite signature generalizing the metric geometry of the Newtonian space. In fact, the explicit physics setting we have in mind is exactly a Lorentz covariant formulation of quantum mechanics, which has been discussed in the literature for over half a century yet without a nice full picture. From the point of view of the Lorentz symmetry, indefiniteness of the norm for a Minkowski vector may be the exact correspondence of the indefiniteness of the norm for a quantum state vector on the relevant Hilbert space. That, of course, poses a challenge to the usual requirement of unitarity. The related issues will be addressed.
Keywords: Lorentz Covariant Quantum Theory, Pseudo-Unitary Representation
Quantum physics with the superposition principle is to be realized with states depicted by vectors on a Hilbert space, a complex vector space, usually endowed with a sesqulinear inner product with a positive definite signature, i.e. giving a positive definite norm. A proper symmetry transformation has to preserve the inner product, hence to be unitary. The latter is of central importance to the standard probability interpretation. However, there has been important theoretical development on understanding quantum mechanics from a symmetry/spacetime and symplecto-geometric perspective that can get around the probability interpretation [1] [2]. After all, for the deterministic Schrödinger dynamics of a quantum system, there is no issue of probability. Measurement, von Neumann measurement, in particular, is a much more involved physical setting, especially more so for a Lorentz covariant quantum theory. The simple bottom line here is that even in the setting of quantum mechanics with the Copenhagen interpretation, the Born probability picture should not be strictly required to be extended to a spacetime description. Maintaining the total probability of finding a particle somewhere in the space, at a particular moment of its existence, to be unity is one thing, asking for the total probability of finding a particle somewhere in spacetime to be unity is quite another. For a particle wavefunction, as a function of the Minkowski spacetime coordinates
{x}^{\mu }
for example, it could be enough that a restriction of it to any particular time value admits the born picture description. Focusing on a formulation of covariant Schrödinger dynamics of a single particle, we present here the case for the consideration of a Hilbert space for state vectors with an indefinite norm. The key notion is the noncompact nature of the Lorentz group
SO\left(1,3\right)
giving all finite dimensional representations as non-unitary, hence failing to preserve any positive definite norm on the representation space. But it is the finite dimensional representations that serve as the natural extensions of the corresponding unitary ones of the
SO\left(3\right)
rotational subgroup. In particular, the Minkowski spacetime as a representation space of
SO\left(1,3\right)
is (1 + 3)-dimensional splitting into the single time space and the 3-dimensional space in the Newtonian limit. Minkowski spacetime, of course, has an invariant inner product on which a Lorentz boost acts as a non-unitary transformation while a rotation acts as a unitary one. It is exactly the kind of pseudo-unitarity we suggest to be incorporated as a basic structure of a fully Lorentz covariant quantum mechanics.
2. The Covariant Harmonic Oscillator
The kind of quantum theory we have in mind can easily be appreciated in the covariant harmonic oscillator problem, which has been among the first studies of a Lorentz covariant quantum mechanics. It is important to note that the problem actually goes beyond the setting of Poincaré symmetry. The proper symmetry behind the problem is that of
{H}_{R}\left(1,3\right)
\begin{array}{l}\left[{J}_{\mu \nu },{J}_{\rho \sigma }\right]=i\hslash \left({\eta }_{\nu \sigma }{J}_{\mu \rho }+{\eta }_{\mu \rho }{J}_{\nu \sigma }-{\eta }_{\mu \sigma }{J}_{\nu \rho }-{\eta }_{\nu \rho }{J}_{\mu \sigma }\right),\\ \left[{J}_{\mu \nu },{X}_{\rho }\right]=i\hslash \left({\eta }_{\mu \rho }{X}_{\nu }-{\eta }_{\nu \rho }{X}_{\mu }\right),\\ \left[{J}_{\mu \nu },{P}_{\rho }\right]=i\hslash \left({\eta }_{\mu \rho }{P}_{\nu }-{\eta }_{\nu \rho }{P}_{\mu }\right),\\ \left[{X}_{\mu },{P}_{\nu }\right]=i\hslash {\eta }_{\mu v}I,\end{array}
where we have adopted
{\eta }_{\mu \nu }=\text{diag}\left\{-1,1,1,1\right\}
. Naively, one wants to think about the operator representation with
{\stackrel{^}{X}}_{\mu }
{x}_{\mu }
{\stackrel{^}{P}}_{\mu }
-i\hslash \frac{\partial }{\partial {x}^{\mu }}
{\stackrel{^}{J}}_{\mu \nu }={\stackrel{^}{X}}_{\mu }{\stackrel{^}{P}}_{\nu }-{\stackrel{^}{X}}_{\nu }{\stackrel{^}{P}}_{\mu }
, while I represented by the identity operator. The representation is unitary and does not work so well as the case of the familiar
{H}_{R}\left(3\right)
setting at all [3]. In fact, unitarity and Lorentz covariance together would force taking only Lorentz invariant states as admissible, while on the technical side the wavefunctions and the integral norm have divergence issues. We emphasize the perspective here of having a formulation and solutions as a natural extension of the
{H}_{\text{}R}\left(3\right)\equiv {H}_{\text{}R}\left(0,3\right)
case without divergence problems. The noncompact nature of
SO\left(1,3\right)
{J}_{\mu \nu }
then points towards its pseudo-unitary representations.
There is a parallel problem for any
{H}_{R}\left(l,m\right)
. The Hermitian operator
\stackrel{^}{N}
\frac{1}{2\hslash }\left({X}_{\mu }{X}^{\mu }+{P}_{\mu }{P}^{\mu }\right)
, plus a constant, commutes with all
{\stackrel{^}{J}}_{\mu \nu }
. Each fixed n-level, for n being the eigenvalue of
\stackrel{^}{N}
, corresponds to a representation of
SO\left(l,m\right)
. We want the
n=0
level to be the trivial representation and the
n=1
level to be the defining vector representation. The latter is to say, the real span of
n=1
Fock states is essentially a
\left(l+m\right)
-dimensional pseudo-Euclidean space of signature
\left(l,m\right)
. The higher n levels then naturally correspond to symmetric Cartesian/pseudo-Euclidean tensors each of which splits into irreducible representations of
SO\left(l,m\right)
corresponding to the rank of the tensors. Of course, all such representations at any finite n are finite dimensional and non-unitary. The
SO\left(l,m\right)
transformations are to be represented by “rotations” on the pseudo-Euclidean space preserving the pseudo-Euclidean inner product. The Hilbert space as the space spanned by all Fock states can be seen as the natural complex extension of it. We will soon report on a detailed analysis with explicit Fock state wavefunctions and the pseudo-unitary inner product along the line.
3. Theory from Symmetry Representation and the Geometric Picture
Basic quantum mechanics is really a representation theory of group
{H}_{R}\left(3\right)
[4], of which
{H}_{R}\left(1,3\right)
is a natural Lorentz covariant extension. In the former case, a natural representation to use is an irreducible component of the regular representation of the Heisenberg-Weyl symmetry
H\left(3\right)
, all of which can be seen as essentially giving the same physics. Such representations are spin zero representations of the full
{H}_{R}\left(3\right)
{J}_{ij}
{X}_{i}{P}_{j}-{X}_{j}{P}_{i}
. The central charge I has to be represented by a multiple of an identity. Taking the latter as a positive real number
\zeta
times the identity
\stackrel{^}{I}
\left[{\stackrel{^}{X}}_{\zeta i},{\stackrel{^}{P}}_{\zeta j}\right]=i\left(\zeta \hslash \right){\delta }_{ij}\stackrel{^}{I}
. We should then identify the true physical position and momentum operators as
\frac{1}{\sqrt{\zeta }}{\stackrel{^}{X}}_{\zeta i}
\frac{1}{\sqrt{\zeta }}{\stackrel{^}{P}}_{\zeta i}
. For a representation with a negative
\zeta
, we should identify
\frac{1}{\sqrt{|\zeta |}}{\stackrel{^}{P}}_{\zeta i}
as the position operators
{\stackrel{^}{X}}_{i}
\frac{1}{\sqrt{|\zeta |}}{\stackrel{^}{X}}_{\zeta i}
as the momentum operators
{\stackrel{^}{P}}_{i}
. The regular representation can be expressed as a direct integral of the irreducible components for all real values of
\zeta
, with a measure vanishing at the
\zeta =0
point, which does not correspond to one such component [5]. Each component then has the natural description with observables given by
\alpha \left({p}^{i}\star ,{x}^{i}\star \right)=\alpha \left({p}^{i},{x}^{i}\right)\star
, like functions of the position and momentum operators
{x}^{i}\star
{p}^{i}\star
, as operators acting on the states with wave functions
\varphi \left({p}^{i},{x}^{i}\right)=〈{p}^{i},{x}^{i}|\varphi 〉
|{p}^{i},{x}^{i}〉
are the coherent states and
\star
is the “product” corresponding to the Moyal star-product of
\alpha \star \beta
One lesson from above is that there is no need at all to think about a negative effective
\hslash
value. We have one theory of quantum mechanics the one particle phase space of which is a Hilbert space for one value of
\zeta
, for which we know
\left[{\stackrel{^}{X}}_{i},{\stackrel{^}{P}}_{j}\right]=i\hslash {\delta }_{ij}\stackrel{^}{I}
. Moreover, the free particle phase space can be seen as the proper quantum model of the physical space on which quantum mechanics is the associated symplectic mechanics. Under the proper formulation, the physical space model and the dynamical theory reduce back exactly to the Newtonian ones at the classical limit [4] [6]. The perspective matches with the intuitive idea that the physical space is the collection of all possible positions for the particle. That is to say, only the single representation with the observed
\hslash
value is physically relevant.
The situation is however different in the case of
{H}_{R}\left(1,3\right)
H\left(1,3\right)
H\left(4\right)
are isomorphic, i.e. really the same so long as we do not have a priori identification of the generators with physical observables.
{H}_{R}\left(1,3\right)
{H}_{R}\left(4\right)
are definitely different as (real) Lie groups/algebras though. The relative sign in
{\eta }_{\mu \nu }
says that the
{X}_{0}\text{-}{P}_{0}
pair maintaining the mathematical nature as the components of the
{X}_{\mu }\text{-}{P}_{\mu }
four-vectors has a commutator of a different sign from the
{X}_{i}\text{-}{P}_{i}
pairs, which has to be preserved in an representation of
{H}_{R}\left(1,3\right)
with the position and momentum operators being Minkowski four-vectors. Hence, we cannot avoid having the commutator
\left[{X}_{0},{P}_{0}\right]
, or actually in terms of the corresponding operators in the physical representation
\left[{\stackrel{^}{X}}_{0},{\stackrel{^}{P}}_{0}\right]
-i\hslash \stackrel{^}{I}
, analogous to an effective
\hslash
value being negative.
Quantum mechanics can completely be described by the symplectic or Kähler geometry of its phase space, the infinite dimensional projective Hilbert space. The observable algebra corresponds to an algebra of the so-called Kählerian functions and Schrödinger dynamics is given by their Hamiltonian flows [7]. Most importantly,
\hslash
, or its effective value, the real parameter in the commutator
\left[\stackrel{^}{X},\stackrel{^}{P}\right]
, characterizes the constant holomorphic sectional curvature of the Kähler geometry. In any
{H}_{R}\left(n\right)
setting then, the projective Hilbert space is compact and positively curved. A natural conclusion is that for a theory with an effective negative value of the commutator, the corresponding phase space would have a negative curvature. Kawamura [8] has indeed discussed such a case as a plausible generalization of quantum mechanics, unfortunately without drawing any connection to the relevant
SO\left(m,n\right)
symmetry. Mathematically, the compact projective Hilbert space can be seen as a coset space
SU\left(N\right)/SU\left(N-1\right)
with N taken to the infinite limit. The noncompact negatively curved analogs are given by
SU\left(M,N\right)/SU\left(M,N-1\right)
SU\left(M,N\right)/SU\left(M-1,N\right)
, from a Hilbert space with a
SU\left(M,N\right)
invariant pseudo-unitary inner product [9].
It is important to note that the covariant harmonic oscillator problem and the formulation of the quantum mechanics itself are much the same. For the usual quantum mechanics, as the unitary representation of
{H}_{R}\left(3\right)
with Hermitian position and momentum operators, for example, the true Hilbert space is not that of the square-integrable functions even for the
\varphi \left({x}^{i}\right)
wavefunction formulation. It is a dense subspace of rapidly decreasing functions, the most ready explicit picture of which is the span of the harmonic oscillator Fock states [10]. Recall that the coherent states can be constructed from the Fock states too. Our analysis points above are in the same direction of pseudo-unitary representation.
We have mentioned above that the projective Hilbert space should be seen as the proper model of the physical space behind quantum mechanics, from which one can retrieve the correct classical limit. It is also true that the submanifold of the coherent states is exactly like a copy of the classical phase space sitting inside the quantum one. The classical phase space is naively a simple product of the space/configuration part and the momentum part with the same Euclidean geometry. In fact, their metrics are simply given by restrictions of the metric for the projective Hilbert space [11]. When one goes to the Lorentz covariant case, the corresponding coherent state submanifold obviously needs to have a metric of Minkowski signature for the spacetime/configuration part and the momentum part. The latter obviously asks for a metric or inner product with an indefinite signature for the quantum Hilbert space. We hope to report on an explicit formulation of such a quantum theory in the near future.
Special thanks go to Suzana Bedi’c for discussions and assistance in editing the manuscript, as well as collaboration on related studies. The author is partially supported by research grant number 107-2119-M-008-011 of the MOST of Taiwan.
Cite this paper: W. Kong, O. (2020) The Case for a Quantum Theory on a Hilbert Space with an Inner Product of Indefinite Signature. Journal of High Energy Physics, Gravitation and Cosmology, 6, 43-48. doi: 10.4236/jhepgc.2020.61005.
[1] Kong, O.C.W. (2019) An Intuitive Geometric Picture of Quantum Mechanics with Noncommutative Values for Observables NCU-HEP-k081.
[2] Kong, O.C.W. (2019) Quantum Spacetime Pictures and Particle Dynamics from a Relativity Perspective. AIP Conference Proceedings, 2075, 100001, 10pp.
[3] Bars, I. (2009) Relativistic Harmonic Oscillator Revisited. Physical Review D, 79, 045009.
[4] Chew, C.S., Kong, O.C.W. and Payne, J. (2019) Observables and Dynamics, Quantum to Classical, from a Relativity Symmetry Perspective. Journal of High Energy Physics, Gravitation and Cosmology, 5, 553-586.
[5] Taylor, M.E. (1986) Noncommutative Harmonic Analysis. American Mathematical Society, Providence, Rhode Island.
https://doi.org/10.1090/surv/022
[6] Chew, C.S., Kong, O.C.W. and Payne, J. (2017) A Quantum Space Behind Simple Quantum Mechanics. Advances in High Energy Physics, 2017, Article ID: 4395918.
[7] Cirelli, R., Manià, A. and Pizzocchero L. (1990) Quantum Mechanics as an Infinite-dimensional Hamiltonian System with Uncertainty Structure: Part I. Journal of Mathematical Physics, 31, 2891-2897.
[8] Kawamura, K. (1997) Quantum Mechanics and Operator Algebras on the Hilbert Ball, arXiv: funct-an/9710002.
[9] Boothby, W.M. and Weiss G.L. (1972) Symmetric Spaces, Marcel Dekker, Inc.
[10] Dubin, D.A., Hennings, M.A. and Smith T.B. (2000) Mathematical Aspects of Weyl Quantization and Phase. World Scientific, Singapore.
[11] Bengtsson, I. and Ζyczkowski, K. (2006) Geometry of Quantum States. Cambridge University Press, Cambridge.
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More calculators to assist you!
Do you have endpoints of a diameter? And are you looking for an easy way to find the equation of a circle? If you are saying YES, then our equation of a circle with diameter endpoints calculator is a perfect fit for you. Go on and insert the diameter endpoints in the indicated fields. You will obtain circle equations in standard, general, and parametric form.
Please read the following article to learn more about:
How to find the equation of a circle with diameter endpoints?; and
How to use our equation of a circle with diameter endpoints calculator?
A circle is a 2-D closed geometrical shape whose boundary points are at an equal distance from the center of the geometry.
To find the equation of a circle with diameter endpoints, you can use the following steps:
Find the distance between the endpoints
(x_1,y_1)
(x_2,y_2)
by using distance formula:
\qquad \scriptsize d=\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}
d
by 2 to get the radius
r
of the circle.
Next, you can use the midpoint formula to find the x coordinates
h
and y coordinates
k
for the circle's center. The equation is given as:
\qquad\qquad \scriptsize \begin{align*} h=\frac{(x_2+x_1)}{2} \\ k=\frac{(y_2+y_1)}{2} \end{align*}
By knowing the center coordinates:
h
k
r
we can get the circle equation in standard form. It is given as:
\qquad \scriptsize (x-h)^2 +(y-k)^2=r^2
We have a fleet of calculators to help you solve any problems related to circles. You can choose the calculators below as per your needs:
What is the equation of the circle whose diameter endpoints are (6,4) and (2,8)?
The equation of a circle is (x-4)² + (y-6)² = 8. You can find this equation by using the following steps:
Find the x-coordinate (h) & y-coordinate (k) of the center of the circle by taking the summation of x-coordinates & y-coordinates of the endpoints of the diameter respectively and dividing by 2.
Calculate the distance between (6,4) and (2,8) using the distance formula and divide by 2 to get the circle's radius.
Substitute the center coordinates and radius into a standard form to get the equation of the circle.
What is the midpoint between the points (3,-6) and (4,7)?
The midpoint is (1.5,0.5). To calculate the midpoint coordinates: sum the corresponding coordinates, and divide by 2.
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Example 29.3.4 (01QW)—The Stacks project
Section 29.3: Immersions
Example 29.3.4. Here is an example of an immersion which is not a composition of an open immersion followed by a closed immersion. Let $k$ be a field. Let $X = \mathop{\mathrm{Spec}}(k[x_1, x_2, x_3, \ldots ])$. Let $U = \bigcup _{n = 1}^{\infty } D(x_ n)$. Then $U \to X$ is an open immersion. Consider the ideals
\[ I_ n = (x_1^ n, x_2^ n, \ldots , x_{n - 1}^ n, x_ n - 1, x_{n + 1}, x_{n + 2}, \ldots ) \subset k[x_1, x_2, x_3, \ldots ][1/x_ n]. \]
Note that $I_ n k[x_1, x_2, x_3, \ldots ][1/x_ nx_ m] = (1)$ for any $m \not= n$. Hence the quasi-coherent ideals $\widetilde I_ n$ on $D(x_ n)$ agree on $D(x_ nx_ m)$, namely $\widetilde I_ n|_{D(x_ nx_ m)} = \mathcal{O}_{D(x_ n x_ m)}$ if $n \not= m$. Hence these ideals glue to a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ U$. Let $Z \subset U$ be the closed subscheme corresponding to $\mathcal{I}$. Thus $Z \to X$ is an immersion.
We claim that we cannot factor $Z \to X$ as $Z \to \overline{Z} \to X$, where $\overline{Z} \to X$ is closed and $Z \to \overline{Z}$ is open. Namely, $\overline{Z}$ would have to be defined by an ideal $I \subset k[x_1, x_2, x_3, \ldots ]$ such that $I_ n = I k[x_1, x_2, x_3, \ldots ][1/x_ n]$. But the only element $f \in k[x_1, x_2, x_3, \ldots ]$ which ends up in all $I_ n$ is $0$! Hence $I$ does not exist.
Comment #4574 by Andy on September 27, 2019 at 12:45
Does anything change if
I_n=(x_1^n,\ldots,x_{n+1}^n,\ldots)
? In other words, does the
x_n-1
matter?
I mean, if
I_n=(x_1^n,\ldots,x_{n-1}^n,x_{n+1}^n,\ldots
x_n-1
is removed? or
I_n=(x_1^n,\ldots,x_{n-1}^n,x_{n+1},x_{n+2},\ldots)
Yes both of those work too. What is especially weird about the closed subscheme
Z
in the example the way we have it now, is that
Z \cap D(x_n)
is some "positive distance" away from the origin because it is a zero dimensional scheme with only one point, namely
(0, \ldots, 0, 1, 0, \ldots)
(and the thickness around this point is finite too). So this way the example is a little bit like the standard way of explaining why the unit sphere in infinite dimensions isn't a compact space.
4 comment(s) on Section 29.3: Immersions
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Shannon–Fano coding - Wikipedia
In the field of data compression, Shannon–Fano coding, named after Claude Shannon and Robert Fano, is a name given to two different but related techniques for constructing a prefix code based on a set of symbols and their probabilities (estimated or measured).
Shannon's method chooses a prefix code where a source symbol
{\displaystyle i}s given the codeword length
{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil }
. One common way of choosing the codewords uses the binary expansion of the cumulative probabilities. This method was proposed in Shannon's "A Mathematical Theory of Communication" (1948), his article introducing the field of information theory.
Fano's method divides the source symbols into two sets ("0" and "1") with probabilities as close to 1/2 as possible. Then those sets are themselves divided in two, and so on, until each set contains only one symbol. The codeword for that symbol is the string of "0"s and "1"s that records which half of the divides it fell on. This method was proposed in a later technical report by Fano (1949).
Shannon–Fano codes are suboptimal in the sense that they do not always achieve the lowest possible expected codeword length, as Huffman coding does.[1] However, Shannon–Fano codes have an expected codeword length within 1 bit of optimal. Fano's method usually produces encoding with shorter expected lengths than Shannon's method. However, Shannon's method is easier to analyse theoretically.
Shannon–Fano coding should not be confused with Shannon–Fano–Elias coding (also known as Elias coding), the precursor to arithmetic coding.
2 Shannon's code: predefined word lengths
2.1 Shannon's algorithm
2.3 Expected word length
3 Fano's code: binary splitting
3.1 Outline of Fano's code
3.2 The Shannon–Fano tree
4 Comparison with other coding methods
4.2 Example with Huffman coding
Regarding the confusion in the two different codes being referred to by the same name, Krajči et al[2] write:
Around 1948, both Claude E. Shannon (1948) and Robert M. Fano (1949) independently proposed two different source coding algorithms for an efficient description of a discrete memoryless source. Unfortunately, in spite of being different, both schemes became known under the same name Shannon–Fano coding.
There are several reasons for this mixup. For one thing, in the discussion of his coding scheme, Shannon mentions Fano’s scheme and calls it “substantially the same” (Shannon, 1948, p. 17). For another, both Shannon’s and Fano’s coding schemes are similar in the sense that they both are efficient, but suboptimal prefix-free coding schemes with a similar performance
Shannon's (1948) method, using predefined word lengths, is called Shannon–Fano coding by Cover and Thomas,[3] Goldie and Pinch,[4] Jones and Jones,[5] and Han and Kobayashi.[6] It is called Shannon coding by Yeung.[7]
Fano's (1949) method, using binary division of probabilities, is called Shannon–Fano coding by Salomon[8] and Gupta.[9] It is called Fano coding by Krajči et al.[2]
Shannon's code: predefined word lengths[edit]
Main article: Shannon coding
Shannon's algorithm[edit]
Shannon's method starts by deciding on the lengths of all the codewords, then picks a prefix code with those word lengths.
Given a source with probabilities
{\displaystyle p_{1},p_{2},\dots ,p_{n}}
the desired codeword lengths are
{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil }
{\displaystyle \lceil x\rceil }
is the ceiling function, meaning the smallest integer greater than or equal to
{\displaystyle x}
Once the codeword lengths have been determined, we must choose the codewords themselves. One method is to pick codewords in order from most probable to least probable symbols, picking each codeword to be the lexicographically first word of the correct length that maintains the prefix-free property.
A second method makes use of cumulative probabilities. First, the probabilities are written in decreasing order
{\displaystyle p_{1}\geq p_{2}\geq \cdots \geq p_{n}}
. Then, the cumulative probabilities are defined as
{\displaystyle c_{1}=0,\qquad c_{i}=\sum _{j=1}^{i-1}p_{j}{\text{ for }}i\geq 2,}
{\displaystyle c_{1}=0,c_{2}=p_{1},c_{3}=p_{1}+p_{2}}
and so on. The codeword for symbol
{\displaystyle i}s chosen to be the first
{\displaystyle l_{i}}
binary digits in the binary expansion of
{\displaystyle c_{i}}
This example shows the construction of a Shannon–Fano code for a small alphabet. There 5 different source symbols. Suppose 39 total symbols have been observed with the following frequencies, from which we can estimate the symbol probabilities.
This source has entropy
{\displaystyle H(X)=2.186}
For the Shannon–Fano code, we need to calculate the desired word lengths
{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil }
{\displaystyle -\log _{2}p_{i}}
Word lengths
{\displaystyle \lceil -\log _{2}p_{i}\rceil }
We can pick codewords in order, choosing the lexicographically first word of the correct length that maintains the prefix-free property. Clearly A gets the codeword 00. To maintain the prefix-free property, B's codeword may not start 00, so the lexicographically first available word of length 3 is 010. Continuing like this, we get the following code:
{\displaystyle \lceil -\log _{2}p_{i}\rceil }
Alternatively, we can use the cumulative probability method.
...in binary
{\displaystyle \lceil -\log _{2}p_{i}\rceil }
Note that although the codewords under the two methods are different, the word lengths are the same. We have lengths of 2 bits for A, and 3 bits for B, C, D and E, giving an average length of
{\displaystyle {\frac {2\,{\text{bits}}\cdot (15)+3\,{\text{bits}}\cdot (7+6+6+5)}{39\,{\text{symbols}}}}\approx 2.62\,{\text{bits per symbol,}}}
which is within one bit of the entropy.
Expected word length[edit]
For Shannon's method, the word lengths satisfy
{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil \leq -\log _{2}p_{i}+1.}
Hence the expected word length satisfies
{\displaystyle \mathbb {E} L=\sum _{i=1}^{n}p_{i}l_{i}\leq \sum _{i=1}^{n}p_{i}(-\log _{2}p_{i}+1)=-\sum _{i=1}^{n}p_{i}\log _{2}p_{i}+\sum _{i=1}^{n}p_{i}=H(X)+1.}
{\displaystyle H(X)=-\textstyle \sum _{i=1}^{n}p_{i}\log _{2}p_{i}}
is the entropy, and Shannon's source coding theorem says that any code must have an average length of at least
{\displaystyle H(X)}
. Hence we see that the Shannon–Fano code is always within one bit of the optimal expected word length.
Fano's code: binary splitting[edit]
Outline of Fano's code[edit]
In Fano's method, the symbols are arranged in order from most probable to least probable, and then divided into two sets whose total probabilities are as close as possible to being equal. All symbols then have the first digits of their codes assigned; symbols in the first set receive "0" and symbols in the second set receive "1". As long as any sets with more than one member remain, the same process is repeated on those sets, to determine successive digits of their codes. When a set has been reduced to one symbol this means the symbol's code is complete and will not form the prefix of any other symbol's code.
The algorithm produces fairly efficient variable-length encodings; when the two smaller sets produced by a partitioning are in fact of equal probability, the one bit of information used to distinguish them is used most efficiently. Unfortunately, Shannon–Fano coding does not always produce optimal prefix codes; the set of probabilities {0.35, 0.17, 0.17, 0.16, 0.15} is an example of one that will be assigned non-optimal codes by Shannon–Fano coding.
Fano's version of Shannon–Fano coding is used in the IMPLODE compression method, which is part of the ZIP file format.[10]
The Shannon–Fano tree[edit]
The left part of the list is assigned the binary digit 0, and the right part is assigned the digit 1. This means that the codes for the symbols in the first part will all start with 0, and the codes in the second part will all start with 1.
Shannon–Fano Algorithm
We continue with the previous example.
All symbols are sorted by frequency, from left to right (shown in Figure a). Putting the dividing line between symbols B and C results in a total of 22 in the left group and a total of 17 in the right group. This minimizes the difference in totals between the two groups.
With this division, A and B will each have a code that starts with a 0 bit, and the C, D, and E codes will all start with a 1, as shown in Figure b. Subsequently, the left half of the tree gets a new division between A and B, which puts A on a leaf with code 00 and B on a leaf with code 01.
After four division procedures, a tree of codes results. In the final tree, the three symbols with the highest frequencies have all been assigned 2-bit codes, and two symbols with lower counts have 3-bit codes as shown table below:
This results in lengths of 2 bits for A, B and C and per 3 bits for D and E, giving an average length of
{\displaystyle {\frac {2\,{\text{bits}}\cdot (15+7+6)+3\,{\text{bits}}\cdot (6+5)}{39\,{\text{symbols}}}}\approx 2.28\,{\text{bits per symbol.}}}
We see that Fano's method, with an average length of 2.28, has outperformed Shannon's method, with an average length of 2.62.
It is shown by Krajči et al[2] that the expected length of Fano's method has expected length bounded above by
{\displaystyle \mathbb {E} L\leq H(X)+1-p_{\text{min}}}
{\displaystyle p_{\text{min}}=\textstyle \min _{i}p_{i}}
is the probability of the least common symbol.
Comparison with other coding methods[edit]
Neither Shannon–Fano algorithm is guaranteed to generate an optimal code. For this reason, Shannon–Fano codes are almost never used; Huffman coding is almost as computationally simple and produces prefix codes that always achieve the lowest possible expected code word length, under the constraints that each symbol is represented by a code formed of an integral number of bits. This is a constraint that is often unneeded, since the codes will be packed end-to-end in long sequences. If we consider groups of codes at a time, symbol-by-symbol Huffman coding is only optimal if the probabilities of the symbols are independent and are some power of a half, i.e.,
{\displaystyle \textstyle 1/2^{k}}
. In most situations, arithmetic coding can produce greater overall compression than either Huffman or Shannon–Fano, since it can encode in fractional numbers of bits which more closely approximate the actual information content of the symbol. However, arithmetic coding has not superseded Huffman the way that Huffman supersedes Shannon–Fano, both because arithmetic coding is more computationally expensive and because it is covered by multiple patents.[11]
A few years later, David A. Huffman (1949)[12] gave a different algorithm that always produces an optimal tree for any given symbol probabilities. While Fano's Shannon–Fano tree is created by dividing from the root to the leaves, the Huffman algorithm works in the opposite direction, merging from the leaves to the root.
Create a leaf node for each symbol and add it to a priority queue, using its frequency of occurrence as the priority.
Remove the two nodes of lowest probability or frequency from the queue
Prepend 0 and 1 respectively to any code already assigned to these nodes
Example with Huffman coding[edit]
We use the same frequencies as for the Shannon–Fano example above, viz:
In this case D & E have the lowest frequencies and so are allocated 0 and 1 respectively and grouped together with a combined probability of 0.282. The lowest pair now are B and C so they're allocated 0 and 1 and grouped together with a combined probability of 0.333. This leaves BC and DE now with the lowest probabilities so 0 and 1 are prepended to their codes and they are combined. This then leaves just A and BCDE, which have 0 and 1 prepended respectively and are then combined. This leaves us with a single node and our algorithm is complete.
The code lengths for the different characters this time are 1 bit for A and 3 bits for all other characters.
This results in the lengths of 1 bit for A and per 3 bits for B, C, D and E, giving an average length of
{\displaystyle {\frac {1\,{\text{bit}}\cdot 15+3\,{\text{bits}}\cdot (7+6+6+5)}{39\,{\text{symbols}}}}\approx 2.23\,{\text{bits per symbol.}}}
We see that the Huffman code has outperformed both types of Shannon–Fano code, which had expected lengths of 2.62 and 2.28.
^ Kaur, Sandeep; Singh, Sukhjeet (May 2016). "Entropy Coding and Different Coding Techniques" (PDF). Journal of Network Communications and Emerging Technologies. 6 (5): 5. Archived from the original (PDF) on 2019-12-03. Retrieved 3 December 2019.
^ a b c Stanislav Krajči, Chin-Fu Liu, Ladislav Mikeš and Stefan M. Moser (2015), "Performance analysis of Fano coding", 2015 IEEE International Symposium on Information Theory (ISIT).
^ Thomas M. Cover and Joy A. Thomas (2006), Elements of Information Theory (2nd ed.), Wiley–Interscience. "Historical Notes" to Chapter 5.
^ Charles M. Goldie and Richard G. E. Pinch (1991), Communication Theory, Cambridge University Press. Section 1.6.
^ Gareth A. Jones and J. Mary Jones (2012), Information and Coding Theory (Springer). Section 3.4.
^ Te Sun Han and Kingo Kobayashi (2007), Mathematics of Information and Coding, American Mathematical Society. Subsection 3.7.1.
^ Raymond W Yeung (2002), A First Course in Information Theory, Springer. Subsection 3.2.2.
^ David Salomon (2013), Data Compression: The Complete Reference, Springer. Section 2.6.
^ Prakash C. Gupta (2006), Data Communications and Computer Networks, Phi Publishing. Subsection 1.11.5.
^ "APPNOTE.TXT - .ZIP File Format Specification". PKWARE Inc. 2007-09-28. Retrieved 2008-01-06. The Imploding algorithm is actually a combination of two distinct algorithms. The first algorithm compresses repeated byte sequences using a sliding dictionary. The second algorithm is used to compress the encoding of the sliding dictionary output, using multiple Shannon–Fano trees.
^ Huffman, D. (1952). "A Method for the Construction of Minimum-Redundancy Codes" (PDF). Proceedings of the IRE. 40 (9): 1098–1101. doi:10.1109/JRPROC.1952.273898.
Fano, R.M. (1949). "The transmission of information". Technical Report No. 65. Cambridge (Mass.), USA: Research Laboratory of Electronics at MIT.
Shannon, C.E. (July 1948). "A Mathematical Theory of Communication". Bell System Technical Journal. 27: 379–423.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Shannon–Fano_coding&oldid=1076520027"
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Will more than five of the ten highest grossing films in 2020 pass the Bechdel test? | Metaculus
Will more than five of the ten highest grossing films in 2020 pass the Bechdel test?
The Bechdel test, made famous by the cartoonist Alison Bechdel in her comic strip “Dykes to Watch Out For”, but credited by her to her friend Liz Wallace, is a three-part test of women’s representation in fiction. To “pass” the test, a film must:
Contain two named female characters.
Who have a conversation with each other.
Which is not about a man.
This question asks: Of the ten highest grossing films of 2020, will more than half satisfy all three of the criteria above?
The ten highest grossing films will be determined by credible media reports, while whether the films pass the bechdel test will be determined by bechdeltest.com.
As passing the test can in some cases be ambiguous, if there is sufficient disagreement in the comments on bechdeltest.com that the resolution might change, a panel of at least two metaculus admins, who have not predicted on the question, will be asked to judge and decide on a resolution. If no metaculus admins are able to judge, or if they do not agree, resolution will be ambiguous. 'Sufficient disagreement' is defined as follows:
Let the number of Top-10 films with disagreement about resolution in the comments be
N
, and the number of Top-10 films passing the test according to the bechdeltest.com be
P
. Disagreement is sufficient if
|P-5| \leq N
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Aggregating values to the Mandelbrot and Julia sets - Nextjournal
Lazaro / Aug 20 2019
Aggregating values to the Mandelbrot and Julia sets
using Datashader, numpy , pandas, numba and colorcet
by Lazaro Alonso, find me on twitter as @LazarusAlon
Since the mandelbrot and julia sets are well known fractal examples in the literature, here only the computational side will be posted.
The traditional approach for plotting this sets is by pre-pixeling the coordinates( on the complex plane) and then computing their values(iteration process). But now, thanks to datashader we can directly perform the iteration test on the complex plane coordinates and then aggregate the iteration outputs into the corresponding pixel.
First let's see the python version installed and conda install some packages.
import sys; sys.version.split()[0]
conda install colorcet
Now, let's work... we are ready to import all that we need.
import numpy as np, pandas as pd, colorcet as cc
import datashader as ds, datashader.transfer_functions as tf
from datashader.colors import colormap_select
from numpy import log, arange
cm = partial(colormap_select, reverse=(background!="black"))
The Mandelbrot set, M, is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map
z_{n + 1} = z_{n}^2 + c
remains bounded. That is, a complex number c is part of the Mandelbrot set if, when starting with z_0 = 0 and applying the iteration repeatedly, the absolute value of z_n remains bounded however large n gets.
c\in M \rightarrow \lim_{n\rightarrow\infty} sup\,|z_{n + 1}| \leq 2.
In code, the iteration process for a complex number c is simply as follows
def mandelbrot(c, b, itmap):
for n in arange(b):
if z.real * z.real + z.imag * z.imag > 4:
return itmap(n, z, b)
where b known as the bailout radius. And itemap will assign a real value to be used later for colouring the whole space of points. Actually, for itemap we will use the following normalized iteration count function:
normIterations(n, z_{n}, b) = n - \frac{\log(log(|z_{n}|)) - \log(\log(b))}{\log(2)}
and in code as follows, (if someone knows the original reference for this normalization please let me know).
def normIterations(n, zn, b):
return n - (log(log(abs(zn))) - log(log(b)))/log(2.0)
Then, a function that works for a whole section of the complex plane will be
def mandelbrot_set(xi, xf, yi, yf, npoints, b):
xycolor = np.zeros((npoints*npoints, 3))
for x in np.linspace(xi, xf, npoints):
for y in np.linspace(yi, yf, npoints):
color = mandelbrot(x + 1j*y, b, normIterations)
xycolor[s,:] = [x, y, color]
return xycolor
Disclaimer: Although this code is pretty fast, probably there are clever ways to do it. Nevertheless, I think this versions are easy to follow and understand.
Testing... datashader magic
xycm = mandelbrot_set(-2.0,0.5,-1.25,1.25,1000,1024)
Please note that the previous array contains one million points.
def get_img(xyc, sx=400, sy = 400, pick_color = "fire", fhow = "eq_hist"):
df = pd.DataFrame(xyc, columns =["x","y","color"]) # because ds takes DataFrames
cvs = ds.Canvas(plot_width = sx, plot_height = sy)
agg = cvs.points(df, 'x', 'y', ds.mean('color')) # ds.mean, ds.min, ds.max
return tf.shade(agg, cmap=cm(palette[pick_color], 0.0), how= fhow) # eq_hist, linear, cbrt, log
We can set different kinds of colors from colorcet as well as scales for colouring.
escalas = ["linear", "cbrt", "log", "eq_hist"]
tf.Images(*[get_img(xycm, fhow = scale) for scale in escalas])
Let's do some more with different colours and scales.
xycm1 = mandelbrot_set(-0.74877,-0.74872,0.06505,0.06510,1000,2500)
xycm2 = mandelbrot_set(0.32642717997233067, 0.3265289052327473,
-0.05451000956418885, -0.05443371561887635,1000,3000)
xycm3 = mandelbrot_set(-1.9963806954442953, -1.996380695443582,
2.628704923646517e-7, 2.62871027270105e-7,1000,1000)
xycm4 = mandelbrot_set(0.3476108223238668926295, 0.3476108223245338122665,
0.0846794087369283253550, 0.0846794087374285150830,1000,10000)
coordxyc = [xycm1, xycm2, xycm3, xycm4]
colores = ['bmw', 'bmy','fire', 'gray', 'kbc']
escalas = ["linear", "eq_hist"]
tf.Images(*[get_img(xycv, pick_color = c, fhow = scale) for xycv in coordxyc
for scale in escalas for c in colores]).cols(5)
# this looks ugly (a lot of numbers) but the pics are nice.
xycm5 = mandelbrot_set(-0.650790400000000001192,-0.648844800000000001192,
0.44539837880859369792, 0.44685757880859369792,1000,10_000)
xycm6 = mandelbrot_set(-0.9548899408372031, -0.9548896813770819,
0.2525416487455764, 0.2525418433406673, 1000,10_000)
xycm7 = mandelbrot_set(0.254828857465066226270, 0.254828889245416226270,
-0.000605561881950000235, -0.000605538046687500235,1000,50_000)
xycm8 = mandelbrot_set(-0.882297664710767940063, -0.882297662380940440063,
0.235365461981556923486,0.235365463728927548486, 1000, 10_000)
xycm9 = mandelbrot_set(-0.6534376561891502063520, -0.6534376520406489856480,
0.3635691455538367401120, 0.3635691486652126556400,1000,10_000)
xycm10 = mandelbrot_set(0.25740289813988496306, 0.25740289814296891154,
coordxycDeeper = [xycm5, xycm6, xycm7, xycm8, xycm9, xycm10]
colores = ['bmw', 'fire', 'kbc']
tf.Images(*[get_img(xycv, sx = 800, sy = 600,
pick_color = c, fhow = "eq_hist")
for xycv in coordxycDeeper
for c in colores]).cols(3)
everything works beautiful ! Now, let's see the julia set.
The Julia Set associated to the function
f_c (z) = z^2 + c
for a constant c the complex plane is the set
\mathcal{J}(f_c)
of points z in the complex plane, such that
|f_{c}^{n}(z)| = |f_{c} (f_{c}(\dots f_{c}(z)))| \leq 2\; \forall_{n}
The code for this condition is
def julia_map(x, y, c, b, itmap):
z = (x + 1j*y)**2 + c
for k in arange(b):
if z.real * z.real + z.imag * z.imag > 4:
return itmap(k, z, b)
and for a region in the complex plane
def create_julia_dots(xi, xf, yi, yf, c, b, npoints):
xycolor = np.zeros((npoints**2, 3))
color = julia_map(x, y, c, b, normIterations)
And the plots are
xyc_julia = create_julia_dots(-1.7, 1.7, -1.7, 1.7,
-0.835 - 0.2321*1j, 500, 2000)
tf.Images(*[get_img(xyc_julia, pick_color = c, fhow ="log")
for c in colores])
cvalues = [0.274 - 0.008 * 1j, 0.285 + 0.01*1j,
-0.70176 - 0.3842*1j, -0.8 + 0.156*1j,
-0.7269 + 0.1889*1j, -0.1 + 0.65*1j,
-0.382 + 0.618*1j, -0.449 + 0.571*1j]
def get_imgJulia(c, escala, color):
xyc = create_julia_dots(-1.5, 1.5, -1.5, 1.5, c, 500, 1000)
df = pd.DataFrame(xyc, columns =["x","y","c"])
agg = cvs.points(df, 'x', 'y', ds.mean('c'))
return tf.shade(agg, cmap=cm(palette[color], 0.0), how=escala)
colores = ['bmw', 'bmy','fire', 'gray', 'kbc', 'bmw',
'bmy','fire', 'gray', 'kbc']
tf.Images(*[get_imgJulia(cvalue, "linear", colores[indx])
for indx, cvalue in enumerate(cvalues)]).cols(4)
One last colormap
tf.Images(*[get_imgJulia(cvalue, "linear", "CET_L17")
For more colors go to colorcet or simply load matplotlib colormaps.
Find me on twitter as @LazarusAlon
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{\displaystyle A\,=\,P\left(1+{\frac {r}{n}}\right)^{nt},}
{\displaystyle A}
{\displaystyle P}
{\displaystyle r}
{\displaystyle n}
{\displaystyle n}
{\displaystyle 365}
{\displaystyle 52}
{\displaystyle 12}
for compounding monthly. As a result, the expone{\displaystyle nt}
{\displaystyle t}
{\displaystyle r/n}
{\displaystyle 7}
{\displaystyle 6\%}
{\displaystyle nt\,=\,12\cdot 7\,=\,84}
{\displaystyle 0.06/12\,=\,0.005}
{\displaystyle \$2100}
{\displaystyle 4}
times per year. Using the formula in 'Foundations', the equation for the account value after 8 years is
{\displaystyle A\,=\,P\left(1+{\frac {r}{n}}\right)^{nt}\,=\,2100\left(1+{\frac {0.06}{4}}\right)^{4\cdot 8}\,=\,2100(1.015)^{32}.}
{\displaystyle A\,=\,2100(1.015)^{32}.}
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Research on Geometric Errors Measurement of Machine Tools Using Auto-Tracking Laser Interferometer
Research on Geometric Errors Measurement of Machine Tools Using Auto-Tracking Laser Interferometer ()
Jr-Rung Chen*, Bing-Lin Ho, Hau-Wei Lee, Shan-Peng Pan, Tsung-Han Hsieh
Center for Measurement Standards, ITRI, Hsinchu City, Taiwan.
For the development of the aviation industry, machine tools are becoming large and travel long distances, making optical alignment setup difficult. An auto-tracking laser interferometer (ATLI) is proposed and researched in this paper for the squareness error measurement of machine tools or coordinate-measuring machines (CMMs). The procedure involves measurement of only one line of an axis, and the measurement results provide us information about not only the positioning errors but also the squareness errors. This specially designed interferometer instrument can be useful in checking industrial machine tools in a short time.
ATLI, CMMs, Laser Interferometer, Machine Tool
Chen, J. , Ho, B. , Lee, H. , Pan, S. and Hsieh, T. (2018) Research on Geometric Errors Measurement of Machine Tools Using Auto-Tracking Laser Interferometer. World Journal of Engineering and Technology, 6, 631-636. doi: 10.4236/wjet.2018.63039.
Machine tools and coordinate measuring machines (CMMs) with 3 - 5 axes have played fundamental roles in industrial development. As seen in Figure 1, a total of 21 geometric errors affect the volumetric accuracy of three-dimensional machine tools and CMMs [1] . These errors include translation, rotation, and squareness errors. Calibration and compensation of these geometric errors are necessary to improve the positioning accuracy of machine tools. When considering the mechanical accuracy of coordinate measuring devices, three primary sources of quasi-static errors can be identified. Geometric errors are due to the limited accuracy of individual machine components, such as guideways and measuring systems. Errors related to the final stiffness of these components mainly arise from the moving parts. Thermal errors originate as a result of expansion and bending of guideways due to temperature gradients.
Geometric errors are caused by straightness errors of the guideways [2] , imperfect alignment of the axis, and flatness errors of the manufactured surfaces. In particular, squareness error of machine tools may arise due to installation, shipping, or heavy cutting. The squareness error between linear motions of two axes can be estimated using diagonal displacement (ISO 230-6) tests. The linear positioning accuracy and the repeatability of the x- , y- , and z-axes and the xy, xz, yz, and xyz diagonal lines can be determined according to the ISO 230-2 and -6 standards [3] [4] . ISO 230-2 defined the minimum measuring point numbers while measuring x, y or z axis and ISO 230-6 defined the minimum point numbers when x-y, x-z, y-z z-x or x-y-z diagonal line is measured.
A laser interferometer and an auto-tracking laser interferometer (ATLI) can be used to perform the seven lines tests. However, the time consumed when using a laser interferometer is approximately three times that when an ATLI is used. The laser tracker [5] [6] and LaserTRACER (LT) [7] are both ATLI types. Dr. Lee [8] has discussed the relationship between ISO 230-2/-6 test results and the positioning accuracy of machine tools using LT. This paper will further discuss the trend of the measurement result when only one ATLI is used for the squareness measurement of machine tools or CMM.
The first step in using ATLI, following ISO 230-2 and 230-6, is to fix the ATLI onto the machine tool carriage [8] . The next step is to determine the ATLI’s location with respect to the home/reference point of the machine tool. The measurement points for each test line are computed when the ATLI’s location is determined. Numerical control (NC) codes are also generated. Users import these NC codes into the machine tool controller. The seven lines test is then performed. Subsequently, a test report can be generated after analyzing the test data.
Figure 1. 21 geometric error terms in three-axis motion.
2.2. Determination of ATLI Coordinate
The second step in the ISO 230-2 and 230-6 test procedures, as mentioned in Section 2.1, is determining the ATLI’s coordinates on the machine tool, which can be calculated using a six-point measurement and the following formula:
\left[\begin{array}{c}{x}_{t}\\ {y}_{t}\\ {z}_{t}\\ {L}_{0}\\ \xi \end{array}\right]={\left[\begin{array}{ccccc}2{x}_{1}& 2{y}_{1}& 2{z}_{1}& 2\Delta {L}_{1}& -1\\ 2{x}_{2}& 2{y}_{2}& 2{z}_{2}& 2\Delta {L}_{2}& -1\\ 2{x}_{3}& 2{y}_{3}& 2{z}_{3}& 2\Delta {L}_{3}& -1\\ 2{x}_{4}& 2{y}_{4}& 2{z}_{4}& 2\Delta {L}_{4}& -1\\ 2{x}_{5}& 2{y}_{5}& 2{z}_{5}& 2\Delta {L}_{5}& -1\\ 2{x}_{6}& 2{y}_{6}& 2{z}_{6}& 2\Delta {L}_{6}& -1\end{array}\right]}^{+}\left[\begin{array}{c}{x}_{1}^{2}+{y}_{1}^{2}+{z}_{1}^{2}-\Delta {L}_{1}^{2}\\ {x}_{2}^{2}+{y}_{2}^{2}+{z}_{2}^{2}-\Delta {L}_{2}^{2}\\ {x}_{3}^{2}+{y}_{3}^{2}+{z}_{3}^{2}-\Delta {L}_{3}^{2}\\ {x}_{4}^{2}+{y}_{4}^{2}+{z}_{4}^{2}-\Delta {L}_{4}^{2}\\ {x}_{5}^{2}+{y}_{5}^{2}+{z}_{5}^{2}-\Delta {L}_{5}^{2}\\ {x}_{6}^{2}+{y}_{6}^{2}+{z}_{6}^{2}-\Delta {L}_{6}^{2}\end{array}\right]
where the residual measurement errors are
\zeta ={x}_{t}^{2}+{y}_{t}^{2}+{z}_{t}^{2}-{L}_{0}^{2}
and the “+” symbol represents the pseudo-inverse operator. The six stop points (
p=1,2,\cdots ,6
) of the cat’s eye reflector are (xp, yp, zp). These points are independent and should be known. (xt, yt, zt) is the LT coordinate to be determined. L0 is the initial distance from the LT to the cat’s eye reflector, which is unknown. ΔLi is the measured distance deviation from the LT.
