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How to Calculate Volume in Litres
1 Finding Volume in Liters from Dimensions
2 Converting Liters from other Metric Units
3 Converting Liters from Imperial Units
A liter (or litre) is a metric unit used to measure volume or capacity.[1] X Research source Liters are a common measurement often used to measure beverages and other liquids, such as a 2 liter bottle of soda. Sometimes you will need to calculate the volume of an object in liters, given the object's dimensions. In other instances, you will need to convert the volume of something that is already given in another unit, such as milliliters or gallons. In all of these instances, through simple multiplication or division, you can easily determine volume in liters.
Finding Volume in Liters from Dimensions
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Convert the dimensions to centimeters. If the dimensions are given in meters, inches, feet, or some other unit of measurement,
convert each dimension to centimeters (cm) before calculating the volume.
This will make it easier to convert to liters.
Always double-check that the dimensions of the shape you're working with have the same unit before you calculate the volume.[2] X Expert Source Grace Imson, MA
1 meter = 100 centimeters.[3] X Research source So, if the length of a cube is 2.5 meters, that converts to 250 centimeters, since
{\displaystyle 2.5\times 100=250}
1 inch = 2.54 centimeters.[4] X Research source So, if the length of a cube is 5 inches, that converts to 12.7 centimeters, since
{\displaystyle 5\times 2.54=12.7}
1 foot = 30.48 centimeters.[5] X Research source So, if the length of a cube is 3 feet, that converts to 91.44 centimeters, since
{\displaystyle 3\times 30.48=91.44}
Find the volume of the shape. How you find the volume will depend on the shape of the three-dimensional object you are measuring, since the volume of each type of shape is calculated differently. To find the volume of a cube, you can use the formula
{\displaystyle {\text{Volume}}={\text{Length}}\times {\text{Width}}\times {\text{Height}}}
[6] X Expert Source Grace Imson, MA
Math Instructor, City College of San Francisco Expert Interview. 1 November 2019. The volume of a three-dimensional shape will be in cubic units, such as cubic centimeters (
{\displaystyle cm^{3}}
For instance, if a fish tank is 40.64 cm long, 25.4 cm wide, and 20.32 tall you would calculate the volume by multiplying these dimensions together:
{\displaystyle {\text{Volume}}={\text{Length}}\times {\text{Width}}\times {\text{Height}}}
{\displaystyle {\text{Volume}}=40.64\times 25.4\times 20.32}
{\displaystyle {\text{Volume}}=20,975cm^{3}}
To find the volume of a cylinder, start by finding the height of the cylinder. Then, find the radius of the circle at the top or bottom. Next, find the area of the circle, which you can find with the formula π
{\displaystyle r^{2}}
, where r is the radius. Finally, multiply the area of the circle by the height of the cylinder to find the volume.[7] X Expert Source Grace Imson, MA
Convert cubic centimeters to liters. To do this, use the conversion rate
{\displaystyle 1\;{\text{liter}}=1,000cm^{3}}
Dividing the volume (in cubic centimeters) of the shape by 1,000 will give you the volume in liters (L).[8] X Research source
If the volume of the fish tank, in cubic centimeters, is 20,975, to find the volume in liters, calculate
{\displaystyle 20,975\div 1,000=20.975}
. So, a fish tank that is 40.64 cm long, 25.4 cm wide, and 20.32 tall has a volume of 20.975 L.
Converting Liters from other Metric Units
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Convert milliliters to liters. There are
1,000 milliliters (mL) in 1 liter (L).
So, to convert milliliters to liters, you would divide the number of milliliters by 1,000.[9] X Research source
If the volume of a carton of almond milk is 1,890 mL, to convert to liters, you would calculate
{\displaystyle 1,890mL\div 1,000=1.89L}
Convert centiliters to liters. There are
100 centiliters (cL) in 1 liter.
So, to convert centiliters to liters, you would divide the number of centiliters by 100.[10] X Research source
If a the volume of a carton of almond milk is 189 cL, to convert to liters, you would calculate
{\displaystyle 189cL\div 100=1.89L}
Convert deciliters to liters. There are
10 deciliters (dL) in 1 liter.
So, to convert deciliters to liters, you would divide the number of deciliters by 10.[11] X Research source
If a the volume of a carton of almond milk is 18.9 dL, to convert to liters, you would calculate
{\displaystyle 18.9dL\div 10=1.89L}
Convert kiloliters to liters. There are
1000 liters in 1 kiloliter (kl).
So, to convert kiloliters to liters, you would multiply the number of kiloliters by 1,000.[12] X Research source
If the volume of a kiddie pool is 240 kl, to convert to liters, you would calculate
{\displaystyle 240kl\times 1000=240,000L}
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Convert hectoliters to liters. There are
100 liters in 1 hectoliter (hl).
So, to convert hectoliters to liters, you would multiply the number of hectoliters by 100.[13] X Research source
If the volume of a kiddie pool is 2,400 hl, to convert to liters, you would calculate
{\displaystyle 2,400hl\times 100=240,000L}
Convert decaliters to liters. There are
10 liters in 1 decaliters (dal).
So, to convert decaliters to liters, you would multiply the number of decaliters by 10.[14] X Research source
If the volume of a kiddie pool is 24,000 dal, to convert to liters, you would calculate
{\displaystyle 24,000dal\times 10=240,000L}
Converting Liters from Imperial Units
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Convert fluid ounces to liters. There are
33.81 fluid ounces (fl oz) in 1 liter.
So, to convert fluid ounces to liters, you would divide the number of fluid ounces by 33.81.[15] X Research source
If a carton of almond milk is 128 fl oz, to convert to liters, you would calculate
{\displaystyle 128fl\;oz\div 33.81=3.786L}
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Convert pints to liters. There are
2.113 fluid pints (fl pt) in 1 liter.
So, to convert fluid pints to liters, you would divide the number of fluid pints by 2.113.[16] X Research source
If a pitcher has a capacity of 8 fl pt, to convert to liters, you would calculate
{\displaystyle 8fl\;pt\div 2.113=3.786L}
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Convert quarts to liters. There are
1.057 quarts (qt) in 1 liter.
So, to convert quarts to liters, you would divide the number quarts of by 1.057.[17] X Research source
If a pitcher has a capacity of 4 quarts, to convert to liters, you would calculate
{\displaystyle 4qt\div 1.057=3.784L}
Convert gallons to liters. There are
3.7854 liters in 1 gallon (gal).
So, to convert gallons to liters, you would multiply the number gallons of by 3.7854.[18] X Research source
If a fish tank has a volume of 120 gallons, to convert to liters, you would calculate
{\displaystyle 120gal\times 3.7854=454.248L}
How do you convert litres into cubed centimeters?
You can use the conversion 1 liter = 1,000 cubic centimeters. To convert from liters to cubic centimeters, you would multiply by 1,000. For example, if a cube has a volume of 34 liters, to find the volume in cubic centimeters, multiply by 1,000: 34 x 1,000 = 34,000 cubic centimeters.
How do I covert liters to meters?
There are 1,000 liters in a cubic meter. So divide the number of liters by 1,000 to convert to cubic meters.
I'm building a round pool, 5 m diameter and 1.5 m deep. How many liters is it?
Find the volume in cubic meters (πr²h). Multiply that number by 1,000.
My tank length is 3 m, width is 4 m, height is 7.5 m. What is the liter capacity of the tank?
3 x 4 x 7.5 = 90 cubic meters = 90,000 liters. (1 cubic meter equals 1,000 liters.)
If I have a volume of 129 liters, what is that in cubic meters?
How many glasses of water is equal to 1 liter?
It depends, of course, on the size of the glass. A typical glass might hold 8 ounces. In that case a liter would consist of slightly more than four glasses of water.
If the diameter is 3 m and it is 3/4 of a meter deep, what is the volume of a Jacuzzi to the nearest liter?
The Jacuzzi is a cylinder. Volume of a cylinder is pi times its radius times its height (or depth, in this case). The radius of a cylinder is half its diameter. So the radius of this cylinder is 3/2. If we plug in the values in the formula, we will get that the volume of the cylinder is 3.534 meter cube (approximately). 1 meter cube can hold 1000 liters of water. Thus this Jacuzzi will hold 3.534 times 1000 liters of water, which is 3534 liters of water.
Length 50 cm, volume 4 liters, what is height and breadth?
If you're inquiring about a rectilinear volume, you cannot calculate the height and breadth if you know only the length and volume.
How do I convert ml to cm^3?
Milliliters and cubic centimeters are equal to each other, so it's a one-to-one conversion.
How do I convert cubic inches to liters?
There are 61.02 cubic inches in a liter, so divide cubic inches by 61.02 to get liters.
This article was co-authored by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 748,000 times.
To calculate volume in litres, first convert the dimensions of the object into centimeters. Then, use the volume formula to calculate the volume of a shape. For example, to calculate the volume of a cube, you would use Volume = length times width times height, and your answer will be in cubic centimeters. Convert the answer to litres by dividing the number by 1,000 because there are 1000 cubic centimeters in 1 liter. For more tips, including how to calculate volume in litres from other metric units, scroll down!
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The Nonexistence of Global Solutions for a Time Fractional Schrödinger Equation with Nonlinear Memory
The Nonexistence of Global Solutions for a Time Fractional Schrödinger Equation with Nonlinear Memory
Yaning Li1*, Quanguo Zhang2
1College of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing, China
2Department of Mathematics, Luoyang Normal University, Luoyang, Henan, China
In this paper, we study the nonexistence of solutions of the following time fractional nonlinear Schrödinger equations with nonlinear memory
\left\{\begin{array}{l}{i}^{\alpha }{}_{0}{}^{C}D{}_{t}^{\alpha }u+\Delta u=\lambda {}_{0}I{}_{t}^{1-\gamma }\left({|u|}^{p}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {ℝ}^{N},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\hfill \\ u\left(0,x\right)=g\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {ℝ}^{N},\hfill \end{array}
0<\alpha <\gamma <1
{i}^{\alpha }
denotes the principal value of
{i}^{\alpha }
p>1
T>0
\lambda \in ℂ\\left\{0\right\}
u\left(t,x\right)
is a complex-value function,
{}_{0}I{}_{t}^{1-\gamma }
denotes left Riemann-Liouville fractional integrals of order
1-\gamma
{}_{0}{}^{C}D{}_{t}^{\alpha }u
\alpha
. We obtain that the problem admits no global weak solution when
1<p<1+\frac{2\left(\alpha +1-\gamma \right)}{\alpha N}
1<p<1+\frac{1-\gamma }{\alpha }
under different conditions for initial data.
Fractional Schrödinger Equation, Nonexistence, Cauchy Problems, Nonlinear Memory
This paper is concerned with the nonexistence of solutions to the Cauchy problem for the time fractional nonlinear Schrödinger equations with nonlinear memory
\left\{\begin{array}{l}{i}^{\alpha }{}_{0}{}^{C}D{}_{t}^{\alpha }u+\Delta u=\lambda {}_{0}I{}_{t}^{1-\gamma }\left({|u|}^{p}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {ℝ}^{N},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\hfill \\ u\left(0,x\right)=g\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {ℝ}^{N},\hfill \end{array}
0<\alpha <\gamma <1
{i}^{\alpha }
denotes principal value of
{i}^{\alpha }
p>1
T>0
\lambda ={\lambda }_{1}+{\lambda }_{2}i\in ℂ\\left\{0\right\},{\lambda }_{1},{\lambda }_{2}\in ℝ
u=u\left(t,x\right)
is a complex-valued function,
g\left(x\right)={g}_{1}\left(x\right)+{g}_{2}\left(x\right)i
{g}_{1}\left(x\right)
{g}_{2}\left(x\right)
are real-valued functions.
{}_{0}I{}_{t}^{1-\gamma }
1-\gamma
{}_{0}{}^{C}D{}_{t}^{\alpha }u=\frac{\partial }{\partial t}{}_{0}I{}_{t}^{1-\alpha }\left(u\left(t,x\right)-u\left(0,x\right)\right)
For the nonlinear Schrödinger equations without gauge invariance (i.e.
\alpha =\gamma =1
\left\{\begin{array}{l}i{u}_{t}+\Delta u=\lambda {|u|}^{p},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {ℝ}^{N},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\hfill \\ u\left(0,x\right)=g\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {ℝ}^{N},\hfill \end{array}
Ikeda and Wakasugi [1] and Ikeda and Inui [2] [3] proved blow-up results of solutions for (2) under different conditions for
1<p<1+\frac{2}{N}
1<p<1+\frac{4}{N}
The main tool they used is test function method. This method is based on rescalings of a compactly support test function to prove blow-up results which is first used by Mitidieri and Pohozaev [4] to show the blow-up results.
Recently, it has been seen that fractional differential equations have better effects in many realistic applications than the classical ones. So, considerable attention has been attracted to time fractional diffusion equation which arises in electromagnetic, acoustic and mechanical phenomena etc. [5] , and is derived from classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order
\alpha
\alpha \in \left(0,1\right]
. Fractional diffusion equation was explicitly applied to physics by Nigmatullin [6] to describe diffusion in media with fractal geometry (special types of porous media). There are many papers about the existence and properties of solutions for fractional differential equation, see for example [7] [8] [9] [10] [11] and the references therein.
For nonlinear time fractional Schrödinger equations (i.e., (1) with
\gamma =1
), Zhang, Sun and Li [12] studied the nonexistence of this problem in
{C}_{0}\left({R}^{N}\right)
and proved that the problem admits no global weak solution with suitable initial
data when
1<p<1+\frac{2}{N}
by using test function method, and also give some
conditions which imply the problem has no global weak solution for every
p>1
In [13] , Cazenave, Dickstein and Weissler considered a class of heat equation with nonlinear memory. They obtained that the solution blows up in finite time and under suitable conditions the solution exists globally. In [14] , using test function method, the authors considered a heat equation with nonlinear memory, they determined Fujita critical exponent of the problem.
Motivated by above results, in present paper, our purpose is to study the nonexistence of global weak solutions of (1) with a condition related to the sign of initial data when
1<p<1+\frac{2\left(\alpha +1-\gamma \right)}{\alpha N}
1<p<1+\frac{1-\gamma }{\alpha }
This paper is organized as follows. In Section 2, some preliminaries and the main results are presented. In Section 3, we give proof of the main results.
2. Preliminaries and the Main Results
For convenience of statement, let us present some preliminaries that will be used in next sections.
{}_{0}{}^{C}D{}_{t}^{\alpha }f\in {L}^{1}\left(0,T\right)
g\in {C}^{1}\left(\left[0,T\right]\right)
g\left(T\right)=0
, then we have the following formula of integration by parts
{\int }_{0}^{T}g{}_{0}{}^{C}D{}_{t}^{\alpha }f\text{d}t={\int }_{0}^{T}\left(f\left(t\right)-f\left(0\right)\right){}_{t}{}^{C}D{}_{T}^{\alpha }g\text{d}t.
We need calculate Caputo fractional derivative of the following function, which will be used in next sections. For given
T>0
n>0
\phi \left(t\right)=\left\{\begin{array}{l}{\left(1-\frac{t}{T}\right)}^{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }t\le T,\hfill \\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>T,\hfill \end{array}
{}_{t}{}^{C}D{}_{T}^{\alpha }\phi \left(t\right)=\frac{\Gamma \left(n+1\right)}{\Gamma \left(n+1-\alpha \right)}{T}^{-\alpha }{\left(1-\frac{t}{T}\right)}^{n-\alpha },\text{\hspace{0.17em}}t\le T,
(see for example [15] ).
Now, we present the definition of weak solution of (1).
g\in {L}_{loc}^{1}\left({R}^{N}\right)
0<\alpha <\gamma <1
T>0
u\in {L}^{p}\left(\left(0,T\right),{L}_{loc}^{\infty }\left({R}^{N}\right)\right)
is a weak solution of (1) if
{\int }_{{R}^{N}}{\int }_{0}^{T}\lambda {}_{0}I{}_{t}^{1-\gamma }\left({|u|}^{p}\right)\phi +{i}^{\alpha }g\left(x\right){}_{t}{}^{C}D{}_{T}^{\alpha }\phi \text{d}t\text{d}x={\int }_{{R}^{N}}{\int }_{0}^{T}u\left(\Delta \phi +{i}^{\alpha }{}_{t}{}^{C}D{}_{T}^{\alpha }\phi \right)\text{d}t\text{d}x
\phi \in {C}_{x\mathrm{,}t}^{\mathrm{2,1}}\left({R}^{N}\times \left[\mathrm{0,}T\right]\right)
sup{p}_{x}\phi \subset \subset {R}^{N}
\phi \left(x,T\right)=0
T>0
can be arbitrarily chosen, then we call u is a global weak solution for of (1).
{G}_{1}\left(x\right)=\mathrm{cos}\frac{\text{π}\alpha }{2}{g}_{1}\left(x\right)-\mathrm{sin}\frac{\text{π}\alpha }{2}{g}_{2}\left(x\right)
{G}_{2}\left(x\right)=\mathrm{cos}\frac{\text{π}\alpha }{2}{g}_{2}\left(x\right)+\mathrm{sin}\frac{\text{π}\alpha }{2}{g}_{1}\left(x\right)
\beta =1-\gamma
The following theorems show main result of this paper.
1<p<1+\frac{2\left(\alpha +\beta \right)}{\alpha N}
g\in {L}^{1}\left({ℝ}^{N}\right)
{\lambda }_{1}{\int }_{{ℝ}^{N}}{G}_{1}\left(x\right)\text{d}x>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{2}{\int }_{{ℝ}^{N}}{G}_{2}\left(x\right)\text{d}x>0,
then problem (1) admits no global weak solution.
1<p<1+\frac{\beta }{\alpha }
\chi \left(x\right)={\left({\int }_{{ℝ}^{N}}{\text{e}}^{-\sqrt{{N}^{2}+{|x|}^{2}}}\text{d}x\right)}^{-1}{\text{e}}^{-\sqrt{{N}^{2}+{|x|}^{2}}}
g\in {L}_{\left({ℝ}^{N}\right)}^{1}
{\lambda }_{1}{\int }_{{ℝ}^{N}}{G}_{1}\left(x\right)\chi \left(x\right)\text{d}x>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{2}{\int }_{{ℝ}^{N}}{G}_{2}\left(x\right)\chi \left(x\right)\text{d}x>0,
3. Proofs of Main Result
In this section, we prove blow-up results and global existence of mild solutions of (1).
Proof of Theorem 2.2. If
1<p<1+\frac{2\left(\alpha +\beta \right)}{\alpha N}
{\lambda }_{1}{\int }_{{ℝ}^{N}}{G}_{1}\left(x\right)\text{d}x>0
, we may as well suppose that
{\lambda }_{1}>0
{\int }_{{ℝ}^{N}}{G}_{1}\left(x\right)\text{d}x>0
\Phi \in {C}_{0}^{\infty }\left({ℝ}^{N}\right)
\Phi \left(s\right)=1
|s|\le 1
\Phi \left(s\right)=0
|s|>2
0\le \Phi \left(s\right)\le 1
T>0
{\phi }_{1}\left(x\right)={\left(\Phi \left({T}^{-\frac{\alpha }{2}}|x|\right)\right)}^{\frac{2p}{p-1}},\text{\hspace{0.17em}}{\phi }_{2}\left(t\right)={\left(1-\frac{t}{T}\right)}^{m},\text{\hspace{0.17em}}m\ge \mathrm{max}\left\{1,\frac{p\left(\alpha +\beta \right)}{p-1}\right\},\text{\hspace{0.17em}}t\in \left[0,T\right].
\phi \left(x,t\right)={}_{t}{}^{C}D{}_{T}^{\beta }{\phi }_{1}\left(x\right){\phi }_{2}\left(t\right)
. Assuming that u is a weak solution of (1), and since
\alpha +\beta <1
for some positive constant C independent of T. Then, by (4), (5) and Hölder inequality, we have
Since , we have . Therefore, if the solution of (1) exists globally, then taking , we obtain
which contradicts with the assumption.
For case , we have
Then by a similar proof as above, we can also obtain a contradiction.
Proof of Theorem 2.3. We only consider the case and , since other cases can be proved by a similar method. Take such that
and , . Let . Suppose that u is a bounded weak solution of (1), taking
and define , then using the definition of weak solution of (1) and since , we derive that
by (6) and dominated convergence theorem, let , we have
Hence, by Jensen’s inequality and (7), we have
Denoting , and , then we have
since , we get by taking , which contradicts with the
assumption. Therefore, if is a solution of (1), then .
Supported by NSF of China (11626132, 11601216).
Li, Y.N. and Zhang, Q.G. (2018) The Nonexistence of Global Solutions for a Time Fractional Schrödinger Equation with Nonlinear Memory. Journal of Applied Mathematics and Physics, 6, 1418-1424. https://doi.org/10.4236/jamp.2018.67118
1. Ikeda, M. and Wakasugi, Y. (2013) Small Data Blow-Up of L2-Solution for the Nonlinear Schrödinger Equation without Gauge Invariance. Differential Integral Equations, 26, 1275-1285.
2. Ikeda, M. and Inui, T. (2015) Small Data Blow-Up of L2 or H1-Solution for the Semilinear Schrodinger Equation without Gauge Invariance. Journal of Evolution Equations, 15, 1-11. https://doi.org/10.1007/s00028-015-0273-7
3. Ikeda, M. and Inui, T. (2015) Some Non-Existence Results for the Semilinear Schrödinger Equation without Gauge Invariance. Journal of Mathematical Analysis and Applications, 425, 758-773. https://doi.org/10.1016/j.jmaa.2015.01.003
4. Mitidieri, E. and Pohozaev, S.I. (2001) A Priori Estimates and Blow-Up of Solutions to Nonlinear Partial Differential Equations and Inequalities. Proceedings of the Steklov Institute of Mathematics, 234, 1-383.
5. Mainardi, F. (1994) On the Initial Value Problem for the Fractional Diffusion-Wave Equation. In Rionero, S. and Ruggeri, T., Eds., Waves and Stability in Continuous Media, World Scientific, Singapore, 246-251.
6. Nigmatullin, R.R. (1986) The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry. Physica Status Solidi, 133, 425-430. https://doi.org/10.1002/pssb.2221330150
7. Andrade, B. and Viana, A. (2017) On a Fractional Reaction-Diffusion Equation. Zeitschrift für angewandte Mathematik und Physik, 68, 59. https://doi.org/10.1007/s00033-017-0801-0
8. Li, Y.N. (2015) Regularity of Mild Solutions for Fractional Abstract Cauchy Problem with Order . Zeitschrift für angewandte Mathematik und Physik, 66, 3283-3298. https://doi.org/10.1007/s00033-015-0577-z
9. Li, Y.N., Sun, H.R. and Feng, Z.S. (2016) Fractional Abstract Cauchy Problem with Order . Dynamics of PDE, 13, 155-177.
10. Vergara, V. and Zacher, R. (2017) Stability, Instability, and Blowup for Time Fractional and Other Nonlocal in Time Semilinear Subdiffusion Equations. Journal of Evolution Equations, 17, 599-626. https://doi.org/10.1007/s00028-016-0370-2
11. Zhang, Q.G. and Sun, H.R. (2015) The Blow-Up and Global Existence of Solutions of Cauchy Problems for a Time Fractional Diffusion Equation. Topological Methods in Nonlinear Analysis, 46, 69-92. https://doi.org/10.12775/TMNA.2015.038
12. Zhang, Q.G., Sun, H.R. and Li, Y.N. (2017) The Nonexistence of Global Solutions for a Time Fractional Nonlinear Schrödinger Equation without Gauge Invariance. Applied Mathematics Letters, No. 64, 119-124. https://doi.org/10.1016/j.aml.2016.08.017
13. Cazenave, T., Dickstein, F. and Weissler, F.B. (2008) An Equation Whose Fujita Critical Exponent Is Not Given by Scaling. Nonlinear Analysis, 68, 862-874. https://doi.org/10.1016/j.na.2006.11.042
14. Fino, A.Z. and Kirane, M. (2012) Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation. Quarterly of Applied Mathematics, 70, 133-157. https://doi.org/10.1090/S0033-569X-2011-01246-9
15. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, Vol 204. Elsevier Science B.V., Amsterdam. https://doi.org/10.1016/S0304-0208(06)80001-0
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Outer mitochondrial membrane - cellbio
From cellbio
The outer mitochondrial membrane is the outer membrane of the mitochondrion, an organelle found in most eukaryotic cells. It is an example of a biological membrane. It comprises a lipid bilayer along with various integral membrane proteins embedded in that bilayer. It helps control the entry and exit of materials between the mitochondrion on the inside and the cytoplasm that surrounds it on the outside. The immediate inside side of the outer mitochondrial membrane is the intermembrane space.
The mitochondrion also has an inner mitochondrial membrane which is also a lipid bilayer; the intermembrane space separates the outer and inner mitochondrial membranes.
Type of organisms whose cells contain the outer mitochondrial membrane Same as the organisms whose cells contain mitochondria: eukaryotic cells only, including plant cells, animal cells, and the cells of protists and fungi
Type of cells within the organisms that contain the outer mitochondrial membrane Same as the cells that contain mitochondria: all cells except red blood cells in mammals (other vertebrates do have mitochondria in their red blood cells).
Number of outer mitochrondrial membranes per cell Same as the number of mitochondria: 1 to 1000s, depending on the energy needs of the cell
{\displaystyle 60-75}
angstrom thickness, compared with mitochondrial diameter of
{\displaystyle 0.5-1.0\mu m}
, so the thickness is about 1% of the diameter of the mitochondrion.
Location within mitochondrion The outer mitochondrial membrane fully encloses the mitochondrion.
What's on both sides of it The outside is the cytoplasm, i.e., the rest of the cell. The inside is the intermembrane space, that separates the outer and inner mitochondrial membranes.
Structural components Similar to any biological membrane, it has a lipid bilayer (comprising phospholipids) as well as large numbers of integral membrane proteins called porins.
Chemical constituents Phosopholipids and integral membrane proteins called porins; the ratio is about 1:1 by weight, similar to the cell membrane.
Evolutionary origin According to the endosymbiotic theory of mitochondrial origin, the mitochondrion descends from endosymbiotic prokaryotes inside the eukaryotic cell. The outer mitochondrial membrane correspondingly descends from a membrane created by the host cell to firewall the endosymbiont's access to the rest of the cell.
Control of the entry and exit of materials The outer mitochondrial membrane pretty freely allows small molecules to pass through, so the intermembrane space has a similar chemical composition as the cytosol. Large molecules are not allowed.
Retrieved from "https://cellbio.subwiki.org/w/index.php?title=Outer_mitochondrial_membrane&oldid=232"
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Pooled Credit Lines - Sublime Docs
Pooled credit lines allows a group of lenders to supply capital to borrower. In this case, the terms of the loan offering are initially set by the borrower. Once the request is created by the borrower, the loan enters a collection stage during which lenders comfortable with the terms of the raise can start depositing capital into the loan request. Pooled credit lines also allows idle capital in the pool as well as any collateral locked in by the borrower to be deployed on a passive yield generator such as Compound, ensuring full utilization of the funds. This also reduces costs for borrowers since they are not required to pay a maintenance fee on any undrawn balance, which is usually the case with lines of credit.
Creating a pooled credit line
A pooled credit line request needs to be created by the borrower. Following is the list of parameters that needs to be supplied during creation:
Borrow Amount: Total capital the borrower wants to raise, e.g. 5M USDC
Borrowed Asset: Asset the borrower wishes to borrow, e.g., USDC
Loan Duration: Duration of the loan period, e.g. 12 months
Interest Rate: The interest rate the borrower is willing to provide, e.g. 10%
Collateral Asset: Asset that will be used as collateral, e.g., WBTC
Minimum Collateral Ratio: The minimum collateral ratio that the borrower will maintain throughout the duration of the loan, e.g. 50% of the loan amount
Minimum Raise Target: The minimum amount collected which would be sufficient for the loan to go active. Thus, minimum raise target <= borrow amount. e.g. 2M USDC
Lender Verifier: Denotes the verifier by whom lenders willing to deposit capital into the pool must be verified, e.g. Twitter verifier. Note that this argument is optional - borrowers can keep participation as free for all
Collateral Savings Strategy: Savings strategy where any idle liquidity in the credit line will be deployed, e.g., all the idle USDC in the pooled credit line can be deployed to Compound
Borrowed Asset Savings Strategy: Savings strategy where any collateral locked in by the borrower will be deployed, e.g., all WBTC deposited by the borrower as collateral could be locked in Aave to earn interest
Grace Period Interest Rate: Interest rate that will be used for calculating interest during the grace period
Grace Period Duration: Grace period after the end of the loan duration in which the borrower can still repay their debt
Token transferability: If set as $true$, lenders in the loan can transfer their position to others by selling their LP tokens
The above list of parameters allows borrowers to create highly customizable offerings. Additionally, if the borrower wishes to set additional checks/restrict participation of lenders (e.g., membership to a private DAO, KYC/AML compliance, etc.), they can do so by choosing one of the existing verifiers or creating a new one that suites their requirements.