2.3. ATLI Measurement Principle and Effects of Positioning and Angular Error
The measurement of ATLI requires only the length information. The motion of a three-axis machine tool comprising 21 error terms results in the ideal and actual distance differences in the ATLI’s coordinate determination. Figure 2 illustrates how the influence of ISO 230-2/-6 tests on the x-axis motion with squareness error is estimated.
In Figure 2, the solid horizontal and vertical lines represent the x- and y-axes of the machine tool, respectively. The dotted lines represent the variation in the straightness deviation of θx in the x-axis motion (i.e., its trajectories), which is recorded as the long the axis length. Δx, Δy, and Δz are the shift distances between the actual and estimated location coordinates of the LT determined from Equation (1). The measurement points are (xi, yi, zi) at different times, i. The length measured by the LT is Lt and t indicates the measuring point at each time. As shown in Equations (2) to (3), ΔLi+1 and
\Delta {{L}^{\prime }}_{i+1}
indicate the ideal and actual distance differences, respectively, at the durations of ti and ti+1, respectively.
\Delta {L}_{i+1,i}=\sqrt{\left({x}_{i+1}^{2}+{y}_{i+1}^{2}+{z}_{i+1}^{2}\right)}-\sqrt{\left({x}_{i}^{2}+{y}_{i}^{2}+{z}_{i}^{2}\right)}
\Delta {{L}^{\prime }}_{i+1,i}=\sqrt{\left({{x}^{\prime }}_{i+1}^{2}+{{y}^{\prime }}_{i+1}^{2}+{{z}^{\prime }}_{i+1}^{2}\right)}-\sqrt{\left({{x}^{\prime }}_{i}^{2}+{{y}^{\prime }}_{i}^{2}+{{z}^{\prime }}_{i}^{2}\right)}
For the effects of positioning and angular error, we consider Figure 2 as a gantry structure. The rigid-body kinematic model of the gantry structure is shown in Figure 3. This gantry is constructed from a moving table as the x-axis and a fixed gantry that represents the y- and z-axes.
O0 is the home position of the machine, O1 is the position of target and Ot is the location of the auto-tracking ranging system, which represents the center of
Figure 2. Influence of ISO 230-2/-6 test results on squareness of machine tools using an ATLI.
Figure 3.CMM appearance and each coordinate frame of the structure.
the reference ball inside the instrument. L0 + ΔL is the measured distance between the ranging instrument and the moving target because, here, incremental interferometer is considered.
Let [sx sy sz]T be the ideal position and LV be the measuring line along the distance on the V-axis; subsequently, the simulated position [ŝx ŝy ŝz]T with errors can be expressed as
\left[\begin{array}{c}{\stackrel{^}{s}}_{x}\\ {\stackrel{^}{s}}_{y}\\ {\stackrel{^}{s}}_{z}\end{array}\right]=\left[\begin{array}{c}{s}_{x}+{\delta }_{xx}\\ {s}_{y}+{\delta }_{yx}\\ {s}_{z}+{\delta }_{zx}\end{array}\right]
Considering a coordinate position shift [εx εy εz]T and scale errors gx of the position errors, the movement can be described as follows:
\left[\begin{array}{c}{\stackrel{^}{s}}_{x}\\ {\stackrel{^}{s}}_{y}\\ {\stackrel{^}{s}}_{z}\end{array}\right]=\left[\begin{array}{c}{g}_{x}{s}_{x}+{\epsilon }_{x}+{g}_{x}{\delta }_{xx}\\ {s}_{y}+{\epsilon }_{y}+{\delta }_{yx}\\ {s}_{z}+{\epsilon }_{z}+{\delta }_{zx}\end{array}\right]
Finally, the effects of positioning and angular error can be described as follows:
\left[\begin{array}{c}{\delta }_{xx}\\ {\delta }_{yx}\\ {\delta }_{zx}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}{\beta }_{x}& 0& \mathrm{sin}{\beta }_{x}\\ 0& 1& 0\\ -\mathrm{sin}{\beta }_{x}& 0& \mathrm{cos}{\beta }_{x}\end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}{\gamma }_{x}& -\mathrm{sin}{\gamma }_{x}& 0\\ \mathrm{sin}{\gamma }_{x}& \mathrm{cos}{\gamma }_{x}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{s}_{x}\\ 0\\ 0\end{array}\right]-\left[\begin{array}{c}{s}_{x}\\ 0\\ 0\end{array}\right]
here, γx is the squareness error of the x-y axis and βx is the squareness angle of the x-z axis.
The experimental setup was based on a three-axis CMM (Leitz PMM-C) in the National Measurement Laboratory in Taiwan; it is a gantry structure with an x-axis of 1400 mm, a y-axis of 700 mm, and a z-axis of 600 mm. The simulation software was developed in VB.NET. Considering Equations (5) and (6), we set the squareness error of the x-y axis to 0.1˚, which is a common scale for squareness in machine tools or CMM assembly lines. The simulated and experimental results are plotted in Figure 4. The plotted line shows an exponential curve because the squareness error will lead to a misalignment error between the ATLI location and measurement path. As shown in Figure 4, the curve of the experimental result is 1.5 μm in the range of −2.5 μm to −4 μm, and the estimated accuracy is 4.5 μm, which is only slightly different from the simulated result. This offset was due to the deviation between the actual and estimated ATLI location coordinates. Thus, we can confirm the ATLI location from the initial measuring point, which is at t1. As shown in Figure 4(b), when the linear positioning errros and squareness errors are simultaneously generated, the deviation first decreases and subsequently increases with the increasing test length.
Figure 4. (a) Simulated and (b) Experimental results when squareness is considered.
This research focused on simulating the measurement results of an auto-tracking ranging system and analyzing trends in the deviation results. The experimental results indicated a significant difference between positioning and squareness error when only one axis was measured using an ATLI. From the plotted trends in the simulated and experimental results, we can identify and quantify linear and squareness errors, and thus ensuring that ATLIs can be used to perform machine tool inspection more easily and conveniently. Compared with traditional interferometers, auto-tracking facilitates easier optic alignment setup, especially for diagonal line measurements of machine tools.
The work was supported by the standard maintenance and services project from Bureau of Standards, Metrology and Inspection (BSMI), Ministry of Economic Affairs (MOEA).
[1] Zhang, G., Ouyang, R., Lu, B., Hocken, R., Veale, R. and Donmez, A. (1988) A Displacement Method for Machine Geometry Calibration. CIRP Annals-Manufacturing Technology, 37, 515-518.
[2] Ptaszynski, W., Gessner A., Frackowiak P. and Staniek, R. (2011) Straightness Measurement of Large Machine Guideways. Metalurgija, 50, 281-284.
[3] International Organization for Standardization (2002) ISO 230: Test Code for Machine Tools, in Part 6 Determination of Positioning Accuracy on Body and Face Diagonals.
[4] International Organization for Standardization (2012) ISO 230: Test Code for Machine Tools, in Part 2: Determination of Accuracy and Repeatability of Positioning Numerically Controlled Axes.
[5] Lau, K., Hocken, R.J. and Haight, W.C. (1986) Automatic Laser Tracking Interferometer System for Robot Metrology. Precision Engineering, 8, 3-8.
[6] Muralikrishnan, B., Phillips, S. and Sawyer, D. (2016) Laser Trackers for Large-Scale Dimensional Metrology: A Review. Precision Engineering, 44, 13-28.
[7] Physikalisch-Technische, B. (2004) Laser Interferometer for Spacing Measurement has Reference Sphere Positioned on Post and Base Plate Made of Thermally Invariant Material (TRACER). German Patent No. 202004007647U1.
[8] Lee, H.-W., Chen, J.-R., Pan, S.-P., Liou, H.-C. and Hsu, P.-E. (2016) Relationship between ISO 230-2/-6 Test Results and Positioning Accuracy of Machine Tools Using Laser TRACER. Applied Sciences, 6, 105.
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Ice shelf - Wikipedia @ WordDisk
Russian ice shelves
Ice shelf disruption
Large floating platform of ice caused by glacier flowing onto ocean surface
Not to be confused with Shelf ice.
Ice shelf extending approximately 6 miles into the Antarctic Sound from Joinville Island
Close-up of Ross Ice Shelf
Panorama of Ross Ice Shelf
In contrast, sea ice is formed on water, is much thinner (typically less than 3 m (9.8 ft)), and forms throughout the Arctic Ocean. It also is found in the Southern Ocean around the continent of Antarctica.
The movement of ice shelves is principally driven by gravity-induced pressure from the grounded ice.[1] That flow continually moves ice from the grounding line to the seaward front of the shelf. Formerly, the primary mechanism of mass loss from ice shelves was thought to have been iceberg calving, in which a chunk of ice breaks off from the seaward front of the shelf. A study by NASA and university researchers, published in the June 14, 2013 issue of Science, found however that warm ocean waters which melt the undersides of Antarctic ice shelves are responsible for most of the continent's ice shelf mass loss.[2]
Typically, a shelf front will extend forward for years or decades between major calving events. Snow accumulation on the upper surface and melting from the lower surface are also important to the mass balance of an ice shelf. Ice may also accrete onto the underside of the shelf.
The density contrast between glacial ice and liquid water means that at least 1/9 of the floating ice is above the ocean surface, depending on how much pressurized air is contained in the bubbles within the glacial ice, stemming from compressed snow. The formula for the denominators above is
{\textstyle 1/[(\rho _{\text{seawater}}-\rho _{\text{glacial ice}})/\rho _{\text{seawater}}]}
, density of cold seawater is about 1028 kg/m3 and that of glacial ice from about 850 kg/m3[3][4] to well below 920 kg/m3, the limit for very cold ice without bubbles.[5][6] The height of the shelf above the sea can be even larger, if there is much less dense firn and snow above the glacier ice.
The world's largest ice shelves are the Ross Ice Shelf and the Filchner-Ronne Ice Shelf in Antarctica.
The term captured ice shelf has been used for the ice over a subglacial lake, such as Lake Vostok.
This article uses material from the Wikipedia article Ice shelf, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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How to Do Conic Sections - wikiHow Life
How to Do Conic Sections
1 Conic Sections: General Information
2 Conic Section 1: Circles
3 Conic Section 2: Parabolas
4 Conic Section 3: Ellipses
5 Conic Section 4: Hyperbolas
Conic sections are an interesting branch of mathematics involving the cutting of a double-napped cone. By cutting the cone in different ways, you can create a shape as simple as a point or as complex as a hyperbola.
Conic Sections: General Information
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Understand what is special about a conic section. Unlike regular coordinate equations, conic sections are general equations and don't necessarily have to be functions. For instance,
{\displaystyle x=5}
, while an equation, is not a function.
Know the difference between a degenerate case and a conic section. The degenerate cases are those where the cutting plane passes through the intersection, or apex of the double-napped cone. Some examples of degenerates are lines, intersecting lines, and points. The four conic sections are circles, parabolas, ellipses and hyperbolas.[1] X Research source
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Realize the idea that conic sections rely on. A conic section on a coordinate plane is just a collection of points which follow a certain rule which relates them all to the direction and focal points of the conic.
Conic Section 1: Circles
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Know what part of the cone you are looking at. A circle is defined as "the collection of points equidistant from a fixed point."[2] X Research source
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Find the coordinates of the center of the circle. For formula sake, we will call the center
{\displaystyle (h,k)}
as is custom when writing the general equation of a conic section.
Find the radius of the circle. The circle is defined as a collection of points which are the same distance away from a set center point
{\displaystyle (h,k)}
. That distance is the radius.
Plug them into the equation of a circle. The equation of a circle is one of the easiest to remember of all the conic sections. Given a center of
{\displaystyle (h,k)}
and a radius of length
{\displaystyle r}
, a circle is defined by
{\displaystyle (x-h)^{2}+(y-k)^{2}}
. Be sure to realize that this isn't a function. If you are trying to graph a circle on your graphing calculator, you will have to do some algebra to separate it into two equations which can be graphed using a calculator or use the "draw" feature.
Graph the circle, if necessary. If the graph is not given to you, graphing can help give you a better idea of what the circle should look like. Plot the point of the center, extend a line the length of the radius from each side, and draw the circle.
Conic Section 2: Parabolas
Understand what a parabola is. By definition, a parabola is "the set of all points equidistant from a line (the directrix) and a fixed point not on the line (the focus)."[3] X Research source
Find the coordinates of the vertex. The vertex,
{\displaystyle (h,k)}
, is the point where the graph has its axis of symmetry. Drawing this point will help you graph the parabola.
Find the focus. The equation for the focus is
{\displaystyle (h,k+p)}
{\displaystyle p}
being the distance between the vertex and the focus.
Plug in to find the directrix. The directrix has an equation of
{\displaystyle y=k-p}
. By using the vertex and focus to create a system of two equations, solve for the variables and plug them into the directrix formula.
Solve for the axis of symmetry. The parabola's axis of symmetry is defined as
{\displaystyle x=h}
. This line shows how the parabola is symmetrical and should cross through the vertex.
Find the equation of the parabola. The formula for the equation of the parabola is
{\displaystyle (y-k)={\frac {1}{4p}}(x-h)^{2}}
. Plug in the variables
{\displaystyle k}
{\displaystyle h}
{\displaystyle p}
to find the equation.
Graph the parabola if the graph is not given to you. This will show how the parabola appears. Plot the point of the vertex and focus, and draw the directrix and axis of symmetry. Draw the parabola either upwards or downwards, depending on if
{\displaystyle p}
is positive or negative, respectively.
Conic Section 3: Ellipses
Know what an ellipse is. An ellipse is defined as "the set of points such that the sum of the distances from any point on the ellipse to two other fixed points is constant."[4] X Research source
Find the center. The center of the ellipse is defined as
{\displaystyle (h,k)}
Find the major axis. The equation for an ellipse is
{\displaystyle {\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1}
{\displaystyle {\frac {(x-h)^{2}}{b^{2}}}+{\frac {(y-k)^{2}}{a^{2}}}=1}
{\displaystyle a>b}
. Whichever denominator has the larger number, the variable in the numerator's (either
{\displaystyle x}
{\displaystyle y}
) corresponding axis is the major axis. The other is the minor axis.
Solve for the vertices. An ellipse has four vertices. To solve for the vertices, let
{\displaystyle x}
{\displaystyle y=0}
and solve for the two variables. These will give you the points on your graph where the ellipse intersects.
Graph the ellipse, if neccesary. Plot the points of the vertices and connect the dots to graph the ellipse. The major axis should appear longer than the minor axis.
Conic Section 4: Hyperbolas
Understand what a hyperbola is. By definition, a hyperbola is "the set of all points such that the difference of the distances between any point on the hyperbola and two fixed points is constant."[5] X Research source This is similar to the ellipse; however, the hyperbola is the difference of the distances, while the ellipse is the sum.
Find the hyperbola's center. The center is defined as
{\displaystyle (h,k)}
and will be the point between the two curves.
Find the transverse axis. The equation of a hyperbola is
{\displaystyle {\frac {(x-h)^{2}}{a^{2}}}-{\frac {(y-k)^{2}}{b^{2}}}=1}
{\displaystyle {\frac {(y-k)^{2}}{a^{2}}}-{\frac {(x-h)^{2}}{b^{2}}}=1}
{\displaystyle a>b}
. Whichever variable is first in the equation and is greater (either
{\displaystyle x}
{\displaystyle y}
) is the transverse axis.
Solve for the vertices. Unlike the ellipse, a hyperbola only has two vertices. To solve for them, let
{\displaystyle x}
{\displaystyle y=0}
and solve for the two variables. The solutions for the variable corresponding with the transverse axis will give you the points on your graph where the hyperbola intersects.
The other two solutions will not be real numbers but eliminating the imaginary component (
{\displaystyle i}
) will give you two other coordinates on the real plane. These points, called the covertices, can help you graph the hyperbola.
Find the asymptotes. The asymptotes are two lines that the hyperbola will never touch but continuously get closer to. You can simply use the slope formula (
{\displaystyle m={\frac {rise}{run}}}
) or solve by factoring to find the asymptotes.
Graph the hyperbola if it isn't given to you. Construct a box using the four points (the two vertices and the two other points found) as the vertices of the box. From here, draw the asymptotes coming out of the corners of the box. Then, draw the two curves coming out of the box, touching the two vertices. Erase the box if you desire.
↑ https://en.wikipedia.org/wiki/Conic_section
↑ https://www.sparknotes.com/math/precalc/conicsections/section3/
|
Mathography: A mathography is a lot like your life history, except that it is focused on mathematics in your life.
Write a letter about yourself to your teacher. The letter will help your teacher get to know you as an individual. The letter should talk about these three general topics: you, you as a student, and you as a math student.
Remember to use complete sentences and make sure that it is neat enough for your teacher to read it easily. Start the letter with “Dear....” Make sure you sign your letter. This assignment should take
15
20
minutes to complete. Parts (b), (c), and (d) below have suggestions for what to write about each of the three topics.
You: Introduce yourself using the name you like to be called. Describe your hobbies, talents, and interests. State your goals or dreams. What are you proud of? What else would you like to share?
You as a Student: State the importance of school in your life. Describe yourself as a student. What kinds of classroom activities do you excel at most? What kinds of activities do you find frustrating? Explain which subjects are your favorites. Tell why you like them. How often do you finish in-class assignments? How faithfully do you do your homework?
You as a Math Student: Describe your most memorable moment in math and explain why you remember it. State your favorite math topic and name your least favorite. Explain how you feel about math this year.
Follow the instructions given to you by your teacher.
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Created by Luciano Mino
Using the trapezoid side calculator
How do I find the missing side of a trapezoid?
With our trapezoid side calculator, you can easily find the missing side of a trapezoid, given the perimeter and the length of its other sides.
Does a trapezoid have one pair of parallel sides?
Can any side of a trapezoid be called a base?
We will answer all those questions in a short text. Keep reading to learn how to use the trapezoid side length calculator!
A trapezoid is a 2D geometric figure with four sides where at least two are parallel to each other. Given that there is no further restriction to what constitutes a trapezoid, different shapes can fit the description such as a parallelogram (two pairs of parallel sides), isosceles trapezoid (two congruent sides), or a right trapezoid (one side perpendicular to the bases).
Trapezoid with a,b,c,d sides.
🙋 You can get a more detailed description of trapezoids in Omni's trapezoid calculator paired with different examples in solving trapezoid problems.
The trapezoid side calculator takes four inputs and calculates the missing side based on the following equation:
a = P - b - c - d
a
b
c
and
are the sides of the trapezoid and
P
is its perimeter. Notice how one of the lengths is on the other side of the equation? That's because we chose
a
to be the missing side in this case.
To use this equation, we would need to leave the missing length on the left side of the expression and then subtract the sum of all the other lengths from the perimeter on the other side.
With the trapezoid side calculator, you just need to leave the missing side empty and input all other parameters. It doesn't matter which side you call 'a', 'b', 'c', or 'd' in your trapezoid. As long as you leave one empty and input the others, the calculator will automatically fill in the blank field.
To find the missing side of a trapezoid:
Identify which side is missing.
Add together all the other sides and write down the result.
Subtract that number from the perimeter. The result is the missing side's length.
As you can see, obtaining the missing side given the length of the other sides and the trapezoid's perimeter is really easy.
Here are other tools that will help you in similar topics as calculating the side length of a trapezoid:
No, only any of the parallel sides can be called a base. The non-parallel sides of a trapezoid are called the 'legs'.
No, but a trapezoid must have at least one pair of parallel sides. It could have up to two pairs of parallel sides. In that case, it would be called a parallelogram.
Luciano Mino
Perfect square trinomial calculator checks if your trinomial is a perfect square of a linear binomial and, if so, finds the binomial.
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ShapValues - Model analysis | CatBoost
v
with contributions of each feature to the prediction for every input object and the expected value of the model prediction for the object (average prediction given no knowledge about the object).
v_{i}
is the contribution of the i-th feature.
v_{feature\_count}
is the expected value of the model prediction.
For a given object the sum
\sum\limits_{i=0}^{feature\_count} v[i]
is equal to the prediction on this object.
This is an implementation of the Consistent Individualized Feature Attribution for Tree Ensembles approach.
See the ShapValues file format.
Use the SHAP package to plot the returned values.
The feature importance
ShapValues_{i}
is calculated as follows for each feature
i
ShapValues_{i} = \displaystyle\sum_{S \subseteq N \backslash \{i\}} \displaystyle\frac{|S|! \left(M - |S| - 1 \right)!}{M!} [f_{x}(S \cup \{i\}) - f_{x}(S)] { , where}
M
is the number of input features.
N
is the set of all input features.
S
is the set of non-zero feature indices (the features that are being observed and not unknown).
f_{x} (S) = E[f(x) | x_{s}]
is the model's prediction for the input
x
E[f(x) | x_{s}]
is the expected value of the function conditioned on a subset S of the input features.
The complexity of computation depends on several conditions:
If the mean leaf count in the tree is less than the number of documents and trees are oblivious:
O(samples\_count \cdot trees\_count \cdot 2 ^ { depth} \cdot dimension \cdot depth ^ 2)
O(trees\_count \cdot {leaves\_in\_tree} \cdot dimension \cdot average\_depth^2 ) +
+O(trees\_count \cdot samples\_count \cdot (features\_in\_tree\_count + dimension)
samples_count is the number of documents in the dataset.
dimension is the dimensionality for Multiclassification and Multiregression.
trees_count is the number of trees.
depth is the depth of tree.
average_depth is the average depth of the trees.
leaves_in_tree is the number of leaves in the tree.
features_in_tree_count is the number of features in the tree.
Detailed information regarding usage specifics for different Catboost implementations.
|
Fabrication of Anode-Supported Tubular Solid Oxide Fuel Cell Using an Extrusion and Vacuum Infiltration Techniques | J. Electrochem. En. Conv. Stor | ASME Digital Collection
, 1500 Illinois Street, Golden, CO 80401
Song, J., and Sammes, N. M. (August 25, 2010). "Fabrication of Anode-Supported Tubular Solid Oxide Fuel Cell Using an Extrusion and Vacuum Infiltration Techniques." ASME. J. Fuel Cell Sci. Technol. December 2010; 7(6): 061013. https://doi.org/10.1115/1.4001324
A simple and mass productive extrusion technique was applied to fabricate anode-supported tubular solid oxide fuel cells (SOFCs). A standard NiO/8YSZ (nickel oxide/8 mol % yttria stabilized zirconia) cermet anode, 8YSZ electrolyte, and lanthanum strontium manganite
(La0.8Sr0.2MnO3)
cathode were used as the material components. Secondary electron microscopy images indicated that vacuum infiltration method successfully generated the thin electrolyte layer (about
15 μm
) with a structurally effective three phase boundaries. Fabricated unit cell showed the open circuit voltage of 1.12 V without any fuel leaking problems. Electrochemical tests showed a maximum power density up to
0.30 W cm−2
800°C
, implying the good performance as tubular SOFCs. This study verified that the extrusion aided by vacuum infiltration process could be a promising technique for mass production of tubular SOFCs.
anodes, cathodes, electrochemical electrodes, extrusion, lanthanum compounds, manganese compounds, nickel compounds, solid oxide fuel cells, strontium compounds, vacuum techniques, yttrium compounds, zirconium compounds, solid oxide fuel cell (SOFC), tubular SOFC, extrusion, infiltration, thin film electrolyte
Anodes, Electrolytes, Extruding, Manufacturing, Solid oxide fuel cells, Vacuum, Electrochemical analysis
Nabielek
Int. J. Appl. Cerm. Technol.
Fabrication and Characterization of Anode-Supported Planar Solid Oxide Fuel Cell Manufactured by a Tape Casting Processes
Fabrication and Characteristics of an Anode-Supported Thin-Film Electrolyte Fabricated by the Tape Casting Method for IT-SOFC
SOFCRoll Development at St. Andrews Fuel Cells Ltd
Development of Solid Oxide Fuel Cells and Shorts Stacks for Mobile Application
Cell and Stack Designs
High Temperature and Solid Fuel Cells
, Chapter 8, pp.
Acumentrics Produces Power From Biogas
Fuel Cells Bull.
Fabrication and Characterization of Tubular Solid Oxide Fuel Cells
Fabrication and Properties of Anode-Supported Tubular Solid Oxide Fuel Cells
Extruded Tubular Strontium- and Magnesium-Doped Lanthanum Gallate, Gadolinium-Doped Ceria, and Yttria-stabilized Electrolyte
Improvement of SOFC Performance Using a Microtubular, Anode-Supported SOFC
Current Collecting Efficiency of Micro Tubular SOFCs
Various Lanthanum Ferrite-Based Cathode Materials With Ni and Cu Substitution for Anode-Supported Solid Oxide Fuel Cells
|
Interactive Query Builder - Maple Help
Home : Support : Online Help : Connectivity : Database Package : Interactive Query Builder
Database query builder Assistant
QueryBuilder( conn, opts )
(optional) database connection
(optional) equation of the form advanced=boolean
The QueryBuilder command opens an Assistant that facilitates building SQL queries. The Assistant queries a database for table and column information. From the returned lists, you can select tables and columns.
If given, the conn argument must specify an instance of a connection that the Assistant can use to contact the database. If conn is not given, you must open a connection using the File>Open Connection menu item before querying the database. The File>Open Connection menu item opens the Database connection builder Assistant.
The QueryBuilder Assistant has two different views for creating queries. The Simple interface is a very basic interface allowing you to select columns from tables with an optional WHERE clause. This view is primarily intended for users unfamiliar with complex query options. This view is opened by default when the QueryBuilder command is executed. The second view is the Advanced interface. This provides access to a far more complex range of query options, including WHERE, GROUP BY, HAVING, and ORDER BY clauses; JOINS, column, and table aliasing; and column expressions.
The structure of the two views is similar. Both have a section for selecting columns (the SELECT area), a section for selecting tables (the FROM area) and a section for selecting clauses. These sections are arranged vertically, with the SELECT area above the FROM area which is above the clauses area.
The SELECT area contains a list of columns that can be selected from the tables currently selected within the FROM area (items in the FROM list must be selected before they appear in the SELECT list). In the Advanced view, the SELECT area may also contain column expressions and the FROM area may contain joins.
Once a query has been constructed, it can be exported or executed. By exporting the query, the text of the query is written to a file or returned to the worksheet as a string. Executing the query will cause the query to be evaluated by the database and the result returned to Maple. The resulting data can be managed in four ways. A browser window can be opened to display the data (this does not close the Assistant). Otherwise the data can be returned to the worksheet as a Result, as a table, or as an Array. The conversion to a table or an Array is performed using the ToMaple command. The Assistant is closed when executing to one of these data types.
For detailed information, see the help menu item in the Query Builder Assistant window.
To start the Query Builder Assistant, use the Database:-QueryBuilder command.
\mathrm{driver},\mathrm{conn}≔\mathrm{Database}\left(\right):
\mathrm{Database}:-\mathrm{QueryBuilder}\left(\mathrm{conn}\right)
|
Metric space - Wikipedia @ WordDisk
{\displaystyle A}
{\displaystyle B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle B}
{\displaystyle A}
{\displaystyle A}
{\displaystyle B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle C}
The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. Some of the non-geometric metric spaces include spaces of finite strings (finite sequences of symbols from a predefined alphabet) equipped with e.g. Hamming distance or Levenshtein distance, a space of subsets of any metric space equipped with Hausdorff distance, a space of real functions integrable on a unit interval with an integral metric
{\displaystyle d(f,g)=\int _{0}^{1}\left\vert f(x)-g(x)\right\vert \,dx}
, or the set of probability measures on the Borel sigma-algebra of any given metric space, equipped with the Wasserstein metric Wp (
{\displaystyle p\in \mathbb {R} _{\geq 1}}
). See also the section § Examples of metric spaces.
This article uses material from the Wikipedia article Metric space, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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MNIST Handwritten Digit Recognition in Keras - Nextjournal
Gregor Koehler / Feb 17 2020
with Micah P. Dombrowski, Joshua Sierles, Andrea Amantini and Philippa Markovics
MNIST Handwritten Digit Recognition in Keras
In this article we'll build a simple neural network and train it on a GPU-enabled server to recognize handwritten digits using the MNIST dataset. Training a classifier on the MNIST dataset is regarded as the hello world of image recognition. To follow along here, you should have a basic understanding of the Multilayer Perceptron class of neural networks.
MNIST contains 70,000 images of handwritten digits: 60,000 for training and 10,000 for testing. The images are grayscale, 28x28 pixels, and centered to reduce preprocessing and get started quicker.
Keras is a high-level neural network API focused on user friendliness, fast prototyping, modularity and extensibility. It works with deep learning frameworks like Tensorflow, Theano and CNTK, so we can get right into building and training a neural network without a lot of fuss.
Let's get started by setting up our environment with Keras using Tensorflow as the backend.
First, we have to install the Tensorflow and Keras packages. We do this in a separate runtime so that we can save it and export as a new environment and never have to install again.
conda install -y -c anaconda \
tensorflow-gpu h5py cudatoolkit=8
Install (Bash)
Python Default
We'll also import the dataset to cache it.
python -c 'from keras.datasets import mnist
mnist.load_data()'
These package imports are pretty standard — we'll get back to the Keras-specific imports further down.
Main (Bash in Python)
# imports for array-handling and plotting
# let's keep our keras backend tensorflow quiet
# for testing on CPU
#os.environ['CUDA_VISIBLE_DEVICES'] = ''
# keras imports for the dataset and building our neural network
importsMain (Python)
Now we'll load the dataset using this handy function which splits the MNIST data into train and test sets.
train-test-splitMain (Python)
Let's inspect a few examples. The MNIST dataset contains only grayscale images. For more advanced image datasets, we'll have the three color channels (RGB).
plt.imshow(X_train[i], cmap='gray', interpolation='none')
plt.title("Digit: {}".format(y_train[i]))
plot-examplesMain (Python)
In order to train our neural network to classify images we first have to unroll the height
\times
width pixel format into one big vector - the input vector. So its length must be
28 \cdot 28 = 784
. But let's graph the distribution of our pixel values.
plt.imshow(X_train[0], cmap='gray', interpolation='none')
plt.title("Digit: {}".format(y_train[0]))
plt.hist(X_train[0].reshape(784))
plt.title("Pixel Value Distribution")
pixel-distributionMain (Python)
As expected, the pixel values range from 0 to 255: the background majority close to 0, and those close to 255 representing the digit.
Normalizing the input data helps to speed up the training. Also, it reduces the chance of getting stuck in local optima, since we're using stochastic gradient descent to find the optimal weights for the network.
Let's reshape our inputs to a single vector vector and normalize the pixel values to lie between 0 and 1.
# let's print the shape before we reshape and normalize
# building the input vector from the 28x28 pixels
# normalizing the data to help with the training
# print the final input shape ready for training
print("Train matrix shape", X_train.shape)
print("Test matrix shape", X_test.shape)
input-formattingMain (Python)
So far the truth (Y in machine learning lingo) we'll use for training still holds integer values from 0 to 9.
y-value-countsMain (Python)
Let's encode our categories - digits from 0 to 9 - using one-hot encoding. The result is a vector with a length equal to the number of categories. The vector is all zeroes except in the position for the respective category. Thus a '5' will be represented by [0,0,0,0,1,0,0,0,0].
# one-hot encoding using keras' numpy-related utilities
print("Shape before one-hot encoding: ", y_train.shape)
print("Shape after one-hot encoding: ", Y_train.shape)
one-hot-encodingMain (Python)
Let's turn to Keras to build a neural network.
Our pixel vector serves as the input. Then, two hidden 512-node layers, with enough model complexity for recognizing digits. For the multi-class classification we add another densely-connected (or fully-connected) layer for the 10 different output classes. For this network architecture we can use the Keras Sequential Model. We can stack layers using the .add() method.
When adding the first layer in the Sequential Model we need to specify the input shape so Keras can create the appropriate matrices. For all remaining layers the shape is inferred automatically.
In order to introduce nonlinearities into the network and elevate it beyond the capabilities of a simple perceptron we also add activation functions to the hidden layers. The differentiation for the training via backpropagation is happening behind the scenes without having to implement the details.
We also add dropout as a way to prevent overfitting. Here we randomly keep some network weights fixed when we would normally update them so that the network doesn't rely too much on very few nodes.
The last layer consists of connections for our 10 classes and the softmax activation which is standard for multi-class targets.
# building a linear stack of layers with the sequential model
nn-setupMain (Python)
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Now that the model is in place we configure the learning process using .compile(). Here we specify our loss function (or objective function). For our setting categorical cross entropy fits the bill, but in general other loss functions are available.
As for the optimizer of choice we'll use Adam with default settings. We could also instantiate an optimizer and set parameters before passing it to model.compile() but for this example the defaults will do.
We also choose which metrics will be evaluated during training and testing. We can pass any list of metrics - even build metrics ourselves - and have them displayed during training/testing. We'll stick to accuracy for now.
# compiling the sequential model
compile-modelMain (Python)
Having compiled our model we can now start the training process. We have to specify how many times we want to iterate on the whole training set (epochs) and how many samples we use for one update to the model's weights (batch size). Generally the bigger the batch, the more stable our stochastic gradient descent updates will be. But beware of GPU memory limitations! We're going for a batch size of 128 and 8 epochs.
To get a handle on our training progress we also graph the learning curve for our model looking at the loss and accuracy.
In order to work with the trained model and evaluate its performance we're saving the model in /results/.
# training the model and saving metrics in history
save_dir = "/results/"
model_name = 'keras_mnist.h5'
plt.legend(['train', 'test'], loc='lower right')
train-modelMain (Python)
keras_mnist.h5
This learning curve looks quite good! We see that the loss on the training set is decreasing rapidly for the first two epochs. This shows the network is learning to classify the digits pretty fast. For the test set the loss does not decrease as fast but stays roughly within the same range as the training loss. This means our model generalizes well to unseen data.
It's time to reap the fruits of our neural network training. Let's see how well we the model performs on the test set. The model.evaluate() method computes the loss and any metric defined when compiling the model. So in our case the accuracy is computed on the 10,000 testing examples using the network weights given by the saved model.
mnist_model = load_model(
loss_and_metrics = mnist_model.evaluate(X_test, Y_test, verbose=2)
print("Test Loss", loss_and_metrics[0])
print("Test Accuracy", loss_and_metrics[1])
evaluateMain (Python)
This accuracy looks very good! But let's stay neutral here and evaluate both correctly and incorrectly classified examples. We'll look at 9 examples each.
# load the model and create predictions on the test set
predicted_classes = mnist_model.predict_classes(X_test)
# see which we predicted correctly and which not
correct_indices = np.nonzero(predicted_classes == y_test)[0]
incorrect_indices = np.nonzero(predicted_classes != y_test)[0]
print(len(correct_indices)," classified correctly")
print(len(incorrect_indices)," classified incorrectly")
# adapt figure size to accomodate 18 subplots
plt.rcParams['figure.figsize'] = (7,14)
figure_evaluation = plt.figure()
# plot 9 correct predictions
for i, correct in enumerate(correct_indices[:9]):
plt.imshow(X_test[correct].reshape(28,28), cmap='gray', interpolation='none')
"Predicted: {}, Truth: {}".format(predicted_classes[correct],
y_test[correct]))
# plot 9 incorrect predictions
for i, incorrect in enumerate(incorrect_indices[:9]):
plt.subplot(6,3,i+10)
plt.imshow(X_test[incorrect].reshape(28,28), cmap='gray', interpolation='none')
"Predicted {}, Truth: {}".format(predicted_classes[incorrect],
y_test[incorrect]))
figure_evaluation
evaluate-examplesMain (Python)
As we can see, the wrong predictions are quite forgiveable since they're in some cases even hard to recognize for the human reader.
In summary we used Keras with a Tensorflow backend on a GPU-enabled server to train a neural network to recognize handwritten digits in under 20 seconds of training time - all that without having to spin up any compute instances, only using our browser.
In the next article of this series (coming soon) we'll harness the power of GPUs even more to train more complex neural networks which include convolutional layers.
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Ceiling function in math
Examples of ceiling function computation
Graph of the ceiling function
Welcome to Omni's ceiling function calculator — the perfect place to fall in love with this popular math operation. In the short article below, we not only give the formal definition of the ceiling function, but also
Explain intuitively what the ceiling function does to a number;
Show how to graph the ceiling function;
Discuss what is the most popular symbol for the ceiling function; and
Go together through some examples of ceiling function computation.
The ceiling function maps a real number x to the smallest integer number that is greater than or equal to x:
\footnotesize \lceil x\rceil =\min \{n\in \mathbb {Z} \colon n\geq x \}
🙋 In the formula above you can see the most widespread ceiling function symbol. It looks like square brackets [ ] with their bottom part missing (so what remains is the... ceiling! Clever, right?). In programming languages, you most often find this function under the command ceil(x).
To get a better understanding of the ceiling function definition, let's go through a few examples together.
Let's compute the ceiling of
11.2
We pose the question dictated by the definition of the ceiling function: what are the integers that are greater than (or equal to)
11.2
12, 13, 14, 15, \ldots
. But we need the smallest one. Clearly, it is
12
\lceil 11.2 \rceil = 12
. Don't forget that you can verify this result with the ceiling function calculator!
-5
The integers that are greater than or equal to
-5
-5, -4, -3, \ldots
. The smallest one is
-5
\lceil -5 \rceil = -5
The last challenge — the ceiling of a non-integer negative number! Let's compute the ceil of
-2.3
What are the integers that are greater than (or equal to)
-2.3
-2, -1, 0, 1\ldots
-2
\lceil -2.3 \rceil = -2
As you can see from what we calculated above, the ceiling function rounds a number up to the nearest integer. If we're already at an integer, there's no need for rounding, and so the ceil function does not affect integers. Logical, right?
🙋 Not enough examples? To generate more, you can put random numbers into our ceiling function calculator and see what comes out!
Once you're done playing with our ceiling function calculator, it's high time we discussed how to graph the ceiling function. Here it is, in all its glory:
Looking at it, you can easily guess why we say that the ceiling function (along with its cousin, the floor function) belongs to the family of so-called step-functions.
What does the ceil function do?
The ceil function transforms a real number into the smallest integer that is greater than or equal to this number. It's like rounding a number up to the nearest integer.
The domain of the floor and ceiling function is the set of all real numbers. The image, in turn, is the set of integers.
How do I type the ceiling function in LaTeX?
The LaTeX code for ⌈ is \lceil, and for ⌉ it's \rceil. Hence, to get ⌈x⌉ you can type \lceil x \rceil.
What is the ceiling of pi?
The ceiling of the number pi is 4. This is because pi is approximately equal to 3.14, and so the smallest integer that is greater than pi is 4.
How do I calculate the ceiling of a number?
To determine the ceiling of a number:
If your number is an integer, then the ceiling is equal to this number. You're done!
Otherwise, write down the integers that are greater than your number.
Pick the smallest of the integers you've written down.
That's it! You've found the ceiling of your number.
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Rectangular Pyramid Surface Area Calculator
Example: Using the surface area of rectangular pyramid calculator.
Other rectangular pyramid calculators
Use this surface area of a rectangular pyramid calculator to deal with rectangular pyramids. Whether rectangular or square, these kinds of pyramids are famously associated with history across the globe! Be it the Egyptian pyramids or the Latin American ones. Pyramids are not just monuments but are also useful in determining the hardness of materials! The Vickers hardness test uses a pyramid indenter for that purpose. This article will explain how you can calculate its surface area.
The total surface area of a rectangular pyramid (
A_t
) is the sum of two areas:
The area of the base; and
The area 4 surrounding triangles, otherwise known as the lateral surface area.
Consider a rectangular pyramid with the following dimensions:
a - Base length;
b - Width of the base; and
h - Height of the pyramid.
Now, the area of the base (
A_b
) is a rather simple calculation. It is the area of a rectangle, i.e.
A_b = a \times b
Whereas, you can calculate the lateral surface area as an area of a triangle. If the slant heights are
l_a
l_b
, for the respective sides, the lateral surface area is given by the equation:
A_l = a \times l_a + b \times l_b
You can write the slant heights in the form of height and base length of the pyramid. Such that the lateral surface area becomes:
\scriptsize A_l = a \times \sqrt{h^2 + \frac{b^2}{4}}+ b \times \sqrt{h^2 + \frac{a^2}{4}}\\ \scriptsize A_t = A_l + A_b
Find the surface area of a rectangular pyramid having base dimensions as 4 and 5 cm. Take the height of the pyramid as 5 cm.
Enter the length of the base as 5 cm.
Insert the width of the base as 4 cm.
Fill in the height of the pyramid as 5 cm.
The surface area of a rectangular pyramid calculator will return the area as:
\scriptsize \begin{align*} \qquad A_b &= 5 \times 4 = 20 \text{ cm}^2 \\ \qquad A_l &= 49.29 \text{ cm}^2\\ \qquad A_t &= 20 + 49.29 = 69.29 \text{ cm}^2 \end{align*}
You can enable the advanced mode to view the slant height and face areas of the pyramids.
There are other tools based on rectangular pyramids that you can refer to learn more cool things about this omnipresent shape, such as:
How do I calculate the slant height of the rectangular pyramid?
To calculate the slant height of the rectangular pyramid:
Find the square of the base length of the pyramid.
Divide the square by 4.
Add the resultant with the square of the height of the pyramid.
Find the square root of the sum to obtain the slant height of the pyramid.
Repeat the steps with the base width to find the other slant height.
How do I calculate the lateral surface area of a rectangular pyramid?
To calculate the lateral surface area of a rectangular pyramid:
Find the slant heights of the pyramid.
Multiply slant height with the base length of the pyramid.
Repeat step 2 with the base width.
Add the two areas to obtain the lateral surface area of the pyramid.
What is the surface area of a pyramid with base length 4 cm and height 6 cm?
The surface area of the pyramid is 66.6 sq. cm. Out of which, the 50.6 sq. cm is the lateral surface area, which is calculated as: 2 × 4 × √((0.25×16) + 36) = 50.6 sq. cm., while 16 sq. cm is the base area.
Base length (a)
Base surface area (A_b)
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Suppose the function f : <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struc
f:\mathbb{R}\to \mathbb{R}
is continuous. For a natural number
k
{x}_{1},{x}_{2},...,{x}_{k}
be points in
\left[a,b\right]
. Prove there is a point
z
\left[a,b\right]
f\left(z\right)=\left(f\left({x}_{1}\right)+f\left({x}_{2}\right)+...+f\left({x}_{k}\right)\right)/k
So I'm thinking about applying the intermediate value theorem:
a<{x}_{1}<b,a<{x}_{2}<b,...,a<{x}_{k}<b
f\left(a\right)<f\left({x}_{1}\right)<f\left(b\right),...,f\left(a\right)<f\left({x}_{k}\right)<f\left(b\right)
k.f\left(a\right)<f\left({x}_{1}\right)+...+f\left({x}_{k}\right)<k.f\left(b\right)
f\left(a\right)<\left(f\left({x}_{1}\right)+f\left({x}_{2}\right)+...+f\left({x}_{k}\right)\right)/k<f\left(b\right)
But I couldn't think of any way to prove that
f\left(a\right)<f\left({x}_{k}\right)<f\left(b\right)
or is it even true?
EDIT: Thanks everyone for your effort.
z\in \left\{{x}_{1},\cdots {x}_{k}\right\}
f\left(z\right)\le f\left({x}_{j}\right)
{x}_{j}
w\in \left\{{x}_{1},\cdots {x}_{k}\right\}
f\left(w\right)\ge f\left({x}_{j}\right)
{x}_{j}
f\left(z\right)\le \frac{f\left({x}_{1}\right)+\cdots +f\left({x}_{k}\right)}{k}\le f\left(w\right)
Now, use the intermediate value theorem.
tswe0uk
m\le \frac{1}{k}\sum _{i=1}^{k}f\left({x}_{i}\right)\le M
M=maxf\left(x\right),m=minf\left(x\right)
\left[a,b\right]
, Now apply IVT
I am trying to understand and prove the fundamental theorem of calculus and I ran into some confusion understanding the intermediate value theorem . several sources online claim that if a function f(x) is continuous on [a,b] let s be a number such that
f\left(a\right)<s<f\left(b\right)
then there exists a number k in the open interval (a,b) such that f(k)=s my question is why do we only assume the open interval shouldn't it also include the closed interval [a,b] and also why does s have to be less than both
f\left(a\right)
f\left(b\right)
In the context of a measure space
\left(X,M,\mu \right)
f
is a bounded measurable function with
a\le f\left(x\right)\le b
\mu
-a.e.
x\in X
. Prove that for each integrable function
g
c\in \left[a,b\right]
{\int }_{X}f|g|d\mu =c{\int }_{X}|g|d\mu
I tried to use the Intermediate value theorem for integral of Riemann but i had no idea. Somebody have any tip?
Use the Intermediate Value Theorem to show that the equation
{x}^{3}+x+1=0
I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) < s for all p or f(p) > s for all s.
-Because f is continuous and the metric space M is connected, f(M) is connected.
-f(M) is connected in the real numbers, so if x,
yϵf\left(M\right)
and x < z < y, then
zϵf\left(M\right)
-I have a feeling the intermediate value theorem will play a part in this, but I'm not sure how.
I'm having trouble relating these three points.
suppose we have following function
f\left(x\right)={x}^{2}+10\mathrm{sin}\left(x\right)
we should show that there exist number
c
f\left(c\right)=1000
clearly we can solve this problem using intermediate value theorem for instance
f\left(0\right)=0
f\left(90\right)={90}^{2}+10\mathrm{sin}\left(90\right)=8100+10\ast 1=8110
and because 1000 is between these two number we can see that there exist such c so that
f\left(c\right)=1000
{x}^{100}+{x}^{99}+{x}^{98}+...+x+1=5
I need to determine whether there exists one solution
x
x>0
. Also, whether I have one solution
x<0
First, I let
a=0
f\left(a\right)=1
b=1
f\left(b\right)=101
f\left(a\right)<5<f\left(b\right)
, thus, by intermediate value theorem, there exits one solution
x\in \left(0,1\right)
. Similarly, I can conclude that there exits one solution smaller than 0. Is my process correct?
Use the Intermediate Value Theorem to show that there is a root of equation
{x}^{3}+2x-\frac{1}{x}=0
\left(\frac{1}{4},1\right)
. Find this root to two decimal places.
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How to find the area of a trapezoid—irregular or not!
How to use out irregular trapezoid area calculator
Discover our other trapezoid calculators!
Quickly calculate the area of an irregular trapezoid with our calculator! Here you will learn:
How to find the area of a trapezoid;
How to use our irregular trapezoid area calculator.
A trapezoid is a four-sided (quadrilateral) two-dimensional shape with a pair of parallel sides, the bases and two not necessarily parallel sides, the legs.
🙋 If there are no parallel sides, the shape is called trapezium: we don't do that here!
The distance between the parallel sides is called height. In the image below you can see all the trapezoid elements:
The elements of a trapezoid.
The two bases,
and
b
The two legs,
c
and
The height,
h
Four angles,
\alpha
\beta
\gamma
\delta
The angles of a trapezoid must satisfy the following equalities:
\alpha+\beta=180\degree
\gamma+delta=180\degree
This means that the total sum must be
360\degree
. Also, the angles associated with the same legs are complementary.
There are many types of trapezoids: here, we will focus on irregular trapezoids: the two legs have different lengths.
Calculating the area of an irregular trapezoid is not a big deal. In the end, you only need the value of the bases and the height!
The formula for the area of an irregular trapezoid is:
A=\frac{a+b}{2}\cdot h
There is an easy—and beautiful proof of this. Remember the formula for the area of a triangle:
A=\frac{b\cdot h}{2}
, and now take your irregular trapezoid, a pair of scissors, and cut!
If you cut on the line connecting one of the angles to the midpoint of the opposite leg, you can then join the two new shapes on the half-side, like in the diagram below:
How to find the area of an irregular trapezoid.
Now you have a triangle with height
h
a+b
. You know how to find its area!
To find the area of an irregular trapezoid, simply insert the values of the bases and the height in our calculator!
You can also use our calculator in reverse, inserting the area and finding either the bases or the height.
🙋 We hid some functionalities in the advanced mode. There are ways to find the values of bases and height when you know the angles or the legs: if you need it, we got your back: insert the parameter you have, and find the area!
For example, assume you have a really "short" trapezoid, with bases
a=10\ \text{cm}
b=6\ \text{cm}
h=2\ \text{cm}
. Input the values in our calculator!
We will compute this expression for you:
\footnotesize A=\frac{a+b}{2}\cdot h=\frac{10+6}{2}\cdot 2\ \text{cm}^2= 16\ \text{cm}^2
We have many other calculators that can help you with your trapezoidal problems! 😉
Try our general trapezoid calculator, or for more specific questions, explore our plethora of tiny calculators:
What is the formula of the area of an irregular trapezoid?
A = (a+b) * h/2
a and b are the bases of the trapezoid; and
h is the height of the trapezoid.
Does the area of a trapezoid depend on the legs?
The oblique sides of a trapezoid, or legs, don't appear in the area formula! Only the height of the trapezoid and the bases affect the value of the area!
This means that it doesn't matter if you consider a scalene trapezoid (all the sides different in value), a right trapezoid, or an isosceles one: as long as bases and height don't change, the area remains the same.
What is the area of an irregular trapezoid with bases a=8, b=2, height h=4, and a side c=5?
Apply the formula for the area of the trapezoid:
A = (a + b) * h/2 = (8 + 2) * 4/2 = 20
The area of this trapezoid is 20. The value of the last side didn't enter the calculations! Find out how to calculate the area of an irregular trapezoid on omnicalculator.com.
How do you find the area of a trapezoid using two triangles?
To find the area of a trapezoid geometrically, cut the trapezoid on a line connecting a vertex to the midpoint of the opposite side.
Rotate the shape you find until you make the two half sides fit together. At this point, you should have a segment of length equal to the sum of the two bases, which is the base of a triangle with height equal to the one of the trapezoid. From here, apply the area of a triangle formula:
A = b * h/2
where b is the base, and h is the height.
Find the perimeter of a trapezoid with any of the possible combination of inputs with our trapezoid perimeter calculator!
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What is the floor function in math?
Properties of floor function
Graph of the floor function
Omni's floor function calculator is here to save your sanity if you have always believed that floors have nothing to do with mathematics and you've suddenly discovered there's something called a floor function in math. In the article below, we will:
Intuitively explain what the floor function does to a number (with examples);
Discuss what the floor function graph looks like; and
Explain various useful properties of the floor function (never again will you have to wonder if the floor function is continuous!).
As a bonus, we explain how to type the floor function in LaTeX. Ready for a ride?
🙋 Satisfied? Looking for more? We also have a ceiling function calculator. Don't forget to visit it once you're done with this floor function calculator.
The floor function of a real number x is the greatest integer number that is less than or equal to x:
\footnotesize \lfloor x\rfloor =\max \{n\in \mathbb {Z} \colon n\leq x \}
It follows that the floor function maps the set of real numbers to the set of integers:
\operatorname{floor} \colon \ \mathbb R \to \mathbb{Z}
. We will now go through some examples so that you can get how this definition works in practice.
🙋 In our floor function calculator, we used the most popular way of denoting the floor function: the square brackets [ ] with their top part missing ⌊x⌋. Sometimes, especially in programming languages, you can see the whole word typed: floor(x).
Let's compute the floor of
21.3
We pose the question dictated by the definition of the floor function: what are the integers that are less than (or equal to)
21.3
There are lots of such integers:
21, 20, 19, 18, \ldots
. But we need the biggest one. Clearly, it is
21
\lfloor 21.3 \rfloor = 21
7
The integers that are less than or equal to
7
7, 6, 5, \ldots
. The biggest one is
7
\lfloor 7 \rfloor = 7
. Note how crucial it is here to remember the "or equal to" part of the definition!
The last challenge — the floor of a negative number! Let's compute the floor of
-1.3
What are the integers that are less than (or equal to)
-1.3
If you think for a bit, you can easily see that the desired integers are
-2, -3, -4, \ldots
-2
\lfloor -1.3 \rfloor = -2
As you can see in the above examples, we can also think of the floor function as rounding the number down to the nearest integer. That's the most intuitive way of understanding what the floor function does to a number.
Now, if you want to see more examples, use our floor function calculator — just throw some numbers at it and see what it spits out.
The floor function has some important properties.
The floor of a number is less than a number but not too much:
\footnotesize \qquad x−1 < \lfloor x \rfloor \leq x
A number is greater than its floor but not too much:
\footnotesize \qquad \lfloor x \rfloor \leq x < \lfloor x \rfloor + 1
Integers can be taken out of the floor freely:
\footnotesize \qquad \lfloor x + n \rfloor =\lfloor x \rfloor + n
The floor function is idempotent:
\footnotesize \qquad \lfloor \lfloor x \rfloor \rfloor = \lfloor x \rfloor
The floor function is non-decreasing:
\footnotesize \qquad x \leq y \Rightarrow \lfloor x \rfloor \leq \lfloor y \rfloor
The floor function is closely related to its sibling, the ceiling function
\lceil x \rceil
\footnotesize \qquad \lfloor x\rfloor = \begin{cases} \lceil x\rceil & \text{if } x \in \mathbb Z \\ \lceil x\rceil - 1 & \text{if } x \notin \mathbb Z \end{cases}
Don't hesitate to test these claims with Omni's floor function calculator! Now, let's discuss the graph of the floor function.
The floor function makes a funny graph: it belongs to the category of the so-called step-functions, and you can easily guess why if you take a look:
Omegatron, CC BY-SA 3.0, via Wikimedia Commons
Just in case, let's recall what the different dot symbols mean in the context of function graphs:
Filled dot means "including";
Empty dot means "not including".
For instance, at
x = 1
we see an empty dot at
y=0
and a filled dot at
y=1
. This means that the value of the floor function at
x = 1
y=1
y=0
Is the floor function continuous?
No, the floor function is not continuous: its points of discontinuity are all integer numbers.
Is the floor function one to one?
No, the floor function is not one-to-one. This is because the floor function maps the whole interval [n, n+1) to n. Hence, many numbers are mapped to one number. Technically speaking, the floor function is not injective.
How do I type floor function in LaTeX?
The LaTeX code for ⌊ is \lfloor and that for ⌋ is \rfloor. Hence, to get ⌊x⌋ you can type \lfloor x \rfloor.
What is the floor of pi?
The floor of the number pi is 3. This is because pi is approximately equal to 3.14, and so the greatest integer that is less than pi is 3.
How do I calculate the floor of a number?
To determine the floor of a number:
If your number is an integer, it is equal to its floor. In other words, you're done!
If you're dealing with a non-integer, then write down the integers that are smaller than your number.
Pick the greatest among the integers you've found in the previous step.
That's it! You've calculated the floor of your number.
The digit sum calculator helps you calculate the total sum of the digits in the given numbers.
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Regression - Objectives and metrics | CatBoost
Regression: objectives and metrics
\frac{\sum\limits_{i=1}^{N} w_{i} | a_{i} - t_{i}| }{\sum\limits_{i=1}^{N} w_{i}}
\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \displaystyle\frac{|a_{i}- t_{i}|}{Max(1, |t_{i}|)}}{\sum\limits_{i=1}^{N}w_{i}}
\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \left(e^{a_{i}} - a_{i}t_{i}\right)}{\sum\limits_{i=1}^{N}w_{i}}
\displaystyle\frac{\sum\limits_{i=1}^{N} (\alpha - 1(t_{i} \leq a_{i}))(t_{i} - a_{i}) w_{i} }{\sum\limits_{i=1}^{N} w_{i}}
\displaystyle\sqrt{\displaystyle\frac{\sum\limits_{i=1}^N (a_{i}-t_{i})^2 w_{i}}{\sum\limits_{i=1}^{N}w_{i}}}
\displaystyle-\frac{\sum_{i=1}^N w_i \log N(t_{i} \vert a_{i,0}, e^{2a_{i,1}})}{\sum_{i=1}^{N}w_{i}} = \frac{1}{2}\log(2\pi) +\frac{\sum_{i=1}^N w_i\left(a_{i,1} + \frac{1}{2} e^{-2a_{i,1}}(t_i - a_{i, 0})^2 \right)}{\sum_{i=1}^{N}w_{i}}
t
is target, a 2-dimensional approx
a_0
is target predict,
a_1
\log \sigma
predict, and
N(y\vert \mu,\sigma^2) = \frac{1}{\sqrt{2 \pi\sigma^2}} \exp(-\frac{(y-\mu)^2}{2\sigma^2})
is the probability density function of the normal distribution.
See the Uncertainty section for more details.
Depends on the condition for the ratio of the label value and the resulting value:
\begin{cases} \displaystyle\frac{\sum\limits_{i=1}^{N} \alpha |t_{i} - e^{a_{i}} | w_{i}}{\sum\limits_{i=1}^{N} w_{i}} & t_{i} > e^{a_{i}} \\ \displaystyle\frac{\sum\limits_{i=1}^{N} (1 - \alpha) |t_{i} - e^{a_{i}} | w_{i}}{\sum\limits_{i=1}^{N} w_{i}} & t_{i} \leq e^{a_{i}} \end{cases}
\displaystyle\frac{\sum\limits_{i=1}^N |a_{i} - t_{i}|^q w_i}{\sum\limits_{i=1}^N w_{i}}
The power coefficient.
Valid values are real numbers in the following range:
[1; +\infty)
L(t, a) = \sum\limits_{i=0}^N l(t_i, a_i) \cdot w_{i} { , where}
l(t,a) = \begin{cases} \frac{1}{2} (t - a)^{2} { , } & |t -a| \leq \delta \\ \delta|t -a| - \frac{1}{2} \delta^{2} { , } & |t -a| > \delta \end{cases}
User-defined parameters:
\delta
parameter of the Huber metric.
\displaystyle\frac{\sum\limits_{i=1}^{N} |\alpha - 1(t_{i} \leq a_{i})|(t_{i} - a_{i})^2 w_{i} }{\sum\limits_{i=1}^{N} w_{i}}
The coefficient used in expectile-based losses.
\displaystyle\frac{\sum\limits_{i=1}^{N}\left(\displaystyle\frac{e^{a_{i}(2-\lambda)}}{2-\lambda} - t_{i}\frac{e^{a_{i}(1-\lambda)}}{1-\lambda} \right)\cdot w_{i}}{\sum\limits_{i=1}^{N} w_{i}} { , where}
\lambda
is the value of the variance_power parameter.
variance_power
The variance of the Tweedie distribution.
Supported values are in the range (1;2).
\frac{\sum_{i=1}^N w_i \log(\cosh(a_i - t_i))}{\sum_{i=1}^N w_i}
\displaystyle\frac{\sum\limits_{i=1}^{N} c^2(\frac{|t_{i} - a_{i} |}{c} - \ln(\frac{|t_{i} - a_{i} |}{c} + 1))w_{i}}{\sum\limits_{i=1}^{N} w_{i}} { , where}
c
is the value of the smoothness parameter.
The smoothness coefficient. Valid values are real values in the following range
(0; +\infty)
The proportion of predictions, for which the difference from the label value exceeds the specified value greater_than.
\displaystyle\frac{\sum\limits_{i=1}^{N} I\{x\} w_{i}}{\sum\limits_{i=1}^{N} w_{i}} { , where}
I\{x\} = \begin{cases} 1 { , } & |a_{i} - t_{i}| > greater\_than \\ 0 { , } & |a_{i} - t_{i}| \leq greater\_than \end{cases}
User-defined parameters: greater_than
Increase the numerator of the formula if the following inequality is met:
|prediction - label|>value
\displaystyle\frac{100 \sum\limits_{i=1}^{N}\displaystyle\frac{w_{i} |a_{i} - t_{i} |}{(| t_{i} | + | a_{i} |) / 2}}{\sum\limits_{i=1}^{N} w_{i}}
1 - \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} (a_{i} - t_{i})^{2}}{\sum\limits_{i=1}^{N} w_{i} (\bar{t} - t_{i})^{2}}
\bar{t}
is the average label value:
\bar{t} = \frac{1}{N}\sum\limits_{i=1}^{N}t_{i}
\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} (\log_{e} (1 + t_{i}) - \log_{e} (1 + a_{i}))^{2}}{\sum\limits_{i=1}^{N} w_{i}}
median(|t_{1} - a_{1}|, ..., |t_{i} - a_{i}|)
MAE + -
MAPE + +
Poisson + +
Quantile + +
RMSE + +
RMSEWithUncertainty + -
LogLinQuantile + +
Lq + +
Huber + +
Expectile + +
Tweedie + +
LogCosh + -
FairLoss - -
NumErrors - +
SMAPE - -
MSLE - -
MedianAbsoluteError - -
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Why 'complete the square'?
How do you know when to apply complete the square formula?
Examples of completing the square
Here, we discuss how you can quickly and easily solve a quadratic equation with the method of completing the square. Examples of solutions are included!
Completing the square is a method in mathematics (in algebra, to be precise) that we use to solve quadratic equations (or equivalently, to factor quadratic trinomials). It's an alternative method to using the quadratic formula. The goal is to obtain a perfect square trinomial on the left side of the equation (where we have the unknown
x
) and a number on the right side. That is, we transform the equation
ax^2 +bx + c = 0
a(x+d)^2 = e
(x+d)^2 = e/a
a
b
c
d
e
are real coefficients. Next, we take a look at the right-hand side:
e/a \ge 0
e/a
is non-negative), we simplify and solve the equation by taking the square root of both sides.
e/a < 0
e/a
is negative), then our equation has no solutions.
In math, a quadratic equation is an equation where a value of
x
is desired so that a quadratic polynomial (a polynomial of degree
2
) is equal to zero:
ax^2+bx+c=0
a
b
c
are real coefficients. If the right side is not zero (i.e.
ax^2+bx+c = w
w \ne 0
), you can always transfer to the left side to get the form given above:
ax^2 + bx + (c-w) = 0
This method is called "complete the square" because we are hunting for perfect square trinomials. Formally, we want to transform the expression
x^2+bx+c
so as to obtain
(x+d)^2
, which is a trinomial that arises from squaring a linear binomial
x+d
. We can then apply the square root to both sides of the equation to solve the initial equation.
In order to solve a quadratic equation by completing the square, follow these steps:
If the leading coefficient of your quadratic equation is not
1
(i.e., if the polynomial is not monic), then divide both sides by
a
Assume we have the expression
x^2 + bx + c = 0
(x+b/2)^2 = x^2 + bx +b^2/4
. The first two terms are the same, but the last terms differ.
To get from
c
b^2/4
, we have to subtract
c
b^2/4
(x^2 + bx + c) - c + b^2/4 = (x+b/2)^2
We have to perform the same operation on the right side! Finally, our equation is equivalent to
(x+b/2)^2 = -c + b^2/4
The next step depends on the sign of the right-hand side:
b^2/4 > c
, then there's no solution.
b^2/4 = c
, then we have one solution, and it is equal to
-b/2
b^2/4 < c
, then there are two solutions and you can find them by taking the square root of both sides.
Completing the square is a method of solving quadratic equations that always works — even if the coefficients are irrational or if the equation does not have real roots!
It's up to you to decide whether you want to deal with a given quadratic expression by using the quadratic formula, or by the method of completing the square. There are many quadratic equations for which the latter is much faster and more elegant — you just need to gain a bit of experience to be able to quickly choose the best method.
Let's discuss a few examples of solving quadratic equations by completing the square.
Example 1. Solve by completing the square:
x^2 + 4x + 4 = 0
We immediately recognize the short multiplication formula working in reverse:
(x+2)^2 =x^2 + 4x + 4
. Thus, our problem can be rewritten as
(x+2)^2 = 0
\sqrt{0} = 0
x+2=0
x = -2
. In fact, in this example we didn't have to complete the square, because the perfect square trinomial was already there, staring at us defiantly!
Example 2. Solve using the completing the square method:
x^2 + 6x + 5 = 0
Let's take a look at the part containing the unknown
x
x^2 + 6x
. To produce these terms by squaring a linear binomial, we can use:
(x + 3)^2 = x^2 + 6x + 9
As you can see, the third term doesn't agree with what we have in our equations, so we need to complete the square. We have
5
in the original equation and
9
in the perfect square. So let's add
4
to both sides of the initial equation:
\begin{split} x^2 + 6x + 5 + 4 &= 0 + 4 \\ x^2 + 6x + 9 &= 4 \\ (x + 3)^2 &= 4 \\ |x + 3| &= 2 \\ x + 3 &= \pm 2 \\ \therefore\qquad x &= -1 \\ \text{ or } x &= -5 \\ \end{split}
x^2 - 2x + 4 = 0
At the left side, we easily recognize
x^2 - 2x
as part of the perfect square trinomial
x^2 - 2x + 1 = (x-1)^2
4
in our equation while we need
1
. So let's subtract
3
\begin{split} x^2 - 2x + 4 - 3 &= -3 \\ x^2 - 2x + 1 &= -3 \\ (x - 1)^2 &= -3 \\ \end{split}
Ouch. The equation says that some number squared (represented by the left-hand side of the equation) should be equal to
-3
. The problem is, squaring always leads to non-negative numbers! Therefore, we can deduce that our equation has no solutions (in real numbers).
What are the special right triangles formulas? How to solve special right triangles? Check out with this special right triangles calculator!
The unit rate calculator is here to give you your fraction's equivalent with one in the denominator (i.e. unit rate).
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Calibrated_airspeed Knowpia
When flying at sea level under International Standard Atmosphere conditions (15 °C, 1013 hPa, 0% humidity) calibrated airspeed is the same as equivalent airspeed (EAS) and true airspeed (TAS). If there is no wind it is also the same as ground speed (GS). Under any other conditions, CAS may differ from the aircraft's TAS and GS.
Calibrated airspeed in knots is usually abbreviated as KCAS, while indicated airspeed is abbreviated as KIAS.
In some applications, notably British usage, the expression rectified airspeed is used instead of calibrated airspeed.[1]
Practical applications of CASEdit
CAS has two primary applications in aviation:
for navigation, CAS is traditionally calculated as one of the steps between indicated airspeed and true airspeed;
for aircraft control, CAS (and EAS) are the primary reference points, since they describe the dynamic pressure acting on aircraft surfaces regardless of density, altitude, wind, and other conditions. EAS is used as a reference by aircraft designers, but EAS cannot be displayed correctly at varying altitudes by a simple (single capsule) airspeed indicator. CAS is therefore a standard for calibrating the airspeed indicator such that CAS equals EAS at sea level pressure and approximates EAS at higher altitudes.
With the widespread use of GPS and other advanced navigation systems in cockpits, the first application is rapidly decreasing in importance – pilots are able to read groundspeed (and often true airspeed) directly, without calculating calibrated airspeed as an intermediate step. The second application remains critical, however – for example, at the same weight, an aircraft will rotate and climb at approximately the same calibrated airspeed at any elevation, even though the true airspeed and groundspeed may differ significantly. These V speeds are usually given as IAS rather than CAS, so that a pilot can read them directly from the airspeed indicator.
Calculation from impact pressureEdit
Since the airspeed indicator capsule responds to impact pressure,[2] CAS is defined as a function of impact pressure alone. Static pressure and temperature appear as fixed coefficients defined by convention as standard sea level values. It so happens that the speed of sound is a direct function of temperature, so instead of a standard temperature, we can define a standard speed of sound.
For subsonic speeds, CAS is calculated as:
{\displaystyle CAS=a_{0}{\sqrt {5\left[\left({\frac {q_{c}}{P_{0}}}+1\right)^{\frac {2}{7}}-1\right]}}}
{\displaystyle q_{c}}
= impact pressure
{\displaystyle P_{0}}
= standard pressure at sea level
{\displaystyle {a_{0}}}
is the standard speed of sound at 15 °C
For supersonic airspeeds, where a normal shock forms in front of the pitot probe, the Rayleigh formula applies:
{\displaystyle CAS=a_{0}\left[\left({\frac {q_{c}}{P_{0}}}+1\right)\times \left(7\left({\frac {CAS}{a_{0}}}\right)^{2}-1\right)^{2.5}/\left(6^{2.5}\times 1.2^{3.5}\right)\right]^{(1/7)}}
The supersonic formula must be solved iteratively, by assuming an initial value for
{\displaystyle CAS}
{\displaystyle a_{0}}
These formulae work in any units provided the appropriate values for
{\displaystyle P_{0}}
{\displaystyle a_{0}}
are selected. For example,
{\displaystyle P_{0}}
= 1013.25 hPa,
{\displaystyle a_{0}}
= 1,225 km/h (661.45 kn). The ratio of specific heats for air is assumed to be 1.4.
These formulae can then be used to calibrate an airspeed indicator when impact pressure (
{\displaystyle q_{c}}
) is measured using a water manometer or accurate pressure gauge. If using a water manometer to measure millimeters of water the reference pressure (
{\displaystyle P_{0}}
) may be entered as 10333 mm
{\displaystyle H_{2}0}
At higher altitudes CAS can be corrected for compressibility error to give equivalent airspeed (EAS). In practice compressibility error is negligible below about 3,000 m (10,000 ft) and 370 km/h (200 kn).
^ Clancy, L. J. (1975) Aerodynamics, pp 31, 32. Pitman Publishing Limited, London. ISBN 0 273 01120 0
^ Some authors in the field of compressible flows use the term dynamic pressure or compressible dynamic pressure instead of impact pressure.
Blake, Walt (2009). Jet Transport Performance Methods. Seattle: Boeing Commercial Airplanes.
Gracey, William (1980), "Measurement of Aircraft Speed and Altitude" (12 MB), p. 15, NASA Reference Publication 1046.
A free windows calculator which converts between various airspeeds (true / equivalent / calibrated) according to the appropriate atmospheric (standard and not standard!) conditions
A free android calculator which converts various airspeeds according to atmospheric characteristics
Newbyte airspeed converter
JavaScript Calibrated Airspeed calculator from True Airspeed and other variables at luizmonteiro.com
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CT Visualization of Cryoablation in Pulmonary Veins | J. Med. Devices | ASME Digital Collection
M. Shenoi,
M. Shenoi
J. Bischof,
, San Jose, CA USA
Shenoi, M., Zhang, X., Bischof, J., and George, L. (July 6, 2009). "CT Visualization of Cryoablation in Pulmonary Veins." ASME. J. Med. Devices. June 2009; 3(2): 027512. https://doi.org/10.1115/1.3135157
Over 2 million adults in the United States are affected by atrial fibrillation (AF), a common cardiac arrhythmia that is associated with decreased survival, increased cardiovascular morbidities, and a decrease in quality of life. AF can be initiated by ectopic beats originating in the myocardial sleeves surrounding the pulmonary viens. Pulmonary vein (PV) isolation via radio frequency ablation is the current gold standard for treating patients with drug-refractory AF. However, cryoablation is emerging as a new minimally-invasive technique to achieve PV isolation. Cryoablation is fast gaining acceptance due to its minimal tissue disruption, decreased thrombogenicity, and reduced complications (RF can lead to low rate of stenosis). One important question in regard to this technology is whether the PV lesion is transmural and circumferential and to what extent adjacent tissues are involved in the freezing process. As ice formation lends itself to image contrast in the body, we hypothesized that intraprocedural CT visualization of the iceball formation would allow us to predict the extent of the cryolesion and provide us with a measure of the adjacent tissue damage. Cryoablation was performed using a prototype balloon catheter cryoablation system (Boston Scientific Corporation). CT visualization of iceball formation was assessed both in vitro and in vivo. Initial in vitro studies were performed in agarose gel phantoms immersed in a
37°C
water bath. Subsequently, in vivo cryoablations were performed in 5 PV ostia in 3 crossbred farm swine. The catheters were positioned in the ostia under fluoroscopic guidance. CT scans of the thoracic region were obtained every 2.5 minutes. Animals were sacrified 6 days after the procedures. Gross pathology and histology of tissues in the region of interest were evaluated. Significant metal artifacts from the catheter and edge artifacts from the tissues surrounding the cryoballoon were observed under CT imaging both in vitro and in vivo. In vitro, it was found that the size of the iceball was comparable to that observed visually during freezing of agarose gel phantoms. In vivo, contrast change consistent with iceball formation was observed during the ablation in two out of five veins. The most clearly delineated iceball also yielded the clearest morbidity. In this case, esophageal injury on the anterior side proximal to the cryoablation site was noticed during necropsy of the animal in which the iceball was visualized. Transmural and circumferential lesions were obtained in all PVs ablated. We have shown that CT can be used to visualize iceball formation in vitro and in vivo (with limitations) using our cryoablation system. While the iceball in vitro is easily visualized, iceball growth in vivo is most evident once the iceball has grown beyond the PV into the adjacent tissues. This suggests that while CT cannot easily visualize iceball growth in the PV wall itself, it may still be an important tool to guide clinicians and reduce potential morbidities in adjacent tissues. The authors acknowledge Dan Busian (Fairview University Medical Center, Minneapolis, MN) and Dr. Erik Cressman for assistance with CT imaging.
blood vessels, cardiovascular system, catheters, computerised tomography, diseases, drugs, freezing, muscle, phantoms, radiofrequency heating, surgery
Cardiovascular system, Catheters, Computerized tomography, Drugs, Freezing, Phantoms, Visualization, Biological tissues, Agar, Imaging, Ablation (Vaporization technology), Arrhythmias, Biomedicine, Blood vessels, Damage, Diseases, Engineering prototypes, Heating, Ice, Metals, Muscle, Radiofrequency ablation, Surgery, Water, Wounds
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Supply and Demand in Investment Markets - Course Hero
Introduction to Finance/Interest Rates/Supply and Demand in Investment Markets
In manufacturing markets, the economic supply-and-demand model follows the basic principle that a consumer demands a product and a manufacturer supplies that product. While manufacturing markets are delicately balancing the amount of supply to meet the demands of the consumer, prices often fluctuate. The manufacturer hopes to achieve equilibrium for optimal performance. Equilibrium is the point at which the supply of currency meets the demand for that currency. Comparatively, in the arena of finance and interest rates, when a consumer purchases a product such as an automobile or house, they are often required to obtain a loan to fund the product. After receiving the loan, consumers must pay the principal amount and a rate of interest that is applied to the remaining amount to pay. An interest rate is the amount due per period, as a percentage of the amount owed by the borrower to the lender at the end of the previous period. Interest rates are generally expressed as an annual percentage. An interest rate for loans can be calculated using either simple or compound interest. Simple interest is an interest rate multiplied by the initial payment. Compound interest is a form of calculating interest in which the interest rate is multiplied by the sum of any remaining unpaid principal and unpaid cumulative interest, as of the previous period.
Banks often use a simple interest formula, and credit cards use a compound interest formula. For example, if the bank is offering a loan of $10,000 loan with a 10 percent interest rate using simple interest, the interest is
\$10{,}000\times10\%=\$1{,}000
. After 5 years, this is $5,000. Credit cards roll this amount back into the loan as compound interest. For the same loan terms, the borrower would pay $6,105 in interest.
\begin{aligned}\text{Compound Interest}&=\$10{,}000\;(1+0.10)^5\\&=\$16{,}105-\$10{,}000\\&=\$6{,}105\end{aligned}
Securities follow the rules of supply and demand. The supply and demand changes as the price makes it more attractive. There is a shift in supply as the price changes.
In finance, supply and demand meet the consumer's needs, but instead of purchase price, the interest rate is the variable. As with manufacturing, in order to gauge when to invest, investors are looking for the rate when the demand for money equals the supply of money, or the equilibrium interest rate. For example, Cogs Inc. needs to raise capital, so it issues 1,000 bonds at a rate of 10 percent. This rate of return is higher than most investors require, making the demand for the bond high. Because the supply is fixed at 1,000 and the demand is high, the value of the bond will be higher than if supply and demand were at equilibrium.
With all loans there is a level of risk in whether or not a loan will be paid back in full. Some institutions desire to sell their loans to other parties, such as a dealer system, to reduce or eliminate risk potential. A dealer system is a practice in which loans are sold to a financial institution or third party as part of the sale of securities. Loanable funds are the total amount of funds that all individuals and institutions supply to borrowers, and the loanable funds theory explains the processes that might attract an investor.
A local bank relies on loanable funds to generate income through interest rates. The interest rate is the cost of money. Where supply is short, money becomes more expensive due to rising interest rates. During the buying and selling process, the interest rate drives decision-making. An investor or secondary market will buy loans using the loanable funds theory. The loanable funds theory is the belief that the interest rate is established by the supply and demand of loanable funds in the marketplace. During this process the investor desires a return on their investment and will use the supply-and-demand model of interest rates to make their purchase determination.
<Vocabulary>Market Interest Rates
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Support vector machine (SVM) for one-class and binary classification - MATLAB - MathWorks Deutschland
f\left(x\right)=\left(x/s\right)\prime \beta +b.
\left\{\begin{array}{l}{\alpha }_{j}\left[{y}_{j}f\left({x}_{j}\right)-1+{\xi }_{j}\right]=0\\ {\xi }_{j}\left(C-{\alpha }_{j}\right)=0\end{array}
f\left({x}_{j}\right)=\varphi \left({x}_{j}\right)\prime \beta +b,
0.5\sum _{jk}{\alpha }_{j}{\alpha }_{k}G\left({x}_{j},{x}_{k}\right)
{\alpha }_{1},...,{\alpha }_{n}
\sum {\alpha }_{j}=n\nu
0\le {\alpha }_{j}\le 1
f\left(x\right)=x\prime \beta +b,
2/‖\beta ‖.
‖\beta ‖
0.5{‖\beta ‖}^{2}+C\sum {\xi }_{j}
{y}_{j}f\left({x}_{j}\right)\ge 1-{\xi }_{j}
{\xi }_{j}\ge 0
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}{x}_{j}\prime {x}_{k}-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}x\prime {x}_{j}+\stackrel{^}{b}.
\stackrel{^}{b}
{\stackrel{^}{\alpha }}_{j}
\stackrel{^}{\alpha }
\text{sign}\left(\stackrel{^}{f}\left(z\right)\right).
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}G\left({x}_{j},{x}_{k}\right)-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}G\left(x,{x}_{j}\right)+\stackrel{^}{b}.
{C}_{j}=n{C}_{0}{w}_{j}^{\ast },
{x}_{j}^{\ast }=\frac{{x}_{j}-{\mu }_{j}^{\ast }}{{\sigma }_{j}^{\ast }},
\begin{array}{c}{\mu }_{j}^{\ast }=\frac{1}{\sum _{k}{w}_{k}^{*}}\sum _{k}{w}_{k}^{*}{x}_{jk},\\ {\left({\sigma }_{j}^{\ast }\right)}^{2}=\frac{{v}_{1}}{{v}_{1}^{2}-{v}_{2}}\sum _{k}{w}_{k}^{*}{\left({x}_{jk}-{\mu }_{j}^{\ast }\right)}^{2},\\ {v}_{1}=\sum _{j}{w}_{j}^{*},\\ {v}_{2}=\sum _{j}{\left({w}_{j}^{*}\right)}^{2}.\end{array}
\sum _{j=1}^{n}{\alpha }_{j}=n\nu .
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networks(deprecated)/indegree - Maple Help
Home : Support : Online Help : networks(deprecated)/indegree
finds vertex indegrees in a graph
indegree(v, G)
indegree(vset, G)
Important:The networks package has been deprecated. Use the superseding command GraphTheory[InDegree]instead.
This routine computes the in-degree of a single vertex.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[indegree](...).
\mathrm{with}\left(\mathrm{networks}\right):
\mathrm{new}\left(G\right):
\mathrm{addvertex}\left(1,2,3,4,G\right):
\mathrm{addedge}\left([{1,2},{1,2},[1,2],[1,3],[4,1]],G\right):
\mathrm{vdegree}\left(1,G\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{indegree}\left(1,G\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{outdegree}\left(1,G\right)
\textcolor[rgb]{0,0,1}{2}
GraphTheory[InDegree]
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Energy and Calorimetry - Course Hero
General Chemistry/Energy and Calorimetry
Energy is defined as the ability to do work, and work is defined as the application of a force over a distance. When energy is transferred from one system to another, some energy is always "lost" as heat. Chemical processes and reactions always involve a transfer of energy. Reactions or processes that require energy, or heat, are endothermic, and those that release heat are exothermic. An exothermic reaction will cause a temperature rise in the surroundings, and an endothermic reaction will cause a temperature drop. The temperature change of a reaction or process can be measured using a calorimeter. This temperature change and the heat capacity of the materials can be used to calculate the energy change of the process.
-P_{\rm{out}}{\Delta V}
The temperature change in the surroundings can be used to calculate the energy change of a chemical process. These changes are described by calorimetry.
\Delta U=q+w
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DEtools,regularsp - Maple Help
Home : Support : Online Help : System : Libraries and Packages : Deprecated Packages and Commands : Deprecated commands : DEtools,regularsp
compute the regular singular points of a second order non-autonomous linear ODE
regularsp(des, ivar, dvar)
second order linear ordinary differential equation or its list form
indicates the independent variable when des is a list with the ODE coefficients
indicates the dependent variable, required only when des is an ODE and the dependent variable is not obvious
Important: The regularsp command has been deprecated. Use the superseding command DEtools[singularities], which computes both the regular and irregular singular points, instead.
The regularsp command determines the regular singular points of a given second order linear ordinary differential equation. The ODE could be given as a standard differential equation or as a list with the ODE coefficients (see DEtools[convertAlg]). Given a linear ODE of the form
p(x) y''(x) + q(x) y'(x) + r(x) y(x) = 0, p(x) <> 0, p'(x) <> 0
a point alpha is considered to be a regular singular point if
1) alpha is a singular point,
2) limit( (x-alpha)*q(x)/p(x), x=alpha ) = 0 and
limit( (x-alpha)^2*r(x)/p(x), x=alpha ) = 0.
The results are returned in a list. In the event that no regular singular points are found, an empty list is returned.
\mathrm{with}\left(\mathrm{DEtools}\right):
An ordinary differential equation (ODE)
\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\left(\frac{\mathrm{\alpha }}{x-1}+\frac{\mathrm{\beta }}{x}+\frac{\mathrm{\gamma }}{{x}^{2}}+\frac{\mathrm{\delta }}{{\left(x-1\right)}^{2}}+{\mathrm{\lambda }}^{2}\right)y\left(x\right)
\textcolor[rgb]{0,0,1}{\mathrm{ODE}}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\left(\frac{\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\beta }}}{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\gamma }}}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\delta }}}{{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{regularsp}\left(\mathrm{ODE}\right)
[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]
\mathrm{singularities}\left(\mathrm{ODE}\right)
\textcolor[rgb]{0,0,1}{\mathrm{regular}}\textcolor[rgb]{0,0,1}{=}{\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}{\mathrm{irregular}}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}
The coefficient list form
\mathrm{coefs}≔[21\left({x}^{2}-x+1\right),0,100{x}^{2}{\left(x-1\right)}^{2}]:
\mathrm{regularsp}\left(\mathrm{coefs},x\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}{\mathrm{\infty }}]
\mathrm{singularities}\left(\mathrm{coefs},x\right)
\textcolor[rgb]{0,0,1}{\mathrm{regular}}\textcolor[rgb]{0,0,1}{=}{\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}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{irregular}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\varnothing }
You can convert convert an ODE to the coefficient list form using DEtools[convertAlg] form
\mathrm{ODE}≔\left(2{x}^{2}+5{x}^{3}\right)\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(5x-{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x\right)+\left(\frac{1}{x}+x\right)y\left(x\right)=0
\textcolor[rgb]{0,0,1}{\mathrm{ODE}}\textcolor[rgb]{0,0,1}{≔}\left(\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\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}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\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}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
L≔\mathrm{convertAlg}\left(\mathrm{ODE},y\left(x\right)\right)
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}[[\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]
\mathrm{regularsp}\left(L,x\right)
[\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}]
\mathrm{singularities}\left(L,x\right)
\textcolor[rgb]{0,0,1}{\mathrm{regular}}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{irregular}}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{0}}
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Orientation of a cone settling in a vertical duct
Yu, Zhaosheng; Pan, Dingyi; Lin, Jianzhong; Shao, Xueming, E-mail: mecsxm@zju.edu.cn
[en] The orientation of a cone settling in a vertical duct has been numerically investigated by using the direct-forcing fictitious domain method. Our results indicate that with the characteristic Reynolds number Re being 30, the cone settles with apex pointing upward if the cone apex angle is smaller than 48°, irrespective of the initial orientation. For the apex angle larger than 48°, the cone also turns upward as long as the releasing angle is smaller than a critical one, otherwise it either turns to an inclined angle if the apex angle is smaller than (or equal to) 52°, or turns downward for the apex angle larger than 52°. The fluid inertial effect helps to turn the cone downward and both the critical apex angle for the unanimous upward-turning and the releasing angle for the turning direction at a fixed apex angle become smaller for a higher Reynolds number, although the effects of the Reynolds number on the critical angles are not enormous. The hairpin-like vortex structures are observed in the wake of the settling cone at Re = 400, accompanied by the persistent oscillation of the orientation angle. Both stable tilted and apex-upward orientations for the cone with the apex angle of 50°, depending on the releasing angle, are confirmed in our experiments. (paper)
Fluid Dynamics Research (Online); ISSN 1873-7005; ; v. 46(1); [15 p.]
CONES, DUCTS, FLUIDS, ORIENTATION, OSCILLATIONS, REYNOLDS NUMBER, VORTICES
Performance analysis of Brillouin optical time domain reflectometry (BOTDR) employing wavelength diversity and passive depolarizer techniques
Lalam, Nageswara; Ng, Wai Pang; Dai, Xuewu; Wu, Qiang; Fu, Yong Qing, E-mail: wai-pang.ng@northumbria.ac.uk
[en] We propose and experimentally validate a wavelength diversity technique combined with a passive depolarizer in order to improve the performance of Brillouin optical time domain reflectometry (BOTDR). The wavelength diversity technique enables the maximization of the launch pump power and suppresses the nonlinear effects, the latter of which limits the conventional BOTDR performance. As a result, the signal-to-noise ratio increases, thus improving the measurement accuracy for strain and temperature. In addition, considering the complexity and expensive methods required for polarization noise suppression in BOTDR system, a simple, low-cost passive depolarizer is employed to reduce the polarization noise. The experimental results show that the signal-to-noise ratio is improved by 4.85 dB, which corresponds to 174% improvement compared to a conventional BOTDR system. (paper)
Available from http://dx.doi.org/10.1088/1361-6501/aa9c6e; Country of input: International Atomic Energy Agency (IAEA)
ACCURACY, COST, NONLINEAR PROBLEMS, PERFORMANCE, POLARIZATION, SIGNAL-TO-NOISE RATIO, WAVELENGTHS
http://dx.doi.org/10.1088/1361-6501/aa9c6e
Turbulent dynamo in a conducting fluid and a partially ionized gas
Xu, Siyao; Lazarian, A., E-mail: syxu@pku.edu.cn, E-mail: lazarian@astro.wisc.edu
[en] By following the Kazantsev theory and taking into account both microscopic and turbulent diffusion of magnetic fields, we develop a unified treatment of the kinematic and nonlinear stages of a turbulent dynamo process, and we study the dynamo process for a full range of magnetic Prandtl number P m and ionization fractions. We find a striking similarity between the dependence of dynamo behavior on P m in a conducting fluid and
\mathcal{R}
(a function of ionization fraction) in a partially ionized gas. In a weakly ionized medium, the kinematic stage is largely extended, including not only exponential growth but a new regime of dynamo characterized by a linear-in-time growth of magnetic field strength, and the resulting magnetic energy is much higher than the kinetic energy carried by viscous-scale eddies. Unlike the kinematic stage, the subsequent nonlinear stage is unaffected by microscopic diffusion processes and has a universal linear-in-time growth of magnetic energy with the growth rate as a constant fraction 3/38 of the turbulent energy transfer rate, showing good agreement with earlier numerical results. Applying the analysis to the first stars and galaxies, we find that the kinematic stage is able to generate a field strength only an order of magnitude smaller than the final saturation value. But the generation of large-scale magnetic fields can only be accounted for by the relatively inefficient nonlinear stage and requires longer time than the free-fall time. It suggests that magnetic fields may not have played a dynamically important role during the formation of the first stars.
DIFFUSION, ENERGY TRANSFER, FLUIDS, GALAXIES, IONIZATION, MAGNETIC FIELDS, NONLINEAR PROBLEMS, PRANDTL NUMBER, SATURATION, STARS, TURBULENCE
Shur, Mikhail L.; Spalart, Philippe R.; Strelets, Mikhail Kh.; Travin, Andrey K., E-mail: strelets@mail.rcom.ru
[en] A CFD strategy is proposed that combines delayed detached-eddy simulation (DDES) with an improved RANS-LES hybrid model aimed at wall modelling in LES (WMLES). The system ensures a different response depending on whether the simulation does or does not have inflow turbulent content. In the first case, it reduces to WMLES: most of the turbulence is resolved except near the wall. Empirical improvements to this model relative to the pure DES equations provide a great increase of the resolved turbulence activity near the wall and adjust the resolved logarithmic layer to the modelled one, thus resolving the issue of 'log layer mismatch' which is common in DES and other WMLES methods. An essential new element here is a definition of the subgrid length-scale which depends not only on the grid spacings, but also on the wall distance. In the case without inflow turbulent content, the proposed model performs as DDES, i.e., it gives a pure RANS solution for attached flows and a DES-like solution for massively separated flows. The coordination of the two branches is carried out by a blending function. The promise of the model is supported by its satisfactory performance in all the three modes it was designed for, namely, in pure WMLES applications (channel flow in a wide Reynolds-number range and flow over a hydrofoil with trailing-edge separation), in a natural DDES application (an airfoil in deep stall), and in a flow where both branches of the model are active in different flow regions (a backward-facing-step flow)
S0142-727X(08)00120-3; Available from http://dx.doi.org/10.1016/j.ijheatfluidflow.2008.07.001; Copyright (c) 2008 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
International Journal of Heat and Fluid Flow; ISSN 0142-727X; ; CODEN IJHFD2; v. 29(6); p. 1638-1649
AIRFOILS, EQUATIONS, FUNCTIONS, LAYERS, MATHEMATICAL SOLUTIONS, PERFORMANCE, REYNOLDS NUMBER, SIMULATION, TURBULENCE, WALLS
Manifestations of the Roton Mode in Dipolar Bose-Einstein Condensates
Wilson, Ryan M.; Ronen, Shai; Bohn, John L.; Pu Han
[en] We investigate the structure of trapped Bose-Einstein condensates (BECs) with long-range anisotropic dipolar interactions. We find that a small perturbation in the trapping potential can lead to dramatic changes in the condensate's density profile for sufficiently large dipolar interaction strengths and trap aspect ratios. By employing perturbation theory, we relate these oscillations to a previously identified 'rotonlike' mode in dipolar BECs. The same physics is responsible for radial density oscillations in vortex states of dipolar BECs that have been predicted previously
Physical Review Letters; ISSN 0031-9007; ; CODEN PRLTAO; v. 100(24); p. 245302-245302.4
ANISOTROPY, ASPECT RATIO, BOSE-EINSTEIN CONDENSATION, OSCILLATIONS, PERTURBATION THEORY, POTENTIALS, TRAPPING, TRAPS
Spatiotemporal evolution of dielectric driven cogenerated dust density waves
Sarkar, Sanjib; Bose, M.; Mukherjee, S.; Pramanik, J.
[en] An experimental observation of spatiotemporal evolution of dust density waves (DDWs) in cogenerated dusty plasma in the presence of modified field induced by glass plate is reported. Various DDWs, such as vertical, oblique, and stationary, were detected simultaneously for the first time. Evolution of spatiotemporal complexity like bifurcation in propagating wavefronts is also observed. As dust concentration reaches extremely high value, the DDW collapses. Also, the oblique and nonpropagating mode vanishes when we increase the number of glass plates, while dust particles were trapped above each glass plates showing only vertical DDWs
BIFURCATION, CONCENTRATION RATIO, DUSTS, GLASS, PARTICLES, PLASMA DENSITY, PLASMA WAVES, PLATES, TRAPPING
Stereoscopic particle image velocimetry of the impinging venous needle jet during hemodialysis
Fulker, David; Forster, Kyle; Simmons, Anne; Barber, Tracie, E-mail: dave_fulker@hotmail.com
[en] Highlights: • A bench top flow rig of hemodialysis is realised. • Dynamic similarity is preserved using dimensional scaling of Reynolds number. • A disturbed flow region develops downstream of the venous needle jet. • Disturbed flows represents a potential site of stenosis on the roof of the vein. • Disturbed flows can be minimized through careful application of the venous needle. - Abstract: Stereoscopic particle image velocimetry (S-PIV) was used to quantify the dynamic flow field caused by the venous needle jet (VNJ) in an idealised model of hemodialysis cannulation. Scaling based on Reynolds number ensured dynamic similarity with physiological conditions. Measurements taken along the center plane indicate the presence of a steady secondary flow region, which develops downstream of the impingement region. Upon impingement, a wall jet forms on the floor of the vein and spreads along the curvature of the vessel. Circulating flow forms due to the interaction between the jet spreading and the wall jet. This secondary flow region represents a potential site of stenosis on the roof of the vein where flow is reversed. The effects of the circulating flows can be minimized by using shallow needle angles, low needle flow rates and placement of the needle away from the walls of the vein.
International Journal of Heat and Fluid Flow; ISSN 0142-727X; ; CODEN IJHFD2; v. 67(Part A); p. 59-68
FLOW RATE, IMAGES, IMPINGEMENT, JETS, PARTICLES, REYNOLDS NUMBER, WALLS
Efficient coupling between a high-Q cavity and a waveguide based on two-dimensional photonic crystal
Benmerkhi, A; Bouchemat, M; Bouchemat, T; Paraire, N, E-mail: hm_Bouchemat@yahoo.fr
[en] In this paper, we investigate the coupling of linear three-hole cavities (L3) into PC waveguides. We choose the L3 cavities owing to the high value of their quality factor (Q) to mode volume (V) ratio and their good match between cavity and waveguide field patterns, which improves the in-plane coupling efficiency. The systems are designed to increase the overlap between the evanescent cavity field and the waveguide mode, and to operate in the linear dispersion region of the waveguide. Our simulations indicate increased coupling when the cavity is tilted by 60° with respect to the waveguide axis. The transmission spectra and the field patterns are obtained by using Fullwave, a commercially available finite-difference time domain (FDTD) code. From a transmission calculation, a very high Q-factor value has been achieved at λ = 1.51 μm.
FQMT'11: Conference on frontiers of quantum and mesoscopic thermophysics; Prague (Czech Republic); 25-30 Jul 2011; Available from http://dx.doi.org/10.1088/0031-8949/2012/T151/014065; Country of input: International Atomic Energy Agency (IAEA)
Physica Scripta (Online); ISSN 1402-4896; ; v. 2012(T151); [4 p.]
CAVITIES, COUPLING, CRYSTALS, DISPERSIONS, EFFICIENCY, QUALITY FACTOR, SIMULATION, SPECTRA, TRANSMISSION, TWO-DIMENSIONAL CALCULATIONS, WAVEGUIDES
Two-degree-of-freedom vortex induced vibration of low-mass horizontal circular cylinder near a free surface at low Reynolds number
Chung, Meng-Hsuan, E-mail: mhsuan@webmail.nkmu.edu.tw
[en] Highlights: • Froude number affects the critical normalized submergence depth. • Proximity to a free surface fosters/weakens VIV for low/high Fr. • Phase lag of transverse displacement behind lift jumps at some reduced velocity. • Free-surface vortex strongly interacts with cylinder-shedding ones for high-Fr VIV. • Frequency of high-amplitude VIV differs from natural structure frequency in fluids. - Abstract: Two-degree-of-freedom vortex induced vibration (VIV) of a low-mass zero-damping circular cylinder horizontally placed near a free surface at Re = 100 was numerically studied with an adaptive Cartesian cut-cell/level-set method. Two Froude numbers and various normalized submergence depths were considered. The results reveal that the Froude number affects the critical normalized submergence depth and possible physical mechanisms are proposed. The in-line vibration amplitude cannot be neglected. Proximity to a free surface strengthens and suppresses the VIV for low and high Froude numbers, respectively; increases the occurrence of amplitude modulation; and in general enhances the magnitude of the time-averaged lift coefficient, which is always negative. The phase lag of the transverse displacement behind the lift coefficient jumps at some reduced velocity, which strongly depends on the Froude number and normalized submergence depth. Regular trajectories exist only in cases with a small vibration amplitude or a large normalized submergence depth. The vortex structures in any case with large transverse amplitude basically originate from the alternative vortex shedding with the negative vortex weaker than the positive one. For the higher Froude number, an extra free surface positive vortex interacts with the vortices from the cylinder surface. The vibration frequency deviates from the natural structure frequency in fluids in the large-amplitude regime.
International Journal of Heat and Fluid Flow; ISSN 0142-727X; ; CODEN IJHFD2; v. 57; p. 58-78
CYLINDERS, DEGREES OF FREEDOM, FLUIDS, FROUDE NUMBER, MECHANICAL VIBRATIONS, MODULATION, REYNOLDS NUMBER, SURFACES, TRAJECTORIES, VELOCITY, VORTICES
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No-hair_theorem Knowpia
The no-hair theorem states that all black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum.[1] All other information (for which "hair" is a metaphor) about the matter that formed a black hole or is falling into it "disappears" behind the black-hole event horizon and is therefore permanently inaccessible to external observers. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair",[1] which was the origin of the name.
In a later interview, Wheeler said that Jacob Bekenstein coined this phrase.[2]
"Richard Feynman, objected to the phrase that seemed to me to best symbolize the finding of one the graduate students: graduate student Jacob Bekenstein had shown that a black hole reveals by nothing outside it what went in, in the way of spinning electric particles. It might show electric charge; yes, mass; yes, but no other features - or as he put it, "A black hole has no hair" - and Richard Feynman thought that was an obscene phrase and he didn't want to use it. But that is a phrase now often used to state this feature of black holes, that they don't indicate any other properties other than a charge and angular momentum and mass." [3]
The first version of the no-hair theorem for the simplified case of the uniqueness of the Schwarzschild metric was shown by Werner Israel in 1967.[4] The result was quickly generalized to the cases of charged or spinning black holes.[5][6] There is still no rigorous mathematical proof of a general no-hair theorem, and mathematicians refer to it as the no-hair conjecture. Even in the case of gravity alone (i.e., zero electric fields), the conjecture has only been partially resolved by results of Stephen Hawking, Brandon Carter, and David C. Robinson, under the additional hypothesis of non-degenerate event horizons and the technical, restrictive and difficult-to-justify assumption of real analyticity of the space-time continuum.
Suppose two black holes have the same masses, electrical charges, and angular momenta, but the first black hole was made by collapsing ordinary matter whereas the second was made out of antimatter; nevertheless, then the conjecture states they will be completely indistinguishable to an observer outside the event horizon. None of the special particle physics pseudo-charges (i.e., the global charges baryonic number, leptonic number, etc., all of which would be different for the originating masses of matter that created the black holes) are conserved in the black hole, or if they are conserved somehow then their values would be unobservable from the outside.[citation needed]
Changing the reference frameEdit
Every isolated unstable black hole decays rapidly to a stable black hole; and (excepting quantum fluctuations) stable black holes can be completely described (in a Cartesian coordinate system) at any moment in time by these eleven numbers:
mass–energy
{\displaystyle M}
{\displaystyle {\textbf {P}}}
(three components),
{\displaystyle {\textbf {J}}}
{\displaystyle {\textbf {X}}}
{\displaystyle Q}
These numbers represent the conserved attributes of an object which can be determined from a distance by examining its gravitational and electromagnetic fields. All other variations in the black hole will either escape to infinity or be swallowed up by the black hole.