Supplying capital into a pooled credit line
Upon creating the request, the loan enters the collection stage during which lenders interested in the offering can begin depositing capital. Any additional measures, such as due diligence by lenders, whitelisting by verifiers must take place during the collection period.
Once a user supplies capital into an offering, they cannot withdraw the initial principal until the end of the loan period. The only exceptions are (i) if the request is cancelled by the borrower, (ii) the minimum raise target is not met, or (iii) the borrower is liquidated, in which case lenders receive any collateral posted by the borrower proportional to their contribution in the pool.
Borrowing from a pooled credit line
If the minimum raise target is met, the credit line enters the active stage during which the borrower can start drawing capital from it.
Once the credit line enters the active stage, all the funds collected in the request are deployed to the borrowed asset savings strategy to earn yield. Whenever the borrower wishes to borrow, the protocol attempts to pull the requested amount from the savings strategy. Borrowing from the credit line requires the borrower to lock in sufficient collateral such that the following condition is satisfied:
\frac{USD(\text{total collateral})}{USD(\text{principal + interest accrued})} \geq \text{minimum collateral ratio}
Borrowers can draw down capital from their pool in parts. Interests are only accrued on amount that the borrower has actively drawn down. For e.g., if a borrower has an active 1M USDC credit line from which they've borrowed 200k USDC, then the borrower is required to pay interest only on the 200k USDC that they've borrowed. This is because the rest of the capital is deployed onto a savings strategy.
Borrowers can repay their debt over multiple repayments before the end of the active stage of the loan. There are no fixed repayment schedules enforced by the smart contract, however, the borrower may share a pre-determined repayment schedule with the lenders through side agreements to ensure confidence. This also allows for sufficient flexibility for the borrower - for e.g., the borrower and the lenders can agree on leniency in case gas costs are extremely high near a repayment deadline. On the other hand, borrowers also have an incentive to tightly stick to pre-determined repayment scheduled, since deviating from it might create a negative impression which would hamper their future borrowing capabilities.
Any amount repaid by the borrower first counts towards repaying any interests accrued till the time of repayment. Once all outstanding interests have been repaid, further repayments by the borrower count towards repaying the principal. Repaying principal replenishes the borrower's borrowing limit, allowing them to drawn down from their credit line again.
Liquidation of the borrower's collateral can take place under two scenarios:
Borrower fails to repay debt by the end of the loan period: In this case, the loan enters a grace period, during which the borrower still has the option of repaying their loan. The borrower continues to accrue interest on their principal during the grace period according to the grace period interest rate. If the borrower fails to repay during the grace period as well, they're considered to have defaulted on their loan. Collateral posted by the borrower is redistributed amongst lenders proportional to their initial deposit.
Borrower's collateral ratio falls below the minimum requirements: Borrowers are required to maintain their collateral ratio above the minimum threshold at all times. Failing to do so makes their collateral liable for liquidation.
Liquidation of a given pooled credit line needs to be executed by one of the lenders of the underwater loan. This is done by calling the liquidate() function. Liquidating a loan allows lenders to take possession of the borrower's collateral proportional to their deposit in the pool, upto the value of their initial deposit. That is, the amount of collateral received by a lender upon liquidation is equal to:
\text{collateral received} = min(\frac{USD(\text{initial deposit})}{\text{collateral asset price in USD}}, \frac{\text{initial deposit by lender}}{\text{total deposit}}\times \text{total collateral})
Interest & principal withdrawal by lenders
Interests earned by lenders come from two sources: (i) interest paid by borrowers based on the principal they borrow, (ii) yield earn by idle capital on the borrowed asset savings strategy. Lenders have the option to withdraw either of them throughout the course of the loan period, as and when they're available. Note that while most savings strategies earn yield continuously, interest repayments by borrowers occur discretely. Thus, those can only be withdrawn by lenders when the borrower transfers them to the pool. Depending on the pre-determined arrangement between the two parties, these repayments can occur over regular intervals.
The initial principal deposited by lenders can only be withdrawn at the end of loan period.
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Equipment level - The RuneScape Wiki
(Redirected from Equipment experience)
This article is about the Invention mechanic. For the level of an item's combat properties, see Equipment tier.
Checking an augmented weapon; experience is displayed, with a pink bar to show progress to the next level.
Equipment level is an attribute of augmented items. Once an item has been augmented, it will gain experience and advance levels when it is used in combat or skilling. Levelling an item increases the experience and materials received when disassembling it. The maximum item level achievable is dependent on research.
2 Equipment experience
2.3 Skilling tables
3 Disassembly and siphoning
4 Perk benefits
Players can complete research to increase the level cap of their augmented items.
Research unlocked
4 Augmented item maximum level 5
27 Augmented item maximum level 10
Equipment experience[edit | edit source]
Equipment experience is gained by using augmented items. Much like skills, the item will level up when it reaches specific experience milestones. Equipment experience is received and stored to at least two decimal places, but is truncated to an integer when displayed in the interface and when logging out.
Items can reach level 5 at 4 Invention, level 10 at 27 Invention, level 15 at 60 Invention, and level 20 at 99 Invention. After reaching one of the mentioned milestone level, don't forget you have to research the item level cap. It is possible to gain item experience past the level cap without leveling the item.
Equipment experience is capped at 1,000,000.
Combat equipment[edit | edit source]
An augmented item receives a proportion of the base combat experience that the player earns when killing a monster. Unlike combat experience, equipment experience is awarded per hit. The total experience
{\displaystyle x}
that a monster granting
{\displaystyle y}
combat experience (excluding Constitution) gives for depleting its life points from full to 0 is:
{\displaystyle x={\begin{cases}0.06y\approx 6\%&{\text{2-handed weapons}}\\0.04y\approx 4\%&{\text{1-handed main-hand weapon}}\\0.04y\approx 4\%&{\text{Body- and leg-slot items}}\\0.02y\approx 2\%&{\text{Off-hand weapons and shields}}\end{cases}}}
Thus, for example, killing one abyssal demon gives:
661 experience to the skill being used
218.1 experience to Constitution
26.4 to any augmented armour and main-hand weapon you are using
13.2 to any augmented off-hand weapon or shield you are using
39.6 to any augmented two-handed weapon you are using
Equipment experience is awarded each time damage is dealt while wielding/wearing the augmented item(s). The amount received each hit is a proportion of the monster's total available experience relative to the damage of the hit - for example, a hit of 1700 damage to an abyssal demon (which has 8500 life points) grants 20% of the total experience for the demon (as noted above), as 1700 is 20% of 8500.
The enlightened perk increases the equipment experience gained for the item(s) it is added to.
Augmented tools receive a portion of the base experience received for gathering resources. Item experience
{\displaystyle x}
gained from a resource that gives
{\displaystyle y}
experience roughly follows:
{\displaystyle x={\begin{cases}{\frac {\lfloor 288\times y\rfloor }{1000}}\approx 28.8\%&{\text{Archaeology}}\\{\frac {\lfloor 118\times y\rfloor }{1000}}\approx 11.8\%&{\text{Fishing, Mining, Woodcutting}}\\{\frac {\lfloor 65\times y\rfloor }{1000}}\approx 6.5\%&{\text{Firemaking}}\\{\frac {\lfloor 78\times y\rfloor }{1000}}\approx 7.8\%&{\text{Smithing}}\\\end{cases}}}
Based on samples the augmented item experience has at least 3 digits not visible to the player, similar to skill experience which has 1 digit not visible to the player.
For example, each large crystal urchin fished grants 350 Fishing experience and 41.3 item experience.
Effects that increase the 'base experience' (i.e. do not count as 'bonus experience' for the popup) also increase the amount of equipment experience gained per action. Such effects include:
Doubled resource gain, via the juju woodcutting potion or fury shark outfit
Resource consumption for additional experience, including the Furnace perk and Crystallise spell
Percentage base experience increases from special aquarium decorations and similar
XP Capacitor 5000 grants double the base experience, provided that it is active.
The enlightened perk also increases the experience gained for each resource gathered.
Skilling urns do not add to the equipment level.
Skilling tables[edit | edit source]
Disassembly and siphoning[edit | edit source]
Levelling a piece of equipment from level 1 to level 10 increases the amount of Invention experience that is contained within it. This experience greatly increases each time the equipment's level increases. Although items can be levelled up to a maximum of level 20 at 99 Invention, further levels beyond 10 do not increase the Invention experience gained from disassembling. Additionally, Invention experience gained from siphoning also stops increasing after item level 12 is reached. For levels 2-10 and 12-20, item effects are unlocked for augmented equipment. Any previously unlocked item effects are carried over each time an item's level increases, unless stated otherwise. Levelling a tier 80 item would work as follows:
Disassembling an item gives different experience based on its tier, with the base value being for tier 80. Tier 90 items give 15% more experience; tier 70, 15% less.
{\displaystyle {\begin{aligned}{\text{XP}}&={\text{Base XP}}\times \left(1+0.015\times ({\text{Tier}}-80)\right)\\\\&={\text{Base XP}}\times {\frac {3\times {\text{Tier}}-40}{200}}\\\end{aligned}}}
While crystal tools are normally tier 70, using a crystal tool siphon grants experience as if they were tier 90 instead - around 35% more experience.
Perk benefits[edit | edit source]
When a piece of a equipment reaches level 20, certain perks on that equipment gain a multiplicative
{\displaystyle 10\%}
boosted chance to activate. This is shown by the on the equipment's tooltip or by checking the equipment. For example, if the Biting 4 perk is equipped and has a
{\displaystyle 8\%}
chance to activate, then at equipment level 20 the activation rate of Biting 4 becomes
{\displaystyle 8\%\times 1.1=8.8\%}
. Known perks that benefit from this boost are as follows:
Biting; Impatient; Scavenging; Spendthrift; Trophy-taker's
Absorbative; Biting; Crystal Shield; Devoted; Enhanced Devoted; Impatient; Lucky; Relentless; Scavenging; Trophy-taker's
Tool perks:
Breakdown; Butterfingers; Charitable; Fortune; Furnace; Honed; Imp Souled; Polishing; Prosper; Pyromaniac; Rapid; Refined; Tinker
Retrieved from ‘https://runescape.wiki/w/Equipment_level?oldid=35816680’
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Theoretical Investigation of Convergence of Max-SINR Algorithm in the MIMO Interference Network
Theoretical Investigation of Convergence of Max-SINR Algorithm in the MIMO Interference Network
Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Design of transceiver for interference channel (IC) is an important research area. Max-SINR algorithm is an iterative algorithm for multi-input multi-output (MIMO) IC. Nodes in the MIMO IC, work in a time division duplex mode, where half of them are equipped with M antennas while the others have N antennas. It is shown how the Max-SINR algorithm proposed by Gomadam et al. converges by considering an equivalent problem, i.e. a constrained maximization problem.
Converge, Interference Channel, MIMO, Max-SINR
To date, different approaches have been developed to address interference management. Beside the conventional methods for interference management [1] , a new method termed “interference alignment” (IA) has been proposed by the researchers. The basic idea behind the IA is to fit undesirable signals into a small portion of the signal space, observed by each receiver (interference subspace), and then leave the remaining signal space free of any interference for the desired signal (signal subspace). In [1] - [12] , and [13] , the authors implement the IA for different scenarios.
Transceiver for multi-input multi-output (MIMO) interference channel (IC) has been designed by progressive minimization of the leakage interference, Algorithm 1 in [4] . In this scheme, the IA is achieved only at very high SNRs. The Max-SINR algorithm, Algorithm 2 in [4] , is another approach to obtain IA. This technique shows significant improvements in terms of sum rate in the range of low-to-intermediate SNRs and achieves the IA at high SNR.
The algorithm 1 seeks perfect interference alignment. In particular, it seeks to create an interference-free subspace of the required number of dimensions that is designated as the desired signal subspace. However, Algorithm 1 makes no attempt to maximize the desired signal power within the desired signal subspace. In fact, Algorithm 1 does not depend at all on the direct channels through which the desired signal arrives at the intended receiver. Therefore, while the interference is eliminated within the desired space, no coherent combining gain (array gain) for the desired signal is obtained with Algorithm 1. While this is optimal as all signal powers approach infinity, it is not optimal in general at intermediate SNR values. Therefore, Algorithm 2 may be designed which will perform better than Algorithm 1 at intermediate SNR values.
Authors show that Algorithm 1 converges and convergence of Max-SINR is not presented. Since all the iterative algorithms have meaning when converges. No convergence proof has been presented in literature for Max SINR algorithm. In this paper, convergence issue of Max-SINR is investigated. It is shown how the Max-SINR algorithm proposed by Gomadam et al. converges by considering an equivalent problem, i.e. a constrained maximization problem.
Convergence of robust MMSE is shown in [14] theoretically. Total fraction of interference leakage to received signal parameter is used to show convergence numerically. To validate the correctness of the proposed proof, numerical convergence behavior of the Max SINR is compared with robust MMSE.
In a K-user MIMO IC, transmitter j and receiver k have M and N antennas, respectively. Independent symbols
{D}^{j}
with power P are sent by the jth transmitter. Channel matrices between transmitter j and receiver k is denoted by
{H}^{kj}
. The received signal at receiver k is expressed by
{Y}^{k}=\sum _{j=1}^{K}\text{ }\text{ }{H}^{kj}{X}^{j}+{Z}^{k}
{X}^{j}
M\times 1
signal vector transmitted by the transmitter j and
{Z}^{k}~CN\left(0,{N}_{0}I\right)
is additive white Gaussian noise (AWGN) vector. Beam-forming strategy is used based on the interference alignment. In particular, transmitter j precodes symbol vector by using the precoder matrix.
{V}^{j}
M\times {D}^{j}
precoder matrix. Columns of
{V}^{j}
{v}_{d}^{j}
, are unit norm vectors. Receiver k estimates the transmitted symbol vector
{s}^{k}
by using the interference suppression matrix
{U}^{k}
. The received signal is filtered by
{U}^{k}
\overline{{Y}^{k}}={U}^{k}{}^{†}{Y}^{k}
Each node works in a time division duplex (TDD) mode. At two consecutive time slots, first, nodes on the left-hand side send the data to the nodes on the right-hand side. Then the role of nodes is switched and the nodes on the left-hand side receive the data, as illustrated in Figure 1.
The relation between the original and reciprocal channel matrices is
\stackrel{←}{{H}^{jk}}={H}^{kj}{}^{†}
[4] . Since the receivers of the reciprocal channel play the roles of
original network’s transmitters and vice versa, then
\stackrel{←}{{V}^{k}}={U}^{k}
\stackrel{←}{{U}^{j}}={V}^{j}
According to the system model, the SINR value for the dth data stream at kth receiver is expressed by (where,
‖\text{ }.\text{ }‖
denotes the Euclidian norm.)
SIN{R}_{d}^{k}=\frac{P{‖{u}_{d}^{k}{}^{{}^{†}}{H}^{kk}{v}_{d}^{k}‖}^{2}}{P{\sum }_{j=1}^{K}{\sum }_{m=1}^{{D}^{j}}{‖{u}_{d}^{k}{}^{{}^{†}}{H}^{kj}{v}_{m}^{j}‖}^{2}-P{‖{u}_{d}^{k}{}^{{}^{†}}{H}^{kk}{v}_{d}^{k}‖}^{2}+{N}_{0}{‖{u}_{d}^{k}‖}^{2}}
Alternatively
SIN{R}_{d}^{k}
SIN{R}_{d}^{k}=\frac{{u}_{d}^{k}{}^{†}{T}_{d}^{k}{u}_{d}^{k}}{{u}_{d}^{k}{}^{†}\left[{S}^{k}-{T}_{d}^{k}+{N}_{0}I\right]{u}_{d}^{k}}
{S}^{k}=P{\sum }_{j=1}^{K}{\sum }_{m=1}^{{D}^{j}}{H}^{kj}{v}_{m}^{j}{v}_{m}^{j}{}^{{}^{†}}{H}^{kj}{}^{{}^{†}}
{T}_{d}^{k}=P{H}^{kk}{v}_{d}^{k}{v}_{d}^{k}{}^{{}^{†}}{H}^{kk}{}^{{}^{†}}
denote, respectively, the covariance matrix of all data streams observed by the receiver k and covariance matrix of the dth desirable data stream.
3. Proof of Convergence of Max-SINR Algorithm
Max-SINR algorithm is presented in algorithm 2 in [4] . It is repeated in Table 1.
Maximizing (3) over
{u}_{d}^{k}
can be written as follow
\mathrm{max}\frac{{u}_{d}^{k}{}^{{}^{†}}G{u}_{d}^{k}}{{u}_{d}^{k}{}^{{}^{†}}F{u}_{d}^{k}}
where matrices are
G={G}^{†}={T}_{d}^{k}\ge 0
F={F}^{†}={S}^{k}-{T}_{d}^{k}+{N}_{0}I>0
. It is shown in [15] that the optimization problem in (4) is equivalent to
\begin{array}{l}\mathrm{max}{u}_{d}^{k}{}^{{}^{†}}G{u}_{d}^{k},\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}{u}_{d}^{k}{}^{{}^{†}}F{u}_{d}^{k}=1.\end{array}
For the equivalent problem, i.e. constrained maximization in (5), Lagrangian function is
l\left({u}_{d}^{k},\lambda \right)={u}_{d}^{k}{}^{{}^{†}}G{u}_{d}^{k}+\lambda \left(1-{u}_{d}^{k}{}^{{}^{†}}F{u}_{d}^{k}\right)
. Lagrange conditions are
Table 1. Max SINR algorithm.
\frac{\partial l\left({u}_{d}^{k},\lambda \right)}{\partial {u}_{d}^{k}}=0
\frac{\partial l\left({u}_{d}^{k},\lambda \right)}{\partial \lambda }=0
. The solution is denoted by
{u}_{d}^{k}{}^{*}
and Lagrange
multiplier by
{\lambda }^{*}
. It is also shown in [15] that
{u}_{d}^{k}{}^{*}
is the eigenvector corresponding to the maximal eigenvalue of
{F}^{-1}G
{\lambda }^{*}
{u}_{d}^{k}{{}^{*}}^{{}^{†}}G{u}_{d}^{k}{}^{*}
Therefore, the unit vector that maximizes (3), is given by
{u}_{d}^{k}=\vartheta \left[{F}^{-1}G\right]
where operator
\vartheta \left[.\right]
denotes the eigenvector corresponding to the maximal eigenvalue of a matrix.
The convergence of the algorithm is proved by considering total Lagrangian function of all data streams in the network
{\sum }_{k=1}^{K}{\sum }_{d=1}^{{D}^{j}}l\left({u}_{d}^{k},\lambda \right)
. The metric is defined in (7). The function is unchanged in the original and reciprocal networks since the transmit and receive filters change their roles. Therefore, each step [4, Algorithm 2] in the algorithm increases the value of the function. This implies that the algorithm converges.
{\mathrm{max}}_{\begin{array}{l}\text{\hspace{0.17em}}{V}^{j}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{U}^{K}\\ \forall j\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}k\in \mathcal{K}\end{array}}metric={\sum }_{k=1}^{K}{\sum }_{d=1}^{{D}^{j}}l\left({u}_{d}^{k},\lambda \right)
{\mathrm{max}}_{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{U}^{K}\\ \forall k\in \mathcal{K}\end{array}}metric={\sum }_{k=1}^{K}{\sum }_{d=1}^{{D}^{j}}{\mathrm{max}}_{{u}_{d}^{k}}l\left({u}_{d}^{k},\lambda \right)
In other words, given
{V}^{j}\text{\hspace{0.17em}}\forall j\in \mathcal{K}
, Step 4 [4, Algorithm 2] increases the value of (7) over all possible choices of
{U}^{k}\text{\hspace{0.17em}}\forall k\in \mathcal{K}
. The filter
\stackrel{←}{{U}^{j}}
computed in Step 7 [4, Algorithm 2], based on
\stackrel{←}{{V}^{k}}={U}^{k}
, also maximizes the metric in the reciprocal channel (9).
{\mathrm{max}}_{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{←}{{U}^{j}}\\ \forall j\in \mathcal{K}\end{array}}\stackrel{←}{metric}={\sum }_{j=1}^{K}{\sum }_{d=1}^{{D}^{j}}\stackrel{←}{l}\left(\stackrel{←}{{u}_{d}^{j}},\stackrel{←}{\lambda }\right)={\sum }_{j=1}^{K}{\sum }_{d=1}^{{D}^{j}}{\stackrel{←}{{u}_{d}^{j}}}^{†}\stackrel{←}{G}\stackrel{←}{{u}_{d}^{j}}+\stackrel{←}{\lambda }\left(1-{\stackrel{←}{{u}_{d}^{j}}}^{†}\stackrel{←}{F}\stackrel{←}{{u}_{d}^{j}}\right)
\stackrel{←}{{V}^{k}}={U}^{k}
\stackrel{←}{{U}^{j}}={V}^{j}
, the metric remains unchanged in the original and reciprocal networks, according to following equation:
\begin{array}{l}\stackrel{←}{metric}={\sum }_{j=1}^{K}{\sum }_{d=1}^{{D}^{j}}{u}_{d}^{j}{}^{{}^{†}}{T}_{d}^{j}{u}_{d}^{j}+{\sum }_{j=1}^{K}{\sum }_{d=1}^{{D}^{j}}{\lambda }_{d}^{j}\left(1+{u}_{d}^{j}{}^{{}^{†}}\left[{T}_{d}^{j}-{N}_{0}I\right]{u}_{d}^{j}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-P{\sum }_{j=1}^{K}{\sum }_{d=1}^{{D}^{j}}{\sum }_{k=1}^{K}{\sum }_{m=1}^{{D}^{j}}{\lambda }_{d}^{j}{u}_{m}^{k}{}^{{}^{†}}{H}^{kj}{v}_{d}^{j}{v}_{d}^{j}{}^{{}^{†}}{H}^{kj}{}^{{}^{†}}{u}_{m}^{k}=metric\end{array}
Therefore, Step 7 [4, Algorithm 2] also can increase the value of (7). Since the value of (7) is monotonically increased after every iteration, convergence of the algorithm is guaranteed.
Simulation results to validate the correctness of the proposed proof are based on fraction of interference leakage to the received signal parameter [14] . Figure 2 and Figure 3 show total fraction of interference leakage to received signal. Since convergence of robust MMSE is shown in [14] , its total parameter is plotted in Figure 2 and Figure 3 as well. Convergence behavior of the algorithm in comparison with robust MMSE implies a convergent scheme.
Figure 2. Sum of fraction of interference leakage to received signal at each user.
M=N=3
D=1
K=4
interference network with
{\sigma }^{2}=0.1
\frac{P}{{N}_{0}}=10\text{\hspace{0.17em}}\text{dB}
M=3
N=4
D=2
K=2
{\sigma }^{2}=0.1
\frac{P}{{N}_{0}}=10\text{\hspace{0.17em}}\text{dB}
Dalir, A. and Aghaeinia, H. (2018) Theoretical Investigation of Convergence of Max-SINR Algorithm in the MIMO Interference Network. Journal of Computer and Communications, 6, 31-37. https://doi.org/10.4236/jcc.2018.64002
1. Jafar, S. and Fakhereddin, M. (2007) Degrees of Freedom for the MIMO Interference Channel. IEEE Transactions on Information Theory, 53, 2637-2642. https://doi.org/10.1109/TIT.2007.899557
2. Gou, T. and Jafar, S.A. (2010) Degrees of Freedom of the K User M × N MIMO Interference Channel. IEEE Transactions on Information Theory, 56, 6040-6057. https://doi.org/10.1109/TIT.2010.2080830
3. Cadambe, V. and Jafar, S.A. (2008) Interference Alignment and the Degrees of Freedom of the K User Interference Channel. IEEE Transactions on Information Theory, 54, 3425-3441. https://doi.org/10.1109/TIT.2008.926344
4. Gomadam, K., Cadambe, V.R. and Jafar, S.A. (2011) A Distributed Numerical Approach to Interference Alignment and Applications to Wireless Interference Networks. IEEE Transactions on Information Theory, 57, 3309-3322. https://doi.org/10.1109/TIT.2011.2142270
5. Zhu, B., Ge, J., Li, J. and Sun, C. (2012) Subspace Optimization-Based Iterative Interference Alignment Algorithm on the Grassmann Manifold. IET Communications, 6, 3084-3090. https://doi.org/10.1049/iet-com.2012.0467
6. Yu, H. and Sung, Y. (2010) Least Squares Approach to Joint Beam Design for Interference Alignment in Multiuser Multi-Input Multi-Output Interference Channels. IEEE Transactions on Signal Processing, 58, 4960-4966. https://doi.org/10.1109/TSP.2010.2051155
7. Papailiopoulos, D.S. and Dimakis, A.G. (2012) Interference Alignment as a Rank Constrained Rank Minimization. IEEE Transactions on Signal Processing, 60, 4278-4288. https://doi.org/10.1109/TSP.2012.2197393
8. Du, H., Ratnarajah, T., Sellathurai, M. and Papadias, C.B. (2013) Reweighted Nuclear Norm Approach for Interference Alignment. IEEE Transactions on Communications, 61, 3754-3765. https://doi.org/10.1109/TCOMM.2013.071813.130065
9. Kumar, K.R. and Xue, F. (2010) An Iterative Algorithm for Joint Signal and Interference Alignment. 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), 13-18 June 2010, Austin, TX, 2293-2297. https://doi.org/10.1109/ISIT.2010.5513646
10. Schmidt, D.A., Shi, C., Berry, R.A., Honig, M.L. and Utschick, W. (2009) Minimum Mean Squared Error Interference Alignment. 2009 Conference Record of the 43 Asilomar Conference on Signals, Systems and Computers, 1-4 November 2009, Pacific Grove, CA, 1106-1110. https://doi.org/10.1109/ACSSC.2009.5470055
11. Santamaria, I., Gonzalez, O., Heath, R.W. and Peters, S.W. (2010) Maximum Sum-Rate Interference Alignment Algorithms for MIMO Channels. 2010 IEEE Global Telecommunications Conference (GLOBECOM 2010), 6-10 December 2010, Miami, FL, 1-6. https://doi.org/10.1109/GLOCOM.2010.5683919
12. Gao, H., Leithon, J., Yuen, C. and Suraweera, H.A. (2013) New Uplink Opportunistic Interference Alignment: An Active Alignment Approach. 2013 IEEE Wireless Communications and Networking Conference (WCNC), 7-10 April 2013, Shanghai, 3099-3104. https://doi.org/10.1109/WCNC.2013.6555057
13. Peters, S.W. and Heath Jr., R.W. (2009) Interference Alignment via Alternating Minimization. 2009 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2009), 19-24 April 2009, Taipei, 2445-2448. https://doi.org/10.1109/ICASSP.2009.4960116
14. Shen, H., Li, B., Tao, M. and Luo, Y. (2010) The New Interference Alignment Scheme for the MIMO Interference Channel. 2010 IEEE Wireless Communications and Networking Conference (WCNC), 18-21 April 2010, Sydney. https://doi.org/10.1109/WCNC.2010.5506770
15. Chong, E.K.P. and Zak, S.H. (2001) An Introduction to Optimization. John Wiley & Sons, Hoboken, 382-383.
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Correspondence to: #E-mail: doyoungk@kyungnam.ac.kr, TEL: +82-55-249-2685
Cryogenic machining, Milling, Compacted graphite iron, Machinability, Surface roughness
극저온 가공, 밀링, CGI 주철, 절삭성, 표면 조도
\left[\begin{array}{c}{\mathrm{F}}_{\mathrm{x}}\left(\mathrm{\phi }\right)\\ {\mathrm{F}}_{\mathrm{y}}\left(\mathrm{\phi }\right)\\ {\mathrm{F}}_{\mathrm{z}}\left(\mathrm{\phi }\right)\end{array}\right]=\left[\begin{array}{ccc}\mathrm{sin}\left(\mathrm{\phi }\right)& -\mathrm{cos}\left(\mathrm{\phi }\right)& 0\\ -\mathrm{cos}\left(\mathrm{\phi }\right)& -\mathrm{sin}\left(\mathrm{\phi }\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathrm{P}}_{1}\left(\mathrm{\phi }\right)\\ {\mathrm{P}}_{2}\left(\mathrm{\phi }\right)\\ {\mathrm{P}}_{3}\left(\mathrm{\phi }\right)\end{array}\right]
Sahm, A., Abele, E., and Schulz, H., “Machining of Compacted Graphite Iron (CGI),” Materialwissenschaft und Werkstofftechnik: Entwicklung, Fertigung, Prüfung, Eigenschaften und Anwendungen Technischer Werkstoffe, Vol. 33, No. 9, pp. 501-506, 2002. [https://doi.org/10.1002/1521-4052(200209)33:9<501::AID-MAWE501>3.0.CO;2-W]
Abele, E., Sahm, A., and Schulz, H., “Wear Mechanism When Machining Compacted Graphite Iron,” CIRP Annals, Vol. 51, No. 1, pp. 53-56, 2002. [https://doi.org/10.1016/S0007-8506(07)61464-4]
Dawson, S. and Hang, F., “Compacted Graphite Iron-A Material Solution for Modern Diesel Engine Cylinder Blocks and Heads,” China Foundry, Vol. 6, No. 3, pp. 241-246, 2009.