By changing the reference frame one can set the linear momentum and position to zero and orient the spin angular momentum along the positive z axis. This eliminates eight of the eleven numbers, leaving three which are independent of the reference frame: mass, angular momentum magnitude, and electric charge. Thus any black hole that has been isolated for a significant period of time can be described by the Kerr–Newman metric in an appropriately chosen reference frame.
The no-hair theorem was originally formulated for black holes within the context of a four-dimensional spacetime, obeying the Einstein field equation of general relativity with zero cosmological constant, in the presence of electromagnetic fields, or optionally other fields such as scalar fields and massive vector fields (Proca fields, etc.).[citation needed]
It has since been extended to include the case where the cosmological constant is positive (which recent observations are tending to support).[7]
Magnetic charge, if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole.
Counterexamples in which the theorem fails are known in spacetime dimensions higher than four; in the presence of non-abelian Yang–Mills fields, non-abelian Proca fields, some non-minimally coupled scalar fields, or skyrmions; or in some theories of gravity other than Einstein's general relativity. However, these exceptions are often unstable solutions and/or do not lead to conserved quantum numbers so that "The 'spirit' of the no-hair conjecture, however, seems to be maintained".[8] It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and solitons.
In 2004, the exact analytical solution of a (3+1)-dimensional spherically symmetric black hole with minimally coupled self-interacting scalar field was derived.[9] This showed that, apart from mass, electrical charge and angular momentum, black holes can carry a finite scalar charge which might be a result of interaction with cosmological scalar fields such as the inflaton. The solution is stable and does not possess any unphysical properties; however, the existence of a scalar field with the desired properties is only speculative.
Observational resultsEdit
The LIGO results provide some experimental evidence consistent with the uniqueness of the no-hair theorem.[10][11] This observation is consistent with Stephen Hawking's theoretical work on black holes in the 1970s.[12][13]
Soft hairEdit
A study by Stephen Hawking, Malcolm Perry and Andrew Strominger postulates that black holes might contain "soft hair", giving the black hole more degrees of freedom than previously thought.[14] This hair permeates at a very low-energy state, which is why it didn't come up in previous calculations that postulated the no-hair theorem.[15] This was the subject of Hawking's final paper which was published posthumously.[16][17]
^ a b Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. pp. 875–876. ISBN 978-0716703341. Archived from the original on 23 May 2016. Retrieved 24 January 2013.
^ Archived at Ghostarchive and the Wayback Machine: "Interview with John Wheeler 2/3" – via YouTube.
^ Transcript: John Wheeler - Feynman and Jacob Bekenstein, Web of Stories. Listeners: Ken Ford, Duration: 1 minute, 19 seconds, Date story recorded: December 1996, Date story went live: 24 January 2008
^ Israel, Werner (1967). "Event Horizons in Static Vacuum Space-Times". Phys. Rev. 164 (5): 1776–1779. Bibcode:1967PhRv..164.1776I. doi:10.1103/PhysRev.164.1776.
^ Israel, Werner (1968). "Event horizons in static electrovac space-times". Commun. Math. Phys. 8 (3): 245–260. Bibcode:1968CMaPh...8..245I. doi:10.1007/BF01645859. S2CID 121476298.
^ Carter, Brandon (1971). "Axisymmetric Black Hole Has Only Two Degrees of Freedom". Phys. Rev. Lett. 26 (6): 331–333. Bibcode:1971PhRvL..26..331C. doi:10.1103/PhysRevLett.26.331.
^ Bhattacharya, Sourav; Lahiri, Amitabha (2007). "No hair theorems for positive Λ". Physical Review Letters. 99 (20): 201101. arXiv:gr-qc/0702006. Bibcode:2007PhRvL..99t1101B. doi:10.1103/PhysRevLett.99.201101. PMID 18233129. S2CID 119496541.
^ Mavromatos, N. E. (1996). "Eluding the No-Hair Conjecture for Black Holes". arXiv:gr-qc/9606008v1.
^ Zloshchastiev, Konstantin G. (2005). "Coexistence of Black Holes and a Long-Range Scalar Field in Cosmology". Phys. Rev. Lett. 94 (12): 121101. arXiv:hep-th/0408163. Bibcode:2005PhRvL..94l1101Z. doi:10.1103/PhysRevLett.94.121101. PMID 15903901. S2CID 22636577.
^ "Gravitational waves from black holes detected". BBC News. 11 February 2016.
^ Pretorius, Frans (2016-05-31). "Viewpoint: Relativity Gets Thorough Vetting from LIGO". Physics. 9. doi:10.1103/physics.9.52.
^ Stephen Hawking.
^ Stephen Hawking celebrates gravitational wave discovery.
^ Hawking, Stephen W.; Perry, Malcolm J.; Strominger, Andrew (2016-06-06). "Soft Hair on Black Holes". Physical Review Letters. 116 (23): 231301. arXiv:1601.00921. Bibcode:2016PhRvL.116w1301H. doi:10.1103/PhysRevLett.116.231301. PMID 27341223. S2CID 16198886.
^ Horowitz, Gary T. (2016-06-06). "Viewpoint: Black Holes Have Soft Quantum Hair". Physics. 9. doi:10.1103/physics.9.62.
^ Haco, Sasha; Hawking, Stephen W.; Perry, Malcolm J.; Strominger, Andrew (2018). "Black Hole Entropy and Soft Hair". Journal of High Energy Physics. 2018 (12): 98. arXiv:1810.01847. Bibcode:2018JHEP...12..098H. doi:10.1007/JHEP12(2018)098. S2CID 119494931.
^ "Stephen Hawking's final scientific paper released". the Guardian. 2018-10-10. Retrieved 2021-09-14.
Hawking, S. W. (2005). "Information Loss in Black Holes". Physical Review D. 72 (8): 084013. arXiv:hep-th/0507171. Bibcode:2005PhRvD..72h4013H. doi:10.1103/PhysRevD.72.084013. S2CID 118893360. , Stephen Hawking's purported solution to the black hole unitarity paradox, first reported in July 2004.
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Ferranti_effect Knowpia
In electrical engineering, the Ferranti effect is the increase in voltage occurring at the receiving end of a very long (> 200 km) AC electric power transmission line, relative to the voltage at the sending end, when the load is very small, or no load is connected. It can be stated as a factor, or as a percent increase.
Illustration of the Ferranti effect; addition of voltages across the line inductance
It was first observed during the installation of underground cables in Sebastian Ziani de Ferranti's 10,000-volt AC power distribution system in 1887.[1]
The capacitive line charging current produces a voltage drop across the line inductance that is in-phase with the sending-end voltage, assuming negligible line resistance. Therefore, both line inductance and capacitance are responsible for this phenomenon. This can be analysed by considering the line as a transmission line where the source impedance is lower than the load impedance (unterminated). The effect is similar to an electrically short version of the quarter-wave impedance transformer, but with smaller voltage transformation.
The Ferranti effect is more pronounced the longer the line and the higher the voltage applied.[2] The relative voltage rise is proportional to the square of the line length and the square of frequency.[3]
The Ferranti effect is much more pronounced in underground cables, even in short lengths, because of their high capacitance per unit length, and lower electrical impedance.
It is interesting to note that an equivalent to the Ferranti effect occurs when inductive current flows through a series capacitance. Indeed, a
{\displaystyle 90^{\circ }}
lagging current
{\displaystyle -jI_{L}}
flowing through a
{\displaystyle -jX_{c}}
impedance results in a voltage difference
{\displaystyle V_{\text{send}}-V_{\text{receive}}=(-jI_{L})(-jX_{c})=-I_{L}X_{c}<0}
, hence in increased voltage on the receiving side.
LC circuit "A series resonant circuit provides voltage magnification."
Failure of the first trans-Atlantic telegraph cable
Characteristic impedance#Transmission line model
^ J. F. Wilson, Ferranti and the British Electrical Industry, 1864-1930, Manchester University Press, 1988 ISBN 0-7190-2369-6 page 44
^ Line-Charging Current Interruption by HV and EHV Circuit Breakers, Carl-Ejnar Sölver, Ph. D. and Sérgio de A. Morais, M. Sc. Archived January 26, 2007, at the Wayback Machine
^ A Knowledge Base for Switching Surge Transients, A. I. Ibrahim and H. W. Dommel Archived May 12, 2006, at the Wayback Machine
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Amplified forecasting: What will Buck's informed prediction of compute used in the largest ML training run before 2030 be? | Metaculus
Amplified forecasting: What will Buck's informed prediction of compute used in the largest ML training run before 2030 be?
Buck Shlegeris is a researcher at the Machine Intelligence Research Institute where he works on existential risk from artificial intelligence. Before joining MIRI, he worked as a software engineer at PayPal and was the first employee at Triplebyte. This talk has some background on his views about AI.
In this competition, your goal is to predict how Buck will predict the following question after reading all comments in this thread and considering the arguments and evidence mentioned.
The most significant update (reasoning or evidence) according to Buck;
The most accurate prediction of Buck’s posterior distribution submitted through an Elicit snapshot.
We’ll give out a $50 prize for each.
What will be the compute used in the largest ML training run before 2030, measured in log
_{10}
(petaflop/s-days)?
A petaflop/s-day consists of performing
10^{15}
neural net operations per second for one day, or a total of about
10^{20}
You can see and build on Buck's prior guess on Elicit. This guess is based on a quick extrapolation from AI and Compute.
Once this Metaculus question closes, Buck will look over the comment thread and any linked Elicit snapshots and build a new posterior distribution. Your goal is to predict that distribution.
This project is similar in spirit to amplifying epistemic spot checks and other work on scaling up individual judgment through crowdsourcing. As in these projects, we’re hoping to learn about mechanisms for delegating reasoning, this time in the forecasting domain.
The objective is to learn whether mechanisms like this could save people like Buck work. Buck wants to know: What would I think if I had more evidence and knew more arguments than I currently do, but still followed the sorts of reasoning principles that I'm unlikely to revise in the course of a comment thread? To get there, participants (a) provide relevant evidence and arguments and (b) predict what Buck’s distribution will be in light of that evidence.
Why not just resolve against the actual outcome? By resolving predictions against Buck's prediction, we can apply the process to questions where we can't observe outcomes. If this initial trial run goes well, we'll run a conditional or counterfactual amplified forecasting experiment next.
We will evaluate both the Metaculus community’s prediction and individuals’ predictions on accuracy by estimating KL divergence between Buck’s final distribution and others. To keep the setup as similar as possible between this run and future (counterfactual or conditional) runs, this question will not resolve as a Metaculus question.
To participate, create your forecast using Elicit, click “Save Snapshot to URL” and post your snapshot URL in a comment below. Share your reasoning and sources in the “Notes” column of the Elicit snapshot.
If you do not want to make your forecast public, you are welcome to forecast as usual on Metaculus. Your prediction will be incorporated into the community prediction.
If you submit multiple predictions, Buck will evaluate the one that you explicitly identify as your final submission, or pick the last submission before the competition closes.
If multiple users’ submissions are very close to Buck’s final distribution, the one submitted first will win.
The prize for “most significant update (reasoning or evidence) according to Buck” will be entirely at Buck’s discretion. For example, Buck may choose to give a prize for best reasoning even if it does not cause Buck to update his beliefs. Or, he may choose to not give out a prize in this category.
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Difference between revisions of "A Symplectic Integrator" - OrbiterWiki
Difference between revisions of "A Symplectic Integrator"
m (minor typo: an->a)
(Added OrbiterWikiLinks.)
On [https://www.orbiter-forum.com/ Orbiter Forum], orbinaut Keithth G has described some of his results comparing the [[Orbiter|Orbiter Space Flight Simulator]] numerical integrator with one of his own. In response to a user question, he provided the following description of the principles underlying a symplectic numeric integrator.
== A second-order symplectic integrator ==
On Orbiter Forum, orbinaut Keithth G has described some of his results comparing the Orbiter Space Flight Simulator numerical integrator with one of his own. In response to a user question, he provided the following description of the principles underlying a symplectic numeric integrator.
A second-order symplectic integrator[edit]
A couple of introductory points:[edit]
So, what is this integrator? If we just focus on a one-dimensional system for a moment, then the integrator maps a pair of points
{\displaystyle \left\{Q_{0},P_{0}\right\}}
to a new pair of points
{\displaystyle \left\{Q_{2},P_{2}\right\}}
{\displaystyle \delta t}
later. We can think of this as taking a pair of numbers,
{\displaystyle \left\{Q_{0},P_{0}\right\}}
, that describe the state of some object at some start time
{\displaystyle t=t_{0}}
and updating this to a new pair of numbers,
{\displaystyle \left\{Q_{2},P_{2}\right\}}
, that describes the new state of the same object at time
{\displaystyle t=t_{0}+\delta t}
{\displaystyle Q_{1}\leftarrow Q_{0}+{\frac {1}{2}}\,\delta t\,P_{0}}
{\displaystyle P_{2}\leftarrow P_{0}+\delta t\,F\left(Q_{1}\right)}
{\displaystyle Q_{2}\leftarrow Q_{1}+{\frac {1}{2}}\,\delta t\,P_{2}}
OK, so what do all of these symbols mean? Let's start with 'Q' and 'P'. In physics, largely because of a longstanding convention in nomenclature, 'Q' is often used to denote a spatial coordinate of something. Here, in our one-dimensional example, you can think of 'Q' as representing the x-coordinate of some object moving in some gravitational field. In the same convention, 'P' is used to denote the momentum of the same object. (Formally, it is the generalised momentum conjugate to 'Q' but that's a nuance that we don't need to worry about here.). Here, we can think of 'P' as representing the x-coordinate of the momentum of a particle. But since momentum is just mass * velocity (in this coordinate system), we can think of 'P' as the x-coordinate of velocity (multiplied by the mass of the object). So, the pair of numbers
{\displaystyle \left\{Q_{0},P_{0}\right\}}
just represents the position and velocity of the object at some initial time. And this is just the state-vector of the object written in cartesian coordinates. In other words, the symplectic integrator is an updating rule that takes an object's state vector at some initial time, and returns a new state vector at some time
{\displaystyle \delta t}
later. (Now, you can extend this idea to three dimensions but for the time being we'll just stick with one-dimension.)
Next, let's focus on the 'F' term. Again by convention, 'F' is used to denote the force acting on an object. Specifically,
{\displaystyle F\left(Q_{1}\right)}
represents the force on the object being modelled when it is at position
{\displaystyle Q_{1}}
. So, to carry out the three steps of the integration updating rule, we need to provide some force function,
{\displaystyle F(Q)}
which allows us to calculate the force on an object at a point in our one-dimensional space (i.e., at any 'Q'). For object moving subject to a gravitational force from a single body, we know that:
{\displaystyle F(Q)=-{\frac {\mu \,m}{(Q-Q^{*})^{2}}}}
(provided that
{\displaystyle Q>Q^{*}}
{\displaystyle \mu }
is the gravitational constant for the body in question; 'm' is the mass of the object that we are modelling, and
{\displaystyle Q^{*}}
is the location of the source of the gravitational field (e.g., the centre of the Sun or the Earth). If we work in Gaussian units where we measure distances in AU (Astronomical Units); velocity in AU/day; and mass in units of the mass of the Sun, then for an object moving in the gravitational potential of the Sun,
{\displaystyle \mu =0.00029591220828559115}
{\displaystyle m=1}
. For an object moving in the gravitational field of the Earth,
{\displaystyle m=1/354710}
. To convert from AU and days to metres and seconds, one needs to know that 1 AU = 149597870700 metres; and that 1 day = 86400 seconds. Unless the mass of the object that we are modelling is changing, it is convenient to set
{\displaystyle m=1}
In this one-dimensional system, the object moving in a gravitational potential is a bit limited in terms of directional options. It can either go up, or it can go down. Clearly, as the object moves towards the location of the gravitational source, the force acting on the object is going to increase without limit and the integration is going to going hay-wire, but so long as we are a reasonable distance away from
{\displaystyle Q^{*}}
then the integration scheme given above will provide a reasonable description of the object's motion.
In three dimensions[edit]
In three dimensions, the integration scheme looks much the same. But now we have to apply it to three spatial coordinates and three velocity components - i.e., the integration scheme becomes one that updates one set of 6 numbers
{\displaystyle \left\{Q_{x,0},Q_{y,0},Q_{z,0},P_{x,0},P_{y,0},P_{z,0}\right\}}
to a new set of six numbers
{\displaystyle \left\{Q_{x,2},Q_{y,2},Q_{z,2},P_{x,2},P_{y,2},P_{z,2}\right\}}
. And the second order integration scheme that does this is as follows:
{\displaystyle Q_{x,1}\leftarrow Q_{x,0}+{\frac {1}{2}}\,\delta t\,P_{x,0}}
{\displaystyle Q_{y,1}\leftarrow Q_{y,0}+{\frac {1}{2}}\,\delta t\,P_{y,0}}
{\displaystyle Q_{z,1}\leftarrow Q_{z,0}+{\frac {1}{2}}\,\delta t\,P_{z,0}}
{\displaystyle P_{x,2}\leftarrow P_{x,0}+\delta t\,F_{x}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle P_{y,2}\leftarrow P_{y,0}+\delta t\,F_{y}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle P_{z,2}\leftarrow P_{z,0}+\delta t\,F_{z}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle Q_{x,2}\leftarrow Q_{x,1}+{\frac {1}{2}}\,\delta t\,P_{x,2}}
{\displaystyle Q_{y,2}\leftarrow Q_{y,1}+{\frac {1}{2}}\,\delta t\,P_{y,2}}
{\displaystyle Q_{z,2}\leftarrow Q_{z,1}+{\frac {1}{2}}\,\delta t\,P_{z,2}}
You should note that we now have three force functions rather than just one. This is because force is really a 'vector' rather than 'scalar' quantity.
{\displaystyle F_{x}\left(Q_{x},Q_{y},Q_{z}\right)}
is the force acting in the x-direction on the object located at position
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{y}\left(Q_{x},Q_{y},Q_{z}\right)}
is the force acting in the y-direction on the object located at position
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{z}\left(Q_{x},Q_{y},Q_{z}\right)}
is the force in the x-direction acting on the object located at position
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
. And if we are in the gravitational field of a single body, then these functions become:
{\displaystyle F_{x}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{x}-Q_{x}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle F_{y}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{y}-Q_{y}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle F_{z}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{z}-Q_{z}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle \left(Q_{x}^{*},Q_{y}^{*},Q_{z}^{*}\right)}
are the coordinates of the gravitating body (when the object whose motion you are modelling is at
{\displaystyle \left(Q_{x},Q_{y},Q_{z}\right)}
Precis[edit]
Retrieved from "https://www.orbiterwiki.org/index.php?title=A_Symplectic_Integrator&oldid=419630"
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Lemma 29.32.13 (01V3)—The Stacks project
Section 29.32: Sheaf of differentials of a morphism
Lemma 29.32.13. Let $f : X \to S$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\Omega _{X/S}$ is an $\mathcal{O}_ X$-module of finite presentation.
Proof. Immediate from Algebra, Lemma 10.131.15, Lemma 29.32.5, Lemma 29.21.2, and Properties, Lemma 28.16.2. $\square$
f
should be assumed locally of finite presentation here I think.
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Solve Partial Differential Equation with LBFGS Method and Deep Learning - MATLAB & Simulink
Train Network Using fmincon
fmincon Objective Function
This example shows how to train a Physics Informed Neural Network (PINN) to numerically compute the solution of the Burger's equation by using the limited-memory BFGS (LBFGS) algorithm.
\left[-1,1\right]×\left[0,1\right]
, this examples uses a physics informed neural network (PINN) [1] and trains a multilayer perceptron neural network that takes samples
\left(x,t\right)
x\in \left[-1,1\right]
t\in \left[0,1\right]
u\left(x,t\right)
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}-\frac{0.01}{\pi }\frac{{\partial }^{2}u}{\partial {x}^{2}}=0,
u\left(x,t=0\right)=-sin\left(\pi x\right)
u\left(x=-1,t\right)=0
u\left(x=1,t\right)=0
Instead of training the network using the trainNetwork function, or using a custom training loop that updates parametes using sgdmupdate or similar functions, this example estimates the learnable parameters by using the fmincon function (requires Optimization Toolbox™). The fmincon function finds the minimum of constrained nonlinear multivariable functions.
\left(x,t\right)
u\left(x,t\right)
fulfills the Burger's equation, the boundary conditions, and the initial condition. To train the model, the exmaple uses the limited-memory BFGS (LBFGS) algorithm which is a quasi-Newton method that approximates the Broyden-Fletcher-Goldfarb-Shanno algorithm.
u\left(x=-1,t\right)=0
u\left(x=1,t\right)=0
u\left(x,t=0\right)=-sin\left(\pi x\right)
x
t
u\left(x,t\right)
Define the parameters for each of the operations and include them in a structure. Use the format parameters.OperationName_ParameterName where parameters is the structure, OperationName is the name of the operation (for example "fc1") and ParameterName is the name of the parameter (for example, "Weights").
The algorithm in this example requires learnable parameters to be in the first level of the stucture, so do not use nested structures in this step. The fmincon function requires the learnable to be doubles.
parameters.fc1_Weights = initializeHe(sz,2,"double");
parameters.fc1_Bias = initializeZeros([numNeurons 1],"double");
parameters.(name + "_Weights") = initializeHe(sz,numIn,"double");
parameters.(name + "_Bias") = initializeZeros([numNeurons 1],"double");
parameters.("fc" + numLayers + "_Weights") = initializeHe(sz,numIn,"double");
parameters.("fc" + numLayers + "_Bias") = initializeZeros([1 1],"double");
fc1_Weights: [20×2 dlarray]
fc1_Bias: [20×1 dlarray]
fc2_Weights: [20×20 dlarray]
fc9_Weights: [1×20 dlarray]
fc9_Bias: [1×1 dlarray]
Create the function modelLoss, listed in the Model Loss Function section at the end of the example, that takes as input the model parameters, the network inputs, and the initial and boundary conditions, and returns the gradients of the loss with respect to the learnable parameters and the corresponding loss.
Define fmincon Objective Function
Create the function objectiveFunction, listed in the fmincon Objective Function section of the example that returns the loss and gradients of the model. The function objectiveFunction takes as input, a vector of learnable parameters, the network inputs, the initial conditions, and the names and sizes of the learnable parameters, and returns the loss to be minimized by the fmincon function and the gradients of the loss with respect to the learnable parameters.
Specify the optimization options:
Optimize using the fmincon optmizer with the LBFGS algorithm for no more than 7500 iterations and function evaluations.
Evaluate with optimality tolerance 1e-5.
Provide the gradients to the algorithm.
options = optimoptions("fmincon", ...
HessianApproximation="lbfgs", ...
MaxIterations=7500, ...
MaxFunctionEvaluations=7500, ...
OptimalityTolerance=1e-5, ...
Train the network using the fmincon function.
The fmincon function requires the learnable parameters to be specified as a vector. Convert the parameters to a vector using the paramsStructToVector function (attached to this example as a supporting file). To convert back to a structure of parameters, also return the parameter names and sizes.
[parametersV,parameterNames,parameterSizes] = parameterStructToVector(parameters);
parametersV = extractdata(parametersV);
Convert the training data to dlarray objects. Specify that the inputs X and T have format "BC" (batch, channel) and that the initial conditions have format "CB" (channel, batch).
X = dlarray(dataX,"BC");
T = dlarray(dataT,"BC");
U0 = dlarray(U0,"CB");
Create a function handle with one input that defines the objective function.
objFun = @(parameters) objectiveFunction(parameters,X,T,X0,T0,U0,parameterNames,parameterSizes);
Update the learnable parameters using the fmincon function. Depending on the number of iterations, this can take a while to run. To enable a detailed verbose output, set the Display optimization option to "iter-detailed".
parametersV = fmincon(objFun,parametersV,[],[],[],[],[],[],[],options);
For prediction, convert the vector of parameters to a structure using the parameterVectorToStruct function (attached to this example as a supporting file).
parameters = parameterVectorToStruct(parametersV,parameterNames,parameterSizes);
t
at 0.25, 0.5, 0.75, and 1, compare the predicted values of the deep learning model with the true solutions of the Burger's equation using the relative
{l}^{2}
% Calcualte true values.
UPred = extractdata(UPred);
err = norm(UPred - UTest) / norm(UTest);
plot(XTest,UPred,"-",LineWidth=2);
plot(XTest, UTest,"--",LineWidth=2)
legend(["Predicted" "True"])
The objectiveFunction function defines the objective function to be used by the LBFGS algorithm.
This objectiveFunction funtion takes as input, a vector of learnable parameters parametersV, the network inputs, X and T, the initial and boundary conditions X0, T0, and U0, and the names and sizes of the learnable parameters parameterNames and parameterSizes, respectively, and returns the loss to be minimized by the fmincon function loss and a vector containing the gradients of the loss with respect to the learnable parameters gradientsV.
function [loss,gradientsV] = objectiveFunction(parametersV,X,T,X0,T0,U0,parameterNames,parameterSizes)
% Convert parameters to structure of dlarray objects.
parametersV = dlarray(parametersV);
[loss,gradients] = dlfeval(@modelLoss,parameters,X,T,X0,T0,U0);
% Return loss and gradients for fmincon.
gradientsV = parameterStructToVector(gradients);
gradientsV = extractdata(gradientsV);
loss = extractdata(loss);
\left(x,t\right)
u\left(x,t\right)
fulfills the Burger's equation, the boundary conditions, and the intial condition. In particular, two quantities contribute to the loss to be minimized:
\text{loss}={\text{MSE}}_{f}+{\text{MSE}}_{u}
{\text{MSE}}_{f}=\frac{1}{{N}_{f}}\sum _{i=1}^{{N}_{f}}{|f\left({x}_{f}^{i},{t}_{f}^{i}\right)|}^{2}
{\text{MSE}}_{u}=\frac{1}{{N}_{u}}\sum _{i=1}^{{N}_{u}}{|u\left({x}_{u}^{i},{t}_{u}^{i}\right)-{u}^{i}|}^{2}
\left\{{x}_{u}^{i},{t}_{u}^{i}{\right\}}_{i=1}^{{N}_{u}}
\left\{{x}_{f}^{i},{t}_{f}^{i}{\right\}}_{i=1}^{{N}_{f}}
{\text{MSE}}_{f}
\frac{\partial u}{\partial t},\frac{\partial u}{\partial x},\frac{{\partial }^{2}u}{\partial {x}^{2}}
u
% Calculate mseF. Enforce Burger's equation.
zeroTarget = zeros(size(f),"like",f);
mseF = l2loss(f, zeroTarget);
% Calculate mseU. Enforce initial and boundary conditions.
mseU = l2loss(U0Pred, U0);
% Calculated loss to be minimized by combining errors.
loss = mseF + mseU;
XT = [X; T];
numLayers = numel(fieldnames(parameters))/2;
weights = parameters.fc1_Weights;
bias = parameters.fc1_Bias;
weights = parameters.(name + "_Weights");
bias = parameters.(name + "_Bias");
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Reviewed by Krishna Nelaturu
How to use our standard form to general form of a circle calculator?
How to convert the circle equation in standard form to general form ?
More calculators to assist you
Are you looking for a quick way to convert the circle equation from standard form to general form? Then our standard form to general form of a circle calculator is a perfect match for you.
Please read this short article to learn more about:
How to use our standard form to the general form of a circle calculator?; and
How to convert the circle equation from standard form to general form?
To use our calculator, you can use the following guide:
Make sure your circle equation is in standard form:
(x-h)^2 +(y-k)^2=\text{C}
Insert the parameters: h, k, and C present in standard form into the respective fields; and
Right away, you will get the circle equation in a general form.
Nice, you are an expert in using our calculator now ;).
To convert the circle equation from the standard to general form, you can use the following steps:
Extract the radius and center information from the standard form. You can see below from the standard form we can observe the variables h, and k, which correspond to the center, and r corresponds to the radius.
\qquad \footnotesize (x-h)^2 +(y-k)^2=r^2
General form of the circle equation is given as:
\qquad \footnotesize x^2 +y^2+Dx+Ey+F=0
Now, use the following formula to compute the coefficients of the general form:
\qquad\footnotesize \begin{align*} D&=-2×h\\ E&=-2×k \\ F&=h^2+k^2-r^2. \end{align*}
Finally, substitute the values obtained to get the circle equation in a general form.
Excellent, you have learned how to convert circle equation from standard form to general form.
Check out our circle-related calculators created to assist you in solving circle-related problems. Whenever you need them, they will be available here:
Equation of a circle with diameter endpoints calculator;
General to standard form of a circle calculator;
What is the general form of a circle equation with diameter endpoints (4,8), (6,6)?
The general form is given as x²+y²-10x-14y+72=0. To find the general form, start with the general form x²+y²+Dx+Ey+F=0, and let's find the coefficients using the following steps:
Find the center (h,k) and distance between the diameter endpoints using the midpoint and distance formulas, respectively.
Divide the distance found in step 1 by 2 to obtain the radius r=1.4142.
Calculate the coefficients of the general form using the following equation:
D=-2×h=-10;
E=-2×k=-14; and
F=h²+k²-r²=72.
substitute the coefficients to obtain the circle equation in a general form:
x²+y²-10x-14y+72=0
How do I find the radius from circle equation in general form?
To find the radius from the general form, use the following steps:
Extract the coefficients from the general form: x²+y²+Dx+Ey+F=0;
Use the formula of F which is given as F=(D/2)²+(E/2)²-r².
Rearrange the formula of F in terms of radius r: r=√((D/2)²+(E/2)²-F); and
Solve the equation in step 3 to obtain the radius of the circle.
Hooray! Now you know how to find the radius if you are given with circle equation in a general form.
Standard form: (x - h)² + (y - k)² = C
General form: x² + y² + Dx + Ey + F = 0
Cube root calculator helps you find the cube root of any positive number.
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Lemma 29.21.12. Let $f : X \to Y$ be a morphism of schemes with diagonal $\Delta : X \to X \times _ Y X$. If $f$ is locally of finite type then $\Delta $ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta $ is of finite presentation.
Proof. Note that $\Delta $ is a morphism of schemes over $X$ (via the second projection $X \times _ Y X \to X$). Assume $f$ is locally of finite type. Note that $X$ is of finite presentation over $X$ and $X \times _ Y X$ is locally of finite type over $X$ (by Lemma 29.15.4). Thus the first statement holds by Lemma 29.21.11. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is a monomorphism, hence separated (Schemes, Lemma 26.23.3). $\square$
Comment #891 by Matthew Emerton on August 08, 2014 at 12:27
At the end of the third sentence, I think it should state that
X\times_Y X
X
Yes, indeed, thanks! Fixed here.
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Pseudosphere Knowpia
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius R is a surface in
{\displaystyle \mathbb {R} ^{3}}
having curvature −1/R2 in each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R2. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.[1]
TractricoidEdit
The same surface can be also described as the result of revolving a tractrix about its asymptote. For this reason the pseudosphere is also called tractricoid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by[2]
{\displaystyle t\mapsto \left(t-\tanh {t},\operatorname {sech} \,{t}\right),\quad \quad 0\leq t<\infty .}
It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.
The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle.
As early as 1693 Christiaan Huygens found that the volume and the surface area of the pseudosphere are finite,[3] despite the infinite extent of the shape along the axis of rotation. For a given edge radius R, the area is 4πR2 just as it is for the sphere, while the volume is 2/3πR3 and therefore half that of a sphere of that radius.[4][5]
Universal covering spaceEdit
The pseudosphere and its relation to three other models of hyperbolic geometry
The half pseudosphere of curvature −1 is covered by the portion of the hyperbolic upper half-plane with y ≥ 1.[6] The covering map is periodic in the x direction of period 2π, and takes the horocycles y = c to the meridians of the pseudosphere and the vertical geodesics x = c to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion y ≥ 1 of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping is
{\displaystyle (x,y)\mapsto {\big (}v(\operatorname {arcosh} y)\cos x,v(\operatorname {arcosh} y)\sin x,u(\operatorname {arcosh} y){\big )}}
{\displaystyle t\mapsto {\big (}u(t)=t-\operatorname {tanh} t,v(t)=\operatorname {sech} t{\big )}}
is the parametrization of the tractrix above.
HyperboloidEdit
In some sources that use the hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere.[7] This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a Minkowski space.
Quasi-sphere
Sine–Gordon equation
^ Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Treatise on the interpretation of non-Euclidean geometry]. Gior. Mat. (in Italian). 6: 248–312.
(Also Beltrami, Eugenio. Opere Matematiche [Mathematical Works] (in Italian). Vol. 1. pp. 374–405. ISBN 1-4181-8434-9. ;
Beltrami, Eugenio (1869). "Essai d'interprétation de la géométrie noneuclidéenne" [Treatise on the interpretation of non-Euclidean geometry]. Annales de l'École Normale Supérieure (in French). 6: 251–288. Archived from the original on 2016-02-02. Retrieved 2010-07-24. )
^ Bonahon, Francis (2009). Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots. AMS Bookstore. p. 108. ISBN 0-8218-4816-X. , Chapter 5, page 108
^ Stillwell, John (2010). Mathematics and Its History (revised, 3rd ed.). Springer Science & Business Media. p. 345. ISBN 978-1-4419-6052-8. , extract of page 345
^ Le Lionnais, F. (2004). Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences (2 ed.). Courier Dover Publications. p. 154. ISBN 0-486-49579-5. , Chapter 40, page 154
^ Weisstein, Eric W. "Pseudosphere". MathWorld.
^ Thurston, William, Three-dimensional geometry and topology, vol. 1, Princeton University Press, p. 62 .
^ Hasanov, Elman (2004), "A new theory of complex rays", IMA J. Appl. Math., 69: 521–537, doi:10.1093/imamat/69.6.521, ISSN 1464-3634, archived from the original on 2013-04-15
Stillwell, J. (1996). Sources of Hyperbolic Geometry. Amer. Math. Soc & London Math. Soc.
Henderson, D. W.; Taimina, D. (2006). "Experiencing Geometry: Euclidean and Non-Euclidean with History". Aesthetics and Mathematics (PDF). Springer-Verlag.
Kasner, Edward; Newman, James (1940). Mathematics and the Imagination. Simon & Schuster. p. 140, 145, 155.
Norman Wildberger lecture 16, History of Mathematics, University of New South Wales. YouTube. 2012 May.
Pseudospherical surfaces at the virtual math museum.
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Home : Support : Online Help : Mathematics : Basic Mathematics : Exponential, Trig, and Hyperbolic Functions : Working in Degrees : Degrees package : diff
resolve degrees trig functions before calling main expand command
resolve degrees trig functions before calling main int command
resolve degrees trig functions before calling main diff command
resolve degrees trig functions before calling main solve command
resolve degrees trig functions before calling main trigsubs command
diff( expr, ... )
expand( expr, ... )
int( expr, ... )
solve( expr, ... )
trigsubs( expr, ... )
The Degrees package provides some "wrapper" commands that process degrees-based trig functions into their radians-based form before calling the main library command of the same name. The result is then processed back into degrees form.
These functions are part of the Degrees package, so they can be used in the short form, for example expand(..), only after executing the command with(Degrees). However, they can always be accessed through the long form of the command by using for example Degrees:-expand(..).
\mathrm{with}\left(\mathrm{Degrees}\right):
\mathrm{int}\left(\mathrm{sind}\left(x\right),x\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{cosd}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{expand}\left(\mathrm{sind}\left(x+y\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{sind}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{cosd}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cosd}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sind}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)
The Degrees[expand], Degrees[int], Degrees[diff], Degrees[solve] and Degrees[trigsubs] commands were introduced in Maple 2021.
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Reviewed by Vishnuvardhan Shakthibala
How to use the width of a rectangle calculator?
How do I calculate the width of a rectangle?
The width of a rectangle calculator can take two dimensions of your rectangle (height, area, diagonal, or perimeter) and work out the width of your rectangle. We'll also show you some formulas for the width of a rectangle, that are based on what information you have available. So don't be a square — read on!
Using the width of a rectangle calculator is easy, with only two steps. Here's how.
Find your rectangle width in the bottom box.
The calculator is omni-directional — if you want to work backwards instead, you can change the rectangle's width and see how its other dimensions react.
And that's it! Now you know how to calculate the width of a rectangle!
Depending on what information you have available, there are many ways to calculate the width of a rectangle.
If you have the area A and length h, its width w is w = A/h.
If you have the perimeter P and length h, its width is w = P/2−h.
If you have the diagonal d and length h, it's width can be found with w = √(d²−h²).
If the rectangle's length is not known, you'd need to do some algebra with the quadratic equation to solve for the width.
That's a lot of different formulas for the width of a rectangle! It's because there are multiple equations that govern the dimensions of a rectangle. Here they are:
\begin{split} A &= \textcolor{red}{w} \times h \\ P &= (\textcolor{red}{w}+h)\times 2 \\ d &= \sqrt{\textcolor{red}{w}^2 + h^2} \end{split}
w
h
A
P
is the rectangle's perimeter; and
is the length of the rectangle's diagonal (as described by Pythagoras).
A rectangle with its length
w
d
P
A
If the width of a rectangle calculator isn't quite what you want, try out our other rectangular calculators:
What is the width of a rectangle?
A rectangle has four sides, but because the sides are paired, there are only two unique dimensions. Conventionally, the width is the shortest of these two dimensions, but when the rectangle is presented as lying on its side, the horizontal side is usually called the width.
What is the width of a rectangle with length 4 m and area 20 m?
Using the formula for a rectangle's area, A = h × w, we can follow these steps:
Rewrite the area formula to make the width w the subject of the equation: w = A/h.
Plug in the values: w = 20/4.
Do the math and find that w = 5 m.
With this orthocenter calculator you'll quickly find out the coordinates of this important triangle center.
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Revision as of 02:54, 3 February 2017 by Are171887 (talk | contribs) (Added Category: Orbiter Math.)
{\displaystyle \left\{Q_{0},P_{0}\right\}}
{\displaystyle \left\{Q_{2},P_{2}\right\}}
{\displaystyle \delta t}
{\displaystyle \left\{Q_{0},P_{0}\right\}}
{\displaystyle t=t_{0}}
{\displaystyle \left\{Q_{2},P_{2}\right\}}
{\displaystyle t=t_{0}+\delta t}
{\displaystyle Q_{1}\leftarrow Q_{0}+{\frac {1}{2}}\,\delta t\,P_{0}}
{\displaystyle P_{2}\leftarrow P_{0}+\delta t\,F\left(Q_{1}\right)}
{\displaystyle Q_{2}\leftarrow Q_{1}+{\frac {1}{2}}\,\delta t\,P_{2}}
{\displaystyle \left\{Q_{0},P_{0}\right\}}
{\displaystyle \delta t}
{\displaystyle F\left(Q_{1}\right)}
{\displaystyle Q_{1}}
{\displaystyle F(Q)}
{\displaystyle F(Q)=-{\frac {\mu \,m}{(Q-Q^{*})^{2}}}}
{\displaystyle Q>Q^{*}}
{\displaystyle \mu }
{\displaystyle Q^{*}}
{\displaystyle \mu =0.00029591220828559115}
{\displaystyle m=1}
{\displaystyle m=1/354710}
{\displaystyle m=1}
{\displaystyle Q^{*}}
{\displaystyle \left\{Q_{x,0},Q_{y,0},Q_{z,0},P_{x,0},P_{y,0},P_{z,0}\right\}}
{\displaystyle \left\{Q_{x,2},Q_{y,2},Q_{z,2},P_{x,2},P_{y,2},P_{z,2}\right\}}
{\displaystyle Q_{x,1}\leftarrow Q_{x,0}+{\frac {1}{2}}\,\delta t\,P_{x,0}}
{\displaystyle Q_{y,1}\leftarrow Q_{y,0}+{\frac {1}{2}}\,\delta t\,P_{y,0}}
{\displaystyle Q_{z,1}\leftarrow Q_{z,0}+{\frac {1}{2}}\,\delta t\,P_{z,0}}
{\displaystyle P_{x,2}\leftarrow P_{x,0}+\delta t\,F_{x}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle P_{y,2}\leftarrow P_{y,0}+\delta t\,F_{y}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle P_{z,2}\leftarrow P_{z,0}+\delta t\,F_{z}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle Q_{x,2}\leftarrow Q_{x,1}+{\frac {1}{2}}\,\delta t\,P_{x,2}}
{\displaystyle Q_{y,2}\leftarrow Q_{y,1}+{\frac {1}{2}}\,\delta t\,P_{y,2}}
{\displaystyle Q_{z,2}\leftarrow Q_{z,1}+{\frac {1}{2}}\,\delta t\,P_{z,2}}
{\displaystyle F_{x}\left(Q_{x},Q_{y},Q_{z}\right)}
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{y}\left(Q_{x},Q_{y},Q_{z}\right)}
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{z}\left(Q_{x},Q_{y},Q_{z}\right)}
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{x}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{x}-Q_{x}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle F_{y}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{y}-Q_{y}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle F_{z}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{z}-Q_{z}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle \left(Q_{x}^{*},Q_{y}^{*},Q_{z}^{*}\right)}
{\displaystyle \left(Q_{x},Q_{y},Q_{z}\right)}
Now, for those interested, it is worthwhile setting up this integrator in, say, an spreadsheet and seeing how it performs under various sizes of time-steps. and initial conditions. Of course, if there is more than one gravitating body, then you have additional terms in the force functions, but the basic scheme of the updating rule remains the same.
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Numerical Analysis of a Cell-Based Indirect Internal Reforming Tubular SOFC Operating With Biogas | J. Electrochem. En. Conv. Stor | ASME Digital Collection
Department of Fundamental Research in Energy Engineering,
, 30-059 Krakow, Poland
e-mail: janusz@agh.edu.pl
Janusz S. Szmyd Professor
Nishino, T., and Szmyd, J. S. (July 14, 2010). "Numerical Analysis of a Cell-Based Indirect Internal Reforming Tubular SOFC Operating With Biogas." ASME. J. Fuel Cell Sci. Technol. October 2010; 7(5): 051004. https://doi.org/10.1115/1.4000998
A numerical study is performed on the thermal and electrochemical characteristics of a tubular solid oxide fuel cell (SOFC) employing the steam reforming of biogas in each individual cell unit but indirectly from the anode. The numerical model used in this study takes account of momentum, heat, and mass transfer in and around the cell, including the effects of radiation, internal reforming, and electrochemical reactions. The biogas, which is fed into the reformer with steam, is assumed to be composed of methane
(CH4)
(CO2)
. The results show that, under the conditions of a constant average current density of
400 mA/cm2
and a constant fuel utilization of 80%, the terminal voltage of the cell decreases but only moderately as the proportion of
CH4
in the fuel supplied to the reformer is reduced. It is also shown that temperature gradients within the cell decrease as the proportion of
CH4
in the supplied fuel is reduced. These results are promising for the future use of biogas for this type of indirect internal reforming SOFC system.
biofuel, electrochemistry, heat transfer, mass transfer, solid oxide fuel cells, steam, solid oxide fuel cells, biogas, indirect internal reforming, thermal management
Biogas, Methane, Solid oxide fuel cells, Steam, Gaseous fuels, Fuels, Computer simulation, Temperature, Current density, Carbon dioxide
Distributed Energy Generation, The Fuel Cell and Its Hybrid Systems
High Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications
Electrochemical and Thermo-Fluid Modeling of a Tubular Solid Oxide Fuel Cell With Accompanying Indirect Internal Fuel Reforming
Influence of the Anodic Recirculation Transient Behaviour on the SOFC Hybrid System Performance
Modelling of Pressurised Hybrid Systems Based on Integrated Planar Solid Oxide Fuel Cell (IP-SOFC) Technology
Running Fuel Cells on Biogas—A Renewable Fuel
Implications for Using Biogas as a Fuel Source for Solid Oxide Fuel Cells: Internal Dry Reforming in a Small Tubular Solid Oxide Fuel Cell
Biogas as a Fuel Source for SOFC Co-Generators
,” Master thesis, Department of Mechanical Engineering, Kyoto University, Kyoto, Japan.
Isotopic and Kinetic Assessment of the Mechanism of Reactions of CH4 With CO2 or H2O to Form Synthesis Gas and Carbon on Nickel Catalysts
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Nucleic acid structure - Wikipedia
Biomolecular structure of nucleic acids such as DNA and RNA
Nucleic acid structure refers to the structure of nucleic acids such as DNA and RNA. Chemically speaking, DNA and RNA are very similar. Nucleic acid structure is often divided into four different levels: primary, secondary, tertiary, and quaternary.
1.1 Complexes with alkali metal ions
Thymine (present in DNA only)
Uracil (present in RNA only)
5-carbon sugar which is called deoxyribose (found in DNA) and ribose (found in RNA).
One or more phosphate groups.[1]
The nitrogen bases adenine and guanine are purine in structure and form a glycosidic bond between their 9 nitrogen and the 1' -OH group of the deoxyribose. Cytosine, thymine, and uracil are pyrimidines, hence the glycosidic bonds form between their 1 nitrogen and the 1' -OH of the deoxyribose. For both the purine and pyrimidine bases, the phosphate group forms a bond with the deoxyribose sugar through an ester bond between one of its negatively charged oxygen groups and the 5' -OH of the sugar.[2] The polarity in DNA and RNA is derived from the oxygen and nitrogen atoms in the backbone. Nucleic acids are formed when nucleotides come together through phosphodiester linkages between the 5' and 3' carbon atoms.[3] A nucleic acid sequence is the order of nucleotides within a DNA (GACT) or RNA (GACU) molecule that is determined by a series of letters. Sequences are presented from the 5' to 3' end and determine the covalent structure of the entire molecule. Sequences can be complementary to another sequence in that the base on each position is complementary as well as in the reverse order. An example of a complementary sequence to AGCT is TCGA. DNA is double-stranded containing both a sense strand and an antisense strand. Therefore, the complementary sequence will be to the sense strand.[4]
Nucleic acid design can be used to create nucleic acid complexes with complicated secondary structures such as this four-arm junction. These four strands associate into this structure because it maximizes the number of correct base pairs, with As matched to Ts and Cs matched to Gs. Image from Mao, 2004.[5]
Complexes with alkali metal ions[edit]
There are three potential metal binding groups on nucleic acids: phosphate, sugar, and base moieties. Solid-state structure of complexes with alkali metal ions have been reviewed.[6]
Secondary structure[edit]
Main article: Nucleic acid secondary structure
Secondary structure is the set of interactions between bases, i.e., which parts of strands are bound to each other. In DNA double helix, the two strands of DNA are held together by hydrogen bonds. The nucleotides on one strand base pairs with the nucleotide on the other strand. The secondary structure is responsible for the shape that the nucleic acid assumes. The bases in the DNA are classified as purines and pyrimidines. The purines are adenine and guanine. Purines consist of a double ring structure, a six-membered and a five-membered ring containing nitrogen. The pyrimidines are cytosine and thymine. It has a single ring structure, a six-membered ring containing nitrogen. A purine base always pairs with a pyrimidine base (guanine (G) pairs with cytosine (C) and adenine (A) pairs with thymine (T) or uracil (U)). DNA's secondary structure is predominantly determined by base-pairing of the two polynucleotide strands wrapped around each other to form a double helix. Although the two strands are aligned by hydrogen bonds in base pairs, the stronger forces holding the two strands together are stacking interactions between the bases. These stacking interactions are stabilized by Van der Waals forces and hydrophobic interactions, and show a large amount of local structural variability.[7] There are also two grooves in the double helix, which are called major groove and minor groove based on their relative size.
An example of RNA secondary structure. This image includes several structural elements, including; single-stranded and double-stranded areas, bulges, internal loops and hairpin loops. Double-stranded RNA forms an A-type helical structure, unlike the common B-type conformation taken by double-stranded DNA molecules.
The antiparallel strands form a helical shape.[3] Bulges and internal loops are formed by separation of the double helical tract on either one strand (bulge) or on both strands (internal loops) by unpaired nucleotides.
Stem-loop or hairpin loop is the most common element of RNA secondary structure.[8] Stem-loop is formed when the RNA chains fold back on themselves to form a double helical tract called the 'stem', the unpaired nucleotides forms single stranded region called the 'loop'.[9] A tetraloop is a four-base pairs hairpin RNA structure. There are three common families of tetraloop in ribosomal RNA: UNCG, GNRA, and CUUG (N is one of the four nucleotides and R is a purine). UNCG is the most stable tetraloop.[10]
Pseudoknot is a RNA secondary structure first identified in turnip yellow mosaic virus.[11] Pseudoknots are formed when nucleotides from the hairpin-loop pair with a single stranded region outside of the hairpin to form a helical segment. H-type fold pseudoknots are best characterized. In H-type fold, nucleotides in the hairpin-loop pair with the bases outside the hairpin stem forming second stem and loop. This causes formation of pseudoknots with two stems and two loops.[12] Pseudoknots are functional elements in RNA structure having diverse function and found in most classes of RNA.
Secondary structure of RNA can be predicted by experimental data on the secondary structure elements, helices, loops, and bulges. DotKnot-PW method is used for comparative pseudoknots prediction. The main points in the DotKnot-PW method is scoring the similarities found in stems, secondary elements and H-type pseudoknots.[13]
Main article: Nucleic acid tertiary structure
DNA structure and bases
A-B-Z-DNA Side View
Difference in size between the major and minor grooves[3]
The tertiary arrangement of DNA's double helix in space includes B-DNA, A-DNA, and Z-DNA. Triple-stranded DNA structures have been demonstrated in repetitive polypurine:polypyrimidine Microsatellite sequences and Satellite DNA.
B-DNA is the most common form of DNA in vivo and is a more narrow, elongated helix than A-DNA. Its wide major groove makes it more accessible to proteins. On the other hand, it has a narrow minor groove. B-DNA's favored conformations occur at high water concentrations; the hydration of the minor groove appears to favor B-DNA. B-DNA base pairs are nearly perpendicular to the helix axis. The sugar pucker which determines the shape of the a-helix, whether the helix will exist in the A-form or in the B-form, occurs at the C2'-endo.[14]
A-DNA, is a form of the DNA duplex observed under dehydrating conditions. It is shorter and wider than B-DNA. RNA adopts this double helical form, and RNA-DNA duplexes are mostly A-form, but B-form RNA-DNA duplexes have been observed.[15] In localized single strand dinucleotide contexts, RNA can also adopt the B-form without pairing to DNA.[16] A-DNA has a deep, narrow major groove which does not make it easily accessible to proteins. On the other hand, its wide, shallow minor groove makes it accessible to proteins but with lower information content than the major groove. Its favored conformation is at low water concentrations. A-DNAs base pairs are tilted relative to the helix axis, and are displaced from the axis. The sugar pucker occurs at the C3'-endo and in RNA 2'-OH inhibits C2'-endo conformation.[14] Long considered little more than a laboratory artifice, A-DNA is now known to have several biological functions.
Z-DNA is a relatively rare left-handed double-helix. Given the proper sequence and superhelical tension, it can be formed in vivo but its function is unclear. It has a more narrow, more elongated helix than A or B. Z-DNA's major groove is not really a groove, and it has a narrow minor groove. The most favored conformation occurs when there are high salt concentrations. There are some base substitutions but they require an alternating purine-pyrimidine sequence. The N2-amino of G H-bonds to 5' PO, which explains the slow exchange of protons and the need for the G purine. Z-DNA base pairs are nearly perpendicular to the helix axis. Z-DNA does not contain single base-pairs but rather a GpC repeat with P-P distances varying for GpC and CpG. On the GpC stack there is good base overlap, whereas on the CpG stack there is less overlap. Z-DNA's zigzag backbone is due to the C sugar conformation compensating for G glycosidic bond conformation. The conformation of G is syn, C2'-endo; for C it is anti, C3'-endo.[14]
A covalently closed, circular DNA (also known as cccDNA) is topologically constrained as the number of times the chains coiled around one other cannot change. This cccDNA can be supercoiled, which is the tertiary structure of DNA. Supercoiling is characterized by the linking number, twist and writhe. The linking number (Lk) for circular DNA is defined as the number of times one strand would have to pass through the other strand to completely separate the two strands. The linking number for circular DNA can only be changed by breaking of a covalent bond in one of the two strands. Always an integer, the linking number of a cccDNA is the sum of two components: twists (Tw) and writhes (Wr).[17]
{\displaystyle Lk=Tw+Wr}
Main article: Nucleic acid quaternary structure
The quaternary structure of nucleic acids is similar to that of protein quaternary structure. Although some of the concepts are not exactly the same, the quaternary structure refers to a higher-level of organization of nucleic acids. Moreover, it refers to interactions of the nucleic acids with other molecules. The most commonly seen form of higher-level organization of nucleic acids is seen in the form of chromatin which leads to its interactions with the small proteins histones. Also, the quaternary structure refers to the interactions between separate RNA units in the ribosome or spliceosome.[18]
Non-helical models of DNA structure
Nucleic acid structure determination (experimental)
Nucleic acid structure prediction (computational)
^ Krieger M, Scott MP, Matsudaira PT, Lodish HF, Darnell JE, Lawrence Z, Kaiser C, Berk A (2004). "Section 4.1: Structure of Nucleic Acids". Molecular cell biology. New York: W.H. Freeman and CO. ISBN 978-0-7167-4366-8.
^ "Structure of Nucleic Acids". SparkNotes.
^ a b c Anthony-Cahill SJ, Mathews CK, van Holde KE, Appling DR (2012). Biochemistry (4th Edition). Englewood Cliffs, N.J: Prentice Hall. ISBN 978-0-13-800464-4.
^ Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Wlater P (2002). Molecular Biology of the Cell (4th ed.). New York NY: Garland Science. ISBN 978-0-8153-3218-3.
^ Mao C (December 2004). "The emergence of complexity: lessons from DNA". PLoS Biology. 2 (12): e431. doi:10.1371/journal.pbio.0020431. PMC 535573. PMID 15597116.
^ Katsuyuki, Aoki; Kazutaka, Murayama; Hu, Ning-Hai (2016). "Chapter 3, section3. Nucleic Acid Constituent complexes". In Astrid, Sigel; Helmut, Sigel; Roland K.O., Sigel (eds.). The Alkali Metal Ions: Their Role in Life. Metal Ions in Life Sciences. Vol. 16. Springer. pp. 43–66. doi:10.1007/978-3-319-21756-7_3. ISBN 978-3-319-21755-0. PMID 26860299.
^ Sedova A, Banavali NK (2017). "Geometric Patterns for Neighboring Bases Near the Stacked State in Nucleic Acid Strands". Biochemistry. 56 (10): 1426–1443. doi:10.1021/acs.biochem.6b01101. PMID 28187685.
^ "RNA structure (Molecular Biology)".
^ Hollyfield JG, Besharse JC, Rayborn ME (December 1976). "The effect of light on the quantity of phagosomes in the pigment epithelium". Experimental Eye Research. 23 (6): 623–35. doi:10.1016/0014-4835(76)90221-9. PMID 1087245.
^ Rietveld K, Van Poelgeest R, Pleij CW, Van Boom JH, Bosch L (March 1982). "The tRNA-like structure at the 3' terminus of turnip yellow mosaic virus RNA. Differences and similarities with canonical tRNA". Nucleic Acids Research. 10 (6): 1929–46. doi:10.1093/nar/10.6.1929. PMC 320581. PMID 7079175.
^ Staple DW, Butcher SE (June 2005). "Pseudoknots: RNA structures with diverse functions". PLoS Biology. 3 (6): e213. doi:10.1371/journal.pbio.0030213. PMC 1149493. PMID 15941360.
^ Sperschneider J, Datta A, Wise MJ (December 2012). "Predicting pseudoknotted structures across two RNA sequences". Bioinformatics. 28 (23): 3058–65. doi:10.1093/bioinformatics/bts575. PMC 3516145. PMID 23044552.
^ a b c Dickerson RE, Drew HR, Conner BN, Wing RM, Fratini AV, Kopka ML (April 1982). "The anatomy of A-, B-, and Z-DNA". Science. 216 (4545): 475–85. doi:10.1126/science.7071593. PMID 7071593.
^ Chen X; Ramakrishnan B; Sundaralingam M (1995). "Crystal structures of B-form DNA-RNA chimers complexed with distamycin". Nature Structural Biology. 2 (9): 733–735. doi:10.1038/nsb0995-733.
^ Sedova A, Banavali NK (2016). "RNA approaches the B-form in stacked single strand dinucleotide contexts". Biopolymers. 105 (2): 65–82. doi:10.1002/bip.22750. PMID 26443416.
^ Mirkin SM (2001). DNA Topology: Fundamentals. Encyclopedia of Life Sciences. doi:10.1038/npg.els.0001038. ISBN 978-0470016176.
^ "Structural Biochemistry/Nucleic Acid/DNA/DNA structure". Retrieved 11 December 2012.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Nucleic_acid_structure&oldid=1070053480"
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Trix (technical analysis) - WikiMili, The Free Encyclopedia
Trix (or TRIX) is a technical analysis oscillator developed in the 1980s by Jack Hutson, editor of Technical Analysis of Stocks and Commodities magazine. It shows the slope (i.e. derivative) of a triple-smoothed exponential moving average. [1] [2] The name Trix is from "triple exponential."
In finance, technical analysis is an analysis methodology for forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the same tools of technical analysis, which, being an aspect of active management, stands in contradiction to much of modern portfolio theory. The efficacy of both technical and fundamental analysis is disputed by the efficient-market hypothesis which states that stock market prices are essentially unpredictable.
An oscillator is a technical analysis indicator that varies over time within a band. Oscillators are used to discover short-term overbought or oversold conditions.
The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
{\displaystyle p_{0}}
{\displaystyle p_{1}}
{\displaystyle f=1-{2 \over N+1}={N-1 \over N+1}}
{\displaystyle TripleEMA_{0}=(1-f)^{3}(p_{0}+3fp_{1}+6f^{2}p_{2}+10f^{3}p_{3}+\dots )}
The coefficients are the triangle numbers, n(n+1)/2. In theory, the sum is infinite, using all past data, but as f is less than 1 the powers
{\displaystyle f^{n}}
become smaller as the series progresses, and they decrease faster than the coefficients increase, so beyond a certain point the terms are negligible.
In statistics, a moving average is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is a type of finite impulse response filter. Variations include: simple, and cumulative, or weighted forms.
In stock and securities market technical analysis, parabolic SAR is a method devised by J. Welles Wilder, Jr., to find potential reversals in the market price direction of traded goods such as securities or currency exchanges such as forex. It is a trend-following (lagging) indicator and may be used to set a trailing stop loss or determine entry or exit points based on prices tending to stay within a parabolic curve during a strong trend.
MACD, short for moving average convergence/divergence, is a trading indicator used in technical analysis of stock prices, created by Gerald Appel in the late 1970s. It is designed to reveal changes in the strength, direction, momentum, and duration of a trend in a stock's price.
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes.
A Savitzky–Golay filter is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the precision of the data without distorting the signal tendency. This is achieved, in a process known as convolution, by fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares. When the data points are equally spaced, an analytical solution to the least-squares equations can be found, in the form of a single set of "convolution coefficients" that can be applied to all data sub-sets, to give estimates of the smoothed signal, at the central point of each sub-set. The method, based on established mathematical procedures, was popularized by Abraham Savitzky and Marcel J. E. Golay who published tables of convolution coefficients for various polynomials and sub-set sizes in 1964. Some errors in the tables have been corrected. The method has been extended for the treatment of 2- and 3-dimensional data.
The mass index is an indicator, developed by Donald Dorsey, used in technical analysis to predict trend reversals. It is based on the notion that there is a tendency for reversal when the price range widens, and therefore compares previous trading ranges.
The Watts–Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering. It was proposed by Duncan J. Watts and Steven Strogatz in their joint 1998 Nature paper. The model also became known as the (Watts) beta model after Watts used to formulate it in his popular science book Six Degrees.
In technical analysis of securities trading, the stochastic oscillator is a momentum indicator that uses support and resistance levels. Dr. George Lane developed this indicator in the late 1950s. The term stochastic refers to the point of a current price in relation to its price range over a period of time. This method attempts to predict price turning points by comparing the closing price of a security to its price range.
The doji is a commonly found pattern in a candlestick chart of financially traded assets in technical analysis. It is characterized by being small in length—meaning a small trading range—with an opening and closing price that are virtually equal.
In baseball, wOBA is a statistic, based on linear weights, designed to measure a player's overall offensive contributions per plate appearance. It is formed from taking the observed run values of various offensive events, dividing by a player's plate appearances, and scaling the result to be on the same scale as on-base percentage. Unlike statistics like OPS, wOBA attempts to assign the proper value for each type of hitting event. It was created by Tom Tango and his coauthors for The Book: Playing the Percentages in Baseball.
In financial technical analysis, the know sure thing (KST) oscillator is a complex, smoothed price velocity indicator developed by Martin J. Pring.
The true strength index (TSI) is a technical indicator used in the analysis of financial markets that attempts to show both trend direction and overbought/oversold conditions. It was first published William Blau in 1991. The indicator uses moving averages of the underlying momentum of a financial instrument. Momentum is considered a leading indicator of price movements, and a moving average characteristically lags behind price. The TSI combines these characteristics to create an indication of price and direction more in sync with market turns than either momentum or moving average. The TSI is provided as part of the standard collection of indicators offered by various trading platforms.
The Triple Exponential Moving Average (TEMA) indicator was introduced in January 1994 by Patrick G. Mulloy, in an article in the Technical Analysis of Stocks & Commodities magazine: "Smoothing Data with Faster Moving Averages"
The Double Exponential Moving Average (DEMA) indicator was introduced in January 1994 by Patrick G. Mulloy, in an article in the "Technical Analysis of Stocks & Commodities" magazine: "Smoothing Data with Faster Moving Averages"
The zero lag exponential moving average (ZLEMA) indicator was created by John Ehlers and Ric Way.
↑ "TRIX Uptrend & Downtrend | Stock Buy & Sell Signal |Technical Analysis". Web.archive.org. Archived from the original on 2016-03-05. Retrieved 2018-03-22.
↑ "TRIX". Web.archive.org. Archived from the original on 2006-01-08. Retrieved 2018-03-22. CS1 maint: BOT: original-url status unknown (link)
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LFSR Calculator | Linear-Feedback Shift
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Fibonacci LFSR and Galois LFSR: a compendium
Examples of the use of Linear-Feedback Shift Registers
How to use our LFSR calculator?
Generating random numbers on a deterministic machine like a computer is complicated — this is where linear-feedback shift registers (LFSR) come in handy, and you can try them out with our LFSR calculator. With a little bit of math and some computer science knowledge, you will learn everything you need about this special type of register.
Here we will guide you through the many questions you might have, like:
What an LFSR is;
Fibonacci LFSR vs Galois LFSR;
How an LFSR works;
The applications of linear-feedback shift registers; and
How to use our LFSR calculator.
First: what is a shift register? It's a type of electronic logic circuit that stores and transmits data by moving one bit in a particular direction of a register at every step: a basic type of computer memory. We already have a bit shift calculator that talks about this principle — go check it out!
A simple representation of a shift register.
Long story short, a shift register is a series of flip-flops connected in a sequential fashion. Every flip-flop corresponds to a bit of memory (they are at the base of every digital memory in use today). In a shift register, the output of each flip-flop is the input of the next one. There are various types of shift registers — the type most commonly used sees the output bit of the register being also the input bit of the next cycle. This is the basis of digital memories.
If you ask a computer scientist what a flip-flop is, they likely won't tell you "a type of summer shoes" — not because they don't spend time outside, but because computer memory is based on an element with the same name. A flip-flop is a circuit with two stable states (bistable): if we call them 1 and 0, it is straightforward to see why they are so valuable for computing. It consists of an input, a pair of complementary outputs (if one of them outputs 1, the other one outputs 0) and a reset. Flip-flops can store one bit of memory, either outputting 0 (low) or 1 (high). When triggered by the input, the circuit switches to the high state until a signal from the reset moves it back to the low state.
If the input bit (the bit entering the register) of the current state of the shift register is the result of some linear operation of the previous state, then we have a linear-feedback shift register or LFSR.
The logic gates XOR (exclusive or) and NXOR (its negation) are the two linear functions used in LFSR: one is the one's complement of the other, meaning that the math underlying is almost the same.
The XOR (left) and NXOR (right) logic gates.
Here is the truth table for the XOR gate. As you can see, it corresponds to a bitwise binary addition operation (modulo 2 addition) — this will make things easier to understand for sure.
If you want to learn more about binary operations, check our calculators! They are not strictly bitwise, so that the output can be different than just 1 and 0, but also 10, 11, etc... Try the binary addition calculator or its sibling, the binary subtraction calculator.
🔎 When referring to linear operations, we mean a transformation of a vector space onto itself. The operation XOR, for example, takes as input two "vectors" spanning the discrete values 1 and 0, giving another vector in the same space as a result.
To define an LFSR, it is necessary to specify the bits used as arguments in the linear operation: their positions are called taps. In mathematical terms, they correspond to a vector of 0s and 1s, with 1s in the positions of the desired taps.
The register's initial state is called seed, and it highly affects the outcome of the operations of the LFSR.
The output of an LFSR is a sequence of bits (0s or 1s) that are "collected" at the end of the register. The sequence depends on the chosen function and the type of LFSR, but it always obeys specific rules.
It is periodic (unless the register collapses into the null vector, which consistently transforms to itself) with a maximum number of steps to return to the initial configuration equal to
2^n-1
n
is the length of the register. This number is called the period of the LFSR.
As said before, if the sequence collapses in the null vector, there it remains with trivial periodicity. In that case, the register is not considered an LFSR, and
0,0,...,0
is not allowed as a seed.
There are two main types of LFSR:
Fibonacci LFSR; and
Galois LFSR.
Each of them is defined using the same set of objects, but they differ in how such objects are used.
In the case of the Fibonacci LFSR, the value of the bits specified by the taps is summed using XOR operations, resulting in a final value that then becomes the new input of the shift register. On the "opposite side" of the register, the last bit becomes part of the output.
🔎 Fibonacci is a well-known name in mathematics — and not only there! Check out our Fibonacci sequence calculator to learn more!
In mathematical terms, we can use a set of two formulas to describe the behavior of a Fibonacci LFSR. Let's introduce the symbols in the formulas:
\vec{X} = [X_1, ..., X_n]
is the shift register itself. It is a vector with elements
X_{i} \in \{0, 1\}
(i.e., either
1
0
The taps are specified by the vector of the connection coefficients, noted as
\vec{c} = [c_1, ..., c_n]
. Both of these vectors have length
n
. We defining the element-wise product of the register and the connection coefficient vectors as
\vec{Y}
Y_i=c_i X_i(t)
. The register at the LFSR obeys the rules:
\footnotesize X_{i}(t+1) = X_{i-1}(t)\ \ \text{for}\ 2\leq i\leq n \\ \footnotesize X_{1}(t+1) = Y_1(t)\oplus ... \oplus Y_n(t)
\oplus
is the XOR operation or binary addition.
As you can see, we shift the register in the first equation, "ejecting"
X_{n}
t
, thus creating the
\vec{X}
vector at the
t+1
state, with an empty space in the first position.
The second equation is how we build that first element: a binary addition of all the elements of
\vec{X}
t
, multiplied by the corresponding connection coefficients, that are equal to
1
only in correspondence of the taps.
A Fibonacci linear-feedback shift register. The taps are marked with a striped background.
To "start" the register, we need to feed it a seed, an initial value on which we first apply the rules of the LFSR. The same holds for a Galois LFSR, but the operating principle differs. Let's take a look!
In a Galois linear-feedback shift register, we first have to shift every bit that is not a tap to the right. This causes the rightmost bit to be "ejected" and moved to the input.
But this is not all for the output bit. It is also added to each tap before the result moves to the space to the right.
Example of a Galois Linear-Feedback Shift Register.
This causes the register to shift unchanged if the output bit is
0
, taking as new input
0
, while in the case of output bit
1
, the tap bits switch, then the register moves to the right. In this case, the input bit is
1
The two types of LFSR produce the same result — minus a reflection and a translation — when the taps are the ones generating a maximally long LFSR. In such a register, all possible states are visited — except the null state, which would make the register collapse in a sequence of 0s. The seed choice is not relevant since it would introduce only a shift in the output.
In order to obtain this kind of coupled outputs, the taps of the Galois register must be the counterparts of the ones of the Fibonacci register. We mean that if the taps of the latter are
1,4,6,12
1,0,0,1,0,1,0,0,0,0,0,1
then the taps we need to use are
1,7,9,12
for a Galois LFSR. The transformation is clear when we look at the corresponding vector:
1,0,0,0,0,0,1,0,1,0,0,1
Let's see an example of a linear-feedback shift register in action.
First, consider a Fibonacci type. The initial state is
1001
, while the taps are
1101
. Here we mark both output and taps on the initial state:
10\textcolor{lightgray}{0}\underline{1} \textcolor{red}{\rightarrow 1}
Now we need to add modulo 2 the "tapped" bits.
1\oplus 0\oplus 1=0
, and thus the new input bit is
0
01\textcolor{lightgray}{0}\underline{0} \textcolor{red}{\rightarrow 0}
And repeat these operations!
10\textcolor{lightgray}{1}\underline{0}\textcolor{red}{\rightarrow\ 0} \\ 11\textcolor{lightgray}{0}\underline{1}\textcolor{red}{\rightarrow\ 1} \\ 11\textcolor{lightgray}{1}\underline{0}\textcolor{red}{\rightarrow\ 0} \\ 01\textcolor{lightgray}{1}\underline{1}\textcolor{red}{\rightarrow\ 1} \\ 00\textcolor{lightgray}{1}\underline{1}\textcolor{red}{\rightarrow\ 1}
Now take a break: the next state of the register would be
1001
, which is equal to the initial state: from now on, the states would repeat, cycling indefinitely.
The sequence we found is
1001011
, with period 7. It would expand this way:
10010111001011100...
Animation of the working principle of a Fibonacci type LFSR. The segments turn red when they are "1".
Do you want to try the same example but with Galois LFSR? We got your back!
Remember: initial state
1001
, taps
1101
The first output would be
1
: it means we are in for a change. Let's see how the register behaves at the next step.
10\textcolor{lightgray}{0}\underline{1}\ \textcolor{red}{\rightarrow\ 1}
The second step has
0
as output. No changes in the register, just a shift:
10\textcolor{lightgray}{1}\underline{0}\ \textcolor{red}{\rightarrow\ 0}
Repeat those steps, and let's see what happens.
01\textcolor{lightgray}{0}\underline{1}\ \textcolor{red}{\rightarrow\ 1} \\ 11\textcolor{lightgray}{0}\underline{0}\ \textcolor{red}{\rightarrow\ 0} \\ 01\textcolor{lightgray}{1}\underline{0}\ \textcolor{red}{\rightarrow\ 0} \\ 00\textcolor{lightgray}{1}\underline{1}\ \textcolor{red}{\rightarrow\ 1} \\ 11\textcolor{lightgray}{1}\underline{1}\ \textcolor{red}{\rightarrow\ 1}
Stop! See the last state? It's the same as the initial state, so we reached the end. This is another period 7 register, with output
1010011
Animation of a Galois LFSR in operation.
You can use our LFSR calculator to find out many things about a selected LFSR. Here we will guide you through it!
First, you have to insert the seed and the taps. Make sure that they have the same length.
Then, choose the type of LFSR you want to use for your register: either Fibonacci or Galois. Choose the desired kind of output too:
Output: this will show you the sequences of output bits. To make it easier to read, we separated the string into groups of 8 bits.
Steps: the calculator will give you the detailed iterations of the register.
Period: you will only get the number of steps repeated in the output.
The last field allows you to specify the desired length of the output/steps you need, but it won't affect the period calculation. If that value is undefined, the calculator will cut the sequence at the end of the first period.
💡 We added the possibility of computing the LFSR using the NXOR gate too! Click on advanced mode and select NXOR: the linear operation applied to the register will change accordingly.
While using our LFSR calculator, you may encounter messages that help you understand the behavior of the register you are using.
For example, if you inserted the correct set of taps, you may find a maximally long LFSR, which means that the period of the LFSR is maximum (equal to
2^n-1
n
is the length of the register). We will tell you when you are doing so, since they are particularly interesting. You can find tables with the taps for such sequences online, like in this file: these "special taps" correspond to the exponents of characteristic polynomials. Place the tap in the position marked by the exponent (if you see
x
, place a
1
in the first position).
If you accidentally end up with a null register, we will stop the calculator and let you know. Change your numbers and try again.
⚠️ The length of the output of an LFSR can grow pretty quickly: the maximal length of a 16 bit register is 65535. Even if our calculator can compute such a long sequence, it would take time: that's why we added a "dangerous calculations" choice in the advanced mode. The default is no, and it stops calculations at 10,000 steps. Are you bold or particularly curious? Select yes and be patient!
Just for fun, let's try it with a mirroring output. Choose a maximally length taps sequence, like
01001
, and a random seed,
11011
. Insert them in our LFSR calculator.
Select "Fibonacci" and "output": the result is
11011000\ 11111001\ 10100100\ 0010101
Now switch to "Galois":
11000110\ 11101010\ 00010010\ 1100111
We need to reverse it — let us do that for you!
11100110\ 10010000\ 10101110\ 1100011
Now let's find the correct shift. Maybe look for a group of clustered
1
s or
0
s, like...
11011000\ \textcolor{red}{11111}001\ 10100100\ 0010101
In the Galois output, the same cluster is broken between the end and the beginning of the sequence:
\textcolor{red}{111}00110\ 10010000\ 10101110\ 11000\textcolor{red}{11}
Shifting the Galois output ten times to the right, we would find the same output of the Fibonacci LFSR.
🔎 Our LFSR generator allows you to use both the Galois and the Fibonacci type. However, the Fibonacci one is the most commonly used: that's why we set it as default!
Now you should know what a LFSR is and how a LFSR works.
LFSR, shift registers, and flip-flops are all parts of a computer we never see, yet they allow for such a complex machine to function as we know it. Go to our Hamming codes calculator if you want to learn more about how computers work behind the keyboard and monitor!
Try our LFSR calculator in all the possible ways and discover which sequences are the longest or which taps give the maximally long outputs!
LFSRs are shift registers that apply linear operations to their bits according to a certain set of rules. There are two types of LFSRs: Galois and Fibonacci. They differ in the way they perform the operation.
What are the uses of LFSR?
LFSR are used in cryptography to generate pseudo-random numbers and in circuit testing to create sequences containing all possible inputs for a given n-bit register. The (almost) random nature of the output of an LFSR allows the simulation of noise in signals and helps overcome interference in signal transmissions.
What are the taps of an LFSR?
The taps of a linear-feedback shift register are the bits that are effectively used in the linear operation. They can be specified as a binary vector. In some instances, they correspond to a maximal-length LFSR, a sequence of all the possible states of the register.
Computers are intrinsically deterministic (only when a physical phenomenon acts on them, it is possible to have a non-deterministic behavior): this makes it difficult to generate truly random numbers on them. Mathematicians developed algorithms that generate sequences of numbers that resemble their random counterparts but are entirely defined by an initial value: the seed.
How do I calculate a Fibonacci LFSR?
In order to calculate a Fibonacci LFSR, add the bits in the register marked by the connection coefficients, modulus 2, and consider the result as new input in the register. The last value exits as an element of the output.
How do I calculate a Galois LFSR?
In a Galois LFSR, shift every bit to the right if the output bit is zero, and use 0 as new input bit too. If the output bit is 1, all of the bits are shifted to the right, but the ones marked by the taps must switch from 1 to 0 and vice-versa. In this case, the next input is 1.
Initial state (seed)
LFSR type
Number of digits in the output/steps
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Class-based programming - Wikipedia
Find sources: "Class-based programming" – news · newspapers · books · scholar · JSTOR (February 2013) (Learn how and when to remove this template message)
Class-based programming, or more commonly class-orientation, is a style of object-oriented programming (OOP) in which inheritance occurs via defining classes of objects, instead of inheritance occurring via the objects alone (compare prototype-based programming).
The most popular and developed model of OOP is a class-based model, instead of an object-based model. In this model, objects are entities that combine state (i.e., data), behavior (i.e., procedures, or methods) and identity (unique existence among all other objects). The structure and behavior of an object are defined by a class, which is a definition, or blueprint, of all objects of a specific type. An object must be explicitly created based on a class and an object thus created is considered to be an instance of that class. An object is similar to a structure, with the addition of method pointers, member access control, and an implicit data member which locates instances of the class (i.e., objects of the class) in the class hierarchy (essential for runtime inheritance features).
3 Critique of class-based models
Encapsulation prevents users from breaking the invariants of the class, which is useful because it allows the implementation of a class of objects to be changed for aspects not exposed in the interface without impact to user code. The definitions of encapsulation focus on the grouping and packaging of related information (cohesion) rather than security issues.
In class-based programming, inheritance is done by defining new classes as extensions of existing classes: the existing class is the parent class and the new class is the child class. If a child class has only one parent class, this is known as single inheritance, while if a child class can have more than one parent class, this is known as multiple inheritance. This organizes classes into a hierarchy, either a tree (if single inheritance) or lattice (if multiple inheritance).
The defining feature of inheritance is that both interface and implementation are inherited; if only interface is inherited, this is known as interface inheritance or subtyping. Inheritance can also be done without classes, as in prototype-based programming.
Critique of class-based models[edit]
Class-based languages, or, to be more precise, typed languages, where subclassing is the only way of subtyping, have been criticized for mixing up implementations and interfaces—the essential principle in object-oriented programming. The critics say one might create a bag class that stores a collection of objects, then extend it to make a new class called a set class where the duplication of objects is eliminated.[1][2] Now, a function that takes an object of the bag class may expect that adding two objects increases the size of a bag by two, yet if one passes an object of a set class, then adding two objects may or may not increase the size of a bag by two. The problem arises precisely because subclassing implies subtyping even in the instances where the principle of subtyping, known as the Liskov substitution principle, does not hold. Barbara Liskov and Jeannette Wing formulated the principle succinctly in a 1994 paper as follows:
{\displaystyle \phi (x)}
{\displaystyle x}
{\displaystyle T}
{\displaystyle \phi (y)}
{\displaystyle y}
{\displaystyle S}
{\displaystyle S}
{\displaystyle T}
Thus, normally one must distinguish subtyping and subclassing. Most current object-oriented languages distinguish subtyping and subclassing, however some approaches to design do not.
Also, another common example is that a person object created from a child class cannot become an object of parent class because a child class and a parent class inherit a person class but class-based languages mostly do not allow to change the kind of class of the object at runtime. For class-based languages, this restriction is essential in order to preserve unified view of the class to its users. The users should not need to care whether one of the implementations of a method happens to cause changes that break the invariants of the class. Such changes can be made by destroying the object and constructing another in its place. Polymorphism can be used to preserve the relevant interfaces even when such changes are done, because the objects are viewed as black box abstractions and accessed via object identity. However, usually the value of object references referring to the object is changed, which causes effects to client code.
Example languages[edit]
See also: Category:Class-based programming languages
Although Simula introduced the class abstraction, the canonical example of a class-based language is Smalltalk. Others include PHP, C++, Java, C#, and Objective-C.
Prototype-based programming (contrast)
^ Kiselyov, Oleg. "Subtyping, Subclassing, and Trouble with OOP". Retrieved 7 October 2012.
^ Ducasse, Stéphane. "A set cannot be a subtype of a bag". Retrieved 7 October 2012.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Class-based_programming&oldid=1066329886"
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The function primroot will compute the first primitive root of n that is greater than g, if possible, otherwise it returns FAIL. The integers that are relatively prime to n form a group of order
\mathrm{\phi }\left(n\right)
under multiplication mod n. If this group is cyclic then a generator of the group is called a primitive root of n (i.e. the order of primroot (g, n) is
\mathrm{\phi }\left(n\right)
). If only one argument n is present (in this case
g=0
) then this function will return the smallest primitive root of the number n.
\mathrm{with}\left(\mathrm{numtheory}\right):
\mathrm{primroot}\left(2\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{primroot}\left(41\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{primroot}\left(0,41\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{primroot}\left(7,41\right)
\textcolor[rgb]{0,0,1}{11}
\mathrm{order}\left(,41\right)
\textcolor[rgb]{0,0,1}{40}
\mathrm{\phi }\left(41\right)
\textcolor[rgb]{0,0,1}{40}
\mathrm{primroot}\left(2,8\right)
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
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How to find the width and length of a rectangle given area and perimeter?
How to use the length and width of a rectangle given area calculator?
More rectangle calculators
If you're struggling to find all those unknowns surrounding rectangles, you're in the right place. Length and width of rectangle given area calculator will answer at least part of your questions. You should provide the area and perimeter. Read on to dig deeper into the topic and learn how to find the length and width of a rectangle given area even without a calculator!
P = perimeter,
A = area,
P (perimeter) = 2L + 2W
A (area) = L × W
Let's determine W from the first equation.
W=\frac{P}{2-L}
And now let's use that in the second equation:
A=L\times\frac{P}{2-L}
Which we can also express as:
L^2 - L\times\frac{P}{2} + A = 0
Solving the equation above, we will obtain L. Then, to find W, we can use
W = A/L or W = (P/2) - L
To determine the length and width of a rectangle, you need to:
Know the area of the rectangle.
Know the perimeter of the rectangle.
Type the area and perimeter values into the calculator. Don't worry if they come in different units - the calculator will deal with it.
Now, you can find both results at the very bottom of the length and width of the rectangle given area calculator.
Still got questions after that length and width of rectangle given area calculator? Check out the rest of our rectangle-related tools:
How to find the length of the rectangle given area and width?
To find the length (L) of a rectangle given area (A) and width (W), you need to:
Know the equation for a rectangle area is A = L × W
Determine L from that previous equation. L = A/Q
To sum up: to find the length of a rectangle, you need to divide its area by the known width.
How to find the width of a rectangle with perimeter and length?
To find the width of a rectangle with a known perimeter and length:
Determine the equation for perimeter
Perimeter (P) = 2 × length (L) + 2 × width (W)
2. Transform the equation:
P - 2 × L = 2 × W
'P/2 - L = W`
so that your final formula is:
width =\frac{perimeter}{2} - lentgh
That way, you can quickly determine the rectangle's width with perimeter and length given.
How to determine the length and width of a rectangle given area and perimeter?
To determine length and width of rectangle given area and perimeter:
State the equations for both area (A) and perimeter (P).
A = length (L) × width (W)
From the first equation, we can also express W as
W = P/(2-L)
Putting this into the second equation will look like this:
A = L × P/(2-L)
L2 - L × P/2 + A = 0
Solving this equation, we will know L - length. Then we can easily determine width as well, knowing that
How to find the width of a rectangle given perimeter 16 in and length 5 in?
To find the width of rectangle given perimeter (16 in) and length (5 in):
Determine the equation for perimeter:
P(perimeter) = 2 × L (length) + 2 ×W (width)
2. As we already know the perimeter and length, we can rewrite the equation:
16 in = 2 × 5 in + 2 × W
3. Let's solve this equation:
16 in = 10 in + 2 × W / - 10 in
6 in = 2 × W / : 2
4. The answer is: width of this rectangle is three inches.
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Calculating Inflation with Index Numbers | Macroeconomics | Course Hero
Explain what a price index is and how to compute one
Calculate inflation rates using price indices
Figure 1. A literal market basket of goods.
If inflation is the percentage change of the price level, what is the "price level"? When economists talk about the price level, what they mean is the average level of prices. To calculate the price level, they begin with the concept of a market basket of goods and services. Imagine a weekly trip to the grocery store. Think about the items you place in your shopping cart (or basket) to buy. That is your market basket. More formally, when economists talk about a market basket of goods and services, they are referring to the different items individuals, businesses, or organizations typically buy.
The next step is to identify the prices of those items, and create a weighted average of the prices. Changes in the prices of goods for which people spend a larger share of their incomes will matter more than changes in the prices of goods for which people spend a smaller share of their incomes. For example, an increase of 10% in the rental rate on housing matters more to most people than whether the price of carrots rises by 10%. To construct an overall measure of the price level, economists compute a weighted average of the prices of the items in the basket, where the weights are based on the actual quantities of goods and services people buy.
The numerical results of a calculation based on a basket of goods can get a little messy. To simplify the task, the price level in each period is typically reported as an index number, rather than as the dollar amount for buying the basket of goods. Index numbers are unit-free measures of economic indicators. Index numbers are based on a value of 100, which makes it easy to measure percent changes. We'll explain this shortly.
Index numbers for prices are called price indices. A price index is essentially the weighted average of prices of a certain type of good or service. Price indices can measure a narrow range of goods and services or a broader range of goods and services. There are price indices for restaurant meals, for groceries, for consumer goods and services, or for everything included in GDP. Figure 2 shows price indices for U.S. higher education, healthcare and groceries, for the period 1990-2015, which are computed by the Bureau of Economic Analysis in the U.S. Commerce Department. Each price index has a base year of 1990 and increases over time. The price index for groceries increased by 70% over the 25-year period. You can see this since the price index increased from a value of 100 in 1990 to a value of 170 in 2015. The price index for healthcare increased by 213% over the same period, and the price index for higher education, which includes tuition, room, board, textbooks and other fees, increased nearly 450% over the period.
Figure 2. Price Indices for U.S. Higher Education, Healthcare & Groceries (1990-2015).
Price indices are created to help calculate the percent change in prices over time. To convert the money spent on the basket to a price index, economists arbitrarily choose one year to be the base year, or starting point from which we measure changes in prices. The base year, by definition, has an index value equal to 100. This sounds complicated, but it is really a simple math trick.
Calculating a Price index
Suppose we look at a simple basket of goods consisting of hamburgers, aspirin and movie tickets, three items that a college student might buy. Say that in any given month, a college student typically purchases 20 hamburgers, one bottle of aspirin, and five movies. Prices for these items over four years are given below. Prices of some goods in the basket may rise while others fall. In this example, the price of aspirin does not change over the four years, while movies increase in price and hamburgers bounce up and down. Each year, the cost of buying the given basket of goods at the prices prevailing at that time is shown.
To calculate the price index in this example, first compute how much money is spent on each good in Year 1.
Year 1 Amount Price Total
Hamburgers 20 $3 $60
Aspirin 1 $10 $10
Movies 5 $6 $30
Next compute the total cost of the market basket in Year 1:
Next, do the same computations for Years 2 through 4.
Hamburgers 20 $3.20 $64
Movies 5 $6.50 $32.50
Total cost of the market basket in Year 2:
$64 + $10 + $32.50 = $106.50
These computations are summarized in Table 1.
Total Cost of Market Basket
Qty 20 1 bottle 5 —
Year 1 Price $3.00 $10.00 $6.00 —
Year 1 Amount Spent $60.00 $10.00 $30.00 $100.00
Now, the total cost of the market basket in each year is not quite a price index, because we haven't established a base year. Say that Year 3 is chosen as the base year. Since the total amount of spending in that year is $107, we divide that amount by itself ($107) and multiply by 100. Mathematically, that is equivalent to dividing $107 by 100, or $1.07. Doing either will give us a value for the price index in the base year of 100. Again, this is because the index number in the base year always has to have a value of 100. Then, to figure out the values of the price index for the other years, we divide the dollar amounts for the other years by 1.07 as well. Note also that the dollar signs cancel out so that price indices have no units. Calculations for the other values of the price index, based on the example presented in Table 1 are shown in Table 2.
Table 2. Calculating Price Indices When Year 3 is the Base Year
\frac{100}{1.07}=93.4
\frac{106.50}{1.07}=99.5
\frac{107}{1.07}=100.0
\frac{117.50}{1.07}=109.8
From Price Indices to Inflation Rates
An inflation rate is just the percentage change in a price index. An inflation rate can be computed for any price index using the general equation for percentage changes between two years, whether in the context of inflation or in any other calculation:
\displaystyle\frac{(\text{Level in new year}-\text{Level in previous year})}{\text{Level in previous year}}=\text{Percentage change}
From Year 1 to Year 2, the price index in Table 2 rises from 93.4 to 99.5. Therefore, the percentage change over this time—the inflation rate—is:
\displaystyle\frac{(99.5-93.4)}{93.4}=0.065=6.5\text{ percent}
From Year 2 to Year 3, the price rises from 99.5 to 100. Thus, the inflation rate over this time, again calculated by the percentage change, is approximately:
\displaystyle\frac{(100-99.5)}{99.5}=0.0047=0.47\text{ percent}
From Year 3 to Year 4, the overall cost rises from 100 to 109.8. The inflation rate is thus:
\displaystyle\frac{(117.50-100)}{100}=0.098=9.8\text{ percent}
These calculations are summarized in Table 3.
Table 3. Calculating the Inflation Rate from the Price index
\displaystyle\frac{(99.5-93.4)}{93.4}=0.065=6.5\text{ percent}
\displaystyle\frac{(100-99.5)}{99.5}=0.0047=0.47\text{ percent}
\displaystyle\frac{(117.50-100)}{100}=0.098=9.8\text{ percent}
This calculation of the change in the total cost of purchasing a basket of goods takes into account how much is spent on each good. The result is equivalent to creating a weighted average of the prices of the three items, with the weights being the percentage of the college student's budget made up by each item. The inflation rate, then, is the percentage change each year in the weighted average of prices.
Closing Thoughts on Price Indices
Three points to remember: first, the inflation rate is the same whether it is based on dollar values or price indices, so then why bother with the price indices?
The inflation calculations we performed above were all based off the the price indices. For example, in year three the inflation rate was calculated this way:
\displaystyle\frac{(100-99.5)}{99.5}=0.0047=0.47\text{ percent}
Note that we could also get to this number by computing inflation rates as the percent change over time in the cost of the market basket. For example, from period 2 to period 3, the overall change in the cost of purchasing the basket rises from $106.50 to $107. Thus, the inflation rate over this time, calculated by the percentage change, is approximately:
\displaystyle\frac{(107-106.50)}{106.50}=0.0047=0.47\text{ percent}
The advantage of using price indices over the costs of the market basket is that indexing allows easier eyeballing of the inflation numbers. If you glance at two annual values for a price index like 107 and 110, you know automatically that the rate of inflation between the two years is about, but not quite exactly equal to, 3%. By contrast, imagine that the price levels were expressed in absolute dollars of a large basket of goods, so that when you looked at the data, the numbers were $19,493.62 and $20,009.32. Most people find it difficult to eyeball those kinds of numbers and say that it is a change of about 3%. However, the two numbers expressed in dollars are exactly in the same proportion of 107 to 110 as the previous example. If you’re wondering why simple subtraction of the index numbers wouldn’t work, read the following feature.
A word of warning: when a price index moves from, say, 107 to 110, the rate of inflation is not exactly 3%. Remember, the inflation rate is not derived by subtracting the index numbers, but rather through the percentage-change calculation. The precise inflation rate as the price index moves from 107 to 110 is calculated as (110 – 107)/107 = 0.028 = 2.8%. When the base year is fairly close to 100, a quick subtraction is not a terrible shortcut to calculating the inflation rate—but when precision matters down to tenths of a percent, subtracting will not give the right answer.
Second, index numbers have no dollar signs or other units attached to them. Although price index numbers are used to calculate a percentage inflation rate, the index numbers themselves do not have percentage signs. Index numbers just mirror the proportions found in other data. They transform the other data so that the data are easier to work with.
Third, the choice of a base year for the index number—that is, the year that is automatically set equal to 100—is arbitrary. It is chosen as a starting point from which changes in prices are tracked. In the official inflation statistics, it is common to use one base year for a few years, and then to update it, so that the base year of 100 is relatively close to the present. But any base year that is chosen for the index numbers will result in exactly the same inflation rate. To see this in the previous example (Table 1), imagine that Year 1, when total spending was $100, was also chosen as the base year, and given an index number of 100. At a glance, you can see that the index numbers would now exactly match the dollar figures, the inflation rate in the first period would be 6.5%, and so on.
Watch the four minute clip from this video to review the distinction between price indices and inflation rates.
arbitrary year whose value as an index number is defined as 100; inflation from the base year to other years can easily be seen by comparing the index number in the other year to the index number in the base year—i.e., 100; so, if the index number for a year is 105, then there has been exactly 5% inflation between that year and the base yearindex numbera unit-free measure of an economic indicators; index numbers are based on a value of 100, which makes it easy to measure percent changesInflation rate: The percentage change in some price indexmarket basket: hypothetical collection of goods and services (or more precisely, the quantities of each good or service) consumers typically buyprice indices: essentially the weighted average of prices of a certain type of good or service; price indices are created to calculate the inflation rate, i.e. the percent change in prices over timeprice level: the average level of prices
Tracking Inflation. Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:[email protected]/Tracking-Inflation. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected]
Green shopping cart. Authored by: polycart. Provided by: Flickr. Located at: https://www.flickr.com/photos/polycart/5783063464/in/photolist-9P2HhL-cWfFTC-7Uy3V8-9PkiXz-F5Xy87-7BJjen-5CP9ce-TckgPq-drJtEN-gHQpx4-ZujbkL-71R9vA-71QXiC-dLEHk3-JBHu5A-dHhXEK-YCzFLW-jQA1mg-qYH7PF-am2frE-9LrezN-9LqveS-9kPuCv-aHrkPc-6TDV5Y-ejcwUR-35qpgj-mTUuPg-ZxvVvo-9e5ehN-gHQvTG-ejc81K-ZxvUZJ-8VbipK-9LkTKx-71LZLM-9LkTAP-7Mjea-9dKTVc-onmHrk-5n7zjw-7knXt9-on9BJ1-c1SMn3-aV1hu2-cWfEbo-cwUthf-5n3hhZ-cWfGr5-KMZjJ. License: CC BY: Attribution
Inflation and CPI Practice- Macro 2.8. Provided by: ACDC Leadership. Located at: https://www.youtube.com/watch?v=JW7IQ45_up8. License: Other. License terms: Standard YouTube License
OpenStax_Economics_CH9_ImageSlideshow.pdf
Notes on Index Numbers for Inflation and Growth
AGEC 4273 Index Numbers and Inflation.pdf
AGEC 4273 • Louisiana State University
1. Calculating inflation using a simple price index Consider a fictio.docx
ECN MICROECONO • Aadinath Mahila Shikshak Prashikshan Mahavidyalaya
Calculating inflation using a simple price index.docx
ECON 2301 • Central Texas College
ECONOMICS MISC • Ahfad University for Women
ECO MISC • Delhi Public School Hyderabad
ACCT 1232 • Adrian College
ECON MISC • Lakhimpur Kendriya Mahavidyalaya
ECN MISC • Ben-Gurion University of the Negev
ECON 179 • Abilene Christian University
1. Calculating inflation using a simple price index.pdf
BUS 2215 • Rochester College
ECON MISC • Taiz University
FINANCE MISC • Delhi Public School, R.K. Puram
1. Calculating inflation using a simple price index 2.pdf
calculating inflation using a simple price index.pdf
AMH 2010 • Pensacola State College
ECONOMICS MISC • Delhi Public School, Hisar
Ch. 13 Index Numbers & Inflation.pptx
MARK 3015 • Texas Christian University
ECON MISC • CCHS
ECO MISC • Dhaka City College
ECON MISC • Uzbekistan State University of World Languages
3.2.1.1-3 TEST [28 marks] - MACROECONOMIC OBJECTIVES, INDICATORS AND INDEX NUMBERS.docx
How to use index numbers to measure the inflation rate Worksheet.docx
ECON MACROECONO • University of Texas, Arlington
PHYSIC 1301 • University of South Asia, Lahore - Campus 2
Returns, Index Numbers, and Inflation.pptx
BBA 101 • National Science College, Kotli
How to use index numbers to measure the inflation rate Worksheet Answers.docx
Macroeconomic Measures_ Inflation and Price Indexes.pdf
ECO ECO2013 • Broward College
Macroeconomic Measures_ Inflation and Price Indexes Notes.pdf
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Implement quaternion representation of six-degrees-of-freedom equations of motion of custom variable mass with respect to wind axes - Simulink - MathWorks Deutschland
\stackrel{˙}{m}
\begin{array}{l}{\overline{F}}_{w}=\left[\begin{array}{l}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{w}+{\overline{\omega }}_{w}×{\overline{V}}_{w}\right)+\stackrel{˙}{m}\overline{V}r{e}_{w}\\ {A}_{be}=DC{M}_{wb}\frac{\left[{\overline{F}}_{w}-\stackrel{˙}{m}{V}_{re}\right]}{m}\\ {\overline{V}}_{w}=\left[\begin{array}{l}V\\ 0\\ 0\end{array}\right],{\overline{\omega }}_{w}=\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\stackrel{˙}{\beta }\mathrm{sin}\alpha \\ {q}_{b}-\stackrel{˙}{\alpha }\\ {r}_{b}+\stackrel{˙}{\beta }\mathrm{cos}\alpha \end{array}\right],{\overline{w}}_{b}=\left[\begin{array}{l}{p}_{b}\\ {q}_{b}\\ {r}_{b}\end{array}\right]\\ {A}_{bb}=DC{M}_{wb}\left[\frac{\overline{F}w-\stackrel{˙}{m}{V}_{re}}{m}-{\overline{\omega }}_{w}×{\overline{V}}_{w}\right]\end{array}
\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{l}L\\ M\\ N\end{array}\right]=I{\stackrel{˙}{\overline{\omega }}}_{b}+{\overline{\omega }}_{b}×\left(I{\overline{\omega }}_{b}\right)+\stackrel{˙}{I}{\overline{\omega }}_{b}\\ {A}_{bb}=\left[\begin{array}{l}{\stackrel{˙}{U}}_{b}\\ {\stackrel{˙}{V}}_{b}\\ {\stackrel{˙}{W}}_{b}\end{array}\right]=DC{M}_{wb}\left[\frac{\overline{F}w-\stackrel{˙}{m}{V}_{re}}{m}-{\overline{\omega }}_{w}×{\overline{V}}_{w}\right]\\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}
\left[\begin{array}{l}{\stackrel{˙}{q}}_{0}\\ {\stackrel{˙}{q}}_{1}\\ {\stackrel{˙}{q}}_{2}\\ {\stackrel{˙}{q}}_{3}\end{array}\right]=-\frac{1}{2}\left[\begin{array}{cccc}0& p& q& r\\ -p& 0& -r& q\\ -q& r& 0& -p\\ -r& -q& p& 0\end{array}\right]\left[\begin{array}{l}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]
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Right Trapezoid Area Calculator
Created by Gabriela Diaz
Reviewed by Luciano Mino
How to find the area of a right trapezoid
How to use the right trapezoid area calculator
More trapezoid calculators
Welcome to the right trapezoid area calculator, where you'll be able to compute the area of any right trapezoid in a blink of an eye! 😉
Here we'll also learn:
What is a right trapezoid; and
How to find the area of a right trapezoid yourself, even if at first you don't know its height.
A right trapezoid is a particular case of a trapezoid, a four-sided geometry with at least one pair of opposite sides parallel to each other. These parallel sides, also known as bases, are identified in the image below as
and
b
In the specific case of a right trapezoid, one of the two remaining sides,
c
from the figure, is perpendicular to the parallel sides
and
b
, creating 90° angles (right angles) between them.
To find the area of a right trapezoid, use the formula: A = ( a + b ) x h/2.
A – Area of the trapezoid;
a and b – Bottom and top bases; and
h – The height.
Meaning that if you know all of these dimensions, you'll be able to calculate the area of your right trapezoid directly.
But what if you don't know the height h? Would you still be able to calculate the area? This is when some good old trigonometry comes to the rescue!
Looking at the image, we can see a right triangle forming from the height h, the difference of the bases (a - b) and the side d.