Dawson, S. and Schroeder, T., “Practical Applications for Compacted Graphite Iron,” AFS Transactions, Vol. 47, No. 5, pp. 1-9, 2004.
Tasdelen, B., Escursell, M., Grenmyr, G., and Nyborg, L., “Machining of Gray Cast Irons and Compacted Graphite Iron,” Proc. of the Swedish Production Symposium, pp. 1-6, 2007.
Altintas, Y., Eynian, M., and Onozuka, H., “Identification of Dynamic Cutting Force Coefficients and Chatter Stability with Process Damping,” CIRP Annals, Vol. 57, No. 1, pp. 371-374, 2008. [https://doi.org/10.1016/j.cirp.2008.03.048]
Dhar, N., Paul, S., and Chattopadhyay, A., “The Influence of Cryogenic Cooling on Tool Wear, Dimensional Accuracy and Surface Finish in Turning AISI 1040 and E4340C Steels,” Wear, Vol. 249, Nos. 10-11, pp. 932-942, 2001. [https://doi.org/10.1016/S0043-1648(01)00825-0]
Dhar, N. and Kamruzzaman, M., “Cutting Temperature, Tool Wear, Surface Roughness and Dimensional Deviation in Turning AISI-4037 Steel under Cryogenic Condition,” International Journal of machine tools and manufacture, Vol. 47, No. 5, pp. 754-759, 2007. [https://doi.org/10.1016/j.ijmachtools.2006.09.018]
Wang, Z. and Rajurkar, K., “Cryogenic Machining of Hard-to-Cut Materials,” Wear, Vol. 239, No. 2, pp. 168-175, 2000. [https://doi.org/10.1016/S0043-1648(99)00361-0]
Wang, Z., Rajurkar, K. P., Fan, J., Lei, S., Shin, Y., et al., “Hybrid Machining of Inconel 718,” International Journal of Machine Tools and Manufacture, Vol. 43, No. 13, pp. 1391-1396, 2003. [https://doi.org/10.1016/S0890-6955(03)00134-2]
Suhaimi, M. A., Park, K.-H., Yang, G.-D., Sharif, S., and Kim, D.-W., “Effect of Cryogenic High-Speed Milling of Compacted Graphite Iron Using Indirect Spray System,” The International Journal of Advanced Manufacturing Technology, Vol. 99, No. 9, pp. 2149-2157, 2018. [https://doi.org/10.1007/s00170-018-2213-5]
Ding, F., Wang, C., Lin, H., Li, S., Zheng, L., et al., “Research on Machining Compacted Graphite Iron under Oil-on-Water Cooling and Lubrication Conditions based on Modified Material Model,” The International Journal of Advanced Manufacturing Technology, Vol. 105, No. 12, pp. 5061-5079, 2019. [https://doi.org/10.1007/s00170-019-04543-y]
Suhaimi, M. A., Park, K.-H., Sharif, S., Kim, D.-W., and Mohruni, A. S., “Evaluation of Cutting Force and Surface Roughness in High-Speed Milling of Compacted Graphite Iron,” Proc. of the MATEC Web of Conferences, Paper No. 03016, 2017. [https://doi.org/10.1051/matecconf/201710103016]
Kim, D. Y., Kim, D. M., and Park, H. W., “Study on Characteristics of Cryogenic Machining Process of Titanium Alloy at a Low Cutting Speed,” Vol. 34, No. 4, pp. 237-241, 2017. [https://doi.org/10.7736/KSPE.2017.34.4.237]
Courbon, C., Pusavec, F., Dumont, F., Rech, J., and Kopac, J., “Tribological Behaviour of Ti6Al4V and Inconel718 under Dry and Cryogenic Conditions-Application to the Context of Machining with Carbide Tools,” Tribology International, Vol. 66, pp. 72-82, 2013. [https://doi.org/10.1016/j.triboint.2013.04.010]
Hong, S. Y., Ding, Y., and Jeong, J., “Experimental Evaluation of Friction Coefficient and Liquid Nitrogen Lubrication Effect in Cryogenic Machining,” Machining Science and Technology, Vol. 6, No. 2, pp. 235-250, 2002. [https://doi.org/10.1081/MST-120005958]
Hong, S. Y., Ding, Y., and Jeong, W.-C., “Friction and Cutting Forces in Cryogenic Machining of Ti-6Al-4V,” International Journal of Machine Tools and Manufacture, Vol. 41, No. 15, pp. 2271-2285, 2001. [https://doi.org/10.1016/S0890-6955(01)00029-3]
Bermingham, M., Kirsch, J., Sun, S., Palanisamy, S., and Dargusch, M., “New Observations on Tool Life, Cutting Forces and Chip Morphology in Cryogenic Machining Ti-6Al-4V,” International Journal of Machine Tools and Manufacture, Vol. 51, No. 6, pp. 500-511, 2011. [https://doi.org/10.1016/j.ijmachtools.2011.02.009]
Kuzu, A. T., Bijanzad, A., and Bakkal, M., “Experimental Investigations of Machinability in the Turning of Compacted Graphite Iron Using Minimum Quantity Lubrication,” Machining Science and Technology, Vol. 19, No. 4, pp. 559-576, 2015. [https://doi.org/10.1080/10910344.2015.1085313]
Guo, Y., Mann, J., Yeung, H., and Chandrasekar, S., “Enhancing Tool Life in High-Speed Machining of Compacted Graphite Iron (CGI) Using Controlled Modulation,” Tribology Letters, Vol. 47, No. 1, pp. 103-111, 2012. [https://doi.org/10.1007/s11249-012-9966-z]
Magnusson, A. and Baldwin J, W., “Low Temperature Brittleness,” Journal of the Mechanics and Physics of Solids, Vol. 5, No. 3, pp. 172-181, 1957. [https://doi.org/10.1016/0022-5096(57)90003-0]
Panin, V. E., Derevyagina, L., Lemeshev, N., Korznikov, A., Panin, A. V., et al., “On the Nature of Low-Temperature Brittleness of BCC Steels,” Physical Mesomechanics, Vol. 17, No. 2, pp. 89-96, 2014. [https://doi.org/10.1134/S1029959914020015]
Chen, R. and Yeun, W., “Review of the High-Temperature Oxidation of Iron and Carbon Steels in Air or Oxygen,” Oxidation of Metals, Vol. 59, No. 5, pp. 433-468, 2003. [https://doi.org/10.1023/A:1023685905159]
Bertrand, N., Desgranges, C., Poquillon, D., Lafont, M.-C., and Monceau, D., “Iron Oxidation at Low Temperature (260-500 C) in Air and the Effect of Water Vapor,” Oxidation of Metals, Vol. 73, No. 1, pp. 139-162, 2010. [https://doi.org/10.1007/s11085-009-9171-0]
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User:The Illusional Ministry/test - Wikisource, the free online library
User:The Illusional Ministry/test
Introduction to Bhagavadgîtâ
English translation by Kāshināth Trimbak Telang (1850-1893) in 1882 (Volume 8, Sacred Books of the East).
567659User:The Illusional Ministry — Introduction to Bhagavadgîtâ
{\displaystyle {\mathfrak {Literary}}}
{\displaystyle {\mathfrak {Criticism,}}}
{\displaystyle {\mathfrak {Discussion,}}}
{\displaystyle {\mathfrak {and}}}
{\displaystyle {\mathfrak {Information.}}}
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Hc Verma I for Class 12 Science Physics Chapter 20 - Dispersion And Spectra
Hc Verma I Solutions for Class 12 Science Physics Chapter 20 Dispersion And Spectra are provided here with simple step-by-step explanations. These solutions for Dispersion And Spectra are extremely popular among Class 12 Science students for Physics Dispersion And Spectra Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Hc Verma I Book of Class 12 Science Physics Chapter 20 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Hc Verma I Solutions. All Hc Verma I Solutions for class Class 12 Science Physics are prepared by experts and are 100% accurate.
Dispersive power depends on angular deviation, and angular deviation is valid only for a small refracting angle and a small angle of incidence. Therefore, dispersive power is not valid for a prism of large refracting angle. It is also not valid for a glass slab or a glass sphere, as it has a large refracting angle.
No, it cannot be negative, as the refractive index for violet light is always greater than that for red light. Also, refractive index is inversely proportional to
{\lambda }^{2}
. The sign of ω will be positive, as
\mu
is still greater than 1 and as
{\mu }_{v}>{\mu }_{r}
No, it is not possible even when prisms are be combined with their refractive angle reversed with respect to each other. There will be at least a net deviation and dispersion equal to the dispersion and deviation produced by a single prism.
No, monochromatic light cannot be used to produce a pure spectrum. A spectrum is produced when a light of different wavelengths is deviated through different angles and gets separated. Monochromatic light, on the other hand, has a single wavelength.
Yes, the focal length of a lens depends on the colour of light.
According to lens-maker's formula,
\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right)
Here, f is the focal length, μ is the refractive index, R is the radius of curvature of lens.
The refractive index (μ) depends on the inverse of square of wavelength.
The focal length of a mirror is independent of the colour of light.
A rainbow can be produced using a prism. Another way of producing a rainbow is to dip a mirror inside water, keeping it inclined along the wall of a tumbler. The light coming from water after reflecting from the mirror will give a rainbow.
(a) increases if the average refractive index increases
If μ is the average refractive index and A is the angle of prism, then the angular dispersion produced by the prism is given by
\delta =\left(\mu -1\right)A
For the crown glass, we have:
Refractive index for red rays = μr
Refractive index for yellow rays = μy
Refractive index for violet rays = μv
For the flint glass, we have:
Refractive index for red rays = μ'r
Refractive index for yellow rays = μ'y
Refractive index for violet rays = μ'v
Let δcy and δfy be the angles of deviation produced by the crown and flint prisms for the yellow light.
Total deviation produced by the prism combination for yellow rays:
δy = δcy − δfy
= 2δcy − δfy
=2(μcy + 1)A − (μfy − 1)A'
Angular dispersion produced by the combination is given by
δv − δr = [(μvc − 1)A − (μvf − 1)A' + (μvc − 1)A −
\left[\left({\mu }_{rc}-1\right)A-\left({\mu }_{rf}-1\right)A\text{'}+\left({\mu }_{rc}-1\right)A\right]
μvc = Refractive index for the violet colour of the crown glass
μvf = Refractive index for the violet colour of the flint glass
{\mu }_{rc}
= Refractive index for the red colour of the crown glass
{\mu }_{rf}
= Refractive index for the red colour of the flint glass
δv − δr = 2(μvc −1)A − (μvf − 1)A'
(a) For zero angular dispersion, we have:
δt − δt = 0 = 2(μvc −1)A − (μvf − 1)A'
⇒ \frac{A\text{'}}{A}=\frac{2\left({\mu }_{\mathrm{vf}}-1\right)}{\left({\mu }_{vc}-1\right)}\phantom{\rule{0ex}{0ex}}=\frac{2\left({\mu }_{r}-{\mu }_{r}\right)}{\left({\mu }_{r}-\mu \right)}
(b) For zero deviation in the yellow ray, δy = 0.
⇒ 2(μcy − 1)A = (μfy − 1)A
⇒ \frac{A\text{'}}{A}=\frac{2\left({\mu }_{\mathrm{cy}}-1\right)}{\left({\mu }_{\mathrm{fy}}-1\right)}\phantom{\rule{0ex}{0ex}}=\frac{2\left({\mu }_{y}-1\right)}{\left(\mu {\text{'}}_{y}-1\right)}
Refractive index of the flint glass, μf = 1.620
Refractive index of the crown glass, μc = 1.518
Refractive angle of the flint prism, Af = 6°
Let the refractive angle of the crown prism be Ac.
For the net deviation of the mean ray to be zero,
Deviation by the flint prism = Deviation by the crown prism
i.e., (μf − 1)Af = (μc − 1)Ae
⇒{A}_{c}=\left(\frac{{\mu }_{\mathrm{f}}-1}{{\mu }_{\mathrm{e}}-1}\right){A}_{\mathrm{f}}
⇒{A}_{\mathrm{c}}=\left(\frac{1.620-1}{1.518-1}\right)×6.0°=7.2°
Thus, the refracting angle of the crown prism is 7.2
°
Refractive index for red light, μr = 1.56
Refractive index for yellow light, μy = 1.60
Refractive index for violet light, μv = 1.68
Angle of prism, A = 6°
(a) Dispersive power
\left(\omega \right)
\omega =\frac{{\mu }_{v}-{u}_{r}}{{\mu }_{y}-1}\phantom{\rule{0ex}{0ex}}\mathrm{On} \mathrm{substituting} \mathrm{the} \mathrm{values} \mathrm{in} \mathrm{the} \mathrm{above} \mathrm{formula}, \mathrm{we} \mathrm{get}:\phantom{\rule{0ex}{0ex}}\omega =\frac{\left(1.68-1.56\right)}{\left(1.60-1\right)}
=\frac{0.12}{0.60}=0.2
(b) Angular dispersion = (μv − μr)A
=(0.12) × 6° = 0.72°
Thus, the angular dispersion produced by the thin prism is 0.72°.
Focal lengths of the convex lens:
For red rays,
{f}_{r}=100 \mathrm{cm}
For yellow rays,
{f}_{y}=98 \mathrm{cm}
For violet rays,
{f}_{v}=96 \mathrm{cm}
{\mu }_{r}
= Refractive index for the red colour
{\mu }_{y}
= Refractive index for the yellow colour
{\mu }_{v}
= Refractive index for the violet colour
Focal length of a lens
\left(f\right)
\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right)
\mu
is the refractive index and R1 and R2 are the radii of curvatures of the lens.
\left(\mu -1\right)=\frac{1}{f}×\frac{1}{\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right)}\phantom{\rule{0ex}{0ex}}⇒ \left(\mu -1\right)=\frac{k}{f} \left[k=\frac{1}{\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right)}\right]
{\mu }_{r}-1=\frac{k}{100}
{\mathrm{\mu }}_{y}-1=\frac{k}{98}
{\mathrm{\mu }}_{v}-1=\frac{k}{96}
Dispersive power (ω) is given by
\omega =\frac{{\mathrm{\mu }}_{v}-{\mathrm{\mu }}_{r}}{{\mathrm{\mu }}_{y}-1}\phantom{\rule{0ex}{0ex}}\mathrm{Or},\omega =\frac{\left({\mathrm{\mu }}_{v}-1\right)-\left({\mathrm{\mu }}_{r}-1\right)}{\left({\mathrm{\mu }}_{y}-1\right)}
\omega =\frac{\frac{k}{96}-\frac{k}{100}}{\frac{k}{98}}=\frac{98×4}{9600}\phantom{\rule{0ex}{0ex}}⇒\omega =0.0408
Thus, the dispersive power of the material of the lens is 0.048.
Difference in the refractive indices of violet and red lights = 0.014
Let μv and μr be the refractive indices of violet and red colours.
μv − μr = 0.014
Real depth of the newspaper = 2.00 cm
Apparent depth of the newspaper = 1.32 cm
\mathrm{Refractive} \mathrm{index}=\frac{\mathrm{Real} \mathrm{depth}}{\mathrm{Apparent} \mathrm{depth}}\phantom{\rule{0ex}{0ex}}\therefore \mathrm{Refractive} \mathrm{index} \mathrm{for} \mathrm{yellow} \mathrm{light} \left({\mu }_{\mathrm{y}}\right) \mathrm{is} \mathrm{given} \mathrm{by}\phantom{\rule{0ex}{0ex}}{\mu }_{y}=\frac{2.00}{1.32}=1.515
\mathrm{Also},\phantom{\rule{0ex}{0ex}}\mathrm{Dispersive} \mathrm{power}, \omega =\frac{{\mu }_{v}-{\mu }_{r}}{{\mu }_{y}-1}\phantom{\rule{0ex}{0ex}} =\frac{0.014}{1.515-1}
\mathrm{Or}, \omega =\frac{0.014}{0.515}=0.027
Thus, the dispersive power of the material is 0.027.
The refractive indices for red and yellow lights are μr = 1.61 and μy = 1.65, respectively.
Dispersive power, ω = 0.07
Angle of minimum deviation, δy = 4°
\mathrm{Now}, \mathrm{using} \mathrm{the} \mathrm{relation} \omega =\frac{{\mu }_{v}-{\mu }_{r}}{{\mu }_{y}-1}, \mathrm{we} \mathrm{get}:\phantom{\rule{0ex}{0ex}}⇒0.07=\frac{1.65-1.61}{{\mu }_{y}-1}
⇒{\mu }_{y}-1=\frac{0.04}{0.07}=\frac{4}{7}
Let the angle of the prism be A.
Angle of minimum deviation, δ = (μ − 1)A
⇒A=\frac{{\delta }_{y}}{{\mu }_{y}-1}=\frac{4}{\left(\frac{4}{7}\right)}=7°
Thus, the angle of the prism is 7
°
Minimum deviations suffered by
Red beam, δr = 38.4°
Yellow beam, δy = 38.7°
Violet beam, δv = 39.2°
If A is the angle of prism having refractive index μ, then the angle of minimum deviation is given by
\delta =\left(\mu -1\right)A
⇒
\left(\mu -1\right)=\frac{\delta }{A}
Dispersive power
\left(\omega \right)
\omega =\frac{{\mu }_{v}-{\mu }_{r}}{{\mu }_{y}-1}\phantom{\rule{0ex}{0ex}} =\frac{\left({\mu }_{v}-1\right)-\left({\mu }_{r}-1\right)}{\left({\mu }_{y}-1\right)}
\omega =\frac{\frac{{\delta }_{v}}{A}-\frac{{\delta }_{r}}{A}}{\frac{{\delta }_{y}}{A}}
⇒\omega =\frac{{\delta }_{v}-{\delta }_{r}}{{\delta }_{y}}=\frac{\left(39.2\right)-\left(38.4\right)}{\left(38.7\right)}\phantom{\rule{0ex}{0ex}}
⇒\omega =\frac{\left(0.8\right)}{38.7}=0.0206
So, the dispersive power of the medium is 0.0206.
Let A be the angle of the prisms.
Refractive indices of the prisms for violet light, μ1 = 1.52 and μ2 = 1.62
Angle of deviation, δ = 1.0°
As the prisms are oppositely directed, the angle of deviation is given by
δ = (μ2 − 1)A − (μ1 − 1)A
δ = (μ2 −μ1 )A
A=\frac{\delta }{{\mu }_{2}-{\mu }_{1}}=\frac{1}{\left(1.62\right)-\left(1.52\right)}=\frac{1}{0.1}\phantom{\rule{0ex}{0ex}}⇒A=10°
So, the angle of the prisms is 10∘.
If μ is the refractive index and A is the angle of prism, then the angular dispersion produced by the prism will be given by
\delta =\left(\mu -1\right)A
Because the relative refractive index of glass with respect to water is small compared to the refractive of glass with respect to air, the dispersive power of the glass prism is more in air than that in water.
In combination (refractive angles of prisms reversed with respect to each other), the deviations through two prisms cancel out each other and the net deviation is due to the third prism only.
(d) Both A and B are correct.
Because line spectra contain wavelengths that are absorbed by atoms and band spectra contain bunch wavelengths that are absorbed by molecules, both statements are correct.
Focal length is inversely proportional to refractive index and refractive index is inversely proportional to
{\lambda }^{2}
. So, keeping other parameters the same, we can say:
f\propto \frac{1}{{\lambda }^{2}} \left(\because {\lambda }_{r}<{\lambda }_{v}\right)\phantom{\rule{0ex}{0ex}}
∴ fv < fr
(b) The emergent beam is white.
(c) The light inside the slab is split into different colours.
White light will split into different colours inside the glass slab because the value of refractive index is different for different wavelengths of light; thus, they suffer different deviations. But the emergent light will be white light. As the faces of the glass slide are parallel, the emerging lights of different wavelengths will reunite after refraction.
(a) have dispersion without average deviation
(b) have deviation without dispersion
(c) have both dispersion and average deviation
Consider the case of prisms combined such that the refractive angles are reversed w.r.t. each other. Then, the net deviation of the yellow ray will be
{\delta }_{y}=\left({\mu }_{y}-1\right)A-\left({\mu }_{y}\text{'}-1\right)A\text{'}
And, the net angular dispersion will be
{\delta }_{y}-{\delta }_{r}=\left({\mu }_{y}-1\right)A\left(\omega -\omega \text{'}\right)
Thus, by choosing appropriate conditions, we can have the above mentioned cases.
(d) allows a more parallel beam when it passes through the lens
To produce a pure spectrum, a parallel light beam is required to be incident on the dispersing element. So, the incident light is passed through a narrow slit placed in the focal plane of an achromatic lens.
(c) Chromatic aberration
The focal length, power and chromatic aberration are dependent on the refractive index of the lens, which itself is dependent on the wavelength of the light.
(b) The focal length of a converging lens
(d) The focal length of a diverging lens
The focal length of a lens is inversely proportional to the refractive index of the lens and the refractive index of the lens is inversely proportional to the square of wavelength. Therefore, the focal length is directly dependent on wavelength; it increases when the wavelength is increased.
For crown glass, we have:
Refractive index for red colour, μcr = 1.515
Refractive index for violet colour, μcv = 1.525
For flint glass, we have:
Refractive index for red colour, μfr = 1.612
Refractive index for violet colour,μfv = 1.632
Refracting angle, A = 5°
δc = Angle of deviation for crown glass
δf = Angle of deviation for flint glass
As prisms are similarly directed and placed in contact with each other, the total deviation produced
\left(\delta \right)
δ = δc + δf
{\mu }_{c}
– 1)A + (
{\mu }_{f}
– 1)A
{\mu }_{c}
{\mu }_{f}
For violet light, δv = (μcv + μfv – 2)A
For red light, δr = (μcr + μfr – 2)A
Angular dispersion of the combination:
δv – δr = (μcv + μfv – 2)A – (μcr + μfr – 2)A
= (μcv + μfv – μcr – μfr) A
= (1.525 + 1.632 – 1.515 – 1.612)5
So, the angular dispersion produced by the combination is 0.15°.
For the first prism,
Angle of prism, A' = 6°
Angle of deviation, ω' = 0.07
Refractive index for yellow colour, μ'y = 1.50
For the second prism,
Angle of deviation, ω = 0.08
Refractive index for yellow colour, μy= 1.60
Let the angle of prism for the second prism be A.
The prism must be oppositely directed, as the combination produces no deviation in the mean ray.
(a) The deviation of the mean ray is zero.
δy = (μy – 1)A – (μ'y – 1)A' = 0
\therefore
(1.60 – 1)A = (1.50 – 1)A'
\frac{0.50×6°}{0.60}=5°
(b) Net angular dispersion on passing a beam of white light:
(μy – 1)ωA – (μy – 1)ω'A'
⇒ (1.60 – 1)(0.08)(5°) – (1.50 – 1)(0.07)(6°)
⇒ 0.24° – 0.21° = 0.03°
(c) For the prisms directed similarly, the net deviation in the mean ray is given by
δy = (μy – 1)A + (μy – 1)A'
= (1.60 – 1)5° + (1.50 – 1)6°
= 3° + 3° = 6°
(d) For the prisms directed similarly, angular dispersion is given by
δv – δr = (μy – 1)ωA – (μy – 1)ω'A'
= 0.24° + 0.21°
If μ'v and μ'r are the refractive indices of material M1, then we have:
μ'v – μ'r = 0.014
If μv and μr are the refractive indices of material M2, then we have:
μv – μr = 0.024
Angle of prism for M1, A' = 5.3°
Angle of prism for M2, A = 3.7°
(a) When the prisms are oppositely directed, angular dispersion
\left({\delta }_{1}\right)
δ1 = (μv – μr)A – (μ'v – μ'r)A'
δ1 = 0.024 × 3.7° – 0.014 × 5.3°
So, the angular dispersion is 0.0146°.
(b) When the prisms are similarly directed, angular dispersion
\left({\delta }_{2}\right)
δ2 = (μv – μr)A + (μ'v – μ'r)A'
δ2 = 0.024 × 3.7° + 0.014 × 5.3°
So, the angular dispersion is 0.163°.
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f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } } & { \text { for } - 2 \leq x < 1 } \\ { 2 - x } & { \text { for } \quad 1 \leq x < 4 } \end{array} \right.
Use the graph to find the range and zeros of the function.
Check the inequality signs carefully to see if the points are included or not.
h\left(x\right) = f\left(x − 1\right)
h\left(x\right)
Write an expression for the piecewise function
h\left(x\right)
. Be sure to change the domain.
\left(x − 1\right)
2 − \left(x − 1\right)
makes the expression
3 − x
1
Find the range and zeros of
h\left(x\right)
. How does this compare to your answer to part (b)?
|
Where do I get additional help?[edit]
Contact our user community using our Octave Discourse.
What is Octave?[edit]
What is Octave Forge?[edit]
Who uses Octave?[edit]
Who develops Octave?[edit]
Why "Octave"?[edit]
Why "GNU" Octave?[edit]
How can I cite Octave?[edit]
URL https://octave.org/doc/v7.1.0/
url = {https://octave.org/doc/v7.1.0/},
What documentation exists for Octave?[edit]
How can I report a bug in Octave?[edit]
Octave does not start[edit]
Operating system: e.g. MS Windows 10 (version 2004) or Ubuntu 20.04
MS Windows[edit]
Solution 1: Octave on MS Windows uses VBS scripts to start the program. You can test whether your system is blocking VBS scripts by doing the following:
Using Notepad or another text editor, create a text file containing only the text:
msgbox("This is a test script, Click OK to close")
Save the file on your Desktop with the name testscript.vbs (be sure that the editor didn't end it in .txt or .vbs.txt).
Double click the file. If scripts can run, a popup window will appear with that message.
If the file opens in notepad or an editor, it means it still ended in .txt. MS Windows insecurely hides file extensions by default. To show file extensions follow these instructions at Microsoft.com.
If both testscript.vbs and octave.vbs open a text- or other editor, it means your MS Windows file associations have .vbs files associated with another program. To fix this, right click the .vbs file, select "Open With", select "Choose Another App", check the box that says "Always use this app to open .vbs files". Finally, select "Microsoft Windows Based Script Host" from the list. If it is not in the list, select "More apps". If still not there, select "Look for Another App on this PC" and navigate to C:\Windows\System32\wscript.exe, select it, and select "Open". If wscript is still not present you will need to seek assistance installing missing MS Windows components.
Try moving testscript.vbs to another location, such as a C:\temp folder. Additionally try to move testscript.vbs in the Octave installation folder containing octave.vbs and see if VBS scripts can be run there.
If testscript.vbs doesn't run in any of those locations, then scripting appears to be disabled or blocked on your system. If testscript.vbs runs in some locations but not others, there there may be other security permissions errors to be solved. If testscript.vbs runs in the same folder as octave.vbs, but directly double-clicking on octave.vbs does not start Octave, then there appears to be some problem other than script permissions.
Solution 4: Is your computer managed by your company? Does your administrator prohibit script execution?
I do not see any output of my script until it has finished?[edit]
When I try plotting from a script, why am I not seeing anything?[edit]
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I have problem X using the latest Octave version[edit]
Why is Octave's floating-point computation wrong?[edit]
Floating-point arithmetic is an approximation in binary to arithmetic on real or complex numbers. Just like you cannot represent 1/3 exactly in decimal arithmetic (0.333333... is only a rough approximation to 1/3 for any finite number of 3s), you cannot represent some fractions like
{\displaystyle 1/10}
{\displaystyle 0.0{\overline {0011}}_{b}}
{\displaystyle 1/6=0.1{\overline {6}}_{d}}
Missing lines when printing under Windows with OpenGL toolkit and Intel integrated GPU[edit]
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|
Seasonally Adjusted Annual Rate (SAAR) Definition
What Is a Seasonally Adjusted Annual Rate (SAAR)?
Understanding a Seasonally Adjusted Annual Rate (SAAR)
A seasonally adjusted annual rate (SAAR) seeks to remove seasonal impacts on a business to gain a deeper understanding of how the core aspects of a business perform throughout the year. For example, the ice cream industry tends to have a large level of seasonality as it sells more ice cream in the summer than in the winter, and by using seasonally adjusted annual sales rates, the sales in the summer can be accurately compared to the sales in the winter. It is often used by analysts in the automobile industry to account for car sales.
Seasonal adjustment is a statistical technique designed to even out periodic swings in statistics or movements in supply and demand related to changing seasons. Seasonal adjustments provide a clearer view of nonseasonal changes in data that would otherwise be overshadowed by the seasonal differences.
Calculating a Seasonally Adjusted Annual Rate (SAAR)
To calculate SAAR, take the un-adjusted monthly estimate, divide by its seasonality factor, and multiply by 12.
Analysts start with a full year of data, and then they find the average number for each month or quarter. The ratio between the actual number and the average determines the seasonal factor for that time period.
Imagine a business earns $144,000 over a course of a year and $20,000 in June. Its average monthly revenue is $12,000, making June's seasonality factor as follows:
\$20,000/\$12,000=1.67
The following year, revenue during June climbs to $30,000. When divided by the seasonality factor, the result is $17,964, and when multiplied by 12, that makes the SAAR $215,568; indicating growth. Alternatively, SAAR can be calculated by taking the unadjusted quarterly estimate, dividing by its seasonality factor, and multiplying by four.
Seasonally Adjusted Annual Rates (SAARs) and Data Comparisons
A seasonally adjusted annual rate (SAAR) helps with data comparisons in a number of ways. By adjusting the current month's sales for seasonality, a business can calculate its current SAAR and compare it to the previous year's sales to determine if sales are increasing or decreasing.
Similarly, if a person wants to determine if real estate prices are increasing in their area, they can look at the median prices in the current month or quarter, adjust those numbers for seasonal variations, and convert them into SAARs which can be compared to numbers for the previous years. Without making these adjustments first, the analyst is not comparing apples with apples, and as a result, cannot make clear conclusions.
For example, homes tend to sell more quickly and at higher prices in the summer than in the winter. As a result, if a person compares summer real estate sales prices to median prices from the previous year, they may get a false impression that prices are rising. However, if they adjust the initial data based on the season, they can see whether values are truly rising or just being momentarily increased by the warm weather.
Seasonally Adjusted Annual Rates (SAARs) vs. Non-Seasonally Adjusted Annual Rates
While seasonally adjusted (SA) rates try to remove the differences between seasonal variations, non-seasonally adjusted (NSA) rates do not take into account seasonal ebbs and flows. Concerning a set of information, NSA data corresponds to the information's annual rate, while SA data corresponds to its SAAR.
|
Chemiluminescence Knowpia
Chemiluminescence (also chemoluminescence) is the emission of light (luminescence) as the result of a chemical reaction. There may also be limited emission of heat. Given reactants A and B, with an excited intermediate ◊,
A chemoluminescent reaction in an Erlenmeyer flask
{\displaystyle {\ce {{\underset {luminol}{C8H7N3O2}}+{\underset {hydrogen\ peroxide}{H2O2}}->3-APA[\lozenge ]->{3-APA}+light}}}
3-APA is 3-aminophthalate
3-APA[◊] is the vibronic excited state fluorescing as it decays to a lower energy level.
The decay of this excited state[◊] to a lower energy level causes light emission.[1] In theory, one photon of light should be given off for each molecule of reactant. This is equivalent to Avogadro's number of photons per mole of reactant. In actual practice, non-enzymatic reactions seldom exceed 1% QC, quantum efficiency.
In a chemical reaction, reactants collide to form a transition state, the enthalpic maximum in a reaction coordinate diagram, which proceeds to the product. Normally, reactants form products of lesser chemical energy. The difference in energy between reactants and products, represented as
{\displaystyle \Delta H_{rxn}}
, is turned into heat, physically realized as excitations in the vibrational state of the normal modes of the product. Since vibrational energy is generally much greater than the thermal agitation, it rapidly disperses in the solvent through molecular rotation. This is how exothermic reactions make their solutions hotter. In a chemiluminescent reaction, the direct product of the reaction is an excited electronic state. This state then decays into an electronic ground state and emits light through either an allowed transition (analogous to fluorescence) or a forbidden transition (analogous to phosphorescence), depending partly on the spin state of the electronic excited state formed.
Chemiluminescence differs from fluorescence or phosphorescence in that the electronic excited state is the product of a chemical reaction rather than of the absorption of a photon. It is the antithesis of a photochemical reaction, in which light is used to drive an endothermic chemical reaction. Here, light is generated from a chemically exothermic reaction. The chemiluminescence might be also induced by an electrochemical stimulus, in this case is called electrochemiluminescence.
Bioluminescence in nature: A male firefly mating with a female of the species Lampyris noctiluca.
The first chemiluminescent compound to be discovered was 2,4,5-triphenylimidazole (lophine), which was reported, in 1877, to emit light when mixed with potassium hydroxide in aqueous ethanol in the presence of air.[2] A standard example of chemiluminescence in the laboratory setting is the luminol test. Here, blood is indicated by luminescence upon contact with iron in hemoglobin. When chemiluminescence takes place in living organisms, the phenomenon is called bioluminescence. A light stick emits light by chemiluminescence.
Liquid-phase reactionsEdit
Chemiluminescence in aqueous system is mainly caused by redox reactions.[3]
Chemiluminescence after a reaction of hydrogen peroxide and luminol
Luminol in an alkaline solution with hydrogen peroxide in the presence of iron or copper,[4] or an auxiliary oxidant,[5] produces chemiluminescence. The luminol reaction is
{\displaystyle {\ce {{\underset {luminol}{C8H7N3O2}}+{\underset {hydrogen\ peroxide}{H2O2}}->3-APA[\lozenge ]->{3-APA}+light}}}
Gas-phase reactionsEdit
Green and blue glow sticks
One of the oldest known chemiluminescent reactions is that of elemental white phosphorus oxidizing in moist air, producing a green glow. This is a gas-phase reaction of phosphorus vapor, above the solid, with oxygen producing the excited states (PO)2 and HPO.[6]
Another gas phase reaction is the basis of nitric oxide detection in commercial analytic instruments applied to environmental air-quality testing. Ozone is combined with nitric oxide to form nitrogen dioxide in an activated state.
The activated NO2[◊] luminesces broadband visible to infrared light as it reverts to a lower energy state. A photomultiplier and associated electronics counts the photons that are proportional to the amount of NO present. To determine the amount of nitrogen dioxide, NO2, in a sample (containing no NO) it must first be converted to nitric oxide, NO, by passing the sample through a converter before the above ozone activation reaction is applied. The ozone reaction produces a photon count proportional to NO that is proportional to NO2 before it was converted to NO. In the case of a mixed sample that contains both NO and NO2, the above reaction yields the amount of NO and NO2 combined in the air sample, assuming that the sample is passed through the converter. If the mixed sample is not passed through the converter, the ozone reaction produces activated NO2[◊] only in proportion to the NO in the sample. The NO2 in the sample is not activated by the ozone reaction. Though unactivated NO2 is present with the activated NO2[◊], photons are emitted only by the activated species that is proportional to original NO. Final step: Subtract NO from (NO + NO2) to yield NO2[7]
Infrared chemiluminescenceEdit
In chemical kinetics, infrared chemiluminiscence (IRCL) refers to the emission of infrared photons from vibrationally excited product molecules immediately after their formation. The intensities of infrared emission lines from vibrationally excited molecules are used to measure the populations of vibrational states of product molecules.[8][9]
The observation of IRCL was developed as a kinetic technique by John Polanyi, who used it to study the attractive or repulsive nature of the potential energy surface for gas-phase reactions. In general the IRCL is much more intense for reactions with an attractive surface, indicating that this type of surface leads to energy deposition in vibrational excitation. In contrast reactions with a repulsive potential energy surface lead to little IRCL, indicating that the energy is primarily deposited as translational energy.[10]
Enhanced chemiluminescenceEdit
Enhanced chemiluminescence (ECL) is a common technique for a variety of detection assays in biology. A horseradish peroxidase enzyme (HRP) is tethered to an antibody that specifically recognizes the molecule of interest. This enzyme complex then catalyzes the conversion of the enhanced chemiluminescent substrate into a sensitized reagent in the vicinity of the molecule of interest, which on further oxidation by hydrogen peroxide, produces a triplet (excited) carbonyl, which emits light when it decays to the singlet carbonyl. Enhanced chemiluminescence allows detection of minute quantities of a biomolecule. Proteins can be detected down to femtomole quantities,[11][12] well below the detection limit for most assay systems.
Gas analysis: for determining small amounts of impurities or poisons in air. Other compounds can also be determined by this method (ozone, N-oxides, S-compounds). A typical example is NO determination with detection limits down to 1 ppb. Highly specialised chemiluminescence detectors have been used recently to determine concentrations as well as fluxes of NOx with detection limits as low as 5 ppt.[13][14][15]
Analysis of organic species: useful with enzymes, where the substrate is not directly involved in the chemiluminescence reaction, but the product is
Detection and assay of biomolecules in systems such as ELISA and Western blots
Lighting objects. Chemiluminescence kites,[16] emergency lighting, glow sticks[17] (party decorations).
Combustion analysis: Certain radical species (such as CH* and OH*) give off radiation at specific wavelengths. The heat release rate is calculated by measuring the amount of light radiated from a flame at those wavelengths.[18]
Biological applicationsEdit
Chemiluminescence has been applied by forensic scientists to solve crimes. In this case, they use luminol and hydrogen peroxide. The iron from the blood acts as a catalyst and reacts with the luminol and hydrogen peroxide to produce blue light for about 30 seconds. Because only a small amount of iron is required for chemiluminescence, trace amounts of blood are sufficient.
In biomedical research, the protein that gives fireflies their glow and its co-factor, luciferin, are used to produce red light through the consumption of ATP. This reaction is used in many applications, including the effectiveness of cancer drugs that choke off a tumor's blood supply[citation needed]. This form of bioluminescence imaging allows scientists to test drugs in the pre-clinical stages cheaply. Another protein, aequorin, found in certain jellyfish, produces blue light in the presence of calcium. It can be used in molecular biology to assess calcium levels in cells. What these biological reactions have in common is their use of adenosine triphosphate (ATP) as an energy source. Though the structure of the molecules that produce luminescence is different for each species, they are given the generic name of luciferin. Firefly luciferin can be oxidized to produce an excited complex. Once it falls back down to a ground state a photon is released. It is very similar to the reaction with luminol.
{\displaystyle {\ce {Luciferin{}+O2{}+ATP->[{\text{Luciferase}}]Oxyluciferin{}+CO2{}+AMP{}+PPi{}+light}}}
Many organisms have evolved to produce light in a range of colors. At the molecular level, the difference in color arises from the degree of conjugation of the molecule, when an electron drops down from the excited state to the ground state. Deep sea organisms have evolved to produce light to lure and catch prey, as camouflage, or to attract others. Some bacteria even use bioluminescence to communicate. The common colors for the light emitted by these animals are blue and green because they have shorter wavelength than red and can transmit more easily in water.
In April 2020, researchers reported to have genetically engineered plants to glow much brighter than previously possible by inserting genes of the bioluminescent mushroom Neonothopanus nambi. The glow is self-sustained, works by converting plants' caffeic acid into luciferin and, unlike for bacterial bioluminescence genes used earlier, has a relatively high light output that is visible to the naked eye.[19][20][21][22]
Chemiluminescence is different from fluorescence. Hence the application of fluorescent proteins such as Green fluorescent protein is not a biological application of chemiluminescence.
^ Vacher, Morgane; Fdez. Galván, Ignacio; Ding, Bo-Wen; Schramm, Stefan; Berraud-Pache, Romain; Naumov, Panče; Ferré, Nicolas; Liu, Ya-Jun; Navizet, Isabelle; Roca-Sanjuán, Daniel; Baader, Wilhelm J.; Lindh, Roland (March 2018). "Chemi- and Bioluminescence of Cyclic Peroxides". Chemical Reviews. 118 (15): 6927–6974. doi:10.1021/acs.chemrev.7b00649. PMID 29493234.
^ Radziszewski, B. R. (1877). "Untersuchungen über Hydrobenzamid, Amarin und Lophin". Berichte der Deutschen Chemischen Gesellschaft (in German). 10 (1): 70–75. doi:10.1002/cber.18770100122.
^ Shah, Syed Niaz Ali; Lin, Jin-Ming (2017). "Recent advances in chemiluminescence based on carbonaceous dots". Advances in Colloid and Interface Science. 241: 24–36. doi:10.1016/j.cis.2017.01.003. PMID 28139217.
^ "Luminol chemistry laboratory demonstration". Retrieved 2006-03-29.
^ "Investigating luminol" (PDF). Salters Advanced Chemistry. Archived from the original (PDF) on September 20, 2004. Retrieved 2006-03-29.
^ Rauhut, Michael M. (1985), Chemiluminescence. In Grayson, Martin (Ed) (1985). Kirk-Othmer Concise Encyclopedia of Chemical Technology (3rd ed), pp 247 John Wiley and Sons. ISBN 0-471-51700-3
^ Air Zoom | Glowing with Pride Archived 2014-06-12 at the Wayback Machine. Fannation.com. Retrieved on 2011-11-22.
^ Atkins P. and de Paula J. Physical Chemistry (8th ed., W.H.Freeman 2006) p.886 ISBN 0-7167-8759-8
^ Steinfeld J.I., Francisco J.S. and Hase W.L. Chemical Kinetics and Dynamics (2nd ed., Prentice-Hall 1998) p.263 ISBN 0-13-737123-3
^ Atkins P. and de Paula J. p.889-890
^ Enhanced CL review. Biocompare.com (2007-06-04). Retrieved on 2011-11-22.
^ High Intensity HRP-Chemiluminescence ELISA Substrate Archived 2016-04-08 at the Wayback Machine. Haemoscan.com (2016-02-11). Retrieved on 2016-03-29.
^ "ECOPHYSICS CLD790SR2 NO/NO2 analyser" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2015-04-30.
^ Stella, P., Kortner, M., Ammann, C., Foken, T., Meixner, F. X., and Trebs, I.: Measurements of nitrogen oxides and ozone fluxes by eddy covariance at a meadow: evidence for an internal leaf resistance to NO2, Biogeosciences, 10, 5997-6017, doi:10.5194/bg-10-5997-2013, 2013.
^ Tsokankunku, Anywhere: Fluxes of the NO-O3-NO2 triad above a spruce forest canopy in south-eastern Germany. Bayreuth, 2014 . - XII, 184 P. ( Doctoral thesis, 2014, University of Bayreuth, Faculty of Biology, Chemistry and Earth Sciences) [1]
^ Kinn, John J "Chemiluminescent kite" U.S. Patent 4,715,564issued 12/29/1987
^ Kuntzleman, Thomas Scott; Rohrer, Kristen; Schultz, Emeric (2012-06-12). "The Chemistry of Lightsticks: Demonstrations To Illustrate Chemical Processes". Journal of Chemical Education. 89 (7): 910–916. Bibcode:2012JChEd..89..910K. doi:10.1021/ed200328d. ISSN 0021-9584.
^ Chemiluminescence as a Combustion Diagnostic Archived 2011-03-02 at the Wayback Machine Venkata Nori and Jerry Seitzman - AIAA - 2008
^ "Sustainable light achieved in living plants". phys.org. Retrieved 18 May 2020.
^ "Scientists use mushroom DNA to produce permanently-glowing plants". New Atlas. 28 April 2020. Retrieved 18 May 2020.
^ "Scientists create glowing plants using mushroom genes". the Guardian. 27 April 2020. Retrieved 18 May 2020.
^ Mitiouchkina, Tatiana; Mishin, Alexander S.; Somermeyer, Louisa Gonzalez; Markina, Nadezhda M.; Chepurnyh, Tatiana V.; Guglya, Elena B.; Karataeva, Tatiana A.; Palkina, Kseniia A.; Shakhova, Ekaterina S.; Fakhranurova, Liliia I.; Chekova, Sofia V.; Tsarkova, Aleksandra S.; Golubev, Yaroslav V.; Negrebetsky, Vadim V.; Dolgushin, Sergey A.; Shalaev, Pavel V.; Shlykov, Dmitry; Melnik, Olesya A.; Shipunova, Victoria O.; Deyev, Sergey M.; Bubyrev, Andrey I.; Pushin, Alexander S.; Choob, Vladimir V.; Dolgov, Sergey V.; Kondrashov, Fyodor A.; Yampolsky, Ilia V.; Sarkisyan, Karen S. (27 April 2020). "Plants with genetically encoded autoluminescence". Nature Biotechnology. 38 (8): 944–946. doi:10.1038/s41587-020-0500-9. ISSN 1546-1696. PMC 7610436. PMID 32341562. S2CID 216559981.
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Convert transfer function filter parameters to zero-pole-gain form - MATLAB tf2zpk - MathWorks
tf2zpk
Poles, Zeros, and Gain of IIR Filter
Convert transfer function filter parameters to zero-pole-gain form
[z,p,k] = tf2zpk(b,a)
[z,p,k] = tf2zpk(b,a) finds the matrix of zeros z, the vector of poles p, and the associated vector of gains k from the transfer function parameters b and a. The function converts a polynomial transfer-function representation
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{1}+{b}_{2}{z}^{-1}\cdots +{b}_{n-1}{z}^{-n}+{b}_{n}{z}^{-n-1}}{{a}_{1}+{a}_{2}{z}^{-1}\cdots +{a}_{m-1}{z}^{-m}+{a}_{m}{z}^{-m-1}}
of a single-input/multi-output (SIMO) discrete-time system to a factored transfer function form
H\left(z\right)=\frac{Z\left(z\right)}{P\left(z\right)}=k\frac{\left(z-{z}_{1}\right)\left(z-{z}_{2}\right)\cdots \left(z-{z}_{m}\right)}{\left(z-{p}_{1}\right)\left(z-{p}_{2}\right)\cdots \left(z-{p}_{n}\right)}.
Use tf2zpk when working with transfer functions expressed in inverse powers (1 + z–1 + z–2). A similar function, tf2zp, is more useful for working with positive powers (s2 + s + 1), such as in continuous-time transfer functions.
Design a 3rd-order Butterworth filter with normalized cutoff frequency
0.4\pi
rad/sample. Find the poles, zeros, and gain of the filter.
Plot the poles and zeros to verify that they are where expected.
text(real(z)-0.1,imag(z)-0.1,'\bfZeros','color',[0 0.4 0])
text(real(p)-0.1,imag(p)-0.1,'\bfPoles','color',[0.6 0 0])
Transfer function numerator coefficients, specified as a vector or matrix. If b is a matrix, then each row of b corresponds to an output of the system. b contains the coefficients in descending powers of z. The number of columns of b must be equal to the length of a. If the numbers differ, make them equal by padding zeros. You can use the function eqtflength to accomplish this.
Transfer function denominator coefficients, specified as a vector. a contains the coefficients in descending powers of z.
Zeros of the system, returned as a matrix. z contains the numerator zeros in its columns. z has as many columns as there are outputs.
Poles of the system, returned as a column vector. p contains the pole locations of the denominator coefficients of the transfer function
Gains of the system, returned as a column vector. k contains the gains for each numerator transfer function.
The complexity of outputs, z and k, might be different in MATLAB® and the generated code.
sos2zp | ss2zp | tf2sos | tf2ss | tf2zp | zp2tf
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16:10 aspect ratio - Wikipedia
(Redirected from 16:10)
Aspect ratio for computer displays
An LG 19-inch LCD monitor with an aspect ratio of 16:10.
16:10 (1.6) is an aspect ratio commonly used for computer displays and tablet computers. This ratio is equal to 8/5, and is close to the golden ratio (
{\displaystyle \varphi }
), which is approximately 1.618.
1.1 Rise in popularity from 2003
1.2 Decline from 2008
LCD computer displays with a 16:10 ratio first rose to mass market prominence in 2003. By 2008, the 16:10 aspect ratio had become the most common aspect ratio for LCD monitors and laptop displays.[1] After 2010, however, 16:9 became the mainstream standard. This shift was driven by lower manufacturing costs and the 16:9 aspect ratio being used as a standard in modern televisions.[2][3]
Rise in popularity from 2003[edit]
Until about 2003, most computer monitors had a 4:3 aspect ratio, with some using a 5:4 ratio. Between 2003 and 2006, monitors with 16:10 aspect ratios became commonly available, first in laptops, and later in display monitors. Such displays were considered better suited for word processing and computer-aided design.[4][5]
From 2005 to 2008, 16:10 overtook 4:3 as the highest-selling aspect ratio for LCD monitors. At the time, 16:10 made up 90% of the notebook market, and was the most commonly used aspect ratio for laptops.[2] However, 16:10 had a short reign as the most common aspect ratio.
Decline from 2008[edit]
Around 2008–2010, computer display manufacturers began a rapid shift to the 16:9 aspect ratio. By 2011, 16:10 had almost disappeared from new mass-market products. By October 2012, the market share of 16:10 displays had dropped to less than 23 per cent, according to Net Applications.[6]
The primary reason for this move was considered to be production efficiency:[3][7] Since display panels for TVs use the 16:9 aspect ratio, it became more efficient for display manufacturers to produce computer display panels in the same aspect ratio.[8] A 2008 report by DisplaySearch also cited several other reasons, including the ability for PC and monitor manufacturers to expand their product ranges by offering products with wider screens and higher resolutions. This helped consumers adopt such products more easily, "stimulating the growth of the notebook PC and LCD monitor market".[2]
The shift from 16:10 to 16:9 was met with a mixed response. The lower cost of 16:9 computer displays was seen as a positive, along with their suitability for gaming and movies, as well as the convenience of having the same aspect ratio in different devices.[3][9] On the other hand, there was criticism towards the lack of vertical screen real estate when compared to 16:10 displays of the same screen diagonal.[9][10] For this reason, some considered 16:9 displays less suitable for productivity-oriented tasks, such as editing documents or spreadsheets and using design or engineering applications, which are mostly designed for taller, rather than wider screens.[9][11][12]
Companies like BenQ, Dell and Eizo, among others, still offer 16:10 aspect ratio monitors as of March 2021. These monitors are intended for photographers, video editors, digital artists, desktop publishers, graphic designers and business customers.[citation needed]
In 2020, Dell released high-end productivity laptops with the 16:10 aspect ratio, and Microsoft launched a new version of its 3:2 Surface Book. The version of the Dell XPS produced around this time was the first that moved away from the classic 16:9 aspect ratio. Other examples include the Acer Swift 3, LG Gram, and Asus ProArt Studiobook.[13] In 2021, the Steam Deck, a handheld gaming computer produced by Valve, was announced, featuring a 16:10 display.[14] More gaming laptops have also been increasingly made available with 16:10 screens. Apple used 16:10 aspect ratios in its MacBook lineup of laptops until late 2021, when they were changed to a taller 1.55:1 ratio.[15]
Tablets started to enjoy mainstream popularity beginning late 2010/early 2011 and remain popular to the present day. Aspect ratios for tablets typically include 16:10, 16:9 and 4:3. Tablets have caused a shift in production away from purely 16:9 aspect ratios and a resurgence of "productivity" aspect ratios (including 16:10 and 4:3) in place of "media" aspect ratios (16:9 and ultra-widescreen formats). The format remains widely popular in the TV and smartphone industries, where it is more suited.
Many Android tablets have a 16:10 aspect ratio, because the 16:10 aspect ratio is suitable for reading books, and many papers have an aspect ratio close to 16:10 (e.g., ISO 216 papers use the 1:1.414 aspect ratio).[16] Apple's iPad uses a 4:3 aspect ratio for similar reasons. Both formats come significantly closer to emulating the aspect ratio of A4 paper (210 mm × 297 mm or 8.27 in × 11.69 in) than 16:9, which has been regarded as the reason for their success.
This is a list of common resolutions with the 16:10 aspect ratio:
WXGA+ 1440 900
WSXGA+ 1680 1050
WQXGA 2560 1600
WQUXGA 3840 2400
^ a b c "Product Planners and Marketers Must Act Before 16:9 Panels Replace Mainstream 16:10 Notebook PC and Monitor LCD Panels, New DisplaySearch Topical Report Advises". DisplaySearch. 2008-07-01. Retrieved 2011-09-08.
^ a b c Ricker, Thomas (2008-07-02). "Widescreen LCDs going widescreen by 2010". Engadget.
^ NEMATech Computer Display Standards "NEMA Specifications". Archived from the original on 2012-03-02. Retrieved 2011-04-29.
^ "Introduction--Monitor Technology Guide". necdisplay.com. Archived from the original on 2007-03-15. (currently offline)
^ Miller, Michael J. (2008-03-21). "Where Displays Are Heading". PC Magazine. Archived from the original on 2012-08-08. Retrieved 2012-07-09.
^ Kowaliski, Cyril (2008-07-02). "DisplaySearch: Transition to 16:9 displays is 'unstoppable'". The Tech Report. Retrieved 2012-07-09.
^ a b c Ulanoff, Lance (2008-08-27). "Stop Shrinking My Laptop Screen". PC Magazine. Archived from the original on 2017-12-17. Retrieved 2012-07-09.
^ "Gateway's 16:10 displays show common sense". The Inquirer. 2010-07-16. Archived from the original on July 19, 2010. Retrieved 2012-07-09. {{cite web}}: CS1 maint: unfit URL (link)
^ Orion, Egan (2010-06-11). "Time to ditch awful HD 1080p widescreens". The Inquirer. Archived from the original on June 12, 2010. Retrieved 2012-07-09. {{cite web}}: CS1 maint: unfit URL (link)
^ Novakovic, Nebojsa (2011-08-26). "Monitor Aspect Ratios — Beyond 16:9, iPad to the rescue?". VR-Zone. Retrieved 2012-07-09.
^ "Best 16:10 and 3:2 Aspect Ratio Laptops - ThunderboltLaptop". ThunderboltLaptop. 2020-06-10. Retrieved 2020-06-23.
^ Peters, Jay (2021-07-15). "Valve's gaming handheld is called the Steam Deck and it's shipping in December". The Verge. Retrieved 2021-08-06.
^ Burnette, Ed. "How to build the perfect Android tablet, part 4: Resolution and aspect ratio". ZDNet. Retrieved 2021-10-24.
Retrieved from "https://en.wikipedia.org/w/index.php?title=16:10_aspect_ratio&oldid=1086847186"
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Edge-matching puzzle - Wikipedia
Edge-matching puzzle
(Redirected from TetraVex)
A partially completed Eternity II edge-matching puzzle
An edge-matching puzzle is a type of tiling puzzle involving tiling an area with (typically regular) polygons whose edges are distinguished with colours or patterns, in such a way that the edges of adjacent tiles match.
Edge-matching puzzles are known to be NP-complete, and capable of conversion to and from equivalent jigsaw puzzles and polyomino packing puzzle.[1]
The first edge-matching puzzles were patented in the U.S. by E. L. Thurston in 1892.[2] Current examples of commercial edge-matching puzzles include the Eternity II puzzle, Tantrix, Kadon Enterprises' range of edge-matching puzzles, and the Edge Match Puzzles iPhone app.