From here if you know:
The two sides (a - b) and d; or
One side and one angle of the right triangle, you're all set!
If you know two sides, the calculation of h revolves around the Pythagorean theorem as:
h = √(d² - (a - b)²)
On the other hand, if you have one side and one angle you can get the value of h by using the arcsine function (the inverse of the sine) as:
h = arcsin(δ) * d
h = arcsin(γ - π/2) * d
To use the right trapezoid area calculator:
Input the bases a and b. For example, let's assume a = 10 and b = 6.
Enter the value of the height h. In our example, suppose h = 4.
The calculator will display the result for the area on the last row. For our calculation, we get A = 32. And that's it! 😀
💡 Don't know the value of the height h? Click on the Advanced mode button to get the value of the area by entering the angles and the slant side.
Now that you've learned how to calculate the area of a right trapezoid, why not read about the area of an irregular trapezoid or ultimately expand your knowledge about trapezoids with our trapezoid calculator.
To learn about specific topics concerning trapezoids, we recommend other of our tools:
Trapezoid height calculator
Isosceles trapezoid area calculator
Area of an irregular trapezoid calculator
Can a trapezoid have exactly one right angle?
No, a trapezoid can't have only one right angle. The minimum amount of right angles a trapezoid can have is two. This configuration is known as a right trapezoid. The following number of right angles is four; this is the case of a rectangle.
How do I find the height of a right trapezoid?
To find the height of a right trapezoid you might meet one of these cases:
If the bases a and b, and the slant side d are known, use the Pythagorean theorem: h = √(d² - (a - b)²).
When knowing the slant side d and the acute angle δ use: h = arcsin(δ) x d.
If knowing the slant side d and the obtuse angle γ utilize: h = arcsin(γ - π/2) x d.
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Generate downlink test model waveform - MATLAB lteTestModelTool - MathWorks España
lteTestModelTool
Generate Downlink E-TM 2a Waveform
Generate Downlink Waveform Using Full E-TM Configuration Structure
Generate downlink test model waveform
[waveform,grid,tm] = lteTestModelTool(tmn,bw,ncellid,duplexmode)
[waveform,grid,tm] = lteTestModelTool(tm)
lteTestModelTool starts the LTE Waveform Generator app for the parameterization and generation of the E-UTRA test model (E-TM) waveforms.
[waveform,grid,tm] = lteTestModelTool(tmn,bw,ncellid,duplexmode) accepts inputs for the test model number and channel bandwidth for the generated waveform. Optionally, accepts inputs for the physical cell identity and duplex mode.
[waveform,grid,tm] = lteTestModelTool(tm) where a user-defined test model configuration structure is provided as an input.
Generate a time domain signal, txWaveform, and a 2-dimensional array of the Resource Elements, txGrid, for Test Model TS 36.141 E-TM 2a with 10MHz bandwidth. This is a 256QAM E-TM.
Specify test model number and bandwidth. Generate txWaveform. Plot the txGrid output.
[txWaveform,txGrid,tm] = lteTestModelTool('2a','10MHz');
plot(txGrid,'.')
The plot of all the complex resource element symbols in the frame is dominated by the 256QAM PDSCH constellation.
Generate a time domain signal, txWaveform, and a 2-dimensional array of the Resource Elements, txGrid, for Test Model TS 36.141 E-TM 3.2 with 15MHz bandwidth.
Specify test model number and bandwidth for tmCfg configuration structure and create it. Generate txWaveform. View the waveform with a spectrum analyzer.
tmn = '3.2';
bw = '15MHz';
tmCfg = lteTestModel(tmn,bw);
[txWaveform,txGrid,tm] = lteTestModelTool(tmCfg);
saScope = dsp.SpectrumAnalyzer('SampleRate', tm.SamplingRate);
tmn — Test model number
'1.1' | '1.2' | '2' | '2a' | '2b' | '3.1' | '3.1a' | '3.1b' | '3.2' | '3.3'
Test model number, specified as a character vector or string scalar. Use double quotes for string. For more information on these test model numbers, see TS 36.141 [1], Section 6.1.
bw — Channel bandwidth
'1.4MHz' | '3MHz' | '5MHz' | '10MHz' | '15MHz' | '20MHz' | '9RB' | '11RB' | '27RB' | '45RB' | '64RB' | '91RB'
Channel bandwidth, specified as a character vector or string scalar. Use double quotes for string. You can set the nonstandard bandwidths, '9RB','11RB','27RB','45RB','64RB', and '91RB', only when tmn is '1.1'. These nonstandard bandwidths specify custom test models.
Example: '15MHz'
1 or 10 (default) | optional | integer
Physical layer cell identity, specified as an integer. If you do not specify this argument, the default is 1 for standard bandwidths and 10 for non-standard bandwidths.
duplexmode — Duplex mode of the generated waveform
Duplex mode of the generated waveform, specified as 'FDD' or 'TDD'. Optional.
Example: 'FDD'
tm — User-defined test model configuration
User-defined test model configuration, specified as a scalar structure. You can use lteTestModel to generate the various tm configuration structures as per TS 36.141, Section 6 [1]. This configuration structure then can be modified as per requirements and used to generate the waveform.
waveform — Generated E-TM time-domain waveform
Generated E-TM time-domain waveform, returned as a T-by-P numeric matrix, where P is the number of antennas and T is the number of time-domain samples. TS 36.141 [1], Section 6 fixes P = 1, making waveform a T-by-1 column vector.
Resource grid, returned as a 2-D numeric array of resource elements for a number of subframes across a single antenna port. The number of subframes (10 for FDD and 20 for TDD), start from subframe zero, across a single antenna port, as specified in TS 36.141 [1], Section 6.1. Resource grids are populated as described in Represent Resource Grids.
tm — Test model configuration
E-UTRA test model (E-TM) configuration, returned as a scalar structure. tm contains the following fields.
Test model configuration, returned as a scalar structure containing information about the OFDM modulated waveform as described in lteOFDMInfo and test model specific configuration parameters as described in lteTestModel. These fields are included in the output structure:
'1.1', '1.2', '2', '2a', '2b', '3.1', '3.1a', '3.1b' '3.2', '3.3'
'1.4MHz', '3MHz', '5MHz', '10MHz', '15MHz', '20MHz', '9RB', '11RB', '27RB', '45RB', '64RB', '91RB',
Channel bandwidth type, in MHz, returned as a character vector. Non-standard bandwidths, '9RB', '11RB', '27RB', '45RB', '64RB', and '91RB', specify custom test models.
{N}_{\text{RB}}^{\text{DL}}
CellRefP 1
Number of cell-specific reference signal antenna ports. This argument is for informational purposes and is read-only.
CyclicPrefix 'Normal'
Cyclic prefix length. This argument is for informational purposes and is read-only.
CFI 1, 2, or 3
'Normal', 'Extended'
This argument is for informational purposes and is read-only.
Number of time-domain samples over which windowing and overlapping of OFDM symbols is applied
CellRSPower
Cell-specific reference symbol power adjustment, in dB
PSSPower
Primary synchronization signal (PSS) symbol power adjustment, in dB
SSSPower
Secondary synchronization signal (SSS) symbol power adjustment, in dB
PBCHPower
PBCH symbol power adjustment, in dB
PCFICHPower
PCFICH symbol power adjustment, in dB
NAllocatedPDCCHREG
Number of allocated PDCCH REGs. This argument is derived from tmn and bw.
PDCCH symbol power adjustment, in dB
PDSCHPowerBoosted
PDSCH symbol power adjustment, in dB, for the boosted physical resource blocks (PRBs)
PDSCHPowerDeboosted
PDSCH symbol power adjustment, in dB, for the de-boosted physical resource blocks (PRBs)
These fields are present only when DuplexMode is set to 'TDD'.
SSC enumerates the special subframe configuration. TS 36.211 [2], Section 4.2 specifies the special subframe configurations (lengths of DwPTS, GP, and UpPTS).
TDDConfig enumerates the subframe uplink-downlink configuration to be used in this frame. TS 36.211 [2], Section 4.2 specifies uplink-downlink configurations (uplink, downlink, and special subframe combinations).
AllocatedPRB
Allocated physical resource block list
Sampling rate of the time-domain waveform
Number of fast Fourier transform (FFT) points
NLayers 1
Number of transmission layers, returned as 1. This argument is for informational purposes and is read-only.
TxScheme 'Port0'
Transmission scheme. The E-TMs have a single antenna port. This argument is for informational purposes and is read-only.
Cell array of one or two character vectors. Valid values of character vectors include: 'QPSK', '16QAM', '64QAM', '256QAM', '1024QAM'
Modulation formats, specifying the modulation formats for boosted and deboosted PRBs. This argument is for informational purposes and is read-only.
In previous releases, the input-free syntaxes of this function opened the LTE Test Model Generator app. Starting in R2019b, input-free calls to this function open the LTE Waveform Generator app for an E-TM waveform.
lteTestModel | lteDLConformanceTestTool | lteRMCDLTool | lteRMCULTool
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Mobile Robot Kinematics Equations - MATLAB & Simulink - MathWorks Italia
Unicycle Kinematics
Bicycle Kinematics
Ackermann Kinematics
Learn details about mobile robot kinematics equations including unicycle, bicycle, differential drive, and Ackermann models. This topic covers the variables and specific equations for each motion model [1]. For an example that simulates the different mobile robots using these models, see Simulate Different Kinematic Models for Mobile Robots.
The robot state is represented as a three-element vector: [
x
y
\theta
For a given robot state:
x
: Global vehicle x-position in meters
y
: Global vehicle y-position in meters
\theta
: Global vehicle heading in radians
For Ackermann kinematics, the state also includes steering angle:
\psi
: Vehicle steering angle in radians
The unicycle, bicycle, and differential drive models share a genrealized control input, which accepts the following:
v
: Vehicle speed in meters/s
\omega
: Vehicle angular velocity in radians/s
Other variables represented in the kinematics equations are:
r
: Wheel radius in meters
\underset{}{\overset{˙}{\varphi }}
: Wheel speed in radians/s
d
: Track width in meters
l
: Wheel base in meters
\psi
The unicycle kinematics equations model a single rolling wheel that pivots about a central axis using the unicycleKinematics object.
The unicycle model state is [
x
y
\theta
x
y
\theta
\underset{}{\overset{˙}{\varphi }}
: Wheel speed in meters/s
r
v
\omega
: Vehicle heading angular velocity in radians/s
Depending on the VehicleInputs name-value argument, you can input only wheel speeds or the vehicle speed and heading rate. This change in input affects the equations.
Wheel Speed Equation
Vehicle Speed and Heading Rate Equation (Generalized)
When the generalized inputs are given as the speed
v=r\underset{}{\overset{˙}{\varphi }}
and vehicle heading angular velocity
\omega
, the equation simplifies to:
The bicycle kinematics equations model a car-like vehicle that accepts the front steering angle as a control input using the bicycleKinematics object.
The bicycle model state is [
x
y
\theta
x
y
\theta
l
: Wheel base, in meters
\psi
v
\omega
Depending on the VehicleInputs name-value argument, you can input the vehicle speed as either the steering angle or heading rate. This change in input affects the equations.
Steering Angle Equation
\left[\begin{array}{c}\underset{}{\overset{˙}{x}}\\ \underset{}{\overset{˙}{y}}\\ \underset{}{\overset{˙}{\theta }}\end{array}\right]=\left[\begin{array}{c}v\mathrm{cos}\left(\theta \right)\\ v\mathrm{sin}\left(\theta \right)\\ \frac{v}{l}\mathrm{tan}\left(\psi \right)\end{array}\right]\left[\begin{array}{c}v\\ \omega \end{array}\right]
In this generalized format, the heading rate
\omega
can be related to the steering angle
\psi
\omega =\frac{v}{l}\mathrm{tan}\psi
. Then, the ODE simplifies to:
The differential drive kinematics equations model a vehicle where the wheels on the left and right may spin independently using the differentialDriveKinematics object.
The differential drive model state is [
x
y
\theta
x
: Global vehicle x-position, in meters
y
: Global vehicle y-position, in meters
\theta
: Global vehicle heading, in radians
{\underset{}{\overset{˙}{\varphi }}}_{L}
: Left wheel speed in meters/s
{\underset{}{\overset{˙}{\varphi }}}_{R}
: Right wheel speed in meters/s
r
d
v
\omega
Depending on the VehicleInputs name-value argument, you can input the wheel speed as either the steering angle or heading rate. This change in input affects the equations.
In the generalized format, the inputs are given as the speed
v=\frac{r}{2}\left({\underset{}{\overset{˙}{\varphi }}}_{R}+{\underset{}{\overset{˙}{\varphi }}}_{L}\right)
\omega =\frac{r}{2d}\left({\underset{}{\overset{˙}{\varphi }}}_{R}-{\underset{}{\overset{˙}{\varphi }}}_{L}\right)
. The ODE simplifies to:
The Ackermann kinematic equations model a car-like vehicle model with an Ackermann-steering mechanism using the ackermannKinematics object. The equation adjusts the position of the axle tires based on the track width so that the tires follow concentric circles. Mathematically, this means that the input has to be the steering heading angular velocity
\underset{}{\overset{˙}{\psi }}
, and there is no generalized format.
x
y
\theta
\psi
x
y
\theta
\psi
l
v
For the Ackermann kinematics model, the ODE is:
\left[\begin{array}{c}\underset{}{\overset{˙}{x}}\\ \underset{}{\overset{˙}{y}}\\ \underset{}{\overset{˙}{\theta }}\\ \underset{}{\overset{˙}{\psi }}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\left(\theta \right)& 0\\ \mathrm{sin}\left(\theta \right)& 0\\ \mathrm{tan}\left(\psi \right)/l& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}v\\ \underset{}{\overset{˙}{\psi }}\end{array}\right]
For an example the simulates the different mobile robot using these models, see Simulate Different Kinematic Models for Mobile Robots.
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Parameter and Signal Conversions - MATLAB & Simulink - MathWorks Deutschland
Online Conversions and Operations
Streamlining Simulations and Generated Code
To completely understand the results generated by fixed-point Simulink® blocks, you must be aware of these issues:
When numerical block parameters are converted from doubles to fixed-point data types
When input signals are converted from one fixed-point data type to another (if at all)
When arithmetic operations on input signals and parameters are performed
For example, suppose a fixed-point Simulink block performs an arithmetic operation on its input signal and a parameter, and then generates output having characteristics that are specified by the block. The following diagram illustrates how these issues are related.
The sections that follow describe parameter and signal conversions. Rules for Arithmetic Operations discusses arithmetic operations.
Parameters of fixed-point blocks that accept numerical values are always converted from double to a fixed-point data type. Parameters can be converted to the input data type, the output data type, or to a data type explicitly specified by the block. For example, the Discrete FIR Filter block converts its Initial states parameter to the input data type, and converts its Numerator coefficient parameter to a data type you explicitly specify via the block dialog box.
Parameters are always converted before any arithmetic operations are performed. Additionally, parameters are always converted offline using round-to-nearest and saturation. Offline conversions are discussed below.
Because parameters of fixed-point blocks begin as double, they are never precise to more than 53 bits. Therefore, if the output of your fixed-point block is longer than 53 bits, your result might be less precise than you anticipated.
An offline conversion is a conversion performed by your development platform (for example, the processor on your PC), and not by the fixed-point processor you are targeting. For example, suppose you are using a PC to develop a program to run on a fixed-point processor, and you need the fixed-point processor to compute
y=\left(\frac{ab}{c}\right)u=Cu
over and over again. If a, b, and c are constant parameters, it is inefficient for the fixed-point processor to compute ab/c every time. Instead, the PC's processor should compute ab/c offline one time, and the fixed-point processor computes only C·u. This eliminates two costly fixed-point arithmetic operations.
Consider the conversion of a real-world value from one fixed-point data type to another. Ideally, the values before and after the conversion are equal.
{V}_{a}={V}_{b},
where Vb is the input value and Va is the output value. To see how the conversion is implemented, the two ideal values are replaced by the general [Slope Bias] encoding scheme described in Scaling:
{V}_{i}={F}_{i}{2}^{{E}_{i}}{Q}_{i}+{B}_{i}.
Solving for the output data type's stored integer value, Qa is obtained:
\begin{array}{c}{Q}_{a}=\frac{{F}_{b}}{{F}_{a}}{2}^{{E}_{b}-{E}_{a}}{Q}_{b}+\frac{{B}_{b}-{B}_{a}}{{F}_{a}}{2}^{-{E}_{a}}\\ ={F}_{s}{2}^{{E}_{b}-{E}_{a}}{Q}_{b}+{B}_{net},\end{array}
where Fs is the adjusted fractional slope and Bnet is the net bias. The offline conversions and online conversions and operations are discussed below.
Both Fs and Bnet are computed offline using round-to-nearest and saturation. Bnet is then stored using the output data type and Fs is stored using an automatically selected data type.
The remaining conversions and operations are performed online by the fixed-point processor, and depend on the slopes and biases for the input and output data types. The conversions and operations are given by these steps:
The initial value for Qa is given by the net bias, Bnet:
{Q}_{a}={B}_{net}.
The input integer value, Qb, is multiplied by the adjusted slope, Fs:
{Q}_{RawProduct}={F}_{s}{Q}_{b}.
The result of step 2 is converted to the modified output data type where the slope is one and bias is zero:
{Q}_{Temp}=convert\left({Q}_{RawProduct}\right).
This conversion includes any necessary bit shifting, rounding, or overflow handling.
The summation operation is performed:
{Q}_{a}={Q}_{Temp}+{Q}_{a}.
This summation includes any necessary overflow handling.
Note that the maximum number of conversions and operations is performed when the slopes and biases of the input signal and output signal differ (are mismatched). If the scaling of these signals is identical (matched), the number of operations is reduced from the worst (most inefficient) case. For example, when an input has the same fractional slope and bias as the output, only step 3 is required:
{Q}_{a}=convert\left({Q}_{b}\right).
Exclusive use of binary-point-only scaling for both input signals and output signals is a common way to eliminate mismatched slopes and biases, and results in the most efficient simulations and generated code.
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Effects of Bulk Flow Pulsations on Phase-Averaged and Time-Averaged Film-Cooled Boundary Layer Flow Structure | J. Fluids Eng. | ASME Digital Collection
I.-S. Jung, Graduate Student,
I.-S. Jung, Graduate Student
Turbo and Power Machinery Research Center, Department of Mechanical Engineering, Seoul National University, Seoul 151-742, Korea
P. M. Ligrani, Fellow ASME Professor,
P. M. Ligrani, Fellow ASME Professor
Convective Heat Transfer Laboratory, Department of Mechanical Engineering, 50 S. Central Campus Drive, University of Utah, Salt Lake City, UT 84112
J. S. Lee, Mem. ASME Professor
Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division August 31, 2000; revised manuscript received March 5, 2001. Associate Editor: K. Zaman.
Jung, I., Ligrani, P. M., and Lee, J. S. (March 5, 2001). "Effects of Bulk Flow Pulsations on Phase-Averaged and Time-Averaged Film-Cooled Boundary Layer Flow Structure ." ASME. J. Fluids Eng. September 2001; 123(3): 559–566. https://doi.org/10.1115/1.1383972
Flow structure in boundary layers film cooled from a single row of round, simple angle holes, and subject to bulk flow pulsations, is investigated, including phase-averaged streamwise velocity variations, and alterations of time-averaged flow structure. The bulk flow pulsations are in the form of sinusoidal variations of velocity and static pressure, and are similar to flow variations produced by potential flow interactions and passing shock waves near turbine surfaces in gas turbine engines. Injection hole length to diameter ratio is 1.6, time-averaged blowing ratio is 0.50, and bulk flow pulsation frequencies range from 0–32 Hz, which gives modified Strouhal numbers from 0–1.02. Profiles of time-averaged flow characteristics and phase-averaged flow characteristics, measured in the spanwise/normal plane at
x/d=5
z/d=0,
show that effects of pulsations are larger as imposed pulsation frequency goes up, with the most significant and dramatic changes at a frequency of 32 Hz. Phase shifts of static pressure (and streamwise velocity) waveforms at different boundary layer locations from the wall are especially important. As imposed pulsation frequency varies, this includes changes to the portion of each pulsation phase when the largest influences of static pressure waveform phase-shifting occur. At a frequency of 32 Hz, these phase shifts result in higher instantaneous injectant trajectories, and relatively higher injectant momentum levels throughout a majority of each pulsation period.
turbines, film flow, boundary layers, pulsatile flow, cooling, shock wave effects, transonic flow
Boundary layers, Flow (Dynamics), Pressure
Rigby, M. J., Johnson, A. B., and Oldfield, M. L. G., 1990, “Gas Turbine Rotor Blade Film Cooling With and Without Simulated NGV Shock Waves and Wakes,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 90-GT-78, Brussels.
Flowfield Measurements in Supersonic Film Cooling Including Effect of Shock-Wave Interaction
Garg, V. K., and Abhari, R. S., 1996, “Comparison Of Predicted And Experimental Nusselt Number For A Film-Cooled Rotating Blade,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 96-GT-223, Birmingham.
Nix, A. C., Reid, T., Peabody, H., Ng, W. F., Diller, T. E., and Schetz, J. A., 1997, “Effects of Shock Wave Passing on Turbine Blade Heat Transfer in a Transonic Cascade,” AIAA Paper No. AIAA-97-0160.
Popp, O., Smith, D. E., Bubb, J. V., Grabowski, H. C. III, Diller, T. E., Schetz, J. A., and Ng, W. F., 1999, “Steady and Unsteady Heat Transfer in a Transonic Film Cooled Turbine Cascade,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 99-GT-259, Indianapolis.
Smith, D. E., Bubb, J. V., Popp, O., Grabowski, H. C. III, Diller, T. E., Schetz, J. A., and Ng, W. F., 2000, “Investigation of Heat Transfer in a Film Cooled Transonic Turbine Cascade, Part I: Steady Heat Transfer,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 2000-GT-202, Munich.
Popp, O., Smith, D. E., Bubb, J. V., Grabowski, H. C. III, Diller, T. E., Schetz, J. A., and Ng, W. F., 2000, “Investigation of Heat Transfer in a Film Cooled Transonic Turbine Cascade, Part II: Unsteady Heat Transfer,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 2000-GT-203, Munich.
Dunn, M. G., Haldeman, C. W., Abhari, R. S., and McMillan, M. L., 2000, “Influence of Vane/Blade Spacing on the Heat Flux for a Transonic Turbine,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 2000-GT-206, Munich.
Bergholz, R. F., Dunn, M. G., and Steuber, G. D., 2000, “Rotor/Stator Heat Transfer Measurements and CFD Predictions for Short-Duration Turbine Rig Tests,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 2000-GT-208, Munich.
Bulk Flow Pulsations and Film Cooling: Part 1, Injectant Behavior
Bulk Flow Pulsations and Film Cooling: Part 2, Flow Structure and Film Effectiveness
The Effect of Injection Hole Length on Film Cooling With Bulk Flow Pulsations
Sohn, D. K., and Lee, J. S., 1997, “The Effects of Bulk Flow Pulsations on Film Cooling From Two Rows of Holes,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 97-GT-129.
Jung, I.-S., and Lee, J. S., 1998, “Effects Of Bulk Flow Pulsations on Film Cooling From Spanwise Oriented Holes,” International Gas Turbine & Aeroengine Congress & Exposition, Paper No. 98-GT-211, Stockholm.
Film Cooling Subject to Bulk Flow Pulsations: Effects of Blowing Ratio, Freestream Velocity, and Pulsation Frequency
Film Cooling Subject to Bulk Flow Pulsations: Effects of Density Ratio, Hole Length-to-Diameter Ratio, and Pulsation Frequency
Al-Asmi
Experimental Study of a Periodic Turbulent Boundary Layer in Zero Mean Pressure Gradient
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Truncated tetraoctagonal tiling - WikiMili, The Free Encyclopedia
Truncated tetraoctagonal tiling
Semiregular tiling in geometry
Vertex configuration 4.8.16
Schläfli symbol tr{8,4} or
{\displaystyle t{\begin{Bmatrix}8\\4\end{Bmatrix}}}
Dual Order-4-8 kisrhombille tiling
The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry.
Truncated tetraoctagonal tiling with *842, , mirror lines
There are 15 subgroups constructed from [8,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,8,1+,4,1+] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4+], (4*4) with gyration points removed, becomes (*4444) or (*44), and another [8*,4], index 16 as [8+,4], (8*2) with gyration points removed as (*22222222) or (*28). And their direct subgroups [8,4*]+, [8*,4]+, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
Small index subgroups of [8,4] (*842)
= [1+,8,4]
= [8,4,1+]
= = [8,1+,4]
= [1+,8,4,1+]
= [8+,4+]
*842 *444 *882 *4222 *4242 42×
Semidirect subgroups
[8,1+,4,1+]
= = [1+,8,1+,4]
= [8,4+]+
= [8+,4]+
= [8,1+,4]+
= [8+,4+]+ = [1+,8,1+,4,1+]
Radical subgroups
[8,4*]
= [8*,4]
[8,4*]+
= [8*,4]+
*4444 *22222222 4444 22222222
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
*nn2
[n,n]
*∞∞2
4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Wikimedia Commons has media related to Uniform tiling 4-8-16 .
In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
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10 CFR § 429.56 - Integrated light-emitting diode lamps. | CFR | US Law | LII / Legal Information Institute
10 CFR § 429.56 - Integrated light-emitting diode lamps.
(a) Determination of Represented Value. Manufacturers must determine the represented value, which includes the certified rating, for each basic model of integrated light-emitting diode lamps by testing, in conjunction with the sampling provisions in this section.
(1) Units to be tested.
(i) The general requirements of § 429.11 (a) are applicable except that the sample must be comprised of production units; and
(ii) For each basic model of integrated light-emitting diode lamp, the minimum number of units tested must be no less than 10 and the same sample comprised of the same units must be used for testing all metrics. If more than 10 units are tested as part of the sample, the total number of units must be a multiple of two. For each basic model, a sample of sufficient size must be randomly selected and tested to ensure that:
(A) Represented values of initial lumen output, lamp efficacy, color rendering index (CRI), power factor, or other measure of energy consumption of a basic model for which consumers would favor higher values are less than or equal to the lower of:
\overline{x}=\frac{1}{n}\sum _{i=1}^{n}{x}_{i}
is the sample mean; n is the number of units; and xi is the measured value for the i th unit; Or,
(2) The lower 99 percent confidence limit (LCL) of the true mean divided by 0.96; or the lower 99 percent confidence limit (LCL) of the true mean divided by 0.98 for CRI and power factor, where:
LCL=\stackrel{—}{x}-{t}_{0.99}\left(\frac{s}{\sqrt{n}}\right)
is the sample mean; s is the sample standard deviation; n is the number of samples; and t0.99 is the t statistic for a 99 percent one-tailed confidence interval with n-1 degrees of freedom (from appendix A to this subpart).
(B) Represented values of input power, standby mode power or other measure of energy consumption of a basic model for which consumers would favor lower values are greater than or equal to the higher of:
\overline{x}=\frac{1}{n}\sum _{i=1}^{n}{x}_{i}
is the sample mean; n is the number of units; and xi is the measured value for the i th unit;
\mathrm{UC}L=\stackrel{—}{x}-{t}_{0.99}\left(\frac{s}{\sqrt{n}}\right)
is the sample mean; s is the sample standard deviation; n is the number of samples; and t0.99 is the t statistic for a 99 percent one-tailed confidence interval with n-1 degrees of freedom (from appendix A to this subpart);
(C) Represented values of correlated color temperature (CCT) of a basic model must be equal to the mean of the sample, where:
\overline{x}=\frac{1}{n}\sum _{i=1}^{n}{x}_{i}
is the sample mean; n is the number of units in the sample; and xi is the measured CCT for the i th unit.
(D) The represented value of lifetime of an integrated light-emitting diode lamp must be equal to or less than the median time to failure of the sample (calculated as the arithmetic mean of the time to failure of the two middle sample units when the numbers are sorted in value order) rounded to the nearest hour.
(2) The represented value of life (in years) of an integrated light-emitting diode lamp must be calculated by dividing the lifetime of an integrated light-emitting diode lamp by the estimated annual operating hours as specified in 16 CFR 305.15(b)(3)(iii).
(3) The represented value of estimated annual energy cost for an integrated light-emitting diode lamp, expressed in dollars per year, must be the product of the input power in kilowatts, an electricity cost rate as specified in 16 CFR 305.15(b)(1)(ii), and an estimated average annual use as specified in 16 CFR 305.15(b)(1)(ii).
(1) The requirements of § 429.12 are applicable to integrated light-emitting diode lamps;
(2) Values reported in certification reports are represented values. Pursuant to § 429.12(b)(13), a certification report must include the following public product-specific information: The testing laboratory's NVLAP identification number or other NVLAP-approved accreditation identification, the date of manufacture, initial lumen output in lumens (lm), input power in watts (W), lamp efficacy in lumens per watt (lm/W), CCT in kelvin (K), power factor, lifetime in years (and whether value is estimated), and life (and whether value is estimated). For lamps with multiple modes of operation (such as variable CCT or CRI), the certification report must also list which mode was selected for testing and include detail such that another laboratory could operate the lamp in the same mode. Lifetime and life are estimated values until testing is complete. When reporting estimated values, the certification report must specifically describe the prediction method, which must be generally representative of the methods specified in appendix BB. Manufacturers are required to maintain records per § 429.71 of the development of all estimated values and any associated initial test data.
(c) Rounding requirements.
(3) Round lamp efficacy to the nearest tenth of a lumen per watt.
(4) Round correlated color temperature to the nearest 100 Kelvin.
(5) Round color rendering index to the nearest whole number.
(8) Round standby mode power to the nearest tenth of a watt.
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Hausdorff distance - Wikipedia
(Redirected from Hausdorff metric)
{\displaystyle [0,1]\to \mathbb {R} ^{3}}
{\displaystyle (M,d)}
{\displaystyle d_{\mathrm {H} }(X,Y)}
{\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X}d(x,Y),\,\sup _{y\in Y}d(X,y)\,\right\},\!}
{\displaystyle d(a,B)=\inf _{b\in B}d(a,b)}
{\displaystyle a\in X}
{\displaystyle B\subseteq X}
{\displaystyle d_{H}(X,Y)=\inf\{\varepsilon \geq 0\,;\ X\subseteq Y_{\varepsilon }{\text{ and }}Y\subseteq X_{\varepsilon }\},\quad }
{\displaystyle X_{\varepsilon }:=\bigcup _{x\in X}\{z\in M\,;\ d(z,x)\leq \varepsilon \},}
{\displaystyle \varepsilon }
{\displaystyle X}
{\displaystyle \varepsilon }
{\displaystyle X}
{\displaystyle \varepsilon }
{\displaystyle X}
{\displaystyle d_{H}(X,Y)=\sup _{w\in M}\left|\inf _{x\in X}d(w,x)-\inf _{y\in Y}d(w,y)\right|=\sup _{w\in X\cup Y}\left|\inf _{x\in X}d(w,x)-\inf _{y\in Y}d(w,y)\right|,}
{\displaystyle d_{\mathrm {H} }(X,Y)=\sup _{w\in M}|d(w,X)-d(w,Y)|}
{\displaystyle d(w,X)}
{\displaystyle w}
{\displaystyle X}
{\displaystyle X,Y\subset M}
{\displaystyle d_{\mathrm {H} }(X,Y)=\varepsilon }
{\displaystyle X\subseteq Y_{\varepsilon }\ {\mbox{and}}\ Y\subseteq X_{\varepsilon }.}
{\displaystyle \mathbb {R} }
{\displaystyle d}
{\displaystyle d(x,y):=|y-x|,\quad x,y\in \mathbb {R} .}
{\displaystyle X:=(0,1]\quad {\mbox{and}}\quad Y:=[-1,0).}
{\displaystyle d_{\mathrm {H} }(X,Y)=1\ }
{\displaystyle X\nsubseteq Y_{1}}
{\displaystyle Y_{1}=[-2,1)\ }
{\displaystyle 1\in X}
{\displaystyle X\subseteq {\overline {Y_{\varepsilon }}}}
{\displaystyle Y\subseteq {\overline {X_{\varepsilon }}}}
{\displaystyle X,Y}
{\displaystyle d_{\mathrm {H} }(X,Y)}
{\displaystyle d_{\mathrm {H} }(X,Y)}
{\displaystyle d_{\mathrm {H} }(X,Y)=0}
{\displaystyle d(x,y)}
{\displaystyle d(x,Y)=\inf\{d(x,y)\mid y\in Y\}.\ }
{\displaystyle d(X,Y)=\sup\{d(x,Y)\mid x\in X\}.\ }
{\textstyle d(\{1,7\},\{3,6\})=\sup\{d(1,\{3,6\}),d(7,\{3,6\})\}=\sup\{d(1,3),d(7,6)\}=2.}
{\displaystyle X\subseteq Y}
{\displaystyle d_{\mathrm {H} }(X,Y)=\max\{d(X,Y),d(Y,X)\}\,.}
{\displaystyle d_{\mathrm {H} }(I(M),J(N))}
{\displaystyle I\colon M\to L}
{\displaystyle J\colon N\to L}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hausdorff_distance&oldid=1038135651"
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Asymptotic Complexity Classes and Comparison | DigitalBitHub
Asymptotic Complexity Classes and Comparison
Time complexity estimates how an algorithm performs, regardless of the programming language and processor used to run the algorithm. We can calculate time complexity simply by counting the number of statements (especially loops and recursive calls) executed by the code written. This time complexity is defined as a function of the input size n. Usually, we use Big-Oh notation to represent the worst-case time complexity of the algorithm.
What is Asymptotic Complexity Classes
As time complexity is the function of input size n. These functions are used to represent the behavior of the algorithm if the input size increases to a very large number.
There are majorly five types of Complexity Classes:
A constant function is a function whose (output) value is the same for every input value.
Example: 1, 10, 50, 1000, 10000, 100000,... (Any positive numerical value)
Decreasing function value decreases if the value of input "n" increases.
\frac{1}{n},\frac{1}{{n}^{2}},\frac{\mathrm{log}n}{n}, \frac{n}{{2}^{n}},...
A polynomial function is a function that involves only non-negative integer powers. Polynomial functions are usually in the form of nC, where C is a positive non-zero integer (C>0).
Example: n0.1, n0.5, n2.5, n50, n10000,...
Linear Function: n
Quadric Function: n2
Cubic Function: n3
Example: log(n), (log(n))2, log(n))10,...
In Exponential functions, the value of function increases drastically if the input value increases. Exponential functions are usually in the form of Cn, where C is a positive non-zero integer (C>1).
Example: 2n, 3n, (1.5)n, 7n,...
The below graph is representing the complexity classes in time vs input value.
Comparison of values of Asymptotic Complexity Classes
Decreasing Functions < Constant Function < Logarithmic Functions < Polynomial Functions < Exponential Functions
Here are some examples of Complexity classes comparisons:
2n<nn
2n<n!
n! < nn
\mathrm{log}\left(n\right)<\sqrt{n}
(log(n))2 < n
(log(n))1000 < n
(log(n))log(n) > n
Ques: Consider one more example with four functions given below and write in increasing order.
\sqrt{n}
F3 = log(n)
\frac{100}{n}
Sol: Increasing order for the given functions is F4 < F1 < F3 < F2
We will see more examples with explanations in the next article.
Asymptotic NotationsComplexity Classes
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George Pólya - Wikiquote
Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.
George Pólya (December 13, 1887 – September 7, 1985) was a Hungarian mathematician and professor of mathematics at ETH Zürich and at Stanford University. His work on heuristics and pedagogy has had substantial and lasting influence on mathematical education, and has also been influential in artificial intelligence.
1.1 How to Solve It (1945)
1.2 Induction and Analogy in Mathematics (1954)
1.3 Mathematical Methods in Science (1977)
1.4 Mathematical Discovery (Volume 1)
2 About George Pólya
"Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result. ...we may wish a better characterization ..."successive trials" or "successive corrections" or "successive approximations." ...You use successive approximations when ...looking for a word in the dictionary ...A mathematician may apply the term ...to a highly sophisticated procedure ...to treat some very advanced problem ...that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation ...may appear as successive approximations to the truth.
Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for ...many ... situations, straightforward algebra is more efficient than successive approximations.
George Pólya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1962)
Jon Fripp; Michael Fripp; Deborah Fripp (2000). Speaking of Science: Notable Quotes on Science, Engineering, and the Environment. Newnes. p. 45. ISBN 978-1-878707-51-2.
How to Solve It (1945)[edit]
Unless otherwise stated, page references are from the Expanded Princeton Science Library Edition (2004) ISBN 0-691-11966-X
There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.
2nd ed. (1957), p. xv
We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.
To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument....
Induction and Analogy in Mathematics (1954)[edit]
Vol. 1. Of Mathematics and Plausible Reasoning
Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs.
Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing.
The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess.
The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life.
The efficient use of plausible reasoning is a practical skill and it is learned... by imitation and practice. ...what I can offer are only examples for imitation and opportunity for practice.
I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation.
I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life.
For many of the stories told the final form resulted from a sort of informal psychological experiment. I discussed the subject with several different classes... Several passages... have been suggested by answers of my students, or... modified... by the reaction of my audience.
Mathematical Methods in Science (1977)[edit]
Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks.
Good approximations often lead to better ones.
The volume of the cone was discovered by Democritus... He did not prove it, he guessed it... not a blind guess, rather it was reasoned conjecture. As Archimedes has remarked, great credit is due to Democritus for his conjecture since this made proof much easier. Eudoxes... a pupil of Plato, subsequently gave a rigorous proof. Surely the labor or writing limited his manuscript to a few copies; none has survived. In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do. However, the substance of what he wrote is nevertheless available to us. ...Euclid's great achievement was the systematization of the works of his predecessors. The Elements preserve several of Eudoxes' proofs.
Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize.
We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us. ...In the solution of a problem ...there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude.
Facing any part of the observable reality, we are never in possession of complete knowledge, nor in a state of complete ignorance, although usually much closer to the latter state.
If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the problem "with bare hands", we have a better chance to understand the scientist's attitude in general, and especially the task of the applied mathematician.
If you cannot solve the proposed problem, try to solve first a simpler related problem.
{\displaystyle {\frac {dy}{dx}}={\frac {\omega ^{2}x}{g}}}
...The first derivative, the result of the differentiation of
{\displaystyle y}
{\displaystyle x}
, was written by Leibniz in the form
{\displaystyle {\frac {dy}{dx}}}
...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative
{\displaystyle {\frac {dy}{dx}}}
was considered as the ratio of two "infinitely small quanitites", of the infinitesimals
{\displaystyle dy}
{\displaystyle dx}
. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...
{\displaystyle {\frac {dy}{dx}}}
is the limit of a ratio of
{\displaystyle dy}
{\displaystyle dx}
... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat
{\displaystyle {\frac {dy}{dx}}}
so as if it were a ratio... and multiply by
{\displaystyle dx}
to achieve the separation of variables. We get
{\displaystyle {dy}={\frac {\omega ^{2}x}{g}}xdx}
Simplicity is worth buying if we do not have to pay too great a loss of precision for it.
Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. Happily here is a commodity of which a little may be made to go a long way. ...the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks.
{\displaystyle {\frac {dy}{dx}}=f(x,y)}
...prescribes the slope
{\displaystyle {\frac {dy}{dx}}}
at each point of the plane (or at each point of a certain region of the plane we call the field"). ...a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines. ...It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction."
Life is full of surprises: our approximate condition for the fall of a body through a resisting medium is precisely analogous to the exact condition for the flow of an electric current through a resisting wire [of an induction coil]. ...
{\displaystyle m{\frac {dv}{dt}}=mg-Kv}
This is the form most convenient for making an analogy with the "fall", i.e., flow, of an electric current.
...in order from left to right, mass
{\displaystyle m}
, rate of change of velocity
{\displaystyle {\frac {dv}{dt}}}
, gravitational force
{\displaystyle mg}
, and velocity
{\displaystyle v}
. What are the electrical counterparts? ...To press the switch, to allow current to start flowing is the analogue of opening the fingers, to allow the body to start falling. The fall of the body is caused by the force
{\displaystyle mg}
due to gravity; the flow of the current is caused by the electromotive force or tension
{\displaystyle E}
due to the battery. The falling body has to overcome the frictional resistance of the air; the flowing current has to overcome the electrical resistance of the wire. Air resistance is proportional to the body's velocity
{\displaystyle v}
; electrical resistance is proportional to the current
{\displaystyle i}
. And consequently rate of change of velocity
{\displaystyle {\frac {dv}{dt}}}
corresponds to rate of change of current
{\displaystyle {\frac {di}{dt}}}
. ...The electromagnetic induction
{\displaystyle L}
opposes the change of current... And doesn't the inertia or mass
{\displaystyle m}
..? Isn't
{\displaystyle L}
, so to speak, an electromagnetic inertia?
{\displaystyle L{\frac {di}{dt}}=E-Ki}
Mathematical Discovery (Volume 1)[edit]
People tell you that wishful thinking is bad. Do not believe it, this is just one of those generally accepted errors.
About George Pólya[edit]
If we could be any mathematician in the history of the world (besides ourselves), who would we rather be? ...we narrowed the choice down to Euler and Pólya, and finally settled on George Pólya because of the sheer enjoyment of mathematics that he has conveyed by so many examples.
Donald E. Knuth, comments at Pólya's 90th birthday celebration quoted by Gerald L. Alexanderson, The Random Walks of George Polya (2000)
For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya.
A. H. Schoenfeld, in "Polya, Problem Solving, and Education" in Mathematics Magazine (1987)
Retrieved from "https://en.wikiquote.org/w/index.php?title=George_Pólya&oldid=3102026"
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Quaternion interpolation between two quaternions - MATLAB quatinterp - MathWorks Switzerland
quatinterp
qi=quatinterp(p,q,f,method)
qi=quatinterp(p,q,f,method) calculates the quaternion interpolation between two normalized quaternions p and q by interval fraction f.
p and q are the two extremes between which the function calculates the quaternion.
Use interpolation to calculate quaternion between two quaternions p=[1.0 0 1.0 0] and q=[-1.0 0 1.0 0] using the SLERP method. This example uses the quatnormalize function to first-normalize the two quaternions to pn and qn.
pn = quatnormalize([1.0 0 1.0 0])
qn = quatnormalize([-1.0 0 1.0 0])
qi = quatinterp(pn,qn,0.5,'slerp')
p — First-normalized quaternion
First normalized quaternion for which to calculate the interpolation, specified as an M-by-4 matrix containing M quaternions. This quaternion must be a normalized quaternion.
Second normalized quaternion for which to calculate the interpolation, specified as an M-by-4 matrix containing M quaternions. This quaternion must be a normalized quaternion.
Interval fraction by which to calculate the quaternion interpolation, specified as an M-by-1 matrix containing M fractions (scalar). f varies between 0 and 1. It represents the intermediate rotation of the quaternion to be calculated.
qi=(qp,qn,qf), where:
If f equals 0, qi equals qp.
If f is between 0 and 1, qi equals method.
If f equals 1, qi equals qn.
method — Quaternion interpolation method
'slerp' (default) | 'lerp' | 'nlerp'
Quaternion interpolation method to calculate the quaternion interpolation. These methods have different rotational velocities, depending on the interval fraction. For more information on interval fractions, see [1].
Quaternion slerp. Spherical linear quaternion interpolation method. This method is most accurate, but also most computation intense.
Slerp\left(p,q,h\right)=p{\left({p}^{*}q\right)}^{h}
h\in \left[0,1\right].
Quaternion lerp. Linear quaternion interpolation method. This method is the quickest, but is also least accurate. The method does not always generate normalized output.
LERP\left(p,q,h\right)=p\left(1-h\right)+qh
h\in \left[0,1\right].
r=LERP\left(p,q,h\right),
NLERP\left(p,q,h\right)=\frac{r}{|r|}.
qi — Interpolation of quaternion
Interpolation of quaternion.
quatlog | quatexp | quatpower | quatconj | quatdivide | quatinv | quatmod | quatmultiply | quatnormalize | quatrotate
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iupac - Maple Help
Home : Support : Online Help : Programming : Data Types : Conversion : iupac
convert an atomic number to and from the temporary International Union of Pure and Applied Chemistry (IUPAC) name or symbol
convert(u, iupac, symbol)
symbol, string, or positive integer
(optional); specify that an atomic number be converted to the IUPAC symbol
The convert(u, iupac) command converts a positive integer representing an atomic number to a string representing its temporary IUPAC name.
The convert(u, iupac, symbol) command converts a positive integer representing an atomic number to a string representing its temporary IUPAC symbol.
The convert(u, iupac) command converts a string or symbol representing an IUPAC name or symbol to a positive integer representing its atomic number.
Temporary IUPAC names are used as placeholders until a proper name and symbol has been chosen for newly discovered elements.
The name of an element is formed by replacing each digit in the atomic number with the corresponding name, and then appending um or ium. The um suffix is used if the last digit name ends in the letter i.
The symbol of an element is formed by replacing each digit in the atomic number with the corresponding symbol, and then capitalizing the first symbol.
\mathrm{e114}≔\mathrm{convert}\left(114,\mathrm{iupac}\right)
\textcolor[rgb]{0,0,1}{\mathrm{e114}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"ununquadium"}
\mathrm{convert}\left(\mathrm{e114},\mathrm{iupac}\right)
\textcolor[rgb]{0,0,1}{114}
\mathrm{e114}≔\mathrm{convert}\left(114,\mathrm{iupac},\mathrm{symbol}\right)
\textcolor[rgb]{0,0,1}{\mathrm{e114}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"Uuq"}
\mathrm{convert}\left(\mathrm{e114},\mathrm{iupac}\right)
\textcolor[rgb]{0,0,1}{114}
\mathrm{convert}\left(112,\mathrm{iupac}\right)
\textcolor[rgb]{0,0,1}{"ununbium"}
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Midsegment of a trapezoid formula
How do I find the midsegment of a trapezoid?
The midsegment of a trapezoid calculator, allows you to obtain the length of the midsegment or median of a trapezoid. The median of a trapezoid is a line parallel to the bases placed in the midpoint between them. With this tool, you will learn the midsegment of a trapezoid formula and how to find the midsegment of any trapezoid.
The median or midsegment of a trapezoid is a line parallel to the trapezoid's bases, which crosses the midpoint between them. It extends from one non-parallel side to the other.
A trapezoid with abcd sides.
Given the length of one base, you can use the midsegment to find the length of the other. Let's take a look at the midsegment of a trapezoid formula to learn how.
The median or midsegment of an ABCD trapezoid formula is straightforward. We just need the length of each of the bases (
AB
CD
), add them, and then divide the result by two:
\text{Midsegment} = \frac{AB+CD}{2}
This is the same as finding the median or average value between the bases, hence the name. If you find any two variables, you can obtain the other with ease by replacing the values in the equation above, or simply use the midsegment of a trapezoid calculator, and it will do the work for you 😉.
To find the midsegment of a trapezoid:
Measure and write down the length of the two parallel bases.
Divide the result by two. This is the length of the midsegment.
You can verify the result with the midsegment of a trapezoid calculator or take a look at our trapezoid calculator to learn more.
In this text, we have covered:
Median of a trapezoid definition;
Median of a trapezoid formula; and
How to find the midsegment of a trapezoid.
Feel free to read through the FAQ section or try other useful tools, similar to the midsegment of a trapezoid calculator:
How many midsegments does a trapezoid have?
A trapezoid has only one midsegment. The midsegment is a line that extends from one non-parallel side to the other, is parallel to the bases, and is placed at the midpoint between them.
How long is the midsegment of a trapezoid with 2 cm bases?
The midsegment of a trapezoid with 2 cm bases is 2 cm. The formula for the midsegment is (AB + CD) / 2, and since AB and CD are identical, the result and the bases' lengths are equal in number.
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3 Ways to Solve One Step Equations - wikiHow
1 Adding or Subtracting to Solve
2 Dividing or Multiplying to Solve
An equation is a mathematical sentence that expresses two equal values.[1] X Research source In algebra you will often work with equations, which have an unknown value represented by a variable. To solve such equations, you need to find the value of the variable. A one-step equation is one in which you only have to perform one operation to determine the unknown value, and so these type of equations are the easiest to solve.
Adding or Subtracting to Solve Download Article
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Write down the equation. It’s easy to solve equations when you understand what they mean. An equation will have a variable (usually
{\displaystyle x}
), which represents an unknown value. The equation will also have a constant, which is a number you need to add or subtract from the variable to equal a certain sum or difference.
For example, you might have the equation
{\displaystyle x-9=5}
. The variable representing the unknown number is
{\displaystyle x}
. When you subtract 9 from the unknown number, the difference is 5.
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Determine how to isolate the variable. To isolate a variable, you need to get it alone on one side of the equation by performing an inverse operation to cancel the constants. Addition and subtraction are inverse operations. So, if the constant is subtracted in the equation, to cancel it you would add.[2] X Research source
{\displaystyle x-9=5}
, 9 is subtracted from the variable, so to isolate the variable you must cancel the 9 by adding it.
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Add or subtract the constant from both sides of the equation. As you manipulate equations to solve them, you must keep both sides balanced. Whatever you do to one side of the equation, you must do to the other side. So, if you need to add a value to isolate the variable, you must also add that same value to the other side of the equation.[3] X Research source
{\displaystyle x-9=5}
, you need to add 9 to the left side to isolate the variable, so you also need to add 9 to the right side of the equation:
{\displaystyle x-9=5}
{\displaystyle x-9+9=5+9}
{\displaystyle x=14}
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Check your work. To make sure your solution is correct, plug the value of
{\displaystyle x}
into the original equation. If the equation is true, your solution is correct.
{\displaystyle x=14}
, substitute 14 for
{\displaystyle x}
in the original equation:
{\displaystyle 14-9=5}
. Since this equation is true, your solution is correct.