1 Notable variations
1.1 MacMahon Squares
1.2 TetraVex
2 Incorporation of edge matching
Notable variations[edit]
MacMahon Squares[edit]
A solution to MacMahon Squares with the largest single-color area[3]
MacMahon Squares is the name given to a recreational math puzzle suggested by British mathematician Percy MacMahon, who published a treatise on edge-colouring of a variety of shapes in 1921.[4] This particular puzzle uses 24 tiles consisting of all permutations of 3 colors for the edges of a square. The tiles must be arranged into a 6×4 rectangular area such that all edges match and, furthermore, only one color is used for the outside edge of the rectangle.[5]
This puzzle can be extended to tiles with permutations of 4 colors, arranged in 10×7.[6] In either case, the squares are a subset of the Wang tiles, reducing tiles that are similar under rotation. Solutions number well into the thousands.[7]
MacMahon Squares, along with variations on the idea, was commercialized as Multimatch.
TetraVex[edit]
TetraVex is a computer game that presents the player with a square grid and a collection of tiles, by default nine square tiles for a 3×3 grid. Each tile has four single-digit numbers, one on each edge. The objective of the game is to place the tiles into the grid in the proper position, completing this puzzle as quickly as possible. The tiles cannot be rotated, and two can be placed next to each other only if the numbers on adjacent edges match.[8][9]
TetraVex was inspired by "the problem of tiling the plane" as described by Donald Knuth on page 382 of Volume 1: Fundamental Algorithms, the first book in his series The Art of Computer Programming. It was named by Scott Ferguson, the development lead and an architect of the first version of Visual Basic, who wrote it for Windows Entertainment Pack 3.[10]
TetraVex is also available as an open source game in the GNOME Games collection.[11]
The possible number of TetraVex can be counted. On a
{\displaystyle n\times {}n}
board there are
{\displaystyle n(n-1)}
horizontal and vertical pairs that must match and
{\displaystyle 4n}
numbers along the edges that can be chosen arbitrarily. Hence there are
{\displaystyle 2n(n-1)+4n=2n(n+1)}
choices of 10 digits, i.e.
{\displaystyle 10^{2n(n+1)}}
possible boards.
Deciding if a TetraVex puzzle has a solution is in general NP-complete.[12] Its computational approach involves the Douglas-Rachford algorithm.[13][14]
Hexagons[edit]
A single-cross serpentile
Serpentiles are the hexagonal tiles used in various abstract strategy games such as Psyche-Paths, Kaliko, and Tantrix. Within each serpentile, the edges are paired, thus restricting the set of tiles in such a way that no edge color occurs an odd number of times within the hexagon.
Mathematically, edge-matching puzzles are two-dimensional. A 3D edge-matching puzzle is such a puzzle that is not flat in Euclidean space, so involves tiling a three-dimensional area such as the surface of a regular polyhedron. As before, polygonal pieces have distinguished edges to require that the edges of adjacent pieces match.
3D edge-matching puzzles are not currently under direct U.S. patent protection, since the 1892 patent by E. L. Thurston has expired.[2] Current examples of commercial puzzles include the Dodek Duo, The Enigma, Mental Misery,[15] and Kadon Enterprises' range of three-dimensional edge-matching puzzles.[16]
Incorporation of edge matching[edit]
Part of a Carcassonne game showing matching edges
The Carcassonne board game employs edge matching to constrain where its square tiles may be placed. The original game has three types of edges: fields, roads and cities.
Wang dominoes
^ Erik D. Demaine, Martin L. Demaine. "Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity" (PDF). Retrieved 2007-08-12.
^ a b "Rob's puzzle page: Edge Matching". Archived from the original on 2007-10-22. Retrieved 2007-08-12.
^ Gardner, Martin (2009). Sphere Packing, Lewis Caroll and Reversi. Cambridge University Press.
^ MacMahon, Percy Alexander (1921). New mathematical pastimes. Gerstein - University of Toronto. Cambridge, University Press.
^ Steckles, Katie. Blackboard Bold: MacMahon Squares. Retrieved 10 March 2021.
^ Guy. Cube Root of 31. Wang Tiles. Retrieved 12 April 2021.
^ Wade Philpott (credited). Kadon Enterprises. Multimatch. Retrieved 12 April 2021.
^ Whittum, Christopher (2013). Energize Education Through Open Source. pg 32.
^ Gagné, Marcel (2006). Moving to Ubuntu Linux.
^ "The Birth of Visual Basic". Forestmoon.com. Retrieved 2010-05-11.
^ "License - README". gnome-games. gnome.org. 2011. Retrieved 2012-10-02.
^ "TetraVex is NP-complete". Information Processing Letters, Volume 99, Issue 5, Pages 171–174. 15 September 2006.
^ Bansal, Pulkit(2010). "Code for solving Tetravex using Douglas–Rachford algorithm". Retrieved 10 March 2021.
^ Linstrom, Scott B.; Sims, Brailey (2020). Survey: Sixty years of Douglas Rachford. Cambridge University Press.
^ "Rob's puzzle page: Pattern Puzzles". Retrieved 2009-06-22.
^ "Kadon Enterprises, More About Edgematching". Retrieved 2009-06-22.
Erich's Matching Puzzles Collection
Color- and Edge-Matching Polygons by Peter Esser[dead link]
Rob's puzzle page by Rob Stegmann
Edge matching squares
This combinatorics-related article is a stub. You can help Wikipedia by expanding it.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Edge-matching_puzzle&oldid=1087275467"
Combinatorics stubs
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(Redirected from Uranium 235)
"U-235" redirects here. For the World War II submarine, see German submarine U-235.
1783870.285±1.996 keV
Uranium-235 (235U) is an isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238, it is fissile, i.e., it can sustain a nuclear chain reaction. It is the only fissile isotope that exists in nature as a primordial nuclide.
Uranium-235 has a half-life of 703.8 million years. It was discovered in 1935 by Arthur Jeffrey Dempster. Its fission cross section for slow thermal neutrons is about 584.3±1 barns.[1] For fast neutrons it is on the order of 1 barn.[2] Most but not all neutron absorptions result in fission; a minority result in neutron capture forming uranium-236.[citation needed]
1 Natural decay chain
2 Fission properties
Natural decay chain[edit]
{\displaystyle {\begin{array}{r}{\ce {^{235}_{92}U->[\alpha ][7.038\times 10^{8}\ {\ce {y}}]{^{231}_{90}Th}->[\beta ^{-}][25.52\ {\ce {h}}]{^{231}_{91}Pa}->[\alpha ][3.276\times 10^{4}\ {\ce {y}}]{^{227}_{89}Ac}}}{\begin{Bmatrix}{\ce {->[98.62\%\beta ^{-}][21.773\ {\ce {y}}]{^{227}_{90}Th}->[\alpha ][18.718\ {\ce {d}}]}}\\{\ce {->[1.38\%\alpha ][21.773\ {\ce {y}}]{^{223}_{87}Fr}->[\beta ^{-}][21.8\ {\ce {min}}]}}\end{Bmatrix}}{\ce {^{223}_{88}Ra->[\alpha ][11.434\ {\ce {d}}]{^{219}_{86}Rn}}}\\{\ce {^{219}_{86}Rn->[\alpha ][3.96\ {\ce {s}}]{^{215}_{84}Po}->[\alpha ][1.778\ {\ce {ms}}]{^{211}_{82}Pb}->[\beta ^{-}][36.1\ {\ce {min}}]{^{211}_{83}Bi}}}{\begin{Bmatrix}{\ce {->[99.73\%\alpha ][2.13\ {\ce {min}}]{^{207}_{81}Tl}->[\beta ^{-}][4.77\ {\ce {min}}]}}\\{\ce {->[0.27\%\beta ^{-}][2.13\ {\ce {min}}]{^{211}_{84}Po}->[\alpha ][0.516\ {\ce {s}}]}}\end{Bmatrix}}{\ce {^{207}_{82}Pb_{(stable)}}}\end{array}}}
Fission properties[edit]
Nuclear fission seen with a uranium-235 nucleus
The fission of one atom of uranium-235 releases 202.5 MeV (3.24×10−11 J) inside the reactor. That corresponds to 19.54 TJ/mol, or 83.14 TJ/kg.[3] Another 8.8 MeV escapes the reactor as anti-neutrinos. When 235
92U nuclides are bombarded with neutrons, one of the many fission reactions that it can undergo is the following (shown in the adjacent image):
36Kr + 3 1
Heavy water reactors and some graphite moderated reactors can use natural uranium, but light water reactors must use low enriched uranium because of the higher neutron absorption of light water. Uranium enrichment removes some of the uranium-238 and increases the proportion of uranium-235. Highly enriched uranium (HEU), which contains an even greater proportion of uranium-235, is sometimes used in the reactors of nuclear submarines, research reactors and nuclear weapons.
If at least one neutron from uranium-235 fission strikes another nucleus and causes it to fission, then the chain reaction will continue. If the reaction continues to sustain itself, it is said to be critical, and the mass of 235U required to produce the critical condition is said to be a critical mass. A critical chain reaction can be achieved at low concentrations of 235U if the neutrons from fission are moderated to lower their speed, since the probability for fission with slow neutrons is greater. A fission chain reaction produces intermediate mass fragments which are highly radioactive and produce further energy by their radioactive decay. Some of them produce neutrons, called delayed neutrons, which contribute to the fission chain reaction. The power output of nuclear reactors is adjusted by the location of control rods containing elements that strongly absorb neutrons, e.g., boron, cadmium, or hafnium, in the reactor core. In nuclear bombs, the reaction is uncontrolled and the large amount of energy released creates a nuclear explosion.
The Little Boy gun-type atomic bomb dropped on Hiroshima on August 6, 1945 was made of highly enriched uranium with a large tamper. The nominal spherical critical mass for an untampered 235U nuclear weapon is 56 kilograms (123 lb),[4] which would form a sphere 17.32 centimetres (6.82 in) in diameter. The material must be 85% or more of 235U and is known as weapons grade uranium, though for a crude and inefficient weapon 20% enrichment is sufficient (called weapon(s)-usable). Even lower enrichment can be used, but this results in the required critical mass rapidly increasing. Use of a large tamper, implosion geometries, trigger tubes, polonium triggers, tritium enhancement, and neutron reflectors can enable a more compact, economical weapon using one-fourth or less of the nominal critical mass, though this would likely only be possible in a country that already had extensive experience in engineering nuclear weapons. Most modern nuclear weapon designs use plutonium-239 as the fissile component of the primary stage;[5][6] however, HEU (highly enriched uranium, in this case uranium that is 20% or more 235U) is often used in the secondary stage as an ignitor for the fusion fuel.
released [MeV][3]
Instantaneously released energy
Kinetic energy of fission fragments 169.1
Kinetic energy of prompt neutrons 4.8
Energy carried by prompt γ-rays 7.0
Energy from decaying fission products
Energy of β−-particles 6.5
Energy of delayed γ-rays 6.3
Energy released when those prompt neutrons which don't (re)produce fission are captured 8.8
Total energy converted into heat in an operating thermal nuclear reactor 202.5
Energy of anti-neutrinos 8.8
Uranium-235 has many uses such as fuel for nuclear power plants and in nuclear weapons such as nuclear bombs. Some artificial satellites, such as the SNAP-10A and the RORSATs were powered by nuclear reactors fueled with uranium-235.[7][8]
^ "#Standard Reaction: 235U(n,f)". www-nds.iaea.org. IAEA. Retrieved 4 May 2020.
^ ""Some Physics of Uranium", UIC.com.au". Archived from the original on July 17, 2007. Retrieved 2009-01-18. {{cite web}}: CS1 maint: bot: original URL status unknown (link)
^ a b Nuclear fission and fusion, and neutron interactions, National Physical Laboratory Archive.
^ "FAS Nuclear Weapons Design FAQ". Archived from the original on 1999-05-07. Retrieved 2010-09-02.
^ FAS contributors (ed.). Nuclear Weapon Design. Federation of American Scientists. Archived from the original on 2008-12-26. Retrieved 2016-06-04. {{cite book}}: |editor= has generic name (help)
^ Miner, William N.; Schonfeld, Fred W. (1968). "Plutonium". In Clifford A. Hampel (ed.). The Encyclopedia of the Chemical Elements. New York (NY): Reinhold Book Corporation. p. 541. LCCN 68029938.
^ Schmidt, Glen (February 2011). "SNAP Overview - radium-219 - general background" (PDF). American Nuclear Society. Retrieved 27 August 2012.
^ "RORSAT (Radar Ocean Reconnaissance Satellite)". daviddarling.info.
uranium-234 Uranium-235 is an
isotope of uranium Heavier:
of uranium-235 Decays to:
Retrieved from "https://en.wikipedia.org/w/index.php?title=Uranium-235&oldid=1075736165"
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Check valve in an isothermal system - MATLAB - MathWorks Italia
Check Valve (IL)
Area Parameterization
Tabulated Volumetric Flow Rate Parameterization
Numerically-Smoothed Pressure
Opening pressure specification
Cracking pressure (gauge)
Maximum opening pressure (gauge)
Check valve in an isothermal system
The Check Valve (IL) block models the flow through a valve from port A to port B, and restricts flow from traveling from port B to port A. When the pressure at port B meets or exceeds the set pressure threshold, the valve begins to open.
You can model valve opening either linearly or by tabulated data, and you can enable faulty behavior by setting Enable faults to On.
When Opening parameterization is set to Linear - Area vs. pressure, the valve open area is linearly related to the opening pressure differential.
There are two options for valve control:
When Opening pressure differential is set to Pressure differential, the control pressure is the pressure differential between ports A and B. The valve begins to open when Pcontrol meets or exceeds the Cracking pressure differential.
When Opening pressure differential is set to Pressure at port A, the control pressure is the pressure difference between port A and atmospheric pressure. When Pcontrol meets or exceeds the Cracking pressure (gauge), the valve begins to open.
Opening Area and Pressure
The linear parameterization of the valve area is
{A}_{valve}=\stackrel{^}{p}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak},
where the normalized pressure,
\stackrel{^}{p}
\stackrel{^}{p}=\frac{{p}_{control}-{p}_{cracking}}{{p}_{\mathrm{max}}-{p}_{cracking}}.
When you set Opening parameterization to Tabulated data - Volumetric flow rate vs. pressure, Avalve is the linearly interpolated value of the Opening area vector parameter versus Pressure differential vector parameter curve for a simulated pressure differential.
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0.
\stackrel{˙}{m}=\frac{{C}_{d}{A}_{valve}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},
\overline{\rho }
\Delta {p}_{crit}=\frac{\pi \overline{\rho }}{8{A}_{valve}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2}.
P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}.
When Opening parameterization is set to Tabulated data - Volumetric flow rate vs. pressure, the valve opens according to the user-provided tabulated data of volumetric flow rate and pressure differential between ports A and B.
Within the limits of the tabulated data, the mass flow rate is calculated as:
\stackrel{˙}{m}=\overline{\rho }\stackrel{˙}{V},
\stackrel{˙}{V}
\overline{\rho }
When the simulation pressure falls below the first element of the Pressure drop vector, ΔpTLU(1), the mass flow rate is calculated as:
\stackrel{˙}{m}={K}_{Leak}\overline{\rho }\sqrt{\Delta p}.
{K}_{Leak}=\frac{{V}_{TLU}\left(1\right)}{\sqrt{|\Delta {p}_{TLU}\left(1\right)|}},
where VTLU(1) is the first element of the Volumetric flow rate vector.
When the simulation pressure rises above the last element of the Pressure drop vector, ΔpTLU(end), the mass flow rate is calculated as:
\stackrel{˙}{m}={K}_{Max}\overline{\rho }\sqrt{\Delta p}.
{K}_{Max}=\frac{{V}_{TLU}\left(end\right)}{\sqrt{|\Delta {p}_{TLU}\left(end\right)|}},
where VTLU(end) is the last element of the Volumetric flow rate vector.
The linear parameterization supports valve opening and closing dynamics. If opening dynamics are modeled, a lag is introduced to the flow response to the modeled control pressure. pcontrol becomes the dynamic control pressure, pdyn; otherwise, pcontrol is the steady-state pressure. The instantaneous change in dynamic control pressure is calculated based on the Opening time constant, τ:
{\stackrel{˙}{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau }.
At the extremes of the control pressure range, you can maintain numerical robustness in your simulation by adjusting the block Smoothing factor. A smoothing function is applied to every calculated control pressure, but primarily influences the simulation at the extremes of this range.
The Smoothing factor, s, is applied to the normalized pressure,
\stackrel{^}{p}
{\stackrel{^}{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{p}}_{}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{p}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}},
{p}_{smoothed}={\stackrel{^}{p}}_{smoothed}\left({p}_{\mathrm{max}}-{p}_{set}\right)+{p}_{set}.
When faults are enabled, the valve open area becomes stuck at a specified value in response to one or both of these triggers:
Maintain at last value
Once triggered, the valve remains at the faulted area for the rest of the simulation. You can set the block to issue a fault report as a warning or error message in the Simulink Diagnostic Viewer with the Reporting when fault occurs parameter.
Faulting in the Linear Parameterization
In the linear parameterization, the fault options are defined by the valve area:
Closed — The valve area freezes at the Leakage area.
Open — The valve area freezes at the Maximum opening area.
Maintain at last value — The valve freezes at the open area when the trigger occurs.
Faulting in the Tabulated Data Parameterization
In the tabulated parameterization, the fault options are defined by the mass flow rate through the valve:
Closed — The valve freezes at the mass flow rate associated with the first elements of the Volumetric flow rate vector and the Pressure drop vector:
\stackrel{˙}{m}={K}_{Leak}\overline{\rho }\sqrt{\Delta p}.
Open — The valve freezes at the mass flow rate associated with the last elements of the Volumetric flow rate vector and the Pressure drop vector:
\stackrel{˙}{m}={K}_{Max}\overline{\rho }\sqrt{\Delta p}.
Maintain at last value — The valve freezes at the mass flow rate and pressure differential when the trigger occurs:
\stackrel{˙}{m}={K}_{Last}\overline{\rho }\sqrt{\Delta p},
{K}_{Last}=\frac{|\stackrel{˙}{m}|}{\overline{\rho }\sqrt{|\Delta p|}}.
Click the "Select a predefined parameterization" hyperlink in the property inspector description.
Predefined block parameterizations use available data sources to supply parameter values. The block substitutes engineering judgement and simplifying assumptions for missing data. As a result, expect some deviation between simulated and actual physical behavior. To ensure accuracy, validate the simulated behavior against experimental data and refine your component models as necessary.
Entry point to the valve.
Exit point to the valve.
This port is visible when Enable faults is set to On and Enable external fault trigger is set to Fault when T >= 0.5.
Opening parameterization — Method of calculating valve opening
Linear - Area vs. pressure (default) | Tabulated data - Area vs. pressure | Tabulated data - Volumetric flow rate vs. pressure
Method of calculating valve opening.
Linear - Area vs. pressure: The valve opening area corresponds linearly to the valve pressure.
Tabulated data - Area vs. pressure: The valve mass flow rate is determined from a table of area values with respect to pressure differential.
Tabulated data - Volumetric flow rate vs. pressure: The valve mass flow rate is determined from a table of volumetric flow rate values with respect to pressure differential.
Opening pressure specification — Pressure differential used for valve control
Pressure differential (default) | Pressure at port A
Specifies the control pressure differential. The Pressure differential option refers to the pressure difference between ports A and B. The Pressure at port A option refers to the pressure difference between port A and atmospheric pressure.
To enable this parameter, set Opening parameterization to Linear - Area vs. pressure.
Cracking pressure differential — Threshold pressure
Pressure beyond which the valve operation is triggered. This is the set pressure when the control pressure is the pressure differential between ports A and B.
To enable this parameter, set Opening pressure specification to Pressure differential and Opening parameterization to Linear - Area vs. pressure.
Cracking pressure (gauge) — Threshold pressure
Gauge pressure beyond which valve operation is triggered when the control pressure is the pressure differential between port A and atmospheric pressure.
To enable this parameter, set Opening parameterization to Linear - Area vs. pressure and Opening pressure specification to Pressure at port A.
Maximum opening pressure differential — Valve differential pressure maximum
Maximum valve differential pressure. This parameter provides an upper limit to the pressure so that system pressures remain realistic.
To enable this parameter, set Opening parameterization to Linear - Area vs. pressure and Opening pressure specification to Pressure differential.
Maximum opening pressure (gauge) — Valve pressure maximum
Maximum valve gauge pressure. This parameter provides an upper limit to the pressure so that system pressures remain realistic.
Maximum opening area — Area of fully opened valve
Leakage area — Valve gap area when in fully closed position
Sum of all gaps when the valve is in its fully closed position. Any area smaller than this value is saturated to the specified leakage area. This contributes to numerical stability by maintaining continuity in the flow.
Pressure differential vector — Differential pressure values for tabular parameterization
[0.01 : 0.01 : 0.1] MPa (default) | 1-by-n vector
Vector of pressure differential values for the tabular parameterization of the valve opening area. The vector elements must correspond one-to-one with the elements in the Opening area vector parameter. The elements are listed in ascending order and must be greater than 0. Linear interpolation is employed between table data points.
Set pressure control to Constant.
Opening parameterization to Linear - Area vs. pressure or Tabulated data - Area vs. pressure.
Opening pressure specification to Pressure differential.
, Tabulated data - Area vs. pressure.
Opening area vector — Valve opening areas for tabular parameterization
[1e-10, 1e-05, 2e-05, 3e-05, 4e-05, 5e-05, 6e-05, 7.5e-05, 8.5e-05, .0001] m^2 (default) | 1-by-n vector
Vector of valve opening areas for the tabular parameterization of the valve opening area. The vector elements must correspond one-to-one with the elements in the Pressure differential vector parameter. The elements are listed in ascending order and must be greater than 0. Linear interpolation is employed between table data points.
To enable this parameter, set Set pressure control to Constant and Opening parameterization to Tabulated data - Area vs. pressure.
Cross-sectional area at the entry and exit ports A and B. These areas are used in the pressure-flow rate equation that determines the mass flow rate through the valve.
To enable this parameter, set Opening parameterization to Linear - Area vs. pressure or Tabulated data - Area vs. pressure.
Whether to account for transient effects to the fluid system due to opening the valve. Setting Opening dynamics to On approximates the opening conditions by introducing a first-order lag in the flow response. The Opening time constant also impacts the modeled opening dynamics.
To enable this parameter, set Opening parameterization to Linear - Area vs. pressure and Opening dynamics to On.
Volumetric flow rate vector — Vector of volumetric flow rate for tabulated data parameterization
[.000358, .045, .11, .191, .284, .39, .505, .63, .764, .905] .* 1e-3 m^3/s (default) | 1-by-n vector
Vector of volumetric flow rate values for the tabular parameterization of valve opening. This vector must have the same number of elements as the Pressure drop vector parameter. The vector elements must be listed in ascending order.
To enable this parameter, set Opening parameterization to Tabulated data - Volumetric flow rate vs. pressure.
Sets the faulted valve area or mass flow rate. You can choose for the valve to seize at the fully closed or fully open position, or at the conditions when faulting is triggered. This parameter sets the area when Opening parameterization is set to Linear - Area vs. pressure and the mass flow rate when Opening parameterization is set to Tabulated data.
Off (default) | Fault when T >= 0.5
Enables fault triggering at a specified time. When the Simulation time for fault event is reached, the valve area will be set to the value specified in the Opening area when faulted parameter.
Pilot-Operated Check Valve (IL) | Pressure Relief Valve (IL) | Counterbalance Valve (IL) | Pressure Compensator Valve (IL)
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An α-particle accelerated through V volt is fired towards a nucleus. Its distance of closest approach…
\alpha
-particle accelerated through V volt is fired towards a nucleus. Its distance of closest approach is r. If a proton accelerated through the same potential is fired towards the same nucleus, the distance of closest approach of proton will be :
Asp Programming MCQ Machine Input MI - set 1 MCQ Excretory System MCQ Kidney cysts and tumors MCQ Drawing Inference MCQ Miscellaneous MCQ Biology and Human Welfare MCQ Planet Kingdom MCQ Percentage MCQ Urology Basic Science MCQ Problems on Ages MCQ IBPS Common Written Exam (PO/MT) Main 2016 Solved Paper MCQ
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The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic
1 October 2016 The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic
We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank
2
to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo
{p}^{2}
. These examples contradict some claims by Xie.
Adrian Langer. "The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic." Duke Math. J. 165 (14) 2737 - 2769, 1 October 2016. https://doi.org/10.1215/00127094-3627203
Received: 29 August 2014; Revised: 12 October 2015; Published: 1 October 2016
Keywords: Bogomolov’s inequality , Bogomolov-Miyaoka-Yau inequality , logarithmic Higgs sheaves , positive characteristic
Adrian Langer "The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic," Duke Mathematical Journal, Duke Math. J. 165(14), 2737-2769, (1 October 2016)
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Isomerization Knowpia
In chemistry isomerization or isomerisation is the process in which a molecule, ion or molecular fragment is transformed into an isomer with a different chemical structure.[1] Enolization is an example of isomerization, as is tautomerization.[2] When the isomerization occurs intramolecularly it may be called a rearrangement reaction.[citation needed]
When the activation energy for the isomerization reaction is sufficiently small, both isomers will exist in a temperature-dependent equilibrium with each other. Many values of the standard free energy difference,
{\displaystyle \Delta G^{\circ }}
, have been calculated, with good agreement between observed and calculated data.[3]
Skeletal isomerization occurs in the cracking process, used in the petrochemical industry. As well as reducing the average chain length, straight-chain hydrocarbons are converted to branched isomers in the process, as illustrated the following reaction.[citation needed]
{\displaystyle CH_{3}CH_{2}CH_{2}CH_{3}}
(n-butane) →
{\displaystyle CH_{3}CH(CH_{3})CH_{3}}
(i-butane)
Fuels containing branched hydrocarbons are favored for internal combustion engines for their higher octane rating.[4]
Terminal alkenes isomerize to internal alkenes in the presence of metal catalysts. This process is employed in the Shell higher olefin process to convert alpha-olefins to internal olefins, which are subjected to olefin metathesis. In certain kinds of alkene polymerization reactions, chain walking is an isomerization process that introduces branches into growing polymers.[citation needed]
The trans isomer of resveratrol can be converted to the cis isomer in a photochemical reaction.[5]
Thermal rearrangement of azulene to naphthalene has been observed.[citation needed]
Aldose-ketose isomerism, aka Lobry de Bruyn–van Ekenstein transformation, provides an example in saccharide chemistry.[citation needed]
An example of an organometallic isomerization is the production of decaphenylferrocene, [(η5-C5Ph5)2Fe] from its linkage isomer.[6][7]
^ Antonov L (2016). Tautomerism: Concepts and Applications in Science and Technology (1st ed.). Weinheim, Germany: Wiley-VCH. ISBN 978-3-527-33995-2.
^ How to Compute Isomerization Energies of Organic Molecules with Quantum Chemical Methods Stefan Grimme, Marc Steinmetz, and Martin Korth J. Org. Chem.; 2007; 72(6) pp 2118 - 2126; (Article) doi:10.1021/jo062446p
^ Karl Griesbaum, Arno Behr, Dieter Biedenkapp, Heinz-Werner Voges, Dorothea Garbe, Christian Paetz, Gerd Collin, Dieter Mayer, Hartmut Höke (2002). "Hydrocarbons". Ullmann's Encyclopedia of Industrial Chemistry. Weinheim: Wiley-VCH. doi:10.1002/14356007.a13_227. {{cite encyclopedia}}: CS1 maint: uses authors parameter (link)
^ Resveratrol Photoisomerization: An Integrative Guided-Inquiry Experiment Elyse Bernard, Philip Britz-McKibbin, Nicholas Gernigon Vol. 84 No. 7 July 2007 Journal of Chemical Education 1159.
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Strong spectral gaps for compact quotients of products of PSL(2,ℝ) | EMS Press
Strong spectral gaps for compact quotients of products of PSL(2,ℝ)
The existence of a strong spectral gap for quotients
\Gamma\bs G
of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If
G
has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice
\Gamma
G=\PSL(2,\bbR)^d
d\geq 2
which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.
Peter Sarnak, Dubi Kelmer, Strong spectral gaps for compact quotients of products of PSL(2,ℝ). J. Eur. Math. Soc. 11 (2009), no. 2, pp. 283–313
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EUDML | Entire functions and -convex structure in commutative Baire algebras. EuDML | Entire functions and -convex structure in commutative Baire algebras.
Entire functions and
m
-convex structure in commutative Baire algebras.
El Kinani, A.; Oudadess, M.
El Kinani, A., and Oudadess, M.. "Entire functions and -convex structure in commutative Baire algebras.." Bulletin of the Belgian Mathematical Society - Simon Stevin 4.5 (1997): 685-687. <http://eudml.org/doc/119768>.
@article{ElKinani1997,
author = {El Kinani, A., Oudadess, M.},
keywords = {unitary commutative locally convex algebra; continuous; Baire space; entire functions operate; -convex; commutative -algebras; -convex; commutative -algebras},
title = {Entire functions and -convex structure in commutative Baire algebras.},
AU - El Kinani, A.
AU - Oudadess, M.
TI - Entire functions and -convex structure in commutative Baire algebras.
KW - unitary commutative locally convex algebra; continuous; Baire space; entire functions operate; -convex; commutative -algebras; -convex; commutative -algebras
Abdellah El Kinani, Equicontinuity of power maps in locally pseudo-convex algebras
unitary commutative locally convex algebra, continuous, Baire space, entire functions operate,
m
-convex, commutative
{B}_{0}
m
{B}_{0}
Structure, classification of commutative topological algebras
Articles by El Kinani
Articles by Oudadess
|
Azimuth - formulasearchengine
The azimuth is the angle formed between a reference direction (North) and a line from the observer to a point of interest projected on the same plane as the reference direction
An azimuth (Template:IPAc-en) (from Arabic al-sumūt, meaning "the directions") is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
An example is the position of a star in the sky. The star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points north. The azimuth is the angle between the north vector and the perpendicular projection of the star down onto the horizon.[1]
Azimuth is usually measured in degrees (°). The concept is used in navigation, astronomy, engineering, mapping, mining and artillery.
1.1 True north-based azimuths
2 Cartographical azimuth
3 Calculating azimuth
6.1 Right ascension
6.2 Horizontal coordinate
In land navigation, azimuth is usually denoted alpha,
{\displaystyle \alpha }
, and defined as a horizontal angle measured clockwise from a north base line or meridian.[2][3] Azimuth has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.[4][5][6]
Today the reference plane for an azimuth is typically true north, measured as a 0° azimuth, though other angular units (grad, mil) can be used. Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, and west 270°. There are exceptions: some navigation systems use south as the reference plane. Any direction can be the plane of reference, as long as it is clearly defined.
Quite commonly, azimuths or compass bearings are stated in a system in which either north or south can be the zero, and the angle may be measured clockwise or anticlockwise from the zero. For example, a bearing might be described as "(from) south, (turn) thirty degrees (toward the) east" (the words in brackets are usually omitted), abbreviated "S30°E", which is the bearing 30 degrees in the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, and the turning direction, stated last, is east or west. The directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be exactly in the direction of one of the cardinal points, a different notation, e.g. "due east", is used instead.
True north-based azimuths
North 0° or 360° South 180°
North-Northeast 22.5° South-Southwest 202.5°
Northeast 45° Southwest 225°
East-Northeast 67.5° West-Southwest 247.5°
East 90° West 270°
East-Southeast 112.5° West-Northwest 292.5°
Southeast 135° Northwest 315°
South-Southeast 157.5° North-Northwest 337.5°
Cartographical azimuth
The cartographical azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):
{\displaystyle az=90.0-180.0/{\pi }\ {atan2\ (X2-X1,Y2-Y1)}}
Remark that the reference axes are swapped relative to the (counterclockwise) mathematical polar coordinate system and that the azimuth is clockwise relative to the north.
Calculating azimuth
We are standing at latitude
{\displaystyle \phi _{1}}
, longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude
{\displaystyle \phi _{2}}
, longitude L (positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth
{\displaystyle \alpha }
{\displaystyle \tan \alpha ={\frac {\sin L}{(\cos \phi _{1})(\tan \phi _{2})-(\sin \phi _{1})(\cos L)}}}
A better approximation assumes the Earth is a slightly-squashed sphere (an oblate spheroid); "azimuth" then has at least two very slightly different meanings. "Normal-section azimuth" is the angle measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; "geodetic azimuth" is the angle between north and the geodesic – that is, the shortest path on the surface of the spheroid from our viewpoint to Point 2. The difference is usually unmeasurably small; if Point 2 is not more than 100 km away the difference will not exceed 0.03 arc second.
Various websites will calculate geodetic azimuth – e.g. GeoScience Australia site. Formulas for calculating geodetic azimuth are linked in the distance article.
Normal-section azimuth is simpler to calculate; Bomford says Cunningham's formula is exact for any distance. I{\displaystyle f}
is the flattening for the chosen spheroid (e.g. 1/298.257223563 for WGS84) then
{\displaystyle e^{2}=f(2-f)\,}
{\displaystyle 1-e^{2}=(1-f)^{2}\,}
{\displaystyle \Lambda =(1-e^{2}){\frac {\tan \phi _{2}}{\tan \phi _{1}}}+e^{2}{\sqrt {\cfrac {1+(1-e^{2})(\tan \phi _{2})^{2}}{1+(1-e^{2})(\tan \phi _{1})^{2}}}}}
{\displaystyle \tan \alpha ={\frac {\sin L}{(\Lambda -\cos L)\sin \phi _{1}}}}
{\displaystyle \phi _{1}}
{\displaystyle \tan \alpha ={\frac {\sin L}{(1-e^{2})\tan \phi _{2}}}}
To calculate the azimuth of the sun or a star given its declination and hour angle at our location, we modify the formula for a spherical earth. Replace
{\displaystyle \phi _{2}}
with declination and longitude difference with hour angle, and change the sign (since hour angle is positive westward instead of east).
A standard Brunton Geo compass, used commonly by geologists and surveyors to measure azimuth
There are a wide variety of azimuthal map projections. They all have the property that directions (the azimuths) from a central point are preserved. Some navigation systems use south as the reference plane. However, any direction can serve as the plane of reference, as long as it is clearly defined for everyone using that system.
Used in celestial navigation, an azimuth is the direction of a celestial body from the observer.[7] In astronomy, an azimuth is sometimes referred to as a bearing. In modern astronomy azimuth is nearly always measured from the north. (The article on coordinate systems, for example, uses a convention measuring from the south.) In former times, it was common to refer to azimuth from the south, as it was then zero at the same time that the hour angle of a star was zero. This assumes, however, that the star (upper) culminates in the south, which is only true if the star's declination is less than (i.e. further south than) the observer's latitude.
If instead of measuring from and along the horizon the angles are measured from and along the celestial equator, the angles are called right ascension if referenced to the Vernal Equinox, or hour angle if referenced to the celestial meridian.
In the horizontal coordinate system, used in celestial navigation and satellite dish installation, azimuth is one of the two coordinates. The other is altitude, sometimes called elevation above the horizon. See also: Sat finder.
In mathematics the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive x-axis and the projection of the vector onto the xy-plane. The angle is the same as an angle in polar coordinates of the component of the vector in the xy-plane and is normally measured in radians rather than degrees. As well as measuring the angle differently, in mathematical applications theta,
{\displaystyle \theta }
, is very often used to represent the azimuth rather than the symbol phi
{\displaystyle \phi }
For magnetic tape drives, azimuth refers to the angle between the tape head(s) and tape.
In sound localization experiments and literature, the azimuth refers to the angle the sound source makes compared to the imaginary straight line that is drawn from within the head through the area between the eyes.
An azimuth thruster in shipbuilding is a propeller that can be rotated horizontally.
The word azimuth is in all European languages today. It originates from medieval Arabic al-sumūt, pronounced as-sumūt in Arabic, meaning "the directions" (plural of Arabic al-samt = "the direction"). The Arabic word entered late medieval Latin in an astronomy context and in particular in the use of the Arabic version of the Astrolabe astronomy instrument. The word's first record in English is in the 1390s in Treatise on the Astrolabe by Geoffrey Chaucer. The first known record in any Western language is in Spanish in the 1270s in an astronomy book that was largely derived from Arabic sources, the Libros del saber de astronomía commissioned by King Alfonso X of Castile.[8]
↑ U.S. Army, Map Reading and Land Navigation, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993), ch. 6, p. 2
↑ U.S. Army, Map Reading and Land Navigation, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (28 March 1956), ch. 3, p. 63
↑ U.S. Army, Advanced Map and Aerial Photograph Reading, Headquarters, War Department, Washington, D.C. (17 September 1941), pp. 24–25
↑ Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota Press (2000), ISBN 0-8166-3661-3, p. 194
↑ "Azimuth" at New English Dictionary on Historical Principles; "azimut" at Centre National de Ressources Textuelles et Lexicales; "al-Samt" at Brill's Encyclopedia of Islam; "azimuth" at EnglishWordsOfArabicAncestry.wordpress.com. In Arabic the written al-sumūt is always pronounced as-sumūt (see pronunciation of "al-" in Arabic).
U.S. Army, Advanced Map and Aerial Photograph Reading, FM 21–26, Headquarters, War Department, Washington, D.C. (17 September 1941)
U.S. Army, Advanced Map and Aerial Photograph Reading, FM 21–26, Headquarters, War Department, Washington, D.C. (23 December 1944)
U.S. Army, Map Reading and Land Navigation, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993)
Retrieved from "https://en.formulasearchengine.com/index.php?title=Azimuth&oldid=222407"
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EUDML | Inversion of bilateral basic hypergeometric series. EuDML | Inversion of bilateral basic hypergeometric series.
Inversion of bilateral basic hypergeometric series.
Schlosser, Michael. "Inversion of bilateral basic hypergeometric series.." The Electronic Journal of Combinatorics [electronic only] 10.1 (2003): Research paper R10, 27 p.-Research paper R10, 27 p.. <http://eudml.org/doc/122933>.
author = {Schlosser, Michael},
keywords = {bilateral basic hypergeometric series; bilateral matrix inverses; Bailey's very-well-poised summation formula},
title = {Inversion of bilateral basic hypergeometric series.},
AU - Schlosser, Michael
TI - Inversion of bilateral basic hypergeometric series.
KW - bilateral basic hypergeometric series; bilateral matrix inverses; Bailey's very-well-poised summation formula
bilateral basic hypergeometric series, bilateral matrix inverses, Bailey's very-well-poised summation formula
Basic hypergeometric functions in one variable,
{}_{r}{\phi }_{s}
Articles by Schlosser
|
EUDML | On a general coarea inequality and applications. EuDML | On a general coarea inequality and applications.
On a general coarea inequality and applications.
Magnani, Valentino. "On a general coarea inequality and applications.." Annales Academiae Scientiarum Fennicae. Mathematica 27.1 (2002): 121-140. <http://eudml.org/doc/124715>.
@article{Magnani2002,
author = {Magnani, Valentino},
keywords = {Carathéodory measure; coarea integrability; coarea factor; Lipschitz map; stratified group; Sard-type theorem},
title = {On a general coarea inequality and applications.},
AU - Magnani, Valentino
TI - On a general coarea inequality and applications.
KW - Carathéodory measure; coarea integrability; coarea factor; Lipschitz map; stratified group; Sard-type theorem
Valentino Magnani, Blow-up of regular submanifolds in Heisenberg groups and applications
Franchi, Bruno,
BV
spaces and rectifiability for Carnot-Carathéodory metrics: an introduction
Carathéodory measure, coarea integrability, coarea factor, Lipschitz map, stratified group, Sard-type theorem
Articles by Magnani
|
To Charles Lyell [2 September 1849]1
It was very good of you to write me so long a letter which has interested me much; I shd. have answered it sooner, but I have not been very well for the few last days. Your letter has, also, flattered me much in many points.
I am very glad you have been thinking over the relation of subsidence & the accumulation of deposits:2 it has to me removed many great difficulties; please to observe that I have carefully abstained from saying that sediment is not deposited during periods of elevation, but only that it is not accumulated to sufficient thickness to withstand subsequent beach action: on both coasts of S. America, the amount of sediment deposited, worn away & redeposited oftentimes must have been enormous, but still there have been no wide formations produced: just read my discussion (p. 135 of my S. American Book) again with this in your mind.—3
I never thought of your difficulty (ie in relation to this discussion) of where was the land whence the 3 miles of S. Wales strata were derived?4 Do you not think that it may be explained, by a form of elevation, which I have always suspected to have been very common (& indeed had once intended getting all facts together). viz thus [DIAGRAM HERE] mountains & continent rising ocean bottom subsiding
The frequency of a deep ocean close to a rising continent, bordered with mountains, seems to indicate these opposite movements of rising & sinking close together: this wd. easily explain the S. Wales & Eocene cases.— I will only add that I shd think there wd be a little more sediment produced during subsidence than during elevation, from the resulting outline of coast after long period of rise.— There are many points in my vols. which I shd. have liked to have discussed with you, but I will not plague you: I shd like to hear whether you think there is anything in my conjecture on Craters of Elevation;5 I cannot possibly believe that St. Jago or Mauritius are the basal fragments of ordinary volcanos; I wd sooner even admit E. de Beaumont’s view than that; much as I wd sooner in my own mind in all cases follow you.— Just look at p. 232 in my S. America for trifling point,6 which however, I remember, to this day releived my mind of a considerable difficulty.—
I remember being struck with your discussion on the Missisippi beds7 in relation to Pampas, but I shd. wish to read them over again, I have, however, relent your work to Mrs Rich,8 who, like all whom I have met, have been much interested by it.— I will stop about my own geology.— But I see I must mention, that Scroope did suggest (& I have alluded to him, p. 118 but without distinct reference & I fear not sufficiently, though I utterly forget what he wrote9 ) the separation of basalt & trachyte, but he does not appear to have thought about the crystals which I believe to be the Keystone of the phenomenon: I cannot but think this separation of the molten elements has played a great part in the metamorphic rocks: how else cd the basaltic dykes come in great granitic districts such as those of Brazil?— What a wonderful book for labour is D’. Archiac!10
We are going on as usual: Emma desires her kind love to Lady Lyell: she boldly means to come to Birmingham with me & very glad she is that Lady Lyell will be there:11 two of our children have had a tedious slow fever.—12 I go on with my aqueous processes & very steadily but slowly gain health & strength. Against all rules13 I dined at Chevening with Ld. Mahon,14 who did me the grt. honour of calling on me, & how he heard of me, I can’t guess— I was charmed with Lady Mahon, & anyone might have been proud at the praises of agreeableness which came from her beautiful lips with respect to you.— I liked old Ld. Stanhope15 very much; though he abused geology & zoology heartily— “To suppose that the omnipotent God made a world, found it a failure, & broke it up & then made it again & again broke it up, as the geologists say, is all fiddle faddle”.— Describing species of birds & shells &c is all “fiddle faddle”.16 But yet I somehow liked him better than Ld Mahon.—
I am heartily glad we shall meet at Birmingham, as I trust we shall if my health will but keep up.— I work now every day at the Cirripedia for 2
\frac{1}{2}
hours & so get on a little but very slowly.— I sometimes after being a whole week employed & having described, perhaps only 2 species agree mentally with Ld. Stanhope that it is all fiddle-faddle: however the other day I got the curious case of a unisexual, instead of hermaphrodite, cirripede, in which the female had the common cirripedial character, & in two of the valves of her shell had two little pockets, in each of which she kept a little husband;17 I do not know of any other case where a female invariably has two husbands.— I have one still odder fact, common to several species, namely that though they are hermaphrodite, they have small additional or as I shall call them Complemental males:18 one specimen itself hermaphrodite had no less than seven of these complemental males attached to it. Truly the schemes & wonders of nature are illimitable.— But I am running on as badly about my Cirripedia as about Geology: it makes me groan to think that probably, I shall never again have the exquisite pleasure of making out some new district,—of evoking geological light out of some troubled, dark region.— So I must make the best of my Cirripedia.—
Remember me most kindly to Mr & Mrs Bunbury.— I am sorry to hear how weak your Father is—19 | Yours most sincerely | C. Darwin
The date is based on the endorsement and the dates of the Birmingham British Association meeting. CD left for Birmingham on 11 September 1849. In his health diary (Down House MS) CD recorded that he was well from 20 to 29 August, but that he was ‘Poorly’ and ‘exhausted’ from 30 August to 1 September.
Lyell was preparing a paper on ‘craters of denudation’ (C. Lyell 1850a), read to the Geological Society on 19 December 1849, in which the effects of elevation and subsidence on volcanic deposits were discussed.
CD’s explanation of the absence of conchiferous deposits as due to denudation and subsidence is in South America, pp. 135–9.
The denudation of South Wales had been the subject of much discussion between CD, Lyell, and Andrew Crombie Ramsay (see Correspondence vol. 3, letter to Charles Lyell, [3 October 1846], and letter to A. C. Ramsay, 10 October [1846]). Lyell did not, however, discuss South Wales in C. Lyell 1850a but rather in his anniversary address to the Geological Society on 15 February 1850 (C. Lyell 1850b, pp. liii, liv–vi).
This was the name given by Christian Leopold von Buch, and adopted by Jean Baptiste Armand Louis Léonce élie de Beaumont, to the theory originally proposed by Alexander von Humboldt that volcanic cones were formed by upward pressure, rather than by the eruption of lava through vents. Such pressure raised originally horizontal layers into a dome that was easily broken through. Lyell had opposed this view as early as 1830 in the first volume of his Principles of geology (C. Lyell 1830–3). For a history of the controversy see Dean 1980. CD speculated that the mountains might still be considered ‘craters of elevation’ by slow elevation, in which the central hollows were formed ‘not by the arching of the surface, but simply by that part having been raised to a less height’ (Volcanic Islands, p. 96).
On p. 232 of South America, CD discussed the ‘Eruptive Sources of the Porphyritic Claystone and Greenstone Lavas’ and suspected that the difficulty of tracing the streams of porphyries to their ancient eruptive sources was because ‘the original points of eruption tend to become the points of injection’.
Lyell delivered a lecture about the delta of the Mississippi River at the Royal Institution on 8 June 1849 and published a detailed account in C. Lyell 1849, 2: 242–56. CD presumably refers to the latter, which is in the Darwin Library–CUL.
Mary Rich, née Mackintosh, Fanny Mackintosh Wedgwood’s half-sister.
Scrope 1825. CD’s copy (unannotated) is in the Darwin Library–Down. See Volcanic islands, p. 118.
Archiac 1847–60, volumes one and two, in which étienne Jules Adolphe Desmier de Saint-Simon, Vicomte d’Archiac, summarised the progress of French geology from 1834–45. CD’s copy of volume one is in the Darwin Library–Down.
The British Association was due to meet in Birmingham, 12–19 September 1849. Lyell was president of section C (geology and physical geography); CD was a vice-president of the association. According to her diary, Emma followed CD to Birmingham on 12 September.
According to Emma Darwin’s diary, Henrietta Darwin developed a fever on 5 July and William Darwin fell ill on 11 July. William did not come ‘down stairs’ until 30 July, and he suffered a relapse on 16 August.
The ‘rules’ set out by James Manby Gully for CD’s water therapy at home.
Philip Henry Stanhope, Viscount Mahon, later 5th Earl Stanhope. Chevening, in Kent, was the family seat.
Philip Henry Stanhope, 4th Earl Stanhope, father of Viscount Mahon.
In the Autobiography CD mentioned that the Earl once said to him, ‘Why don’t you give up your fiddle-faddle of geology and zoology, and turn to the occult sciences?’ (p. 112).
Ibla cumingii (see Living Cirripedia (1851): 189–203).
The complemental males of Scalpellum, first mentioned in letter to J. L. R. Agassiz, 22 October 1848. See Living Cirripedia (1851): 231–43 and 281–93.
Charles Lyell Sr died on 8 November 1849.
Archiac, Etienne Jules Adolphe Desmier de Saint-Simon, Vicomte d’. 1847–60. Histoire des progrès de la géologie de 1834 à 1845. 8 vols. Paris.
Dean, Dennis R. 1980. Graham Island, Charles Lyell, and the craters of elevation controversy. Isis 71: 571–88.
Scrope, George Poulett. 1825. Considerations on volcanos, the probable causes of their phenomena, the laws which determine their march, the disposition of their products, and their connexion with the present state and past history of the globe; leading to the establishment of a new theory of the earth. London: W. Phillips.
Discusses effect of subsidence and elevation on deposits. Cites examples along coasts of South America and Wales. Proposes theory to explain thickness of deposits in south Wales.
Asks CL’s opinion of his theory of "craters of elevation" described in Volcanic islands.
Mentions CL’s comparison of Mississippi beds to the Pampas.
Comments on Poulett Scrope’s views on the separation of basalt and trachyte.
Describes his cirripede work.
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MainDegree - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : MainDegree
main degree of a nonconstant polynomial
MainDegree(p, R)
The function call MainDegree(p,R) returns the main degree of p, that is, the degree of p with respect to its main variable.
This command is part of the RegularChains package, so it can be used in the form MainDegree(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[MainDegree](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
R≔\mathrm{PolynomialRing}\left([x,y,z]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔\left(y+1\right){x}^{3}+\left(z+4\right)x+3
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{MainVariable}\left(p,R\right)
\textcolor[rgb]{0,0,1}{x}
\mathrm{Initial}\left(p,R\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{MainDegree}\left(p,R\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Rank}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{Tail}\left(p,R\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
R≔\mathrm{PolynomialRing}\left([z,y,x]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔\mathrm{expand}\left(\left(y+1\right){x}^{3}+\left(z+4\right)x+3\right)
\textcolor[rgb]{0,0,1}{p}\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}{y}\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}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{MainVariable}\left(p,R\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{Initial}\left(p,R\right)
\textcolor[rgb]{0,0,1}{x}
\mathrm{MainDegree}\left(p,R\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{Rank}\left(p,R\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{Tail}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
R≔\mathrm{PolynomialRing}\left([z,y,x],3\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔{\left(x+y\right)}^{3}{z}^{3}+3{z}^{2}+2z+y+4
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}
\mathrm{MainVariable}\left(p,R\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{Initial}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{MainDegree}\left(p,R\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Rank}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{Tail}\left(p,R\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
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2.1 Rational and irrational numbers | Rational and irrational numbers | Siyavula
A set is a collection of objects that have something in common. A set could be a group of things that we use together, or that have similar properties. Numbers can be organised into number sets. In previous years you learnt about the following number sets:
Natural numbers are the positive counting numbers.
Whole numbers are the positive counting numbers plus zero.
Integers are the whole numbers plus the negative counting numbers.
We can use set notation to write these sets using symbols:
You will learn more about set notation in later years.
number set A number set is a group of numbers that have similar properties.
natural numbers The set of natural numbers includes all the positive counting numbers.
whole numbers The set of whole numbers includes all the positive counting numbers and zero.
integers The set of integers includes all the negative counting numbers, zero, and all the positive counting numbers.
In this chapter you will learn about the set of numbers called rational numbers. A rational number is any number that can be written in the form
\frac{a}{b}
and
b
are both integers and
b\neq 0
. In set notation we write:
In set notation, the symbol
\mid
means "such that". The symbol
\in
means "is an element of". Therefore,
a \in \mathbb{Z}
a
is an element of the set of integers".
Any numbers that are not part of the set of rational numbers are called irrational numbers. This set of numbers is sometimes also called the set of non-rational numbers. You will learn more about irrational numbers further along in this chapter.
rational numbers The set of rational numbers include all numbers that can be written in the form
\frac{a}{b}
and
b
b\neq 0
irrational numbers Irrational numbers are all the numbers that are not part of the set of rational numbers. The set can also be called non-rational numbers.
The following diagram shows the relationship between the number sets discussed above.
All integers are rational numbers. This is because any integer can be written as a fraction with
1
as the denominator. Here are some examples:
All common fractions are rational numbers. In previous years you learnt that a common fraction is written in the form:
where the numerator and the denominator are both whole numbers.
In proper fractions, the numerator is smaller than the denominator. Here are some examples:
In improper fractions, the numerator is larger than the denominator. Here are some examples:
All mixed numbers are rational numbers. In previous years you learnt that a mixed number consists of a whole number and a proper fraction. You also learnt how to write mixed numbers as improper fractions. The same method can be used for positive and negative mixed numbers. Here are some examples:
\begin{align} 2\frac{3}{4} & =\frac{11}{4}\\ 15\frac{7}{8} & = \frac{127}{8}\\ 100\frac{1}{100} & = \frac{101}{100} \\ -5\frac{3}{20}& =\frac{-103}{20}\\ -50\frac{9}{10}& =\frac{-509}{10} \end{align}
Numbers written in decimal notation
All terminating decimals are rational numbers. In a terminating decimal, the digits that come after the decimal point come to an end. Last year you learnt how to convert numbers written in decimal notation to common fractions. Here are some examples:
\begin{array}{rcl} 0.8 & = \frac{8}{10} = & \frac{4}{5} \\ 2.5 & = \frac{25}{10} = & \frac{5}{2} \\ 0.12 & = \frac{12}{100} = & \frac{3}{25} \\ 4.08 & = \frac{408}{100} = & \frac{102}{25} \\ 0.025 & = \frac{25}{1,000} = & \frac{1}{40} \end{array}
All recurring decimals are rational numbers. In a recurring decimal, one or more decimal digits repeat in the same pattern forever. If only one or two digits repeat, we write a dot above the repeating digits. If more than two digits repeat, we write a dot above the first and last digits of the repeating pattern. Here are some examples:
\begin{align} 0.3333333 \ldots & = 0.\dot{3} \\ 0.1555555 \ldots & = 0.1\dot{5} \\ 10.27272727 \ldots & = 10.\dot{2}\dot{7} \\ 2.578578578 \ldots & = 2.\dot{5}7\dot{8} \\ 15.32656565 \ldots & = 15.32\dot{6}\dot{5} \end{align}
Recurring decimals can be written as common fractions.
terminating decimal In a terminating decimal, the digits that come after the decimal point come to an end.
recurring decimal In a recurring decimal, one or more decimal digits repeat in the same pattern forever.
Worked example 2.1: Writing recurring decimals as common fractions (Method 1)
2.\dot{1}\dot{5}
as a common fraction.
Step 1: Write the number as the sum of its whole number and its decimal fraction.
Step 2: Write the decimal fraction as a proper fraction. The recurring digits are the numerator. The denominator has the number 9 repeated as many times as there are digits in the numerator.
The numerator must be 15. It has two digits.
The denominator must therefore be 99.
Step 3: Find the simplest form of the fraction from Step 2.
Step 4: Rewrite Step 1, but now with the decimal fraction as a proper fraction. Add the whole number and the fraction.
\begin{align} &2+\frac{5}{33} \\ =&\frac{2}{1}\times\frac{33}{33}+\frac{5}{33}\\ =&\frac{66}{33}+\frac{5}{33} \\ =&\frac{71}{33} \end{align}
This method only works if there are no non-recurring digits after the decimal point.
2.\dot{1}\dot{5}
Step 1: Write out the recurring pattern at least three times. Let this be equal to
x
. Label the equation as (1).
Step 2: Multiply the written out decimal by multiples of ten. Start with 10, then try 100, then 1,000 and so on. You must get the same digits after the decimal point as those in equation (1). You must also get the repeating digits before the decimal point.
Start with 10:
\begin{align} 10 \times 2.151515 \ldots &= 10 \times x \\ 21.51515 \ldots &= 10x \\ \end{align}
The digits after the decimal point are not the same as in equation (1), and only the number 1 of the repeating digits is before the decimal point.
Now try 100:
\begin{align} 100 \times 2.151515 \ldots &= 100 \times x \\ 215.1515 \ldots &= 100x \\ \end{align}
The digits after the decimal point are the same as in equation (1), and both recurring digits, 15, appear before the decimal point.