Dividing or Multiplying to Solve Download Article
Evaluate the equation. The variable, usually
{\displaystyle x}
, represents an unknown value. Solving an equation means finding the unknown value. The equation may also have a coefficient, which is a number you need to multiply by the variable to equal a certain product. The variable may also be the numerator of a fraction. This means you need to divide the variable by the number in the denominator to equal a certain quotient.
{\displaystyle 3x=24}
{\displaystyle x}
. When you multiply the unknown number and 3, the product is 24.
Determine how to isolate the variable. Isolating a variable means getting it by itself on one side of the equation. To do this you must perform an inverse operation to cancel coefficients or fractions. Multiplication and division are inverse operations. If the variable has a coefficient, to cancel it you would divide by the coefficient, since any number divided by itself is equal to 1. If the variable is the numerator of a fraction, to isolate it you would multiply by the denominator, since multiplying by a number cancels the division by that number.[4] X Research source
{\displaystyle 3x=24}
the variable is multiplied by 3, so to isolate the variable you must cancel the 3 by dividing by 3.
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Multiply or divide from both sides of the equation. As you solve an equation the most important thing to remember is that you must keep both sides of the equation balanced. This means that whatever you do to one side of the equation, you must do to the other side, too.[5] X Research source So, if you need to divide by a value to isolate the variable, you must also divide by the same value on the other side of the equation.
{\displaystyle 3x=24}
, you need to divide by 3 on the left side to isolate the variable, so you also need to divide by 3 on the right side of the equation:
{\displaystyle 3x=24}
{\displaystyle {\frac {3x}{3}}={\frac {24}{3}}}
{\displaystyle x=8}
Check your solution. To make sure your answer is correct, plug the value of
{\displaystyle x}
{\displaystyle x=8}
, substitute 8 for
{\displaystyle x}
{\displaystyle 3(8)=24}
Solve this equation with a fraction:
{\displaystyle {\frac {x}{4}}=8}
Since the variable is divided by 4, to isolate it you need to multiply by 4.
{\displaystyle {\frac {x}{4}}=8}
{\displaystyle 4({\frac {x}{4}})=(4)8}
{\displaystyle x=32}
Checking your work, since
{\displaystyle {\frac {32}{4}}=8}
, your solution is correct.
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Solve this equation with a negative constant:
{\displaystyle -16+x=29}
Since the constant is negative, adding it to both sides will isolate the variable.
{\displaystyle -16+x=29}
{\displaystyle -16+x+16=29+16}
{\displaystyle x=45}
{\displaystyle -16+45=29}
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Solve this equation with a negative coefficient:
{\displaystyle -5x=45}
Since the variable is multiplied by -5, to isolate the variable, you must divide each side by -5. Remember that dividing a positive number by a negative number equals a negative quotient.
{\displaystyle -5x=45}
{\displaystyle {\frac {-5x}{-5}}={\frac {45}{-5}}}
{\displaystyle x=-9}
{\displaystyle -5(-9)=45}
↑ https://www.mathsisfun.com/definitions/equation.html
↑ http://www.purplemath.com/modules/solvelin.htm
↑ https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq/alg-one-step-add-sub-equations/v/solving-one-step-equations
↑ http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_OneVariableOneStep.xml
↑ https://www.khanacademy.org/math/algebra/one-variable-linear-equations/alg1-solving-equations/v/why-we-do-the-same-thing-to-both-sides-simple-equations
Español:resolver ecuaciones de un paso
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GEXF - Maple Help
Home : Support : Online Help : Programming : Input and Output : File Formats : GEXF
GEXF (.gexf) Graph Format
GEXF (Graph Exchange XML Format) is an XML-based file format for storing a single undirected or directed graph.
Import a GEXF file encoding the Petersen graph.
\mathrm{Petersen}≔\mathrm{Import}\left("example/petersen.gexf",\mathrm{base}=\mathrm{datadir}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Petersen}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 10 vertices and 15 edge\left(s\right)}}
\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{Petersen}\right)
Export a Kneser graph to a GEXF file in the home directory of the current user.
\mathrm{KG}≔\mathrm{GraphTheory}:-\mathrm{SpecialGraphs}:-\mathrm{KneserGraph}\left(3,2\right)
\textcolor[rgb]{0,0,1}{\mathrm{KG}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 3 vertices and 0 edge\left(s\right)}}
\mathrm{Export}\left("kneser.gexf",\mathrm{KG},\mathrm{base}=\mathrm{homedir}\right)
\textcolor[rgb]{0,0,1}{524}
GEXF Format, www.gexf.net
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Estimation of Storm Peak and Intra-Storm Directional-Seasonal Design Conditions in the North Sea | OMAE | ASME Digital Collection
Shell Projects & Technology, Aberdeen, UK
Yanyun Wu,
Feld, G, Randell, D, Wu, Y, Ewans, K, & Jonathan, P. "Estimation of Storm Peak and Intra-Storm Directional-Seasonal Design Conditions in the North Sea." Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering. Volume 4A: Structures, Safety and Reliability. San Francisco, California, USA. June 8–13, 2014. V04AT02A014. ASME. https://doi.org/10.1115/OMAE2014-23157
Specification of realistic environmental design conditions for marine structures is of fundamental importance to their reliability over time. Design conditions for extreme waves and storm severities are typically estimated by extreme value analysis of time series of measured or hindcast significant wave height, HS. This analysis is complicated by two effects. Firstly, HS exhibits temporal dependence. Secondly, the characteristics of
HSsp
are non-stationary with respect to multiple covariates, particularly wave direction and season.
We develop directional-seasonal design values for storm peak significant wave height (
HSsp
) by estimation of, and simulation under a non-stationary extreme value model for
HSsp
. Design values for significant wave height (HS) are estimated by simulating storm trajectories of HS consistent with the simulated storm peak events. Design distributions for individual maximum wave height (Hmax) are estimated by marginalisation using the known conditional distribution for Hmax given HS. Particular attention is paid to the assessment of model bias and quantification of model parameter and design value uncertainty using bootstrap resampling. We also outline existing work on extension to estimation of maximum crest elevation and total extreme water level.
Design, North Sea, Storms, Significant wave heights, Waves, Marine structures, Reliability, Simulation, Time series, Uncertainty, Water
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0.1 Short gaps
Linear interpolation is hypothesized to work quite well in bridging short time gaps. This represents a reduction in the density of the points, but when the gaps do not exceed time intervals of 5-10 minutes, the total level of acceptable sparsity could be much higher before it impacts travel metrics of interest.
In the first study, missingness was introduced to the data at random, representing a situation in which the missingness was not functionally related to the content of the data or the user, and in which the gaps. Each period was divided into 288 five-minute time intervals. Sparsity was introduced at ten percent intervals, ranging from
q=0
q = 0
, where no data were removed, to
q=.9
q = .9
in which 90% of the five minute intervals were excluded. For each period, this process was repeated 20 times at each interval to allow for different portions of the data to be removed.
Analysis steps proceeded in this order:
For each 24 hour period
q
q
Sample without replacement the number of intervals to be removed
Replace gaps in the data longer than 3 minutes and larger than 80 meters by linear interpolation between the preceding and subsequent point at one second intervals.
Apply the stop detection algorithm
Mark as a candidate stop all points for which all points in the subsequent 180 seconds are within 80 meters.
Mark as being within a stop all points falling within the previous requirement
Determine state switches (move -> stop, stop -> move, stop -> stop)
Merge stops
If a stop adjacent to another stop is not more than 100 meters distant from the mean center of the previous stop, merge all locations within this stop to the previous stop
If two stops are separated by a move state whose summed distance between all points is less than 300 meters, merge into the previous stop.
Repeat until no further changes are made.
Calculate aggregate measures on both states
If there is at least one move state
Apply top down time ratio algorithm to divide move states into segments
Calculate aggregate measures on move states
Apply top down time ratio algorithm to divide data set into segments and calculate aggregate measures
Repeat step 2 20 times
The data with induced missingness were compared to the complete data set in order to evaluate which metrics were impacted. Table 1 shows the decrease in moved distance and number of stops with increasing sparsity. At 30% sparsity, the mean distance retained is almost 90%, and the median distance retained is 93%. Only as sparsity levels exceed 60% does the median distance lost reach 20%. Similarly, we see that linear interpolation doesn’t reduce the median number of stops until sparsity reaches 50%. In fact, these short gaps become problematic only when they become long gaps, as two or more adjacent short gaps merge into one.
A table of full results can be found in the appendix. Results from this simulation study suggest a fairly robust response from the data with increasing levels of
q
q
. See ??.
Table 1: Selected MCAR Short Gap Results (Median)
Delta Moved Distance (Km)
Delta Radius of Gyration
Delta Number of Stops
0.1 -0.4 (-1.2%) 1.2 (5%) 0 (0%)
0.2 -1.1 (-3.8%) 2.8 (11.3%) 0 (0%)
0.4 -2.6 (-10.2%) 7.1 (28.1%) 0 (0%)
0.5 -3.8 (-14.2%) 10 (40.2%) -1 (-9.1%)
0.6 -4.9 (-18.4%) 13.7 (55.3%) -1 (-16.7%)
0.7 -6.5 (-24%) 19.1 (77.5%) -1 (-20%)
0.8 -8.5 (-31.7%) 26.9 (111.4%) -2 (-30%)
0.9 -11.9 (-47.7%) 39.8 (173.1%) -2 (-43.7%)
We see relatively large increases in the calculated Radius of Gyration,
{r}_{g}\left(t\right)
r_{g}(t)
, both of the stops and of the movements. This metric indicates the mobility tendency of a person, and is presented in kilometers. The increase is due to … ?
0.2 Long gaps
A second simulation study was designed to investigate whether or not the same method of linear interpolation worked for gaps of increasing length. It was hypothesized that, in comparison to short gaps, longer gaps would exhibit a more linear relationship between sparsity and the travel metrics of interest. Rather than decreasing the location density uniformly, gaps of increasing length have a greater potential to eliminate entire trips, greatly distorting metrics such as travel distance and number of stops.
For each of 20 iterations
Select a random time point from the 24 hours
Remove 10 percent of the remaining data following from this time point
Return to Step 3A
Table 2: Selected MCAR Long Gap Results (Median)
0.1 0 (0%) 1.2 (2.5%) 0 (0%)
0.3 -2.3 (-8.7%) 3.4 (8.8%) 0 (0%)
0.4 -6.5 (-25.3%) 4.2 (14.4%) -1 (-25%)
0.5 -11 (-46%) 3.1 (14.6%) -2 (-33.3%)
0.6 -17 (-52.3%) 1.3 (9.2%) -2 (-50%)
0.7 -24.3 (-68.3%) -0.2 (-12.7%) -3 (-54.5%)
0.8 -31.9 (-97.3%) -7.8 (-65%) -4 (-66.7%)
0.9 -38 (-100%) -25.9 (-96.2%) -4 (-75%)
Radius of gyration and urbanicity
0.3 Finding a cutoff point
Short gaps that react favorably to linear interpolation must be distinguished from long gaps, which do not. A more in-depth look at the variation between gap lengths varying from one minute to fifteen minutes was conducted. For each complete data set, 15 iterations were conducted, in which gaps were created within each hour of varying length, after which the comparison of metrics against the full data was made.
0.4 Covariates allow varying cutoff points
Finally, we sought to establish a set of covariates that could impact this relationship. This could allow for extending the boundaries of what we consider the maximum acceptable gap time.
Of central importance was the investigation of time as a metric. Both Android and iOS operating systems employ mechanisms for reducing device activity during times of lesser activity levels, contributing to long gaps during nights that are unlikely to contain travel behavior.
0.4.2 Urbanicity
The urbanicity of a person’s home might also impact the cutoff point decision. Persons living in more rural areas are more likely to make longer trips, which may allow for a longer gap time before significant data losses are incurred.
Figure 1: Radius
The age of a participant may impact the tendency towards certain travel behaviors.
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Section 59.5 (03N7): Feats of the étale topology—The Stacks project
Section 59.5: Feats of the étale topology (cite)
59.5 Feats of the étale topology
For a natural number $n \in \mathbf{N} = \{ 1, 2, 3, 4, \ldots \} $ it is true that
\[ H_{\acute{e}tale}^2 (\mathbf{P}^1_\mathbf {C}, \mathbf{Z}/n\mathbf{Z}) = \mathbf{Z}/n\mathbf{Z}. \]
More generally, if $X$ is a complex variety, then its étale Betti numbers with coefficients in a finite field agree with the usual Betti numbers of $X(\mathbf{C})$, i.e.,
\[ \dim _{\mathbf{F}_ q} H_{\acute{e}tale}^{2i} (X, \mathbf{F}_ q) = \dim _{\mathbf{F}_ q} H_{Betti}^{2i} (X(\mathbf{C}), \mathbf{F}_ q). \]
This is extremely satisfactory. However, these equalities only hold for torsion coefficients, not in general. For integer coefficients, one has
\[ H_{\acute{e}tale}^2 (\mathbf{P}^1_\mathbf {C}, \mathbf{Z}) = 0. \]
By contrast $H_{Betti}^2(\mathbf{P}^1(\mathbf{C}), \mathbf{Z}) = \mathbf{Z}$ as the topological space $\mathbf{P}^1(\mathbf{C})$ is homeomorphic to a $2$-sphere. There are ways to get back to nontorsion coefficients from torsion ones by a limit procedure which we will come to shortly.
Comment #1700 by Yogesh More on November 23, 2015 at 11:52
Very minor remark: I think it would be helpful to add, after the sentence "For integer coefficients, one has
H^2_{etale}(P^1_C, \mathbb{Z})=0
", the following:
By contrast
H^2_{Betti}(P^1_C, \mathbb{Z})=\mathbb{Z}
P^1_C
S^2
Thanks for the suggestion. Added something here.
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Wikijunior:Introduction to Mathematics/Decimals - Wikibooks, open books for an open world
Wikijunior:Introduction to Mathematics/Decimals
2.1 Why use Decimals?
2.3 Almost Equal
2.4 Which is the most?
Almost-equal sign
Why use Decimals?Edit
Some fractions have big numbers and are difficult to use. For example, which of the next fractions is the most?
{\displaystyle {\frac {44467}{38973}}}
{\displaystyle {\frac {82489}{71035}}}
{\displaystyle {\frac {8993}{7873}}}
Show the fractions as mixed fractions and it might be easier to answer which is most.
{\displaystyle {\frac {44467}{38973}}}
{\displaystyle 1{\frac {5494}{38973}}}
{\displaystyle {\frac {82489}{71035}}}
{\displaystyle 1{\frac {11454}{71035}}}
{\displaystyle {\frac {8993}{7873}}}
{\displaystyle 1{\frac {1120}{7873}}}
It is still difficult to answer which is most. A different way to show fractions makes answering easier.
Decimals Numbers have two parts, the same way that mixed numbers have two parts. The two parts are separated by a small circle called a decimal point. The part of the decimal number to the right of the decimal point is the fraction part. The part to the left is the whole part. The way you find the decimal form of a number is by dividing by hand or using a calculator. These are examples of fractions in decimal form:
{\displaystyle {\frac {3}{2}}=1.5}
{\displaystyle {\frac {1}{5}}=0.2}
{\displaystyle {\frac {1}{10}}=0.1}
{\displaystyle {\frac {51}{25}}=2.04}
{\displaystyle {\frac {1}{40}}=0.025}
{\displaystyle {\frac {1}{100}}=0.01}
Just like place amounts made it easy to show big normal numbers, decimal places make it easy to show fractions with big numbers. Note: one tenth =
{\displaystyle {\frac {1}{10}}}
, one hundredth =
{\displaystyle {\frac {1}{100}}}
{\displaystyle .}
{\displaystyle 62897}
{\displaystyle .4}
{\displaystyle 2897}
{\displaystyle .46}
{\displaystyle 897}
{\displaystyle .462}
{\displaystyle 97}
8 ten thousandths
{\displaystyle .4628}
{\displaystyle 7}
9 hundred thousandths
{\displaystyle .46289}
7 7 millionths
Almost EqualEdit
Sometimes when you change a fraction to a decimal the number will never end. For example,
{\displaystyle {\frac {1}{3}}}
becomes 0.33333333333333333333333… and the threes repeat forever. Because you can not write a number that goes on forever, you use only as much as you need for answering your math question.
When you remove a part of a number it is no longer the same number, but because the new number is almost the same as the old number, you can still use it to answer questions about the old number. The almost equal sign (≈) is a special sign you use for when two numbers are almost the same. For example:
{\displaystyle {\frac {1}{3}}\approx 0.333333}
Which is the most?Edit
Using decimal numbers it is now easy to find which fraction is the most.
{\displaystyle {\frac {44467}{38973}}\approx 1.140969389064}
{\displaystyle {\frac {82489}{71035}}\approx 1.161244456958}
{\displaystyle {\frac {8993}{7873}}\approx 1.142258351327}
Because there is a 6 in the hundredths place you know the second fraction is most.
Dividing FractionsEdit
Retrieved from "https://en.wikibooks.org/w/index.php?title=Wikijunior:Introduction_to_Mathematics/Decimals&oldid=3070995"
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Relativistic Newtonian Gravitation That Gives the Correct Prediction of Mercury Precession
Abstract: In the past, there was an attempt to modify Newton’s gravitational theory, in a simple way, to consider relativistic effects. The approach was “abandoned” mainly because it predicted only half of Mercury’s precession. Here we will revisit this method and see how a small logical extension can lead to a relativistic Newtonian theory that predicts the perihelion precession of Mercury correctly.
Keywords: Newton Gravity, Relativistic Adjustment, Precession of Mercury
In 1981 and 1986, Bagge [1] and Phillips [2] each suggested an ad-hoc modification of Newton by simply replacing the smaller mass in the formula with a relativistic mass
F=G\frac{M\frac{m}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}}{{r}^{2}}
The velocity v is the relative velocity between the two gravitational objects: the velocity of Mercury relative to the Sun, for example. Phillips initially claimed that his derivation, based on this, led to a prediction of the perihelion precession of Mercury equal to that of Einstein’s general relativity theory [3]. However, according to criticism from Ghosal in 1987, this approach leads to a perihelion precession of Mercury that is too low. The method has also been criticized by Chow [4] for the same reason. Peters [5] claims that Philipps made a mistake in his Mercury perihelion derivation and that, in reality, his prediction only gives half of the prediction as GR (the GR prediction has been observed). Philipps openly admitted this and discussed his mistakes in detail [6]. He was clear that his theory underestimated the perihelion precession of Mercury, but noted that further adjustments to the theory could potentially be done in the future. Biswas [7] published an interesting paper titled “Special Relativistic Newtonian Gravity” where he claimed:
The resulting theory is significantly different from the general theory of relativity. However, all known experimental results (precession of planetary orbits, bending of the path of light near the Sun, and gravitational spectral shift) are still explained by this theory.
However, Peters [8] then pointed out that Biswas had also made a mistake in his derivation, something Biswas agreed to in correspondence with Peters. Ghosal and Cakraborty [9] agree on the criticism of Biswas, but claim his idea was still interesting. Here we will follow up on this discussion and show that there is a simple and logical way to extend this approach in a fruitful manner.
2. Modified Relativistic Newtonian Gravity That Gives the Correct Prediction of the Precession of Mercury
In the relativistic extension of Newton given by [1] [2],
F=G\frac{M\frac{m}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}}{{r}^{2}}
The velocity v must be interpreted as the velocity between the large and small masses. This extension is, in our view, only valid when the gravity phenomenon is observed from the frame of the large gravitational object, such as predicting the orbital velocity of the moon relative to the Earth, for example. In this case under consideration, however, the small relativistic mass will fall out and we get the same predictions as in standard Newtonian gravity. When it comes to gravity phenomena between two masses as observed from a third frame, we claim it is logical to complete additional ad hoc modifications to the formula above. When observing the Sun’s gravitational influence on Mercury, for example, we must also consider the Sun’s velocity relative to us as we observe it from Earth. We suggest the following modification
F=G\frac{\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}\frac{M}{\sqrt{1-\frac{{v}_{M}^{2}}{{c}^{2}}}}}{{r}^{2}\left(1-\frac{{v}_{M}^{2}}{{c}^{2}}\right)}
{v}_{m}
{v}_{M}
are the velocities of the large and small masses as observed from the observer frame, that is to say, in our case, from Earth. As can be seen in our formula, we are suggesting that r (center to center between the two gravitational masses) should be the length contracted depending on the velocity of the two objects relative to the observer; this is best approximated by the velocity of the large gravitational object relative to the observer frame. For example, assume a galaxy with distance r between the galactic center and one of the stars in the arm of the galaxy, as observed from the galactic center. We claim that this distance likely will appear to be contracted, as observed from Earth and as measured with Einstein-Poincaré synchronized clocks. Its contracted length will follow standard Lorentz length contraction, in our formulation, and will be
r\sqrt{1-\frac{{v}_{M}^{2}}{{c}^{2}}}
. That is to say, for fast-moving galaxies we have two effects that lead to stronger gravity than predicted by the Newtonian theory. The first effect is that the relativistic mass is relevant for gravity (and this mass is larger than the rest-mass), and the second effect is that the distance center to center between the gravity objects must appear to be contracted, as observed from the laboratory (typically the Earth).
In 1859, LeVerrier pointed out that the perihelion of Mercury evidently precesses at a slightly faster rate than predicted by Newtonian mechanics. The Lagrangian is given by
L=T-V
L=\frac{m{c}^{2}}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}+G\frac{M\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}}{r\left(1-\frac{{v}_{M}^{2}}{{c}^{2}}\right)}
{v}_{M}\ll c
, we can use a Taylor series expansion and get
L=\frac{m{c}^{2}}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}+G\frac{M\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}}{r}+\frac{{v}_{M}^{2}}{{c}^{2}}G\frac{M\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}}{r}
L=\frac{m{c}^{2}}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}+\frac{GMm}{r\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}+\frac{{v}_{M}^{2}}{{c}^{2}}\frac{GMm}{r\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}
And to simplify further, we can set
k=GMm
L=\frac{m{c}^{2}}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}+\frac{k}{r\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}+\frac{{v}_{M}^{2}}{{c}^{2}}\frac{k}{r\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}
Next assume that
{v}_{m}\ll c
{v}_{m}\approx {v}_{M}
, we can then use a Taylor series expansion and we get
L=m{c}^{2}+\frac{1}{2}m{v}_{M}^{2}+\frac{k}{r}+\frac{3}{2}\frac{{v}_{M}^{2}}{{c}^{2}}\frac{k}{r}+O\left({c}^{-4}\right)
given extensive calculations, this seems to lead to the same prediction as GR for Mercury precession, that is
\delta =\frac{6\pi m}{{c}^{2}a\left(1-{e}^{2}\right)}
Einstein’s [10] equivalence principle basically states that inertial mass and gravitational mass are the same thing. It is considered a well-tested concept, at least inside a wide range of observational values. In the case of two reference frames, we have
F=\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}a=G\frac{M\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}}{{r}^{2}}
F=ma=G\frac{Mm}{{r}^{2}}
a=G\frac{M}{{r}^{2}}
So, we see it gives exactly the same result as standard theory in this case. A discussion of whether or not this is only valid for a weak gravitational field is outside the scope of this paper. All direct measurements of the equivalence principle have, to our knowledge, been done in a two frame observational setting, so our theory predicts the same here as standard theory. In the case of three reference frames, we have
F=\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}a=G\frac{M\frac{m}{\sqrt{1-\frac{{v}_{m}^{2}}{{c}^{2}}}}}{{r}^{2}{\left(1-\frac{{v}_{M}^{2}}{{c}^{2}}\right)}^{3/2}}
a=G\frac{M}{{r}^{2}{\left(1-\frac{{v}_{M}^{2}}{{c}^{2}}\right)}^{3/2}}
This will indeed give a different predicted acceleration of an object m relative to object M, as observed from a third frame. Such experiments have not been done directly to our knowledge, but possibly indirectly through cosmological observations. Therefore, this should be of interest for possible alternative interpretations of cosmological observations. This would also be significantly different from standard theory when
{v}_{M}
has a significant velocity relative to Earth. Intuitively, this should mean the red-shift may be likely higher than expected when excluding the hypothesis of expanding space. All in all, we think our theory therefore should be highly relevant for further studies. Our model also indicates that galaxy arms should rotate somewhat faster than predicted from standard theory by taking only baryonic matter into account. However, we do not claim that this is enough alone to account for the missing dark matter. This is however a discussion outside the scope of this article. The focus here is that Newton relativistic modifications may have been rejected too early in relation to predictions of Mercury’s precession. Still, we think our theory does not conflict with observations that have been completed in relation to the equivalence principle. Naturally, we are open to further discussions on this. After all, physics can only progress by exploring, testing, and scrutinizing any new ideas carefully.
In the past, several ad hoc modifications of Newton’s gravity theory have been proposed and discussed. These approaches have been criticized for predicting only half of the perihelion of the precession of Mercury. Taking that work as a start, however, we have suggested some logical extensions to this theory. If we are looking at relativistic effects, they should be evaluated from the observer frame. In this case, when the gravity phenomenon is not observed from the large gravity mass itself, but rather from an outside frame such as the Earth, then we also must take into account the velocity of the Sun relative to the Earth. After completing an ad hoc adjustment accordingly, we find the same prediction of precession of Mercury as general relativity theory predicts. Although we have not tested these results further, we think this is interesting enough to require further investigation, and hope this paper will highlight the way for future research.
Cite this paper: Haug, E. (2020) Relativistic Newtonian Gravitation That Gives the Correct Prediction of Mercury Precession. Journal of High Energy Physics, Gravitation and Cosmology, 6, 238-243. doi: 10.4236/jhepgc.2020.62017.
[1] Bagge, E.R. (1981) Relativistic Effects in the Solar System. Atomkernenergie Kerntechnik, 39, 223-228.
[2] Phipps, T.E. (1986) Mercury’s Precession According to Special Relativity. American Journal of Physics, 54, 245-247.
[4] Chow, T. (1992) On Relativistic Newtonian Gravity. European Journal of Physics, 13, 198.
[5] Peters, P.C. (1986) Comment on “Mercury’s Precession According to Special Relativity”. American Journal of Physics, 55, 757.
[6] Phipps, T.E. (1986) Response to “Comment on ‘Mercury’s Precession According to Special Relativity”. American Journal of Physics, 55, 758-759.
[7] Biswas, T. (1988) Minimally Relativistic Newtonian Gravity. American Journal of Physics, 56, 1032.
[8] Peters, P.C. (1990) Comment on “Minimally Relativistic Newtonian Gravity”. American Journal of Physics, 56, 188.
[9] Ghosal, S.K. and Chakraborty, P. (1991) Relativistic Newtonian Gravity: An Improved Version. European Journal of Physics, 12, No. 6.
[10] Einstein, A. (1922) The Meaning of Relativity. Princeton University Press, Princeton, NJ.
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Exponentiation - Balancer
The main formulas used in Balancer protocol make use of a form of exponentiation where both the base and exponent are fixed-point (non-integer) values. Take for example the swap functions, where the weights in both the exponent and the base are fractions:
A_o = \left(1 - \left(\frac{B_i}{B_i+A_i}\right)^{\frac{W_i}{W_o}}\right).B_o
\begin{equation} \begin{gathered} A_i = \left(\left(\frac{B_o}{B_o-A_o}\right)^{\frac{W_o}{W_i}}-1\right).B_i \end{gathered} \end{equation}
Since solidity does not have fixed point algebra or more complex functions like fractional power we use the following binomial approximation:
\begin{equation} \begin{gathered} \left(1+x\right)^\alpha=1+\alpha x+\frac{(\alpha)(\alpha-1)}{2!}x^2+ \frac{(\alpha)(\alpha-1)(\alpha-2)}{3!}x^3+ \cdots = \sum_{k=0}^{\infty}{\alpha \choose k}x^k \end{gathered} \end{equation}
{|x| < 1}
\alpha>1
we split the calculation into two parts for increased accuracy, the first is the exponential with the integer part of
\alpha
(which we can calculate exactly) and the second is the exponential with the fractional part of
\alpha
\begin{equation} \begin{gathered} A_i = \left(1 - \left(\frac{B_o}{B_o-A_o}\right)^{int\left(\frac{W_o}{W_i}\right)}\left(\frac{B_o}{B_o-A_o}\right)^{\frac{W_o}{W_i}\%1}\right).B_i \end{gathered} \end{equation}
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Two-axle vehicle with longitudinal dynamics and motion and adjustable mass, geometry, and drag properties - MATLAB - MathWorks 日本
The vehicle axles are parallel and form a plane. The longitudinal, x, direction lies in this plane and perpendicular to the axles. If the vehicle is traveling on an incline slope, β, the normal, z, direction is not parallel to gravity but is always perpendicular to the axle-longitudinal plane.
β Incline angle
Ï Mass density of air
m{\stackrel{Ë}{V}}_{x}={F}_{x}â\text{â}{F}_{d}âmgâ
\mathrm{sin}\mathrm{β}
{F}_{x}=n\left({F}_{xf}+{F}_{xr}\right)
{F}_{d}=\frac{1}{2}{C}_{d}\mathrm{Ï}A{\left({V}_{x}+{V}_{w}\right)}^{2}â
\mathrm{sgn}\left({V}_{x}+{V}_{w}\right)
{F}_{zf}=\frac{âh\left({F}_{d}+mg\mathrm{sin}\mathrm{β}+m{\stackrel{Ë}{V}}_{x}\right)+bâ
mg\mathrm{cos}\mathrm{β}}{n\left(a+b\right)}
{F}_{zr}=\frac{+h\left({F}_{d}+mg\mathrm{sin}\mathrm{β}+m{\stackrel{Ë}{V}}_{x}\right)+aâ
mg\mathrm{cos}\mathrm{β}}{n\left(a+b\right)}
{F}_{zf}+{F}_{zr}=mg\frac{\mathrm{cos}\mathrm{β}}{n}
\mathrm{α}=\frac{\left(fâ
h\right)+\left({F}_{zf}a\right)â\left({F}_{zr}b\right)}{J}
É‘ is the pitch acceleration.
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Natural logarithm of quaternion - MATLAB quatlog - MathWorks Switzerland
quatlog
Calculate the Natural Logarithm of Quaternion
Natural logarithm of quaternion
ql=quatlog(q)
ql=quatlog(q) calculates the natural logarithm, ql, for a normalized quaternion, q.
This function uses the relationships.
q=\left[\mathrm{cos}\left(\theta \right),\mathrm{sin}\left(\theta \right)v\right],
\mathrm{log}\left(q\right)=\left[0,\theta v\right].
Calculate the natural logarithm of quaternion matrix q=[1.0 0 1.0 0].
qlog = quatlog(quatnormalize([1.0 0 1.0 0]))
qlog =
Quaternions for which to calculate the natural logarithm, specified as an M-by-4 matrix containing M quaternions. This quaternion must be a normalized quaternion.
ql — Natural logarithm of quaternion
Natural logarithm of quaternion.
quatinterp | quatexp | quatpower | quatconj | quatdivide | quatinv | quatmod | quatmultiply | quatnormalize | quatrotate
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Starting kit | The Higgs Machine Learning Challenge
This python script trains a simple naive Bayes classifier on training.csv, classifies the events in test.csv, and creates a file called submission.csv that you can submit at the Kaggle submission site. You can execute this standalone python script (in the directory where you copied training.csv and test.csv), you can execute this iPython notebook, or you can read the same notebook below (or here in a nicer format).
Starting kit for the Higgs boson machine learning challenge¶
This notebook contains a starting kit for the Higgs boson machine learning challenge. Download the training set (called training.csv) and the test set (test.csv), then execute cells in order.
import random,string,math,csv
Reading an formatting training data¶
all = list(csv.reader(open("training.csv","rb"), delimiter=','))
Slicing off header row and id, weight, and label columns.
xs = np.array([map(float, row[1:-2]) for row in all[1:]])
(numPoints,numFeatures) = xs.shape
Perturbing features to avoid ties. It's far from optimal but makes life easier in this simple example.
xs = np.add(xs, np.random.normal(0.0, 0.0001, xs.shape))
Label selectors.
sSelector = np.array([row[-1] == 's' for row in all[1:]])
bSelector = np.array([row[-1] == 'b' for row in all[1:]])
Weights and weight sums.
weights = np.array([float(row[-2]) for row in all[1:]])
sumWeights = np.sum(weights)
sumSWeights = np.sum(weights[sSelector])
sumBWeights = np.sum(weights[bSelector])
Training and validation cuts¶
We will train a classifier on a random training set for minimizing the weighted error with balanced weights, then we will maximize the AMS on the held out validation set.
randomPermutation = random.sample(range(len(xs)), len(xs))
numPointsTrain = int(numPoints*0.9)
numPointsValidation = numPoints - numPointsTrain
xsTrain = xs[randomPermutation[:numPointsTrain]]
xsValidation = xs[randomPermutation[numPointsTrain:]]
sSelectorTrain = sSelector[randomPermutation[:numPointsTrain]]
bSelectorTrain = bSelector[randomPermutation[:numPointsTrain]]
sSelectorValidation = sSelector[randomPermutation[numPointsTrain:]]
bSelectorValidation = bSelector[randomPermutation[numPointsTrain:]]
weightsTrain = weights[randomPermutation[:numPointsTrain]]
weightsValidation = weights[randomPermutation[numPointsTrain:]]
sumWeightsTrain = np.sum(weightsTrain)
sumSWeightsTrain = np.sum(weightsTrain[sSelectorTrain])
sumBWeightsTrain = np.sum(weightsTrain[bSelectorTrain])
xsTrainTranspose = xsTrain.transpose()
Making signal and background weights sum to
1/2
each to emulate uniform priors
p(s)=p(b)=1/2
weightsBalancedTrain = np.array([0.5 * weightsTrain[i]/sumSWeightsTrain
if sSelectorTrain[i]
else 0.5 * weightsTrain[i]/sumBWeightsTrain\
for i in range(numPointsTrain)])
Training naive Bayes and defining the score function¶
Number of bins per dimension for binned naive Bayes.
logPs[fI,bI] will be the log probability of a data point x with binMaxs[bI - 1] < x[fI] <= binMaxs[bI] (with binMaxs[-1] = -
\infty
by convention) being a signal under uniform priors
p(\text{s}) = p(\text{b}) = 1/2
logPs = np.empty([numFeatures, numBins])
binMaxs = np.empty([numFeatures, numBins])
binIndexes = np.array(range(0, numPointsTrain+1, numPointsTrain/numBins))
for fI in range(numFeatures):
# index permutation of sorted feature column
indexes = xsTrainTranspose[fI].argsort()
for bI in range(numBins):
# upper bin limits
binMaxs[fI, bI] = xsTrainTranspose[fI, indexes[binIndexes[bI+1]-1]]
# training indices of points in a bin
indexesInBin = indexes[binIndexes[bI]:binIndexes[bI+1]]
# sum of signal weights in bin
wS = np.sum(weightsBalancedTrain[indexesInBin]
[sSelectorTrain[indexesInBin]])
# sum of background weights in bin
wB = np.sum(weightsBalancedTrain[indexesInBin]
[bSelectorTrain[indexesInBin]])
# log probability of being a signal in the bin
logPs[fI, bI] = math.log(wS/(wS+wB))
The score function we will use to sort the test examples. For readability it is shifted so negative means likely background (under uniform prior) and positive means likely signal. x is an input vector.
# linear search for the bin index of the fIth feature
# of the signal
while bI < len(binMaxs[fI]) - 1 and x[fI] > binMaxs[fI, bI]:
logP += logPs[fI, bI] - math.log(0.5)
Optimizing the AMS on the held out validation set¶
The Approximate Median Significances and b are the sum of signal and background weights, respectively, in the selection region.
def AMS(s,b):
assert b >= 0
bReg = 10.
return math.sqrt(2 * ((s + b + bReg) *
math.log(1 + s / (b + bReg)) - s))
Computing the scores on the validation set
validationScores = np.array([score(x) for x in xsValidation])
Sorting the indices in increasing order of the scores.
tIIs = validationScores.argsort()
Weights have to be normalized to the same sum as in the full set.
wFactor = 1.* numPoints / numPointsValidation
s
b
to the full sum of weights, we start by having all points in the selectiom region.
s = np.sum(weightsValidation[sSelectorValidation])
b = np.sum(weightsValidation[bSelectorValidation])
amss will contain AMSs after each point moved out of the selection region in the sorted validation set.
amss = np.empty([len(tIIs)])
amsMax will contain the best validation AMS, and threshold will be the smallest score among the selected points.
amsMax = 0
We will do len(tIIs) iterations, which means that amss[-1] is the AMS when only the point with the highest score is selected.
for tI in range(len(tIIs)):
# don't forget to renormalize the weights to the same sum
# as in the complete training set
amss[tI] = AMS(max(0,s * wFactor),max(0,b * wFactor))
if amss[tI] > amsMax:
amsMax = amss[tI]
threshold = validationScores[tIIs[tI]]
#print tI,threshold
if sSelectorValidation[tIIs[tI]]:
s -= weightsValidation[tIIs[tI]]
b -= weightsValidation[tIIs[tI]]
amsMax
plt.plot(amss)
Computing the permutation on the test set¶
Reading the test file, slicing off the header row and the id column, and converting the data into float.
test = list(csv.reader(open("test.csv", "rb"),delimiter=','))
xsTest = np.array([map(float, row[1:]) for row in test[1:]])
testIds = np.array([int(row[0]) for row in test[1:]])
testScores = np.array([score(x) for x in xsTest])
Computing the rank order.
testInversePermutation = testScores.argsort()
testPermutation = list(testInversePermutation)
for tI,tII in zip(range(len(testInversePermutation)),
testInversePermutation):
testPermutation[tII] = tI
Computing the submission file with columns EventId, RankOrder, and Class.
submission = np.array([[str(testIds[tI]),str(testPermutation[tI]+1),
's' if testScores[tI] >= threshold else 'b']
for tI in range(len(testIds))])
submission = np.append([['EventId','RankOrder','Class']],
submission, axis=0)
Saving the file that can be submitted to Kaggle.
np.savetxt("submission.csv",submission,fmt='%s',delimiter=',')
|
Volume of a parallelepiped formula
How do I calculate the volume of a parallelepiped?
How do I calculate the volume of a parallelepiped from its sides?
How do I calculate the surface area of a parallelepiped?
How to use this volume of a parallelepiped calculator – And parallelepiped area calculator
This volume of a parallelepiped calculator will help you calculate the volume of a parallelepiped from its three vectors, four vertices, or edge lengths. Additionally, it will also calculate the area of the parallelepiped. Are you wondering how to find the volume of a parallelepiped formed by three vectors? Do you want to learn the formula for the volume of a parallelepiped with four vertices? Read on to find out the answers to all of these questions and more.
A parallelepiped is a polyhedron whose six faces are parallelograms. To describe a parallelepiped, we need its three adjacent sides and their angles, or the three adjacent vectors.
Three co-initial vectors in space describe a parallelepiped.
The formula for the volume of a parallelepiped is given by:
V = \lvert (\vec{a}\times\vec{b})\cdot\vec{c}\, \rvert
V
– Volume of the parallelepiped formed by the three vectors; and
\vec{a}
\vec{b}
\vec{c}
– Three vectors that describe the three adjacent (and unique) sides of a parallelepiped.
The vector multiplication used is called a scalar triple product (or simply, a triple product). It involves the cross product of the vectors
\vec{a}
\vec{b}
, which results in a vector
\vec{a}\times\vec{b}
perpendicular to both
\vec{a}
\vec{b}
. Note that the magnitude of the resultant vector,
\lvert\vec{a} \times \vec{b}\rvert
, is equal to the area of a parallelogram described by these two vectors.
The subsequent dot product between
\vec{a}\times\vec{b}
\vec{c}
denotes the projection of
\vec{a}\times\vec{b}
\vec{c}
. In other words, it sweeps the base parallelogram along
\vec{c}
, analogous to multiplying the base area with height.
We can further simplify the formula and reduce it to one determinant:
\scriptsize\begin{align*} V &= \lvert (\vec{a}\times\vec{b})\cdot\vec{c}\, \rvert\\\\ &= \begin{vmatrix} \bm{i} & \bm{j} & \bm{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix}\cdot (c_1\bm{i} + c_2\bm{j} + c_3\bm{k})\\\\ \implies V & = \begin{vmatrix} c_1 & c_2 & c_3\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix} \end{align*}
a_1
a_2
a_3
– Components of
\vec{\bm{a}}
b_1
b_2
b_3
\vec{\bm{b}}
c_1
c_2
c_3
\vec{\bm{c}}
\bm{i}
\bm{j}
\bm{k}
– Unit vectors along the coordinate axes.
Now that you know how to find the volume of a parallelepiped with vectors, let's learn what to do if the only given values are the vertices of the parallelepiped.
Once we know the vertices of the adjacent sides, we can determine the vectors.
You can find any vector between two points so long as you know the coordinates of these points. Once you determine the vectors, you can calculate the parallelepiped volume using the formula above.
To calculate the volume of a parallelepiped formed by the vectors a, b, and c, follow these simple steps:
Find the cross-product between the vectors a and b to get a × b.
Calculate the dot-product between the vectors a × b and c to get the scalar value (a × b) ∙ c.
Determine the volume of the parallelepiped as the absolute value of this scalar, given by ∣(a × b) ∙ c∣.
For example, consider a parallelepiped formed by the vectors
\vec{\bm{a}} = \bm{i} + 2\bm{j} + 3\bm{k}
\vec{\bm{b}} = 5\bm{i} - 4\bm{j} + 7\bm{k}
\vec{\bm{c}} = -5\bm{i} + \bm{j} + 12\bm{k}
. Then, the volume of the parallelepiped described by these vectors would be
\begin{align*} V &= \lvert (\vec{a}\times\vec{b})\cdot\vec{c}\, \rvert\\\\ &= \begin{vmatrix} -5 & 1 & 12\\ 1 & 2 & 3\\ 5 & -4 & 7 \end{vmatrix}\\\\ V & = 290 \end{align*}
To calculate the volume of a parallelepiped from its sides (or edge lengths), use the formula V = a∙b∙c∙√(1 + 2∙cos(α)∙cos(β)∙cos(γ) - cos²(α) - cos²(β) - cos²(γ)), where:
V – Volume of the parallelepiped;
a, b, and c – Three adjacent sides of the parallelepiped;
α – Angle between the sides b and c;
β – Angle between the sides a and c; and
γ – Angle between the sides a and b.
A parallelepiped with edge lengths and angles.
For example, consider a parallelepiped ABCDEFGH with edge lengths
a = 5
b = 4
c = 7
\angle \text{DAE} = 45\degree
\angle \text{BAD} = 63\degree
\angle \text{BAE} = 50\degree
, then the volume of the parallelepiped would be V = 5·4·7 √(1 + 2·cos(45)·cos(50)·cos(63) - cos²(45) - cos²(50) - cos²(63) = 75.83.
To calculate the surface area of a parallelepiped formed by the vectors a, b, and c, use the formula A = 2 × (∣a × b∣ + ∣b × c∣ + ∣a × c∣), where:
∣a × b∣ – Magnitude of the cross-product between a and b;
∣b × c∣ – Magnitude of the cross-product between b and c; and
∣a × c∣ – Magnitude of the cross-product between a and c.
Alternatively, you can find the surface area from the edge lengths a, b, and c , using the formula A = 2 × (a∙b∙sin(γ) + b∙c∙sin(α) + a∙c∙sin(β)), where:
α – Angle between b and c;
β – Angle between a and c; and
γ – Angle between a and b.
Note that the magnitude of the cross-product between two vectors
\vec{a}
\vec{b}
\lvert \vec{a} \times \vec{b} \rvert
is equal to the area of the parallelogram spanned by these vectors. Hence, adding the magnitudes of the cross-products of the three vectors describing the parallelepiped and multiplying the same by two shall produce its surface area.
a = 5
b = 4
c = 7
\angle \text{DAE} = 45\degree
\angle \text{BAD} = 63\degree
\angle \text{BAE} = 50\degree
, then the surface area would be given by
\scriptsize \begin{align*} A =&\ 2 \times (ab\sin(α) + bc\sin(β) + ac\sin(γ))\\ =&\ 2 \times (5\cdot 4 \sin(63\degree) + 4\cdot7\sin(45\degree) \\ &+ 5\cdot 7 \sin(50 \degree))\\ =&\ 132.6 \end{align*}
This volume of a parallelepiped calculator is a simple tool and easy to use. It has three different modes of calculation to find the volume of a parallelepiped with 3 vectors, 4 vertices, or using the edge lengths and angles:
To calculate the volume of a parallelepiped given 3 vectors:
Select the option vectors
a
b
c
in the Calculate using field.
Enter the values of the components of each vector.
This volume of a parallelepiped calculator will display the calculated volume and surface area of the parallelepiped in the corresponding fields under the Results section.
To calculate the volume of a parallelepiped with 4 vertices:
Select the option vertices p, q, r, and s in the Calculate using field.
Enter the coordinates of each vertex in the corresponding field.
The calculator will display the calculated volume and surface area of the parallelepiped in the corresponding fields under the Results section.
To calculate the volume of a parallelepiped using the edge lengths:
Select the option edge lengths and angles in the Calculate using field.
Enter the edge lengths and angles in their corresponding fields.
⚠️ If your input is not being accepted in any calculation mode, it is because you have entered values in another calculation mode that makes values in the current one impossible to process. To solve this, click the Reload button at the bottom left corner, and enter values again by selecting the desired calculation mode.
How do I determine whether three vectors are coplanar or collinear?
If the volume of a parallelepiped described by the vectors a, b, and c is equal to zero, then the vectors are coplanar. In other words, the vectors a, b, and c are lying on the same plane if ∣(a × b) ∙ c∣ = 0.
If the surface area of a parallelepiped formed by the vectors a, b, and c is equal to zero, then the vectors are collinear. In other words, the vectors a, b, and c are collinear if 2 × (∣a × b∣ + ∣b × c∣ + ∣a × c∣) = 0.
How many parallel faces are in a parallelepiped?
There are three pairs of parallel faces in a parallelepiped. Try saying this sentence fast: Three pairs of parallel parallelograms in a parallelepiped!
Calculate using...
vectors 𝗮, 𝗯, and 𝗰.
Components of vector 𝗮 (a₁𝗶 + a₂𝗷 + a₃𝗸)
Components of vector 𝗯 (b₁𝗶 + b₂𝗷 + b₃𝗸)
Components of vector 𝗰 (c₁𝗶 + c₂𝗷 + c₃𝗸)
Surface area of the parallelepiped
Inequality to interval notation
Use our inequality to interval notation calculator whenever you need to convert between inequalities and intervals. It is a two-way converter!
|
Overview - How training is performed | CatBoost
The goal of training is to select the model
y
, depending on a set of features
x_{i}
, that best solves the given problem (regression, classification, or multiclassification) for any input object. This model is found by using a training dataset, which is a set of objects with known features and label values. Accuracy is checked on the validation dataset, which has data in the same format as in the training dataset, but it is only used for evaluating the quality of training (it is not used for training).
CatBoost is based on gradient boosted decision trees. During training, a set of decision trees is built consecutively. Each successive tree is built with reduced loss compared to the previous trees.
The number of trees is controlled by the starting parameters. To prevent overfitting, use the overfitting detector. When it is triggered, trees stop being built.
Building stages for a single tree:
Preliminary calculation of splits.
(Optional) Transforming categorical features to numerical features.
(Optional) Transforming text features to numerical features.
Choosing the tree structure. This stage is affected by the set Bootstrap options.
Calculating values in leaves.
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Percentiles of data set - MATLAB prctile - MathWorks América Latina
Percentiles of Data Vector
Percentiles of All Values
Percentiles of Data Matrix
Percentiles of Multidimensional Array
Percentiles of Tall Vector for Given Percentage
Percentiles of Tall Matrix Along Different Dimensions
Percentiles of data set
P = prctile(A,p)
P = prctile(A,p,"all")
P = prctile(A,p,dim)
P = prctile(A,p,vecdim)
P = prctile(___,"Method",method)
P = prctile(A,p) returns percentiles of elements in input data A for the percentages p in the interval [0,100].
If A is a vector, then P is a scalar or a vector with the same length as p. P(i) contains the p(i) percentile.
If A is a matrix, then P is a row vector or a matrix, where the number of rows of P is equal to length(p). The ith row of P contains the p(i) percentiles of each column of A.
If A is a multidimensional array, then P contains the percentiles computed along the first array dimension of size greater than 1.
P = prctile(A,p,"all") returns percentiles of all the elements in x.
P = prctile(A,p,dim) operates along the dimension dim. For example, if A is a matrix, then prctile(A,p,2) operates on the elements in each row.
P = prctile(A,p,vecdim) operates along the dimensions specified in the vector vecdim. For example, if A is a matrix, then prctile(A,p,[1 2]) operates on all the elements of A because every element of a matrix is contained in the array slice defined by dimensions 1 and 2.
P = prctile(___,"Method",method) returns either exact or approximate percentiles based on the value of method, using any of the input argument combinations in the previous syntaxes.
Calculate the percentile of a data set for a given percentage.
Calculate the 42nd percentile of the elements of A.
P = prctile(A,42)
P = -0.1026
Find the percentiles of all the values in an array.
Find the 40th and 60th percentiles of all the elements of A.
P = prctile(A,[40 60],"all")
P(1) is the 40th percentile of A, and P(2) is the 60th percentile of A.
Calculate the percentiles along the columns and rows of a data matrix for specified percentages.
Calculate the 25th, 50th, and 75th percentiles for each column of A.
P = prctile(A,[25 50 75],1)
Each column of matrix P contains the three percentiles for the corresponding column in matrix A. 7, 12, and 17 are the 25th, 50th, and 75th percentiles of the third column of A with elements 4, 8, 12, 16, and 20. P = prctile(A,[25 50 75]) returns the same result.