Step 3: Label the equation from Step 2 as (2). Subtract equation (1) from equation (2).
\begin{align} 2.151515 \ldots &= x \phantom{00000}(1) \\ 215.1515 \ldots &= 100x \phantom{00}(2) \end{align}
(2)-(1)
\begin{align} 215.1515 \ldots - 2.151515 \ldots &= 100x - x \\ 213 &= 99x \\ \end{align}
Step 4: Solve the equation from Step 3.
\begin{align} 213 &= 99x \\ \frac{213}{99} &= \frac{99x}{99}\\ x&=\frac{213}{99} \\ &= \frac{71}{33} \end{align}
If there is a non-recurring digit after the decimal point, multiply equation (1) by 10 before you carry on with Step 2.
Exercise 2.1: Write recurring decimals as common fractions
Write each of the following recurring decimals as common fractions.
\begin{align} &4+\frac{2}{9}\\ =&\frac{36+2}{9}\\ =&\frac{38}{9} \end{align}
\begin{align} &33+\frac{3}{9}\\ =&\frac{297+3}{9}\\ =&\frac{300}{9} \end{align}
There is a non-recurring digit after the decimal point, so you need to multiply equation (1) by 10 before you get to the values you can subtract.
You still don't have the repeating digit before the decimal point, so multiply by 100.
The repeating digits are now only the same in equations (2) and (3), so you subtract (2) from (3).
\begin{align} 100x-10x&=2.22222-0.22222 \\ 90x&=2 \\ \frac{90x}{90}&=\frac{2}{90}\\ \therefore x&=\frac{1}{45} \end{align}
All percentages are rational numbers. A percentage is a fraction of which the denominator is 100, and where only the numerator is written down, followed by a percentage symbol. Last year you learnt how to convert percentages to common fractions. Here are some examples:
percentage A percentage is a fraction of which the denominator is 100, and where only the numerator is written down, followed by a percentage symbol.
The square root of a number is a factor of the number that, when multiplied by itself, gives the number. Only square roots that have whole numbers, terminating decimals or common fractions as answers are rational numbers. Here are examples of square roots that are rational numbers:
square root The square root of a number is a factor of the number that, when multiplied by itself, gives the number.
Exercise 2.2: Prove that numbers are rational
Prove that each of the following numbers are rational numbers.
Remember that a rational number is any number that can be written in the form
\frac{a}{b}
and
b
b\neq 0
Irrational numbers cannot be written in the form
\frac{a}{b}
and
b
b\neq 0
Irrational numbers do not have exact values. Examples of irrational numbers are:
decimals that have an infinite number of decimal digits of which none are recurring, for example
2.51689753469 \ldots
square roots that do not have whole numbers, terminating decimals or common fractions as answers (roots), for example
\sqrt{11}
Exercise 2.3: Classify numbers as rational or irrational
Classify each of the following as a rational number or a irrational number.
\sqrt{100}=10
, and whole numbers are rational numbers.
Percentages are rational numbers.
Decimals that are not terminating or recurring are irrational numbers.
Mixed numbers are rational numbers.
\sqrt{27}
does not have a whole number, terminating decimal or common fraction as its root (or answer).
Recurring decimals are rational numbers.
\sqrt{\frac{36}{121}}=\frac{6}{11}
Terminating decimals are rational numbers.
\sqrt{\frac{4}{20}}
does not have a whole number, terminating decimal or common fraction as its root.
\sqrt{0.1}
-\sqrt{4}=-2
2.4 Pi (
\pi
Pi is an example of a irrational number. It is represented by the symbol
\pi
. It represents different ratios in a circle. You should be familiar with the following dimensions of a circle:
For any circle, pi represents the following ratios:
Pi does not have a fixed value. To use it in calculations, we round it to 3.142 or
\frac{22}{7}
(\pi)
is the constant ratio of the circumference to the diameter of any circle.
Worked example 2.3: Confirming the value of
\pi
The circle below has a circumference (
C
62.832\text{ cm}
and an area (
A
314.16\text{ cm}^2
Calculate the ratio of the circumference to the diameter (
d
Calculate the ratio of the area to the square of the radius (
r
What do your answers tell you about the value of
\pi
Step 1: Calculate the ratio of the circumference to the diameter.
\begin{align} &\frac{C}{d} \\ =&\frac{62.832 \text{ cm}}{2 \times 10 \text{ cm}} \\ =&\frac{62.832 \text{ cm}}{20 \text{ cm}} \\ =&3.1416 \end{align}
Step 2: Calculate the ratio of the area to the square of the radius.
\begin{align} &\frac{A}{r^2} \\ =&\frac{314.16\text{ cm}^2}{10\text{ cm} \times 10\text{ cm}} \\ =&\frac{314.16\text{ cm}^2}{100\text{ cm}^2} \\ =&3.1416 \end{align}
Step 3: State what the answers tell you about the value of
\pi
Both ratios are used in calculating the value of
\pi
, and the answers are the same
-
3.1416. This confirms that the value of
\pi
is close to 3.1416.
Exercise 2.4: Calculate the value of
\pi
For each of the following circles, calculate the value of
\pi
to four decimal places.
The circle has a circumference of
25.133\;\text m
\begin{align} \pi&=\frac{C}{d} \\ &=\frac{25.133\; \text m}{8\; \text m} \\ &=3.1416 \end{align}
12.566\text{ cm}^2
\begin{align} \pi=&\frac{A}{r^2} \\ =&\frac{12.566\text{ cm}^2}{2\text{ cm} \times 2\text{ cm}} \\ =&\frac{12.566\text{ cm}^2}{4\text{ cm}^2} \\ =&3.1415 \end{align}
For the circle below,
C=14.14\;\text m
r=2.25\;\text m
\begin{align} \pi&=\frac{C}{d} \\ &=\frac{14.14\; \text m}{2\times 2.25\; \text m} \\ &=\frac{14.14\; \text m}{4.5\; \text m} \\ &=3.1422 \end{align}
A number set is a group of numbers that have similar properties.
The set of natural numbers includes all the positive counting numbers.
The set of whole numbers includes all the positive counting numbers and zero.
The set of integers includes all the negative counting numbers, zero and all the positive counting numbers.
The set of rational numbers include all numbers that can be written in the form
\frac{a}{b}
and
b
b\neq 0
square roots that have whole numbers, terminating decimals or common fractions as answers.
Irrational numbers (or non-rational numbers) are all the numbers that are not part of the set of rational numbers.
The following are irrational numbers:
decimals that have an infinite number of decimal digits of which none are recurring
square roots that do not have whole numbers, terminating decimals or common fractions as their roots.
\pi
) is the constant ratio of the circumference to the diameter of any circle.
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A Brief Primer on Navigating TokenSpace · TokenSpace
TokenSpace may be considered by analogy with our own spatio-temporal conception of reality, consisting of a three-dimensional space delineated (for convenience and visual clarity) by orthogonal axes
\mathcal{{\overline{S}}}
\mathcal{{\overline{M}}}
\mathcal{{\overline{C}}}
. Assets may possess a score or range on each axis between 0 and 1 inclusive giving rise to an object inhabiting a region of TokenSpace described by the (x, y, z) co-ordinates (
\mathcal{{\overline{C}}}
\mathcal{{\overline{M}}}
\mathcal{{\overline{S}}}
). Time-dependence of object properties may also be incorporated to reflect the dynamic nature of cryptocurrency protocol networks and their native assets, tokens issued atop them and network fragmentations such as ledger forks.
\mathcal{{\overline{S}}}
\mathcal{{\overline{M}}}
\mathcal{{\overline{C}}}
correspond to intuitively reasoned assignments of subjective classificatory meta-characteristics Securityness, Moneyness and Commodityness which together form the basis of TokenSpace classification
methods currently in development. Each asset’s location in TokenSpace is intended to be derived from a weighted scoring system based upon taxonomy, typology, intuitive, elicited and/or quantitative methods depending on the choices and assertions of the user — which may or may not be identical to those proposed in this work.
TokenSpace visual impression. Yes, those branches coming out of the axes represent taxonomies!
Definitions of the proposed meta-characteristics:
\mathcal{{\overline{S}}}
— Securityness. The extent to which an item or instrument qualifies as or exhibits characteristics of a securitised asset. For the purposes of clarity this meta-characteristic does not refer to how secure (robust/resistant) a particular network or asset is from adversarial or malicious actions.
\mathcal{{\overline{M}}}
— Moneyness. The extent to which an item or instrument qualifies as or exhibits characteristics of a monetary asset.
\mathcal{{\overline{C}}}
— Commodityness. The extent to which an item or instrument qualifies as or exhibits characteristics of a commoditised asset.
Example scores for a range of assets are outlined in the tables below with visual depictions. Ideal types are postulated canonical examples of particular asset types and are discussed in Section 2 of the manuscript. It is the aim of this and future research to provide suggestions for classification approaches and some examples on how TokenSpace may be utilised to comparatively characterise assets from the perspective of various ecosystem stakeholders. Time-dependence may also be significant in certain instances and can be incorporated into this framework by evaluating an asset’s location in TokenSpace at different points in time and charting asset trajectories.
TokenSpace is expected to be useful to regulators, investors, researchers, token engineers and exchange operators who may construct their own scoring systems based on these concepts. Careful review of territory-specific regulatory guidance and judicious consideration of boundary functions for example delineating “safe”, “marginal” or “dangerous” likely compliance of assets with respect to particular regulatory regimes are recommended and an example is presented in Figure 3. Parallel Industries has developed hybrid multi-level hybrid categorical/numerical taxonomies for each meta-characteristic alongside time-dependent and probability distribution functions for anisotropic score modelling.
Example of cryptographic & legacy assets inhabiting TokenSpace.
Example of a regulatory boundary function. Arbitrary polynomial for illustrative purposes.
TokenSpace, a Parallel Industries research project
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If you were looking for the Empire of Antarctica, sorry, but it was eaten by a polar bear.
Motto: 'Proletarii vsekh stran, soyedinyaytes'! (Workers of the world, unite!)
Anthem: My vodka is frozen again, and I can't find my matches.
The Democratic Penguin's Republic of Antarctica (Russian: Анархия, translit. Anarchia, the "ch" pronounced as in "loch"), is one of the 16 republics, which appeared after the collapse of the Soviet Union in 1991. It is part of the Empire of Antarctica, a large land mass, whose only difference from the Arctic is that it is located on the pole opposite to it. Nobody lives there, except for the people who do. It's really cold, so nobody needs a refrigerator. However, a few members of the elite own refrigerators to show off and to sleep inside.
Antarctica is the fourth largest continent in the world. It has been infiltrated by some Soviet Communists, who maintain a secret base there ostensibly for "scientific research", but the likely reason is to gather intelligence on the U.S.'s top-secret weapons programs from a relatively undetectable spot. They do not get paid and are therefore considered volunteers.
Antarctica was settled in 1990 by Yuri Yarov, a millionaire from Murmansk, Russia. Yarov brought a band of 100,000 men, women and aphids to the tiny King Porge Island. They began what was known as the Great Failed Experiment. No one knows why it is quoted as a failure; this is thought to be anti-Antarctican propaganda by the rest of the world's governments. Yuri Yarov was an executive secretary of the Commonwealth of Independent States.
Yarov received the Order of the Democratic Penguin's Republic of Antarctica on June 2, 1998, and he hung it in the Kremlin of Antarctica, which is build entirely from ice.
Aware that countries like the Soviet Union and the United States of America were planning to claim Antarctica for their exploitative purposes, Yarov declared himself the "leader" of the small continent. Antarctica continues to this day to be a completely anti-authoritarian non-state. Its media rarely issues important reports because the wendigos prevent reporters from going out at night and besides that, it's cold. An exception to this is the yearly punk rock festival in Yarov, when all the gilded youth of Antarctica sits on an iceberg close to the Republic's capital, waiting for the musicians to arrive, which has only happened once so far, entirely by accident.
{\displaystyle \pi \times \infty }
One notable citizen of the Democratic Penguins' Republic of Antarctica is Jack Pen, a globally recognised wildlife expert who was reported missing on his trip to the continent, where he planned to study penguins. Reportedly, once he entered the country, he was no longer allowed to leave it, and was granted a honorary Antarctican citizenship, due to his achievements in the rest of the world. He is currently retired in his luxurious villa in Côte de Glace, where he occasionally produces a film to boost the country's economy.
The official languages of Antarctica are Russian, Ukrainian, Polar and Greenlandese, though the aristocrats often learn other languages.
After Los Angeles, Bollywood, India and Kiev (in Ukraine) Antarctica produces more movies than any other place in the world. Among its best known works are April of the Puffins (produced by a famous Antarctican movie director Jack Pen), Scott of the Sahara (produced by a famous Antarctican movie director Jack Pen), Antarctica minus Five-O (produced by a famous Antarctican movie director Jack Pen), and Beverly Tundra 90210 (produced by a famous Antarctican movie director Jack Pen). Once, an Antarctican film made it to America on a wooden raft and had wild box-office success: that movie is known only as March of the Penguins. Considering the amount of wood rafts sent to the USA, the chances were not low that at least one would have gotten there.
Retrieved from "https://uncyclopedia.com/w/index.php?title=The_Democratic_Penguin%27s_Republic_of_Antarctica&oldid=6138085"
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Linearize Simscape Networks - MATLAB & Simulink - MathWorks í•œêµ
Find Steady-State Operating Point
Troubleshoot Simscape Network Linearizations
Zero Linearization Example
You can linearize models with Simscape™ components using Simulink® Control Design™ software. Typically, some states in a Simscape network have dependencies on other states through constraints.
To find a steady-state operating point at which to linearize a Simscape model, you can use:
Optimization-based trimming — Specify constraints on model inputs, outputs, or states, and compute a steady-state operating point that satisfies these constraints.
To produce better trimming results for Simscape models, you can use projection-based trim optimizers. These optimizers enforce the consistency of the model initial condition at each evaluation of the objective function or nonlinear constraint function.
Simulation snapshots — Specify model initial conditions near an expected equilibrium point, and simulate the model until it reaches steady state.
For more information, see Find Steady-State Operating Points for Simscape Models.
To linearize your model, you must specify the portion of the model you want to linearize using linear analysis points; that is, linearization inputs and outputs, and loop openings. You can only add analysis points to Simulink signals.
To add a linearization input or loop opening to the input of a Simscape component, first convert the Simulink signal using a Simulink-PS Converter (Simscape) block.
To add a linearization output or loop opening to the output of a Simscape component, first convert the Simscape signal using a PS-Simulink Converter (Simscape) block.
For more information on adding linear analysis points, see Specify Portion of Model to Linearize.
After you specify a steady-state operating point and linear analysis points, you can linearize your Simscape model using:
The Model Linearizer.
The linearize function.
An slLinearizer interface.
For general linearization examples, see Linearize Simulink Model at Model Operating Point and Linearize at Trimmed Operating Point.
Simscape networks can commonly linearize to zero when a set of the system equation Jacobians are zero at a given operating condition. Usually, poor initial conditions of the network states cause these zero linearizations.
Consider a system where the mass flow rate from a variable orifice is controlling the position of a piston. The mass flow rate equation of the variable orifice is:
q={C}_{d}A\sqrt{\frac{2}{\mathrm{μ}}}{\left(\frac{p}{{p}^{2}+{p}_{cr}^{2}}\right)}^{0.25}
q is the mass flow rate.
A is the area of the variable orifice opening.
μ is the fluid density.
p is the pressure drop across the orifice, p = pa - pb.
pcr is the critical pressure, which is a function of pa and pb.
The control variable for this system is the orifice area, A, which controls the mass flow rate. The Jacobian of the mass flow rate with respect to the control variable is:
\frac{âq}{âA}={C}_{d}\sqrt{\frac{2}{\mathrm{μ}}}{\left(\frac{p}{{p}^{2}+{p}_{cr}^{2}}\right)}^{0.25}
The linearized mass flow rate equation is:
\begin{array}{l}\stackrel{¯}{q}={C}_{d}\sqrt{\frac{2}{\mathrm{μ}}}{\left(\frac{p}{{p}^{2}+{p}_{cr}^{2}}\right)}^{0.25}\stackrel{¯}{A}+\frac{âq}{â\mathrm{μ}}\stackrel{¯}{\mathrm{μ}}\\ +\left(\frac{âq}{â{p}_{cr}}\frac{â{p}_{cr}}{â{p}_{a}}+\frac{âq}{âp}\right){\stackrel{¯}{p}}_{a}+\left(\frac{âq}{â{p}_{cr}}\frac{â{p}_{cr}}{â{p}_{b}}â\frac{âq}{âp}\right){\stackrel{¯}{p}}_{b}\end{array}
\stackrel{¯}{·}
represents a deviation from the nominal variable.
In the linearized equation, if the nominal pressure drop p across the orifice is zero, then
\stackrel{¯}{A}
has no influence on
\stackrel{¯}{q}
. That is, if the instantaneous pressure drop across the orifice is zero, the orifice area has no influence on the mass flow rate. Therefore, you cannot control the piston position using the orifice area control variable.
To avoid this condition, linearize the model about an operating point where the pressure drop over the orifice is nonzero (pa ≠pb).
To fix linearization problems caused by poor initial conditions of network states, you can:
Linearize the system at a snapshot operating point or trimmed operating point. When possible, this approach is recommended.
Find and modify the problematic states of the operating point. This option can be difficult for models with many states.
Using the first approach, you can ensure that the model states are consistent via the Simulink and Simscape simulation engine. Simscape initial conditions are not necessarily in a consistent state. The Simscape engine places them in a consistent state during simulation and for trimming using the Simscape trim solvers.
A common workflow is to simulate your model, observe at what time the model satisfies the operating condition at which you want to linearize, then take a simulation snapshot. Alternatively, you can trim the model about the condition you are interested in. In either case, the network states are in a consistent condition, which solves most poor linearization issues.
Using the second approach, you search through the physical network states to find conditions that can create a zero Jacobian. This approach can require some intuition about the dynamics of the physical components in your model. As a starting point, search for states that are zero and that interact directly with nonlinear physical elements, such as the variable orifice in the preceding example.
To search the physical states, you can use the Linearization Advisor, which collects diagnostic information during linearization. The Linearization Advisor does not provide diagnostic information on a component-level basis for Simscape networks. Instead, it groups diagnostic information for multiple Simscape components together.
Linearize your model with the Linearization Advisor enabled, and extract the LinearizationAdvisor object.
[linsys,linop,info] = linearize(mdl,io,op,opt);
Create a custom query object, and search the diagnostic information for Simscape blocks.
qSS = linqueryIsBlockType('simscape');
advSS = find(advisor,qSS);
To find problematic state values, check the block operating point in each BlockDiagnostic object.
advSS.BlockDiagnostics(i).OperatingPoint.States
Once you find a problematic state, you can change the value of the state in the model operating point, or create an operating point using operpoint.
You can also search the Linearization Advisor in the Model Linearizer. For more information, see Find Blocks in Linearization Results Matching Specific Criteria.
linearize | slLinearizer
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Mr. Greer solved an equation below. However, when he checked his solution, it did not make the original equation true. Find his error and then find the correct solution.
\begin{aligned}[t] 4x &= 8\left(2x - 3\right) \\ 4x &= 16x - 3 \\ -12x &= -3 \\ x &= \frac{-3}{-12} \\ x &= \frac{1}{4} \end{aligned}
Check Mr. Greer's distribution. Did he do it correctly?
8
-3
-3
Solve the equation by distributing correctly.
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Meaning of life - Uncyclopedia, the content-free encyclopedia
Though the self-proclaimed experts at Wikipedia have created an article called Meaning of life, it's extremely unlikely you'll find the meaning of life in there.
People believe in many answers to "what is the meaning of life?" The following is a list of propositions for the meaning of life:
...to procreate and have Greek incestuous orgies.
...to reprove instruction.
...to eat pizza and buy as many lava lamps as possible.
...to reach the highest level.
...to catch 'em all.
...to be the very best that no one ever was.
...to get epic phat lewt.
...to become the chosen one and tell Gretzky you stole his spot
...to throw a bagel party.
...to eat the bagels at the bagel party.
...to praise Raptor Jesus, for he went extinct for your sins.
...to praise the great Flying Spaghetti Monster and hope to be touched by His Noodly Appendage.
...to slay the mightiest dragon in the darkest of all dungeons.
...to gain complete control over reality using Garry's Mod.
...to acquire as much material wealth, sleep with as many attractive women, and acquire as much power as possible.
...to be or not to be; but choose wisely.
...to lick your elbow. Yeah, just cut your arm off and lick it, then you will have finally achieved something in your miserable life.
...to crush your enemies, to see them driven before you, and to hear the lamentations of their women.
...to possess peace, love, joy, and all others' base.
...to eat all the pies.
...to pronounce "*".
...to know the meaning of life.
...Chicken McNuggets, and some fries with that.
...to find out what the hell we've got a spleen for.
...to find an exception to rule 34; or prove that there are no exceptions.
...to eat your face.
...to finish a game of Monopoly, with savings!
...to sleep with your gran.
...the condition which distinguishes animals and plants from inorganic objects and dead organisms.
...to follow word-for-word the scriptural teachings of a religious doctrine, and (Just like in SHIPWRECKED) get as many other people to join your religion's 'team' as possible. When the Apocalypse comes (The next one's in 2012) the team (religion) with the most members/followers wins, and become God's chosen children who will live in paradise forever. Everyone else will burn in eternal damnation.
...to hunt down gypsies and poke them with a stick. If there are no gypsies available, just about anyone you don't like can be used as an alternative substitute.
...to watch Noddy, and survive.
...to rid the world of fax machines and the people operating them.
...to eat as much cake as possible.
...to see as many semi or fully naked women as possible.
...to find the holy pie of truth.
...to have as many fictional friends as possible.
...to fuck an infinite number of monkeys with typewriters.
...to have so much sex your dick falls off.
...is not this.
...to find out what the point of "Lost" is
Money, while not the scientifically definitive meaning of life, is certainly the most widely accepted substitute. Money is the drive behind such wonders of human achievement as wars, religions, and pie (Apple pie has recently publicly decried any connection with money - an investigation by members of the UN is underway). Money is also Time, hence Time is money, which gets a little confusing, since this postulates that everything equal to Money is actually made of Time. Some people have speculated either that happiness and money are the same (within the limits of the Heisenberg Uncertainty Principle), or that happiness and money have nothing to do with each other (being non-zero parameters of a chaotic system), and others have noted that money cannot buy happiness outright, (but rental rates start at a reasonable figure). These people are spending too much time philosomophizing, and should devote more of it to making money. It is completely acceptable to philosomophize on the meaning of pie, however.
It has been hypothesized that money is not the answer to the meaning of life but is, in fact, the root of all evil. Others have speculated that, indeed, lack of money is the root of all evil. This is merely clap-trap made up by people who don't have any. The challenge that "money kills people" is easily dismissed by the fact that "money doesn't kill people, poverty kills people."
Using the first hypothesis, all women have been proven to be pure evil. Women take time and money. Time is money(as stated before), so therefore women are money times money. Money is the root of all evil, so women equal the root of all evil multiplied by itself, or simply evil.
While money, happiness, & pie have all driven people--and their wallets--to the point of no return, some (mainly scholars and those who didn't have the time of their life during high school) believe the meaning of life is to prepare us for another life. Only this time, our success will depend on what we learn and what experiences we sacrifice for the propelling of learning. This is, of course, crap written by a couple of very angsty and deprived teens.
Though widely discredited as an invention of Hollywood and The Beatles, "love" is actually considered by some jerks to be a "real" human emotion capable "of" bonding the whole universe. However much bullshit this so-called "love" may be, it has gained a following in some religious circles, who preach to "love thy neighbor" and pray to "the Loving One." These are mostly small groups that you've never heard of, so don't worry your pretty little head about it.
Aglets (the plastic things at the end of your shoelace)
One of the few theories with scientific merit is that the plastic things at the end of your shoelace, also called Aglets, are a vital part to our existence, as many would argue. We believe that inside the modest shoelace contains an impossibly powerful bomb, capable of blowing The Universe into bite-sized chunks (tasty with BBQ sauce). This theory was proven by sir Shoelaceobsesser Von David the Third, while he was mowing his lawn. He noticed that his shoelaces were about to be destroyed by his evil and horribly poorly built lawn mower, but they were saved because his shoes had extremely cool Silver Aglets. Later, when he shared this information with his colleagues, they considered the idea completely heinous and illogical. Therefore, using the Alternate Inverse of Reality Postulate, it has been proven that Aglets are essential and vital, as well as important, especially when they are Silver.
The recent updated version of the popular television show may in fact be the true meaning of life; everything from the tall hot blond to the crazy little asian brobot. Executive producer Ron Moore is quoted as saying, "I'm not just trying to create a show here, I am trying to create the meaning of life. We are going to have tall blond hypersexual robots that look like humans, and shorter crazy hypersexual asian robots who look like humans, there will be explosions, booze. Meaning of life."
The Knights who say Ni are currently debating over wheather The Great and Almighty Shrubbery is the Meaning of Life, or wheather it is 42, or wheather it is Cornflakes. A Shertain Shcottish Shpy complies with the "Cornflake Theory:"
"Cornflakesh ish shurley the Meaning of Life, ash it wash, ish, and alwaysh will be there, shtanding in the cereal ishle necksht to the Froshted Flakesh."
The Meaning of Life timeline II:The timeline returns
22 October 2006 Group Of Democrats for Finding Out the Meaning Of Life (GODFOMOL) begins.
23 October 2006 GODFOMOL sends White House its records. Some include Black Eyed Peas.
24 October 2006 GODFOMOL sends White House the meaning of life.
26 October 2006 The letter was a fake. Unfortunately, George Bush's head was already in Bill Clinton's butt.
As the leader of the Tibetan people and moderately successful sideman to James Brown, The Dalai Lama once claimed to hold the truth to the meaning of life. However, this wise and jazzy man's theory was later discredited when it was revealed that he had lied about his identity and was proven to in fact be an alpaca.
Try and be nice to people, avoid eating fat, read a good book every now and then, get some walking in, and try and live together in peace and harmony with people of all creeds and nations and above all, avoid the salmon mousse.
This, of course, is absolute rubbish.
"your mum is dead. I'm sorry. Look if there's anything I can do here's my card....you can call me Daddy.
Basically every human is connected like a set of dominoes, every breath you take affects some thing else. You are as important as a rock, that rock is as important as the planet Mars, and Mars is as important as a cow named Pete (and that cow is as important as the meaning of life). However because the universe is huge and all, you only affect Earth, and Earth is as important as.... absolutely nothing. The other bad thing about this is that every thing affects you, so basically every thought you have is just from other people, so you have no free will, fate controls every thing. So think of the universe as a bunch of domino sets, in the "Earth" set a rock domino is the same as a tree domino and a king domino, but like other domino sets, you can predict the fate of all the dominoes.
Your dad got it on with your mom thus there you are. Fnord. What, you want more details? Okay let's go. Insert part A ( long sausage thing) in to extension slot B (hole). Vibrate, then put in the air to cool down for 9 months or so. Raise the little bugger when it's growing up and then you should have a perfect little pet, after at least 7 scoldings a year. Fnord.
Cats should be pet every moment of every day. If you see a cat that is not being pet, there is something wrong here and the universe is about to cave in on itself. It's okay though, it is so easy for you to perform your godly service and pet that cat till he or she is satisfied. What do you mean there has to be more to life then this? Obviously you've never met a cat before. People think walking dogs is more important, but the truth is, dogs don't like being walked. It's true. Have you ever asked your dog if he really enjoys those little trips? What do you mean you think he enjoys it? Listen, just...SHUT UP and pet a damn cat. At least someone will be happy for a few minutes.
Trapping A Mouse
It was found out on September 5th, 2008, that trapping a mouse was the key to everything. It was thus proven by this formula.
{\displaystyle Trap(Mouse)=KeytoEverything}
“Well in my opinion the meaning of life is to write one or two good poems and get as much bum as one can”
Does this text reveal anything about the meaning of life? Maybe, but even if it does, it's probably like way too profound for you to understand.
So you're looking for the meaning of life and you think you'll find it here.
For long you must have been pondering the typical age-old question
WHAT THE FUCK IS THE MEANING OF LIFE?!
But before we get to the issue of the meaning of life, we must first consider several other questions.
Why do you want to know the meaning to life?
Probably because, you suck. Hey, don't act like I should care. In a society like this one, it's completely delusional to expect a thing like that. Does this mean you should give up right away? Not unless you really want to. I don't care what you do anyway, it's your obligation choice. Now, let's move along to the more important stuff, as you seem to be very determined to discover the meaning of life.
Are you ready for the meaning of life?