Calculate the 25th, 50th, and 75th percentiles along the rows of A.
Each row of matrix P contains the three percentiles for the corresponding row in matrix A. 2.75, 4, and 5.25 are the 25th, 50th, and 75th percentiles of the first row of A with elements 2, 3, 4, 5, and 6.
Find the percentiles of a multidimensional array along multiple dimensions.
Calculate the 40th and 60th percentiles for each page of A by specifying dimensions 1 and 2 as the operating dimensions.
Ppage = prctile(A,[40 60],[1 2])
Ppage =
Ppage(:,:,1) =
Ppage(1,1,1) is the 40th percentile of the first page of A, and Ppage(2,1,1) is the 60th percentile of the first page of A.
Calculate the 40th and 60th percentiles of the elements in each A(:,i,:) slice by specifying dimensions 1 and 3 as the operating dimensions.
Pcol = prctile(A,[40 60],[1 3])
Pcol = 2×5
Pcol(1,4) is the 40th percentile of the elements in A(:,4,:), and Pcol(2,4) is the 60th percentile of the elements in A(:,4,:).
Calculate exact and approximate percentiles of a tall column vector for a given percentage.
Calculate the exact 50th percentile of A. Because A is a tall column vector and p is a scalar, prctile returns the exact percentile value by default.
Pexact = prctile(A,p)
Pexact =
Calculate the approximate 50th percentile of A. Specify the "approximate" method to use an approximation algorithm based on T-Digest for computing the percentile.
Papprox = prctile(A,p,"Method","approximate")
[Pexact,Papprox] = gather(Pexact,Papprox)
Pexact = 1522
Papprox = 1.5220e+03
The values of the exact percentile and the approximate percentile are the same to the four digits shown.
Calculate exact and approximate percentiles of a tall matrix for specified percentages along different dimensions.
Create a tall matrix A containing a subset of variables stored in varnames from the airlinesmall data set. See Percentiles of Tall Vector for Given Percentage for details about the steps to extract data from a tall array.
When operating along a dimension that is not 1, the prctile function calculates exact percentiles only so that it can compute efficiently using a sorting-based algorithm (see Algorithms) instead of an approximation algorithm based on T-Digest.
Calculate the exact 25th, 50th, and 75th percentiles of A along the second dimension.
p = [25 50 75];
Pexact = prctile(A,p,2)
When the function operates along the first dimension and p is a vector of percentages, you must use the approximation algorithm based on t-digest to compute the percentiles. Using the sorting-based algorithm to find percentiles along the first dimension of a tall array is computationally intensive.
Calculate the approximate 25th, 50th, and 75th percentiles of A along the first dimension. Because the default dimension is 1, you do not need to specify a value for dim.
[Pexact,Papprox] = gather(Pexact,Papprox);
Show the first five rows of the exact 25th, 50th, and 75th percentiles along the second dimension of A.
Pexact(1:5,:)
Each row of the matrix Pexact contains the three percentiles of the corresponding row in A. 30.5, 347.5, and 688.5 are the 25th, 50th, and 75th percentiles, respectively, of the first row in A.
Show the approximate 25th, 50th, and 75th percentiles of A along the first dimension.
Papprox
Papprox = 3×4
Each column of the matrix Papprox contains the three percentiles of the corresponding column in A. The first column of Papprox contains the percentiles for the first column of A.
p — Percentages for which to compute percentiles
Percentages for which to compute percentiles, specified as a scalar or vector of scalars from 0 to 100.
Consider an input matrix A and a vector of percentages p:
P = prctile(A,p,1) computes percentiles of the columns in A for the percentages in p.
P = prctile(A,p,2) computes percentiles of the rows in A for the percentages in p.
Dimension dim indicates the dimension of P that has the same length as p.
The size of the output P in the smallest specified operating dimension is equal to the length of p. The size of P in the other operating dimensions specified in vecdim is 1. The size of P in all dimensions not specified in vecdim remains the same as the input data.
Consider a 2-by-3-by-3 input array A and the percentages p. prctile(A,p,[1 2]) returns a length(p)-by-1-by-3 array because 1 and 2 are the operating dimensions and min([1 2]) = 1. Each page of the returned array contains the percentiles of the elements on the corresponding page of A.
method — Method for calculating percentiles
Method for calculating percentiles, specified as one of these values:
"exact" — Calculate exact percentiles with an algorithm that uses sorting.
"approximate" — Calculate approximate percentiles with an algorithm that uses T-Digest.
y=f\left(x\right)={y}_{1}+\frac{\left(x-{x}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}\left({y}_{2}-{y}_{1}\right).
Similarly, if the 100(1.5/n)th percentile is y1.5/n and the 100(2.5/n)th percentile is y2.5/n, then linear interpolation finds the 100(2.3/n)th percentile, y2.3/n as
{y}_{\frac{2.3}{n}}={y}_{\frac{1.5}{n}}+\frac{\left(\frac{2.3}{n}-\frac{1.5}{n}\right)}{\left(\frac{2.5}{n}-\frac{1.5}{n}\right)}\left({y}_{\frac{2.5}{n}}-{y}_{\frac{1.5}{n}}\right).
For data that is distributed in different partitions, t-digest computes quantile estimates (and percentile estimates) for each data partition separately, and then combines the estimates while maintaining a constant-memory bound and constant relative accuracy of computation (
q\left(1-q\right)
for the qth quantile). For these reasons, t-digest is practical for working with tall arrays.
k\left(q,\delta \right)=\delta \cdot \left(\frac{{\mathrm{sin}}^{-1}\left(2q-1\right)}{\pi }+\frac{1}{2}\right),
For an n-element vector A, prctile returns percentiles by using a sorting-based algorithm:
The sorted elements in A are taken as the 100(0.5/n)th, 100(1.5/n)th, ..., 100([n – 0.5]/n)th percentiles. For example:
For a data vector of five elements such as {6, 3, 2, 10, 1}, the sorted elements {1, 2, 3, 6, 10} respectively correspond to the 10th, 30th, 50th, 70th, and 90th percentiles.
For a data vector of six elements such as {6, 3, 2, 10, 8, 1}, the sorted elements {1, 2, 3, 6, 8, 10} respectively correspond to the (50/6)th, (150/6)th, (250/6)th, (350/6)th, (450/6)th, and (550/6)th percentiles.
prctile uses linear interpolation to compute percentiles for percentages between 100(0.5/n) and 100([n – 0.5]/n).
prctile assigns the minimum or maximum values of the elements in A to the percentiles corresponding to the percentages outside that range.
prctile treats NaNs as missing values and removes them.
P = prctile(A,p) returns the exact percentiles (using a sorting-based algorithm) only if A is a tall column vector.
P = prctile(A,p,dim) returns the exact percentiles only when one of these conditions exists:
A is a tall array and dim is not 1. For example, prctile(A,p,2) returns the exact percentiles along the rows of the tall array A.
If A is a tall array and dim is 1, then you must specify method as "approximate" to use an approximation algorithm based on T-Digest for computing the percentiles. For example, prctile(A,p,1,"Method","approximate") returns the approximate percentiles along the columns of the tall array A.
P = prctile(A,p,vecdim) returns the exact percentiles only when one of these conditions exists:
A is a tall array and vecdim does not include 1. For example, if A is a 3-by-5-by-2 array, then prctile(A,p,[2,3]) returns the exact percentiles of the elements in each A(i,:,:) slice.
A is a tall array and vecdim includes 1 and all the dimensions of A with a size greater than 1. For example, if A is a 10-by-1-by-4 array, then prctile(A,p,[1 3]) returns the exact percentiles of the elements in A(:,1,:).
If A is a tall array and vecdim includes 1 but does not include all the dimensions of A with a size greater than 1, then you must specify method as "approximate" to use the approximation algorithm. For example, if A is a 10-by-1-by-4 array, you can use prctile(A,p,[1 2],"Method","approximate") to find the approximate percentiles of each page of A.
If the output P is a vector, the orientation of P differs from MATLAB® when all of these conditions are true:
In this case, the output P matches the orientation of A, not the orientation of p.
Previously, prctile required Statistics and Machine Learning Toolbox™.
quantile | median | iqr
|
Types of Chemical Reactions - Course Hero
General Chemistry/Reactions in Chemistry/Types of Chemical Reactions
When two or more reactants combine to form a product, this is known as an addition reaction or synthesis reaction. The general equation for this reaction is:
\rm A+\rm B\rightarrow\rm{AB}
For example, when magnesium ribbon is burned in air, magnesium (Mg) reacts with oxygen gas (O2) in air to form magnesium oxide.
{2{\rm{Mg}}}{(s)}+{\rm O}_2{(g)}\rightarrow2{\rm{MgO}}{(s)}
Note that this reaction is also a combustion reaction, a reaction in which a substance is burned in the presence of an oxidizer, and an oxidation-reduction reaction, a reaction in which oxidation states of two or more atoms change. The same reaction can be classified as multiple types. Another example is the reaction between nitrogen gas (N2) and hydrogen gas (H2), forming ammonia gas (NH3).
{\rm N}_2{(g)}+3{\rm H}_2{(g)}\rightarrow2{\rm{NH}}_3{(g)}
This reaction, forming ammonia gas from atmospheric nitrogen, occurs only under very specific conditions. Ammonia is a nitrogen-based compound that is used to produce many important nitrogen-containing chemicals, such as artificial fertilizers.
A reaction in which a single compound breaks apart into two or more substances is called a decomposition reaction. The general equation for decomposition reaction is:
\rm{AB}\rightarrow\rm A+\rm B
There are three types of decomposition reactions. When a substance decomposes with application of heat, it is called thermal decomposition. For example, calcium carbonate (CaCO3), when heated, decomposes into calcium oxide (CaO) and carbon dioxide (CO2).
{\rm{CaCO}}_3{(s)}+\rm{heat}\rightarrow{\rm{CaO}}{(s)}+{\rm{CO}}_2{(g)}
When a substance decomposes because electricity passes through it, electrolytic decomposition occurs. Water (H2O) decomposes into oxygen and hydrogen when electric current passes through it.
2{\rm H}_2{\rm O}{(l)}+\rm{electricity}\rightarrow2{\rm H}_2{(g)}+{\rm O}_2{(g)}
A substance can also decompose when exposed to light by a process called photodecomposition. Silver chloride (AgCl), for example, decomposes into silver (Ag) and chlorine gas (Cl2) on exposure to sunlight.
2{\rm{AgCl}}{(s)}\rightarrow2{\rm{Ag}}{(s)}+{\rm{Cl}}_2{(g)}
An oxidation state, also called an oxidation number, is a hypothetical charge assigned to an atom, ion, or polyatomic ion, indicating how many electrons have been lost (or gained). Chemical reactions all involve electrons. However, in a lot of chemical reactions, the oxidation states of atoms do not change. Chemical reactions in which an oxidation state of one or more atoms change are called oxidation-reduction reactions or redox reactions.
Oxidation-reduction reactions form the basis of electrochemistry, the branch of chemistry that studies batteries and other power cells. Combustion, burning a substance (often called fuel) with an oxidizer (often oxygen), is also a type of oxidation-reduction reaction. Consider the burning of methane (CH4) with oxygen gas (O2) to produce carbon dioxide (CO2) and water (H2O).
{\rm{CH}_4}({g})+{\rm{O}_2}({g})\rightarrow{\rm{CO}_2}({g})+2{\rm{H}_2\rm{O}}(l)
The oxidation state of each atom in an oxidation-reduction reaction is determined by its position in the periodic table and its bonding behavior in the reaction.
Oxidation States in
{\rm{CH}_4}({g})+{\rm{O}_2}({g})\rightarrow{\rm{CO}_2}({g})+2{\rm{H}_2\rm{O}}({l})
Hydrogen atoms are +1 when bonded to nonmetals.
Carbon is –4 because it is coupled with four hydrogen atoms.
Since this is an elemental molecule, the oxidation state is zero.
Oxygen in most compounds has an oxidation state of –2.
Carbon is +4 because it is coupled with two oxygen atoms.
Oxygen in most compounds is –2, which works out with two hydrogen atoms.
Overall, carbon's oxidation state went from –4 to +4. Hydrogen's oxidation state did not change and stayed +1. Oxygen's oxidation went from zero to –2.
Oxidation is a reaction that involves the removal of an electron from an atom. The carbon in methane in the example is oxidized. The opposite of oxidation is reduction. Reduction is a reaction that involves the addition of an electron to an atom. Oxygen gas in the example is reduced.
An oxidizing agent causes oxidation of another substance by stripping electrons from it, and it is reduced in the process. In the example, oxygen gas is the oxidizing agent. A reducing agent causes the reduction of another substance by giving it electrons, and it is oxidized in the process. In the example, carbon in methane is the reducing agent.
Oxidation always occurs with reduction. When an atom is oxidized, another must be reduced. The name oxidation comes from oxygen because oxygen is a common and strong oxidizing agent. Oxygen typically causes oxidation in other compounds and gets reduced in turn. There are other oxidizing agents, and some are stronger than oxygen. Fluorine, for example, is the most electronegative element and can cause oxidation in oxygen.
Oxidation-reduction reactions, such as combustion, can release energy as heat. It is possible to set up an oxidation-reduction reaction so that it releases energy as electricity. An oxidation-reduction reaction involves a transfer of electrons. If the reaction is separated into two parts, the electron transfer can occur over a wire. A flow of electrons causes electricity. Setting up an oxidation-reduction reaction this way involves setting it up as two half-reactions. A half-reaction is either the oxidation or the reduction part of an oxidation-reduction reaction.
Consider the reaction between magnesium (Mg) and copper oxide (CuO):
\rm{Mg}+\rm{CuO}\rightarrow\rm{Cu}+\rm{MgO}
The oxidation states of each element in the reaction are written above each atom in the chemical formulas.
\overset0{\rm M}\rm g+\overset{+2\;-2}{\rm{CuO}}\rightarrow\overset0{\rm Cu} +\overset{+2\;\;\;-2}{\rm{MgO}}
Notice how the oxidation states change. The oxidation state of magnesium changes from zero to +2. Because magnesium loses electrons, it is oxidized. The oxidation state of copper changes from +2 to zero. Because copper gains electrons, it is reduced. The oxidation state of oxygen remains as –2. Oxygen is not oxidized or reduced in this reaction.
It is possible to write the two half-reactions as:
\rm{Mg}\rightarrow\rm{Mg}^{2+}+2\rm e^-\;\;\;\text{(oxidation reaction)}
2\rm e^-+\rm{Cu}^{2+}\rightarrow\rm{Cu}\;\;\;\text{(reduction reaction)}
The electrons in this reaction move from magnesium to copper. Magnesium loses electrons, and copper gains them.
Neutralization (Acid-Base) Reactions
In neutralization reactions, acids and bases react with one another, forming salt and water.
The Arrhenius definition of acids and bases, proposed by Swedish chemist Svante Arrhenius, characterizes acids and bases depending on the ions produced by each in aqueous solution. An acid is a substance that produces hydrogen (H+) in solution, whereas a base produces hydroxide (OH–) ions in solution.
A neutralization reaction is a reaction between an acid and a base. Acid-base reactions produce a salt and water.
\text{Acid}+\text{Base}\rightarrow\text{Salt}+\text{Water}
Controlling acidity is important in living organisms. Neutralization reactions happen frequently in nature. Neutralization reactions are important in industry and agriculture as well.
For example, hydrochloric acid (HCl) reacts with sodium hydroxide (NaOH), forming sodium chloride (NaCl) and water (H2O).
{\rm {HCl}}{({aq})}+{\rm{NaOH}}{({aq})}\rightarrow{\rm{NaCl}}{({aq})}+{\rm H}_2{\rm O}{({l})}
In the example, hydrochloric acid is the acid. Sodium hydroxide is the base. The salt that forms is sodium chloride, or table salt.
Another example is the reaction of sulfuric acid (H2SO4) and potassium hydroxide (KOH), forming potassium sulfate (K2SO4) and water.
{\rm H}_2{\rm{SO}}_4{({aq})}+{2\rm{KOH}}{({aq})}\rightarrow{\rm K}_2{\rm{SO}}_4{({aq})}+2{\rm H}_2{\rm O}{(l)}
The acid in this example is sulfuric acid, the base is potassium hydroxide, and the salt is potassium sulfate.
A third example is the reaction of nitric acid (HNO3) with sodium hydroxide, forming sodium nitrate (NaNO3) and water.
{\rm {HNO}}_3{({aq})}+{\rm{NaOH}}{({aq})}\rightarrow{\rm {NaNO}}_3{({s})}+{\rm H}_2{\rm O}{(l)}
Single- and Double-Displacement Reactions
When an atom or a group of atoms from a reactant is replaced with another atom or group of atoms in the products, such a reaction is called a single-displacement reaction. The general form of a single-displacement reaction is:
\rm{AB}\;+\;\rm C\;\rightarrow\;\rm{AC}\;+\;\rm B
In a single-displacement reaction, C is more reactive than B. Otherwise, the reaction will not move forward. When zinc (Zn) reacts with hydrochloric acid (HCl), hydrogen (H) is replaced by zinc, forming zinc chloride (ZnCl2).
{\rm{Zn}}{(s)}+2{\rm{HCl}}{(l)}\rightarrow{\rm{ZnCl}}_2{(s)}+{\rm H}_2{(g)}
In this reaction, the zinc replaces the hydrogen atom.
Another example is when hydrogen iodide (HI) reacts with chlorine gas (Cl2), forming hydrogen chloride (HCl) and iodine (I2)
2{\rm{HI}}{(g)}+{\rm{Cl}}_2{(g)}\rightarrow2{\rm{HCl}}{(g)}+{\rm I}_2{(s)}
In the reaction, chlorine gas replaces iodine.
When two reactants exchange atoms or groups, such a reaction is called a double-displacement reaction. These reactions usually take place in an aqueous state. The general form of this reaction is:
\rm{AB}+\rm{CD}\rightarrow\rm{AD}+\rm{BC}
Consider the reaction between sodium sulfide (Na2S) and hydrochloric acid. Sodium chloride (NaCl), which is table salt, and hydrogen sulfide (H2S) are produced by this reaction.
{\rm{Na}}_2{\rm S}({aq})+2{\rm{HCl}}({aq})\rightarrow2{\rm{NaCl}}({aq})+{\rm H}_2{\rm S}({g})
Sodium chloride is a soluble salt in this example. Single- and double-displacement reactions can produce a wide variety of salts. Some salts are not soluble in water and will precipitate. A precipitate is an insoluble product that settles as a residue at the bottom of the reaction vessel. A reaction that forms an insoluble salt, which forms as a solid in the reaction container, is called a precipitation reaction.
Solubility and Precipitate Formation
Double-displacement reactions do not always result in precipitates. Precipitation reactions are closely connected to solubility, the maximum amount of a substance that can be dissolved in another substance at specific conditions. A substance or dissolved material in a solution is called a solute. A substance that dissolves a material to form a solution is called a solvent. Water is a very common solvent. Solubility depends on the solute, on the solvent, and on conditions such as pressure and temperature.
A substance that has high solubility in a solvent under specific conditions is said to be soluble. Some substances are insoluble, which means they are incapable of being dissolved in another substance.
For example, when silver nitrate (AgNO3) and sodium iodide (NaI) react with each other in aqueous forms, insoluble silver iodide (AgI) is formed as a precipitate.
{\rm A\rm g\rm N{\rm O}_3}{(aq)}+{\rm N\rm a\rm I}{(aq)}\rightarrow{\rm A\rm g\rm I}{(s)}+{\rm N\rm a\rm N{\rm O}_3}{(aq)}
This reaction can be written in ionic form as:
{\rm{Ag}^+}{({aq})}+{\rm{NO}_3}^-{({aq})}+{\rm{Na}^+}{(aq)}+{\rm I^-}{(aq)}\rightarrow{\rm{AgI}}{(s)}+{\rm{Na}^+}{(aq)}+{\rm{NO}_3}^-{(aq)}
When spectator ions (which are present in both the reactants and the products) are removed, the net ionic equation becomes:
{\rm{Ag}^+}{({aq})}+{\rm I^-}{({aq})}\rightarrow{\rm{AgI}}{(s)}
Similarly, when the solution of barium chloride (BaCl2) reacts with sodium sulfate (Na2SO4), an insoluble precipitate of barium sulfate (BaSO4) is formed.
{\rm{BaCl}}_2{({aq})}+{\rm{Na}}_2{\rm{SO}}_4{({aq})}\rightarrow{\rm{BaSO}}_4{(s)}+2{\rm{NaCl}}{({aq})}
{\rm{Ba}^{2+}}({aq})+{\rm{SO}_4}^{2-}({aq})\rightarrow{\rm{BaSO}_4}({s})
The solid insoluble precipitate of barium sulfate is formed at the bottom of the test tube.
Magnesium reacts with copper sulfate to form copper as a precipitate.
Chemists have come up with a set of rules to determine if a salt is soluble in water or not. To predict if a precipitate will form, consider the solubility rules for salts in water. If two rules appear to contradict each other, the rule with the lower number takes precedence.
1. Group 1 element salts are soluble.
2. Salts with nitrate ions are soluble.
3. Salts of chloride, bromide, and iodide are soluble except for silver, lead, and mercury halides.
4. All silver salts are insoluble, except AgNO3 and a few other rare exceptions.
5. Sulfate salts are soluble except for calcium, barium, lead, strontium, and silver salts.
6. Hydroxide salts of Group 1 elements are soluble. Hydroxide salts of Group 2 elements (calcium, strontium, and barium) are partially soluble. Hydroxide salts of transition metals are insoluble.
7. Sulfides of transition metals are insoluble.
8. Carbonates, phosphates, fluorides, and chromates are insoluble.
There are exceptions to these rules. However, one can predict precipitate formation based on the rules with fair success.
Multiple-Classifications Reactions
Reactions can be classified into different types, but often a reaction can belong to more than one category. For example, almost all acid-base reactions are also double-displacement reactions. Many redox reactions are also combination reactions. Consider the reaction in which nitric acid (HNO3) reacts with sodium hydroxide (NaOH), a base, and forms a salt, sodium nitrate (NaNO3).
{\rm{HNO}}_3{({aq})}+{\rm{NaOH}}\,{({aq})}\rightarrow{\rm{NaNO}}_3{({s})}+{\rm H}_2{\rm O}{({l})}
This is an acid-base reaction and a double-displacement reaction at the same time.
Another example is the reaction of calcium (Ca) with fluorine gas (F2) to form calcium fluoride (CaF2).
{\rm{Ca}}({s})+{\rm F}_2({g})\rightarrow{\rm{CaF}}_2({s})
This is a combination, as well as an oxidation-reduction reaction, because calcium's oxidation state has changed from zero to +2 and fluorine's oxidation state has changed from zero to –1.
Zinc (Zn) reacts with hydrochloric acid (HCl) to form zinc chloride (ZnCl2) and hydrogen gas (H2).
{\rm{Zn}}({s})+2{\rm{HCl}}({aq})\rightarrow{{\rm{ZnCl}}_2}({aq})+{\rm H}_2({g})
This is a single-displacement reaction, as well as an oxidation-reduction reaction, because zinc's oxidation state has changed from zero to +2 and hydrogen's oxidation state has changed from +1 to zero.
Sodium hydroxide (NaOH), a strong base, reacts with hydrochloric acid (HCl) and forms sodium chloride (NaCl) and water (H2O).
{\rm{NaOH}}({aq})+{\rm{HCl}}({aq})\rightarrow{\rm{NaCl}}({aq})+{\rm {H}_2}{\rm O}({l})
This is a neutralization reaction as well as a double-displacement reaction.
<Ionic Equations>Suggested Reading
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Quaternion interpolation between two quaternions - Simulink - MathWorks í•œêµ
Quaternion interpolation between two quaternions
The Quaternion Interpolation block calculates the quaternion interpolation between two normalized quaternions by an interval fraction. Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention. Select the interpolation method from SLERP, LERP, or NLERP. For equations used for the interpolation methods, see Algorithms.
The two normalized quaternions are the two extremes between which the block calculates the quaternion.
q0 — First normalized quaternion
First normalized quaternion for which to calculate the interpolation. This quaternion must be a normalized quaternion
q1 — Second normalized quaternion
Second normalized quaternion for which to calculate the interpolation, specified as a 4-by-1 vector or 1-by-4 vector. This quaternion must be a normalized quaternion.
f — Interval fraction
Interval fraction by which to calculate the quaternion interpolation . This value varies between 0 and 1. It represents the intermediate rotation of the quaternion to be calculated. This fraction affects the interpolation method rotational velocities.
The interval fraction affects the rotational velocities of the interpolation methods for the Methods parameter. For more information on interval fractions, see [1].
qf — Natural logarithm
Natural logarithm of quaternion, returned as a vector.
Methods — Quaternion interpolation method
SLERP (default) | LERP | NLERP
Quaternion interpolation method to calculate the quaternion interpolation, specified as:
Quaternion slerp. Spherical linear quaternion interpolation method.
Quaternion lerp. Linear quaternion interpolation method.
Normalized quaternion linear interpolation method.
These methods have different rotational velocities, depending on the interval fraction from input port f. For more information on interval fractions, see [1].
Block Parameter: method
Values: 'SLERP' | 'LERP' | 'NLERP'
Default: 'SLERP'
Error (default) | None | Warning
Slerp\left(p,q,h\right)=p{\left({p}^{*}q\right)}^{h}
hâ\left[0,1\right].
LERP\left(p,q,h\right)=p\left(1âh\right)+qh
hâ\left[0,1\right].
r=LERP\left(p,q,h\right),
NLERP\left(p,q,h\right)=\frac{r}{|r|}.
[1] Dam, Erik B., Martin Koch, Martin Lillholm. "Quaternions, Interpolation, and Animation." University of Copenhagen, København, Denmark, 1998.
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Algebraic Method Definition
What Is the Algebraic Method?
The algebraic method refers to various methods of solving a pair of linear equations, including graphing, substitution and elimination.
What Does the Algebraic Method Tell You?
The graphing method involves graphing the two equations. The intersection of the two lines will be an x,y coordinate, which is the solution.
With the substitution method, rearrange the equations to express the value of variables, x or y, in terms of another variable. Then substitute that expression for the value of that variable in the other equation.
\begin{aligned} &8x+6y=16\\ &{-8}x-4y=-8\\ \end{aligned}
8x+6y=16−8x−4y=−8
First, use the second equation to express x in terms of y:
{-8}x=-8+4yx=\frac{-8+4y}{{-8}x}=1-0.5y
−8x=−8+4yx=−8x−8+4y=1−0.5y
Then substitute 1 - 0.5y for x in the first equation:
\begin{aligned} &8\left(1-0.5y\right)+6y=16\\ &8-4y+6y=16\\ &8+2y=16\\ &2y=8\\ &y=4\\ \end{aligned}
8(1−0.5y)+6y=168−4y+6y=168+2y=162y=8y=4
Then replace y in the second equation with 4 to solve for x:
\begin{aligned} &8x+6\left(4\right)=16\\ &8x+24=16\\ &8x=-8\\ &x=-1\\ \end{aligned}
8x+6(4)=168x+24=168x=−8x=−1
The second method is the elimination method. It is used when one of the variables can be eliminated by either adding or subtracting the two equations. In the case of these two equations, we can add them together to eliminate x:
\begin{aligned} &8x+6y=16\\ &{-8}x-4y=-8\\ &0+2y=8\\ &y=4\\ \end{aligned}
8x+6y=16−8x−4y=−80+2y=8y=4
Now, to solve for x, substitute the value for y in either equation:
\begin{aligned} &8x+6y=16\\ &8x+6\left(4\right)=16\\ &8x+24=16\\ &8x+24-24=16-24\\ &8x=-8\\ &x=-1\\ \end{aligned}
8x+6y=168x+6(4)=168x+24=168x+24−24=16−248x=−8x=−1
The algebraic method is a collection of several methods used to solve a pair of linear equations with two variables.
The most-commonly used algebraic methods include the substitution method, the elimination method, and the graphing method.
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Classical modal logic - Wikipedia
Find sources: "Classical modal logic" – news · newspapers · books · scholar · JSTOR (January 2009) (Learn how and when to remove this template message)
In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of the modal operators
{\displaystyle \Diamond A\equiv \lnot \Box \lnot A}
that is also closed under the rule
{\displaystyle A\equiv B\vdash \Box A\equiv \Box B.}
Alternatively one can give a dual definition of L by which L is classical if and only if it contains (as axiom or theorem)
{\displaystyle \Box A\equiv \lnot \Diamond \lnot A}
and is closed under the rule
{\displaystyle A\equiv B\vdash \Diamond A\equiv \Diamond B.}
The weakest classical system is sometimes referred to as E and is non-normal. Both algebraic and neighborhood semantics characterize familiar classical modal systems that are weaker than the weakest normal modal logic K.
Every regular modal logic is classical, and every normal modal logic is regular and hence classical.
Chellas, Brian. Modal Logic: An Introduction. Cambridge University Press, 1980.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Classical_modal_logic&oldid=995355994"
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Ranking - Objectives and metrics | CatBoost
Ranking: objectives and metrics
PairAccuracy
Groupwise metrics
QueryCrossEntropy
FilteredDCG
AverageGain
RecallAt
QueryAUC
Pairwise metrics use special labeled information — pairs of dataset objects where one object is considered the winner and the other is considered the loser . This information might be not exhaustive (not all possible pairs of objects are labeled in such a way). It is also possible to specify the weight for each pair.
If GroupId is specified, then all pairs must have both members from the same group if this dataset is used in pairwise modes.
Read more about GroupId
The identifier of the object's group. An arbitrary string, possibly representing an integer.
If the labeled pairs data is not specified for the dataset, then pairs are generated automatically in each group using per-object label values (labels must be specified and must be numerical). The object with a greater label value in the pair is considered the winner .
The following variables are used in formulas of the described pairwise metrics:
p
is the positive object in the pair.
n
is the negative object in the pair.
See all common variables in Variables used in formulas.
\displaystyle\frac{-\sum\limits_{p, n \in Pairs} w_{pn} \left(log(\displaystyle\frac{1}{1 + e^{- (a_{p} - a_{n})}})\right)}{\sum\limits_{p, n \in Pairs} w_{pn}}
The object weights are not used to calculate and optimize the value of this metric. The weights of object pairs are used instead.
Usage information See more.
use_weights
Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false .
max_pairs
The maximum number of generated pairs in each group. Takes effect if no pairs are given and therefore are generated without repetition.
Default: All possible pairs are generated in each group
\displaystyle\frac{-\sum\limits_{p, n \in Pairs} w_{pn} \left(log(\displaystyle\frac{1}{1 + e^{- (a_{p} - a_{n})}})\right)}{\sum\limits_{p, n \in Pairs} w_{pn}}
This metric may give more accurate results on large datasets compared to PairLogit but it is calculated significantly slower.
This technique is described in the Winning The Transfer Learning Track of Yahoo!’s Learning To Rank Challenge with YetiRank paper.
\displaystyle\frac{\sum\limits_{p, n \in Pairs} w_{pn} [a_{p} > a_{n}] }{\sum\limits_{p, n \in Pairs} w_{pn} }
The object weights are not used to calculate the value of this metric. The weights of object pairs are used instead.
Can't be used for optimization. See more.
The calculation of this metric is disabled by default for the training dataset to speed up the training. Use the hints=skip_train~false parameter to enable the calculation.
An approximation of ranking metrics (such as NDCG and PFound). Allows to use ranking metrics for optimization.
The value of this metric can not be calculated. The metric that is written to output data if YetiRank is optimized depends on the range of all N target values (
i \in [1; N]
) of the dataset:
target_{i} \in [0; 1]
— PFound
target_{i} \notin [0; 1]
— NDCG
This metric gives less accurate results on big datasets compared to YetiRankPairwise but it is significantly faster.
The object weights are not used to optimize this metric. The group weights are used instead.
This objective is used to optimize PairLogit. Automatically generated object pairs are used for this purpose. These pairs are generated independently for each object group. Use the Group weights file or the GroupWeight column of the Columns description file to change the group importance. In this case, the weight of each generated pair is multiplied by the value of the corresponding group weight.
The probability of search continuation after reaching the current object.
The number of permutations.
i \in [1; N]
target_{i} \in [0; 1]
target_{i} \notin [0; 1]
This metric gives more accurate results on big datasets compared to YetiRank but it is significantly slower.
Directly optimize the FilteredDCG metric calculated for a pre-defined order of objects for filtration of objects under a fixed ranking. As a result, the FilteredDCG metric can be used for optimization.
FilteredDCG = \sum\limits_{i=1}^{n}\displaystyle\frac{t_{i}}{i} { , where}
t_{i}
is the relevance of an object in the group and the sum is computed over the documents with
a > 0
The filtration is defined via the raw formula value:
Zeros correspond to filtered instances and ones correspond to the remaining ones.
The ranking is defined by the order of objects in the dataset.
Sort objects by the column you are interested in before training with this loss function and use the --has-timefor the Command-line version option to avoid further objects reordering.
For optimization, a distribution of filtrations is defined:
\mathbb{P}(\text{filter}|x) = \sigma(a) { , where}
\sigma(z) = \displaystyle\frac{1}{1 + \text{e}^{-z}}
The gradient is estimated via REINFORCE.
Refer to the Learning to Select for a Predefined Ranking paper for calculation details.
The scale for multiplying predictions.
num_estimations
The number of gradient samples.
Directly optimize the selected metric. The value of the selected metric is written to output data
Refer to the StochasticRank: Global Optimization of Scale-Free Discrete Functions paper for details.
The metric that should be optimized.
Default: Obligatory parameter
Supported values: DCG, NDCG, PFound.
The number of gradient estimation iterations.
Controls the penalty for coinciding predictions (aka ties).
Metric-specific parameters:
Available if the corresponding metric is set in the metric parameter.
The number of top samples in a group that are used to calculate the ranking metric. Top samples are either the samples with the largest approx values or the ones with the lowest target values if approx values are the same.
Default: –1 (all label values are used).
Metric calculation principles.
Default: Base.
Possible values: Base, Exp.
Metric denominator type.
Default: Default: LogPosition.
Possible values: LogPosition, Position.
Default: LogPosition.
QueryCrossEntropy(\alpha) = (1 - \alpha) \cdot LogLoss + \alpha \cdot LogLoss_{group}
See the QueryCrossEntropy section for more details.
The coefficient used in quantile-based losses.
\displaystyle\sqrt{\displaystyle\frac{\sum\limits_{Group \in Groups} \sum\limits_{i \in Group} w_{i} \left( t_{i} - a_{i} - \displaystyle\frac{\sum\limits_{j \in Group} w_{j} (t_{j} - a_{j})}{\sum\limits_{j \in Group} w_{j}} \right)^{2}} {\sum\limits_{Group \in Groups} \sum\limits_{i \in Group} w_{i}}}
- \displaystyle\frac{\sum\limits_{Group \in Groups} \sum\limits_{i \in Group}w_{i} t_{i} \log \left(\displaystyle\frac{w_{i} e^{\beta a_{i}}}{\sum\limits_{j\in Group} w_{j} e^{\beta a_{j}}}\right)} {\sum\limits_{Group \in Groups} \sum_{i\in Group} w_{i} t_{i}}
The input scale coefficient.
PFound(top, decay) =
= \sum_{group \in groups} PFound(group, top, decay)
See the PFound section for more details
nDCG(top) = \frac{DCG(top)}{IDCG(top)}
See the NDCG section for more details.
Default: Position.
DCG(top)
See the FilteredDCG section for more details.
Represents the average value of the label values for objects with the defined top
M
label values.
See the AverageGain section for more details.
Default: This parameter is obligatory (the default value is not defined).
The calculation of this function consists of the following steps:
The objectsare sorted in descending order of predicted relevancies (
a_{i}
The metric is calculated as follows:
PrecisionAt(top, border) = \frac{\sum\limits_{i=1}^{top} Relevant_{i}}{top} { , where}
Relevant_{i} = \begin{cases} 1 { , } & t_{i} > {border} \\ 0 { , } & {in other cases} \end{cases}
The label value border. If the value is strictly greater than this threshold, it is considered a positive class. Otherwise it is considered a negative class.
a_{i}
RecalAt(top, border) = \frac{\sum\limits_{i=1}^{top} Relevant_{i}}{\sum\limits_{i=1}^{N} Relevant_{i}}
Relevant_{i} = \begin{cases} 1 { , } & t_{i} > {border} \\ 0 { , } & {in other cases} \end{cases}
a_{i}
MAP(top, border) = \frac{1}{N_{groups}} \sum\limits_{j = 1}^{N_{groups}} AveragePrecisionAt_{j}(top, border) { , where}
N_{groups}
is the number of groups
AveragePrecisionAt(top, border) = \frac{\sum\limits_{i=1}^{top} Relevant_{i} * PrecisionAt_{i}}{\sum\limits_{i=1}^{top} Relevant_{i} }
The value is calculated individually for each j-th group.
Relevant_{i} = \begin{cases} 1 { , } & t_{i} > {border} \\ 0 { , } & {in other cases} \end{cases}
PrecisionAt_{i} = \frac{\sum\limits_{j=1}^{i} Relevant_{j}}{i}
ERR = \frac{1}{|Q|} \sum_{q=1}^{|Q|} ERR_q
ERR_q = \sum_{i=1}^{top} \frac{1}{i} t_{q,i} \prod_{j=1}^{i-1} (1 - t_{q,j})
Targets should be from the range [0, 1].
t_{q,i} \in [0, 1]
MRR = \frac{1}{|Q|} \sum_{q=1}^{|Q|} \frac{1}{rank_q}
rank_q
refers to the rank position of the first relevant document for the q-th query.
The type of AUC. Defines the metric calculation principles.
\displaystyle\frac{\sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} {\sum w_{i} \cdot w_{j}}
The sum is calculated on all pairs of objects
(i,j)
t_{i} = 0
t_{j} = 1
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
Refer to the Wikipedia article for details.
If the target type is not binary, then every object with target value
t
w
is replaced with two objects for the metric calculation:
o_{1}
t \cdot w
and target value 1
o_{2}
(1 – t) \cdot w
and target value 0.
Target values must be in the range [0; 1].
\displaystyle\frac{\sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} {\sum w_{i} * w_{j}}
(i,j)
t_{i} < t_{j}
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
The type of AUC. Defines the metrics calculation principles.
Default: Classic.
Possible values: Classic, Ranking.
Examples: AUC:type=Classic, AUC:type=Ranking.
Default: False for Classic type, True for Ranking type.
Examples: AUC:type=Ranking;use_weights=False.
\displaystyle\frac{ \sum_q \sum_{i, j \in q} \sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} { \sum_q \sum_{i, j \in q} \sum w_{i} \cdot w_{j}}
(i,j)
t_{i} = 0
t_{j} = 1
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
t
w
o_{1}
t \cdot w
o_{2}
(1 – t) \cdot w
\displaystyle\frac{ \sum_q \sum_{i, j \in q} \sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} { \sum_q \sum_{i, j \in q} \sum w_{i} * w_{j}}
(i,j)
t_{i} < t_{j}
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
The type of QueryAUC. Defines the metric calculation principles.
Default: Ranking.
Examples: QueryAUC:type=Classic, QueryAUC:type=Ranking.
Examples: QueryAUC:type=Ranking;use_weights=False.
PairLogit + +
PairLogitPairwise + +
PairAccuracy - -
YetiRank + +
YetiRankPairwise + +
StochasticFilter + -
StochasticRank + -
QueryCrossEntropy + +
QueryRMSE + +
QuerySoftMax + +
PFound - -
NDCG - -
DCG - -
FilteredDCG - -
AverageGain - -
PrecisionAt - -
RecallAt - -
ERR - -
MRR - -
AUC - -
QueryAUC - -
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Section 59.27 (03PF): Étale coverings—The Stacks project
Section 59.27: Étale coverings (cite)
59.27 Étale coverings
We recall the definition.
Definition 59.27.1. An étale covering of a scheme $U$ is a family of morphisms of schemes $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ such that
each $\varphi _ i$ is an étale morphism,
the $U_ i$ cover $U$, i.e., $U = \bigcup _{i\in I}\varphi _ i(U_ i)$.
Lemma 59.27.2. Any étale covering is an fpqc covering.
Proof. (See also Topologies, Lemma 34.9.6.) Let $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ be an étale covering. Since an étale morphism is flat, and the elements of the covering should cover its target, the property fp (faithfully flat) is satisfied. To check the property qc (quasi-compact), let $V \subset U$ be an affine open, and write $\varphi _ i^{-1}(V) = \bigcup _{j \in J_ i} V_{ij}$ for some affine opens $V_{ij} \subset U_ i$. Since $\varphi _ i$ is open (as étale morphisms are open), we see that $V = \bigcup _{i\in I} \bigcup _{j \in J_ i} \varphi _ i(V_{ij})$ is an open covering of $V$. Further, since $V$ is quasi-compact, this covering has a finite refinement. $\square$
So any statement which is true for fpqc coverings remains true a fortiori for étale coverings. For instance, the étale site is subcanonical.
Definition 59.27.3. (For more details see Section 59.20, or Topologies, Section 34.4.) Let $S$ be a scheme. The big étale site over $S$ is the site $(\mathit{Sch}/S)_{\acute{e}tale}$, see Definition 59.20.4. The small étale site over $S$ is the site $S_{\acute{e}tale}$, see Definition 59.20.4. We define similarly the big and small Zariski sites on $S$, denoted $(\mathit{Sch}/S)_{Zar}$ and $S_{Zar}$.
Loosely speaking the big étale site of $S$ is made up out of schemes over $S$ and coverings the étale coverings. The small étale site of $S$ is made up out of schemes étale over $S$ with coverings the étale coverings. Actually any morphism between objects of $S_{\acute{e}tale}$ is étale, in virtue of Proposition 59.26.2, hence to check that $\{ U_ i \to U\} _{i \in I}$ in $S_{\acute{e}tale}$ is a covering it suffices to check that $\coprod U_ i \to U$ is surjective.
The small étale site has fewer objects than the big étale site, it contains only the “opens” of the étale topology on $S$. It is a full subcategory of the big étale site, and its topology is induced from the topology on the big site. Hence it is true that the restriction functor from the big étale site to the small one is exact and maps injectives to injectives. This has the following consequence.
Proposition 59.27.4. Let $S$ be a scheme and $\mathcal{F}$ an abelian sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. Then $\mathcal{F}|_{S_{\acute{e}tale}}$ is a sheaf on $S_{\acute{e}tale}$ and
\[ H^ p_{\acute{e}tale}(S, \mathcal{F}|_{S_{\acute{e}tale}}) = H^ p_{\acute{e}tale}(S, \mathcal{F}) \]
In accordance with the general notation introduced in Section 59.20 we write $H_{\acute{e}tale}^ p(S, \mathcal{F})$ for the above cohomology group.
Comment #1340 by yogesh more on March 10, 2015 at 20:47
minor typo:03PF penultimate sentence of the proof of lemma 45.27.2, line 2708 of the code: "...is an open covering of
V
" (instead of
U
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LIFO Calculator for Inventory
What is LIFO (last-in, first-out)?
How to use LIFO for costs of goods sold calculation
How to calculate ending inventory by LIFO
How to use our LIFO method calculator
The LIFO calculator for inventory and costs of goods sold (COGS) is an intelligent tool that can help you calculate your current inventory value and the amount you have to report as COGS by considering the LIFO method.
This article will cover how to determine ending inventory by LIFO after selling in contrast to the FIFO method. Also, we will see how to calculate its cost of goods sold using LIFO, and show how to use our LIFO calculator online to make more profits.
LIFO stands for last-in, first-out, and it's an accounting method for measuring the COGS (costs of goods sold) based on inventory prices. The particularity of the LIFO method is that it takes into account the price of the last acquired items whenever you sell stock.
Let's consider this example. The company acquired T-shirts as per the following scheme:
T-shirt buying price
The company receives a sell order of 10 T-shirts. What will be the related costs? If you use our LIFO calculator, you will see the result is 144 USD.
This is because we took the last seven items at 15 USD and added the three items we bought at 13 USD:
7 \times 15 + 3 \times 13 = 144
That is LIFO. Last-in the inventory, first-out when the sell occurs.
It is quite different from the FIFO method (first-in, first-out), where we would have taken the two t-shirts bought at 10 USD, then the other five t-shirts at 13 USD, and finally the last three ones at 15 USD. COGS, in this case, would be 130 USD. A percentage decrease of 9.7%.
Keep in mind the LIFO vs. FIFO difference, as we will explain it more in the following paragraphs.
As mentioned above, companies have to define their cost of goods sold for determining a selling price that can keep their profit margins. Let's explain how they do it when using the LIFO method. We adopt the following notation:
q_1
= Number of units purchased 1st time.
p_1
= 1st units purchased price.
q_2
= Number of units purchased 2nd time.
p_2
= 2nd units purchased price.
q_i
= Number of units purchased last time.
p_i
= Last units purchased price.
Then the inventory value
\text{InvVal}
formula reads:
\text{InvVal} = p_1 q_1 + p_2 q_2 + \ldots + p_i q_i
Here, we are assuming the company has not sold any product yet. Please note how increasing/decreasing inventory prices through time can affect the inventory value.
The LIFO method assumes that it will take the last acquired items when the company sells its inventory. Assuming the company sells
n
q_i
= Number of units purchased the last time.
q_{i-1}
= Number of units purchased one time before the last one.
q_{i-2}
= Number of units purchased two times before the last one.
...and so on, until the number of sold items is equal to n. Assuming,
n = q_i + q_{i-1} + q_{i-2}
COGS would be:
\text{COGS} = q_i p_i + q_{i-1} p_{i-1} + q_{i-2} p_{i-2}
Following our example above:
\text{COGS} = 3 \times 13+ 7 \times 15 = 144
Notice how the cost of goods sold could increase if the last prices of the items the company bought also increase. What happens during inflationary times, and by rising COGS, it would reduce not only the operating profits but also the tax payment.
Continuing with out formulas above, we would not have the last items because we sold them accordingly to the LIFO method. Then, the ending inventory/remaining inventory
\text{InvVal}
\text{InvVal} = q_1 p_1 + q_2 p_2 \ldots + q_{i-3} p_{i-3}
In the T-shirt example we mentioned above, the initial inventory value was 190 USD. You can use our LIFO calculator or go through all the T-shirts we bought and multiply them by their respective price. Then after selling the last ten items, the inventory value is:
\footnotesize \text{InvVal} = 2 \times 10 + 2 \times 13 + 0 \times 15 = 46
Thus, we end up with an inventory value of 46 USD.
Here we are going to determine the cost of goods sold using the LIFO calculator, our revenue, and our profit margin:
Add the number of items you bought and their respective prices. Our LIFO calculator will indicate the current amount of your existing inventory.
In the next section, add the number of total units sold. Our tool will indicate the COGS.
Finally, include the selling price. Our LIFO calculator online will indicate the total revenues and the profit margin.
Continuing with our example above, assume we sell the 10 items at a 16 USD each:
\text{Revenues} = 16 \times 10 = 160
\text{COGS} = 144
\text{Profits} = 160 - 144 = 16
Note that we have income. Besides, the formula for the profit margin is:
\text{Profit margin} = \frac{\text{Profits}}{\text{Revenue}} \cdot 100\%
\text{Profit margin} = \frac{16}{160} \cdot 100\% = 10\%
How does the LIFO method affect taxable profits?
When you compare the cost of goods sold using the LIFO calculator, you see that COGS increases when the prices of acquired items rise. Such a situation will reduce the profits on which the company pays taxes. Consequently, LIFO can help lower taxable income.
Which one is better FIFO or LIFO?
LIFO is only allowed in the USA, whereas, in the world, companies use FIFO. In the USA, companies prefer to use LIFO because it can help them reduce their taxable income. Furthermore, when USA companies have operations outside their country of origin, they present a section where the overseas inventory registered by FIFO is modified to LIFO. You can also check FIFO and LIFO calculators at the Omni Calculator website to learn what happens in inflationary/deflationary environments.
How do I calculate ending inventory using LIFO?
To determine the ending inventory using LIFO follows these steps:
Determine the existing inventory by multiplying each acquisition price per the amount bought.
Define how many items you are going to sell.
Subtract the items you sold from the existing inventory. Start removing the last ones.
Multiply the remaining ones (which are the ones you bought first) per their respective prices. Then, you have the ending inventory amount using LIFO. You can also try our LIFO calculator online.
How do I calculate COGS using LIFO?
To calculate COGS using LIFO:
Keep a record of each acquisition price per the amount bought.
Define how many items you are going to sell. Our LIFO method calculator would bring a result here.
Take the last items and their respective prices. Select only the ones you sold.
Multiply their prices by their amount. There you have your COGS as per the LIFO method.
How does inflation affect FIFO ending inventory calculation?
If you use a LIFO calculator as an ending inventory calculator, you will see that you keep the cheapest inventory in your accounts with inflation (and rising prices through time). The most expensive items would go to the COGS calculation. In that sense, we will see a smaller ending inventory during inflation compared to a non-inflationary period.
Which financial ratios does LIFO ending inventory calculation affect?
If LIFO affects COGS and makes it more significant during inflationary times, we will have a reduced net income margin. Besides, inventory turnover will be much higher as it will have higher COGS and smaller inventory. Also, all the current asset-related ratios will be affected because of the change in inventory value.
How does deflation affect LIFO ending inventory calculation?
Considering that deflation is the item's price decrease through time, you will see a smaller COGS with the LIFO method. Also, you will see a more significant remaining inventory value because the most expensive items were bought and kept at the very beginning.
Inventories bought
1st units purchased price
Number of units purchased 1st time
Enter at least your 1st inventory buy and its price.
Revenues and margin
Gratuity calculator is a tool that helps you calculate the amount that you will receive after at least 5 years of work.
Use the price elasticity of supply calculator to determine the responsiveness of the supplied quantity of a good to changes in its price.
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