What if some Deus ex Machina suddenly appeared out of nowhere and told you the meaning of life, would you be ready for it? Probably not. Even if you would be able to comprehend even the littlest bit of the meaning of life, it would most likely be too much to bear, and lead to an occurrence of Sudden Instant Death Syndrome. And in the case you survive by some kind of miraculous coincidence, you would probably misinterpret it none the less.
Is it even possible to discover the meaning of life?
Sure, if some divine entity lowered itself to your pathetic level and revealed it, you'd know the meaning of life, but what are the odds of that actually happening? Seriously though, is there a way to discover the meaning of life without divine intervention? Science has so far succeeded in absolutely not discovering the meaning of life, and we shouldn't expect it to do so anytime soon. Religion offers some views, though they tend to be highly paradoxical. Maybe through deductive reasoning, but that doesn't seem to be an option for you. You simply lack the required insight and understanding of the workings of reality.
So maybe you'll never discover the meaning of life, in the present situation it's the most realistic prospect. In fact, we shouldn't bother looking for it anyway. It's far more beneficial to just assume that your life is in some way meaningful, than to be completely obsessed with finding the meaning of life. It's not even so strange to think that we require to be lacking in knowledge of the meaning of life, as otherwise some heavenly being should have exposed it long ago. If one believes a reason is needed to live, then the belief that it must be so is reason enough on its own to believe that life is meaningful, there's no need for a further explanation of the exact meaning. And now it's about time you get a life or get lost.
In conclusion, the meaning of life is rather elusive and open to interpretation and shouldn't be bothered with. However, an even more absurd philosophy was recently discovered, The Meaning of Death, though I'm not really sure it's worth all the trouble. As it turns out, it's a waste of time as well and it, in turn, caused many people to tell philosophers "next time you want to know the meaning of something, read a bloody dictionary!"
A Retrospective: How the concept of the meaning of life relates to one's status in modern society
What the fuck is wrong with people asking what the meaning of life is?
They're pathetic losers, or asking too many stupid questions, probably both.
What the fuck is wrong with people saying that life has no meaning?
They're cynical know-it-all bastards.
What the fuck is wrong with people claiming to know what the meaning of life is?
They're wrong, horribly wrong... and delusional.
How can one take advantage of this situation?
Use commonly held truths and simple reasoning to support your own opinion. Describe everything not corresponding with your opinion as being doubtful, inaccurate, too simplistic or irrelevant. If you can convincingly explain why, describe such positions as being utterly false, completely absurd and simply ignoring the real issue.
But make sure your own position has the potential of becoming popular or is already widely supported. And try to come over as tolerant to some alternative beliefs, but intolerant to the people who interpret such beliefs to support terrorism or some otherwise unpopular spare-time occupation.
Then, when you've gained some respect and proved your sincerity, you can start claiming that you can "make a difference", that you will "make it right", that you'll bring "the change everybody's been hoping for", and all that quasi-messianic "saving the world" stuff. And if you don't make too many mistakes, you can end up as the leader of some important organization (like the government of your nation or somekind of religion).
However, you would probably need a lot of luck to actually pull that off, especially with all those blatantly unrealistic expectations you'd be creating.
I'm sorry, I don't know. See your mom.
If you still haven't realised what the meaning of life is, maybe you should look it up in a dictionary, and possibly take some time to think it over and look for some help.
Other stuff you don't understand
Ancient and Greek
Above the starry-sky judges God,
the way we judged
HowTo:Start a Religion
It doesn't matter what your answer is
as long as you feel good about it
What was God thinking!?
Retrieved from "https://uncyclopedia.com/w/index.php?title=Meaning_of_life&oldid=5946787"
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A Sequential Linear Programming Matrix Method to Insensitive H∞ Output Feedback for Linear Discrete-Time Systems | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
A Sequential Linear Programming Matrix Method to Insensitive
H∞
Output Feedback for Linear Discrete-Time Systems
Xiang-Gui Guo,
Tianjin Key Laboratory for Control Theory & Applications in Complicated System,
e-mail: guoxianggui@163.com
State Key Laboratory of Synthetical
Automation for Process Industries,
e-mail: yangguanghong@mail.neu.edu.cn
Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received January 20, 2013; final manuscript received September 12, 2013; published online October 15, 2013. Assoc. Editor: Jongeun Choi.
Guo, X., and Yang, G. (October 15, 2013). "A Sequential Linear Programming Matrix Method to Insensitive
H∞
Output Feedback for Linear Discrete-Time Systems." ASME. J. Dyn. Sys., Meas., Control. January 2014; 136(1): 014506. https://doi.org/10.1115/1.4025458
This paper studies the problem of designing insensitive
H∞
output-feedback controllers for linear discrete-time systems. The designed controllers are insensitive to additive/multiplicative controller coefficient variations. An LMI-based procedure, which is a sequential linear programming matrix method (SLPMM), is proposed to solve the considered problem which is a nonconvex problem itself. It is worth mentioning that the nonfragile control design method is adopted to obtain an effective solution for accelerating convergence of SLPMM algorithm due to the fact that a good starting point for the iteration is very important.
Control equipment, Design, Feedback, Algorithms, Discrete time systems
Reduction of Controller Fragility by Pole Sensitivity Minimization
On the Absence of Limit Cycles in State-Space Digital Filters With Minimum L2-Sensitivity
IEEE Trans. Circuits Syst. Express Briefs
On the Structure of Digital Controllers With Finite Word Length Consideration
H∞ Filter Design for Delta Operator Formulated Systems With Low Sensitivity to Filter Coefficient Variations
Non-Fragile Dynamic Output Feedback H∞ Control for Discrete-Time Systems
Non-Fragile Dynamic Output Feedback H∞ Control for Continuous-Time Systems With Controller Coefficient Sensitivity Consideration
Proceedings of 2011 Chinese Control Decision and Conference
, Mianyang, China, pp.
Non-Fragile H∞ and H2 Filter Designs for Continuous-Time Linear Systems Based on Randomized Algorithms
An LMI-Based Algorithm for Designing Suboptimal Static H2
H∞ Output Feedback Controllers
–1005.10.1049/iet-cta.2008.0008
, “Optimal Finite Wordlength Digital Control With Skewed Sampling,”
Proceeding of 1994 American Control Conference
, Baltimore, MD, 3, pp. 3482–3486.10.1109/ACC.1994.735226
Bemussou
Extended H2 and H∞ Norm Characterizations and Controller Parametrizations for Discrete-Time Systems
Proceedings of 2004 IEEE International Symposium on Computer Aided Control Systems Design
, Taipei, Taiwan, China, pp.
SeDumi interface 1.02: A Tool for Solving LMI Problems With SeDumi
Proceedigns of 2002 International Symposium on Computer Aided Control System Design
H 2 Guaranteed Cost Control Design and Implementation for Dual-Stage Hard Disk Drive Track-Following Servos
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EUDML | A combinatorial construction of . EuDML | A combinatorial construction of .
A combinatorial construction of
{G}_{2}
Wildberger, N.J.. "A combinatorial construction of .." Journal of Lie Theory 13.1 (2003): 155-165. <http://eudml.org/doc/123286>.
@article{Wildberger2003,
author = {Wildberger, N.J.},
keywords = {simple exceptional Lie algebra of type ; 7 dimensional representation; directed multigraph; simple exceptional Lie algebra of type },
title = {A combinatorial construction of .},
AU - Wildberger, N.J.
TI - A combinatorial construction of .
KW - simple exceptional Lie algebra of type ; 7 dimensional representation; directed multigraph; simple exceptional Lie algebra of type
simple exceptional Lie algebra of type
{G}_{2}
, 7 dimensional representation, directed multigraph, simple exceptional Lie algebra of type
{G}_{2}
Articles by Wildberger
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EUDML | Perturbed integral equations in modular function spaces. EuDML | Perturbed integral equations in modular function spaces.
Perturbed integral equations in modular function spaces.
Hajji, A.; Hanebaly, E.
Hajji, A., and Hanebaly, E.. "Perturbed integral equations in modular function spaces.." Electronic Journal of Qualitative Theory of Differential Equations [electronic only] 2003 (2003): Paper No. 20, 7 p., electronic only-Paper No. 20, 7 p., electronic only. <http://eudml.org/doc/123781>.
author = {Hajji, A., Hanebaly, E.},
keywords = {integral equation; modular space; Musielak-Orlicz space; Lipschitz operator},
title = {Perturbed integral equations in modular function spaces.},
AU - Hajji, A.
AU - Hanebaly, E.
TI - Perturbed integral equations in modular function spaces.
KW - integral equation; modular space; Musielak-Orlicz space; Lipschitz operator
integral equation, modular space, Musielak-Orlicz space, Lipschitz operator
{L}^{p}
Articles by Hajji
Articles by Hanebaly
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EUDML | Certain properties of parabolic starlike and convex functions of order . EuDML | Certain properties of parabolic starlike and convex functions of order .
Certain properties of parabolic starlike and convex functions of order
\rho
Aghalary, R.; Kulkarni, S.R.
Aghalary, R., and Kulkarni, S.R.. "Certain properties of parabolic starlike and convex functions of order .." Bulletin of the Malaysian Mathematical Sciences Society. Second Series 26.2 (2003): 153-162. <http://eudml.org/doc/50894>.
@article{Aghalary2003,
author = {Aghalary, R., Kulkarni, S.R.},
keywords = {starlike functions; convex function; Hadamard product; subordination},
title = {Certain properties of parabolic starlike and convex functions of order .},
AU - Aghalary, R.
AU - Kulkarni, S.R.
TI - Certain properties of parabolic starlike and convex functions of order .
KW - starlike functions; convex function; Hadamard product; subordination
starlike functions, convex function, Hadamard product, subordination
Articles by Aghalary
Articles by Kulkarni
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Base color - zxc.wiki
Color wheel according to Johannes Itten (1961)
Color valence = the color of how the light-emitting diodes are perceived
Color stimulus = spectrum of light-emitting diodes in red, green, blue and white
In the narrower sense, basic colors are the color valences theoretically used as a reference value in a selected color space . In a broader sense, it is the for mixing usable colorants to a certain color perception to achieve.
1.2 Spectral color
1.3 color valence
1.4 Primary valences
1.5 Non-color
1.6 basic color
1.7 primary colors and secondary colors
1.8 Optimal colors
1.9 light-dark, achromatic colors
1.11 primordial colors
2 basic colors in languages
3 color spaces and models
A “ three-color theory ” is based on the experience of the artists with the basic colors red, yellow and blue, from which all other colors can be mixed. For the RGB color space, on the other hand, it is the three valences red, green and blue to which the luminescent substances of the monitor are optimally matched. The basic colors of a multicolor print, for example an inkjet print, are: yellow, magenta (red-blue / fuchsia) and cyan (blue-green). Three basic colors are sufficient to describe the color space , but perception is based on pairs of opposites ( four-color theory ). In principle, different color triples are possible as basic colors for the description of the colors .
Main article : Spectral color
Spectral colors are the bright, pure colors as they appear in the solar spectrum, on the edge of CDs or in the rainbow. Because of the sacred number seven, Newton assigned seven basic colors to this continuum: violet , indigo , blue , green , yellow , yellow- red , red , although the continuum offers a continuous color sequence. The "white" light is broken down into the "bright" colors of the spectrum by means of diffraction or interference effects. More precisely, it is color stimuli that are perceived as color valences due to the wavelength-dependent splitting. A spectral color is typically the appearance of a “single” wavelength or (real) monochromatic light. "Mixtures" of several spectral colors are called valence colors , so the valence color magenta is an "overlay" of the spectral colors violet and red.
Color valence
Main article : color valence
The triggering event of the color impression is the color stimulus , the calculated quantity (numerical value or vector ) resulting from this is the color valence .
Primary valences
When developing the CIE standard valence system , three primary valences were determined as calibration color values, which are derived from the sensitivities of the three cones . The primary valences correspond to the LMS color space , the L cone (valence) is denoted by, the primary valence is assigned to the M cone and the primary valence derived from the sensitivity spectrum of the S cone is denoted by. These primary valences are used as basis vectors of a three-dimensional color space. The letters L , M and S for the cones stand for long, medium and shortwave.
{\ displaystyle {\ vec {R}}}
{\ displaystyle {\ vec {G}}}
{\ displaystyle {\ vec {B}}}
At the beginning of color measurement, these primary valences were measured indirectly. With this measuring technique, light was subtracted (by changing the comparison light), so to speak, color was removed. In order to avoid such negative color values, virtual basic valences were derived according to the calculation rules for vectors , which span the color space, these are
{\ displaystyle {\ vec {X}}}
as red, green and blue valence.
{\ displaystyle {\ vec {Y}}}
{\ displaystyle {\ vec {Z}}}
The parameters required for the 3D model display are selected in various color spaces in such a way that the color valence obtained does not correspond to any visible color or that the result is outside the gamut . In colorimetry, such (imperceptible) color locations are referred to as “non-color” or, better, as imaginary colors .
Although the visible spectrum and the variety of all color nuances practically form a continuum, a restriction to a few color names is necessary for communication. Depending on the language and culture, there are two to six names for colors that are considered basic colors.
The term basic color was developed in American linguistics in the 1960s in a (still controversial) language universalist hypothesis. In the influential Basic Color Terms (1969), Brent Berlin and Paul Kay suggested that all languages have a minimum occupation of two color categories in their vocabulary (such as white / light and black / dark); in addition, the third category is red, the fourth yellow or green, etc., up to a maximum of 11 categories. The Berlin / Kaysche model was later applied with considerable modifications by German linguists, who mostly recognize 8 to 11 basic color words (see color ).
In his theoretical writings, Goethe used the word basic color (mostly as plural) in the sense of "primary color valences" (in painting, dyeing, chemistry and optics) ", for example blue, yellow and red. The Goethe dictionary also documents Goethe's varied use of elementary color , main color and primary color , often with reference to his different color schemes. About Goethe's understanding of color and the development of his color theories. (See color theory and color theory (Goethe) )
In the experience of the painters and subsequently theoretically with the French Jacques-Christophe Le Blond and in Young's three-color theory , these are: red, yellow, blue. They are "basic" colors from which all others can be mixed. According to Ewald Hering's counter-color theory, there are four basic colors , these form the pairs green-red and blue-yellow (in addition to light-dark). Ostwald , Itten and Küppers founded their color theories from the possibility of basically finding other basic color triples.
For the RGB color space it is the three valences red, green and blue to which the luminescent substances of the monitor can be optimally matched. As an alternative, in newer devices a yellow is also selected under conversion in order to better simulate the LMS color space of the eye.
Primary colors and secondary colors
(Subtractive) primary colors in color printing
Historically, a subdivision into primary and secondary colors has emerged, the former being fundamental and the secondary colors being mixed directly from them.
Primary colors are the initial colors of an imaginary or actual mixing process. For the additive mixture are light colors used (usually red, green and blue), for the subtractive mixing of the body colors cyan, yellow and magenta, often with the aid of an additional black pigment, such as a process color in color printing .
Secondary colors are mixtures of two primary colors.
Rudolf Arnheim recommends distinguishing between “generative” and “fundamental” primary colors. Generative primary colors are colors that are used for mixing, i.e. for generating secondary colors. Fundamental primary colors, on the other hand, are the primary colors of the psychological level. It turns out that test subjects can describe any color well by describing the colors as mixtures of the four colors red, yellow, green and blue (this is what the NCS is based on ). On the other hand, it is psychologically hardly possible to imagine yellow as a mixture of red and green (i.e. in an additive mixture). A “bluish yellow” can hardly be imagined as a yellow-green (as a subtractive mixture).
According to this structure, the term tertiary color should be understood to mean those that can be explained by further mixing; in this sense it is the broken colors ( clouded colors ). Typical representatives are the range of brown tones , which can be a broken yellow, red or orange. Tertiary colors from cold shades lead to the group of olive tones. In the real sense, the achromatic colors white, gray and black also belong to the tertiary colors. From Ostwald , the term "veiled colors" was coined in his color theory, since them the clarity of saturated colors is missing.
Spectrum of an optimal color over a spectral-like color design
Enhanced by Wilhelm Ostwald and Robert Luther , the term "optimal colors" arose for idealized colors that are based on sections of the spectrum, whereby the intensities only assume the values 0 and 1. Depending on the location of the jump wavelengths, there are blue short-end colors, green middle colors, red long-end colors and purple middle-wrong colors.
Chiaroscuro, achromatic colors
Somewhat apart from the bright colors, there are black and white , the “extreme” cases of neutral gray . These achromatic colors play a separate role because they are (precisely) not colored.
Wilhelm Ostwald uses the terms veiled (that is blackened) and whitish colors, which he opposed to the optimal colors. Siegfried Rösch achieved the breakthrough for this by deriving the concept of relative brightness from the optimal colors .
In Ostwald's color wheel, full color is the name for the color. They are the most saturated and purest (because narrowly limited) optimal colors . In this color system, the colors are more cloudy and less saturated by black, and are called blackened light. The addition of white, the whitening, causes an increase in the brightness of the color. If the white component of the full color component is displaced, the achromatic color white is obtained. The total part sum " color = full color part v + white part w + black part s " is always 100%, because more than color is not possible.
Primordial colors
Ewald Hering based his counter-color theory on the four primary colors red, yellow, blue and green , whereby the color pairs red / green and blue / yellow exclude each other as opposing colors.
For his color theory, Küppers uses the term primordial colors for the color sensations orange-red (R), green (G) and violet-blue (B). These primordial colors result from the (symbolized) "sensory forces" of the visual organ, as they ultimately form the basis of the LMS color space .
Basic colors in languages
The color wheel , which is perceived as continuous, can be divided differently by basic colors. What is referred to or perceived as the basic color depends on cultural traditions and conventions.
In the European system (Indo-European language area) four (or six) colors are used as a basis: In addition to "black" and "white", the four basic "bright" colors "red", "blue", "yellow" and "green" are known . However, this naming system is relatively new. In ancient times , completely different basic colors were used.
Germanic color names penetrated the Romance languages after the Migration Period : The Germanic color word gel (yellow, English yellow ) can be found as giallo in Italian and similarly in other Romance languages. The Old High German blao (blue) was also adopted from several Romance languages: French bleu , Italian blu and Catalan blue .
The word blanc ( Catalan and French for “white”, also Italian bianco , Spanish blanco and Portuguese branco ) also has a Germanic origin that is still recognizable in the German word “blank”. The actual Latin word for “white”, however, was albus (compare album), which lives on in Portuguese alvo (“white”, “pure”), in Romanian alb (“white”) and in Spanish alba (“dawn”). Compare with blanco and oscuro (for light and dark).
The color word “azur” (cf. “ Côte d'Azur ”) known from older (southern) French varieties can also be found in Italian azzurro and Spanish: azul . Originally Italian speakers perceive blu (dark blue) and azzurro (sky blue) as completely different basic colors, such as how yellow and green represent different colors for a German speaker . For the Romans, the sky was not “blue” but “bright”.
In Greek , χλωρός (chloros) stands for “yellow-green” (compare the element chlorine and chlorophyll ), γλαυκός (glaucos) is a dull “blue-gray-green” (compare glaucoma ).
The Japanese language has other than the borrowing of "gurin" (green) from English No Category Green , but "green" is (yellow) as a shade of blue viewed ( development language ).
Main article : Green and blue in different languages
The Chinese language differentiates between two types of green: 綠色 (lü se) or just 綠 (lü) for a light, more yellowish green and 青色 (qing se) for a rich, bluish green, turquoise or cyan.
Harald Küppers : The basic law of color theory (= dumont paperbacks. 65). 10th, revised and updated edition. DuMont, Cologne 2002, ISBN 3-8321-1057-7 .
Wiktionary: basic color - explanations of meanings, word origins, synonyms, translations
Bruce MacEvoy: Do "primary" colors exist? Own homepage, January 8, 2015 (English; information on the topic of "basic colors").
^ Brent Berlin , Paul Kay : Basic Color Terms. Their Universality and Evolution. University of California Press, Berkeley, Los Angeles 1969 (English).
↑ Lexicon entry: basic color. In: Goethe dictionary. Volume 4, 2nd delivery: Gestaltberkeit – bald head. Kohlhammer, Berlin a. a. 1999 ( woerterbuchnetz.de ).
^ Rudolf Arnheim : Art and Seeing. A psychology of the creative eye. 3rd, unchanged edition. de Gruyter, Berlin a. a. 2000, ISBN 3-11-016892-8 (preface by Michael Diers ).
↑ Compare the children's song text Backe ,backe Kuchen : "Saffron makes the cake gehl" (alternatively: "gel").
This page is based on the copyrighted Wikipedia article "Grundfarbe" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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EUDML | Alexander's capacity for intersections of ellipsoids in . EuDML | Alexander's capacity for intersections of ellipsoids in .
Alexander's capacity for intersections of ellipsoids in
{ℂ}^{N}
Jȩdrzejowski, Mieczysław
Jȩdrzejowski, Mieczysław. "Alexander's capacity for intersections of ellipsoids in .." Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica 1258(40) (2002): 39-44. <http://eudml.org/doc/123292>.
@article{Jȩdrzejowski2002,
author = {Jȩdrzejowski, Mieczysław},
keywords = {ellipsoid; projective capacity; extremal function},
title = {Alexander's capacity for intersections of ellipsoids in .},
AU - Jȩdrzejowski, Mieczysław
TI - Alexander's capacity for intersections of ellipsoids in .
KW - ellipsoid; projective capacity; extremal function
ellipsoid, projective capacity, extremal function
Capacity theory and generalizations
Articles by Jȩdrzejowski
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EUDML | Completion of a Cauchy space without the -restriction on the space. EuDML | Completion of a Cauchy space without the -restriction on the space.
Completion of a Cauchy space without the
{T}_{2}
-restriction on the space.
Rath, Nandita. "Completion of a Cauchy space without the -restriction on the space.." International Journal of Mathematics and Mathematical Sciences 24.3 (2000): 163-172. <http://eudml.org/doc/48722>.
keywords = {Cauchy map; -map; stable completion; completion in standard form; regular completion; -map},
title = {Completion of a Cauchy space without the -restriction on the space.},
TI - Completion of a Cauchy space without the -restriction on the space.
KW - Cauchy map; -map; stable completion; completion in standard form; regular completion; -map
Boris G. Averbukh, On finest unitary extensions of topological monoids
Cauchy map,
s
-map, stable completion, completion in standard form, regular completion,
s
-map
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MRControlLimits - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : ProcessControl Package : MRControlLimits
MRControlLimits(X, options)
The MRControlLimits command computes the upper and lower control limits for the MR chart. Unless explicitly given, the average of the moving ranges of two observations of the underlying quality characteristic is computed based on the data.
ignore=truefalse -- This option controls how missing values are handled by the MRControlLimits command. Missing values are represented by undefined or Float(undefined). So, if ignore=false and X contains missing data, the MRControlLimits command returns undefined. If ignore=true, all missing items in X are ignored. The default value is true.
rbar=deduce or realcons -- This option specifies the average of the moving ranges of two observations.
\mathrm{with}\left(\mathrm{ProcessControl}\right):
\mathrm{infolevel}[\mathrm{ProcessControl}]≔1:
A≔[33.75,33.05,34.00,33.81,33.46,34.02,33.68,33.27,33.49,33.20,33.62,33.00,33.54,33.12,33.84]:
\mathrm{MRControlLimits}\left(A\right)
[\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.57127089652357}]
l≔\mathrm{MRControlLimits}\left(A,\mathrm{confidencelevel}=0.95\right)
\textcolor[rgb]{0,0,1}{l}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.57127089652357}]
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Linear Kalman Filters - MATLAB & Simulink - MathWorks Benelux
\begin{array}{l}m\stackrel{¨}{x}=f\\ \stackrel{¨}{x}=\frac{f}{m}=a\end{array}
\begin{array}{l}{x}_{1}=x\\ {x}_{2}=\stackrel{˙}{x},\end{array}
\frac{d}{dt}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]a
\frac{d}{dt}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]a+\left[\begin{array}{c}0\\ 1\end{array}\right]{v}_{k}
\frac{d}{dt}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {y}_{1}\\ {y}_{2}\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {y}_{1}\\ {y}_{2}\end{array}\right]+\left[\begin{array}{c}0\\ {a}_{x}\\ 0\\ {a}_{y}\end{array}\right]+\left[\begin{array}{c}0\\ {v}_{x}\\ 0\\ {v}_{y}\end{array}\right]
\left[\begin{array}{c}{x}_{1,k+1}\\ {x}_{2,k+1}\end{array}\right]=\left[\begin{array}{cc}1& T\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{1,k}\\ {x}_{2,k}\end{array}\right]+\left[\begin{array}{c}\frac{1}{2}{T}^{2}\\ T\end{array}\right]a+\left[\begin{array}{c}\frac{1}{2}{T}^{2}\\ T\end{array}\right]\stackrel{˜}{v}
where xk+1 is the state at discrete time k+1, and xk is the state at the earlier discrete time k. If you include noise, the equation becomes more complicated, because the integration of noise is not straightforward. For details on how to obtain the discretized process noise from a continuous system, See [1].
{x}_{k+1}={A}_{k}{x}_{k}+{B}_{k}{u}_{k}+{G}_{k}{v}_{k}
{z}_{k}={H}_{k}{x}_{k}+{w}_{k}
{x}_{k+1|k}={F}_{k}{x}_{k|k}+{B}_{k}{u}_{k}.
{P}_{k+1|k}={F}_{k}{P}_{k|k}{F}_{k}^{T}+{G}_{k}{Q}_{k}{G}_{k}^{T}.
{z}_{k+1|k}={H}_{k+1}{x}_{k+1|k}
{S}_{k+1}={H}_{k+1}{P}_{k+1|k}{H}_{k+1}^{T}+{R}_{k+1}
{K}_{k+1}={P}_{k+1|k}{H}_{k+1}^{T}{S}_{k+1}^{-1}
{x}_{k+1|k+1}={x}_{k+1|k}+{K}_{k+1}\left({z}_{k+1}-{z}_{k+1|k}\right)
{P}_{k+1|k+1}={P}_{k+1|k}-{K}_{k+1}{S}_{k+1}{K}_{k+1}^{T}
\left[\begin{array}{c}{x}_{k+1}\\ {v}_{x,k+1}\\ {y}_{k+1}\\ {v}_{y,k+1}\\ {z}_{k+1}\\ {v}_{z,k+1}\end{array}\right]=\left[\begin{array}{cccccc}1& T& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& T& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& T\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{k}\\ {v}_{x,k}\\ {y}_{k}\\ {v}_{y,k}\\ {z}_{k}\\ {v}_{z,k}\end{array}\right]
\left[\begin{array}{c}{x}_{k+1}\\ {v}_{x,k+1}\\ {a}_{x,k+1}\\ {y}_{k+1}\\ {v}_{y,k+1}\\ {a}_{y,k+1}\\ {z}_{k+1}\\ {v}_{z,k+1}\\ {a}_{z,k+1}\end{array}\right]=\left[\begin{array}{ccccccccc}1& T& \frac{1}{2}{T}^{2}& 0& 0& 0& 0& 0& 0\\ 0& 1& T& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& T& \frac{1}{2}{T}^{2}& 0& 0& 0\\ 0& 0& 0& 0& 1& T& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& T& \frac{1}{2}{T}^{2}\\ 0& 0& 0& 0& 0& 0& 0& 1& T\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{k}\\ {v}_{x,k}\\ {a}_{x,k}\\ {y}_{k}\\ {v}_{y,k}\\ {a}_{y,k}\\ {z}_{k}\\ {v}_{z,k}\\ {a}_{z,k}\end{array}\right]
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After-Tax Real Rate of Return Definition
The after-tax real rate of return is the actual financial benefit of an investment after accounting for the effects of inflation and taxes. It is a more accurate measure of an investor’s net earnings after income taxes have been paid and the rate of inflation has been adjusted for. Both of these factors will impact the gains an investor receives, and so must be accounted for. This can be contrasted with the gross rate of return and the nominal rate of return of an investment.
The after-tax real rate of return takes into consideration inflation and taxes to determine the true profit or loss of an investment.
The opposite of the after-tax real rate of return is the nominal rate of return, which only looks at gross returns.
Tax-advantaged investments, such as Roth IRAs and municipal bonds, will see less of a discrepancy between nominal rates of return and after-tax rates of return.
Understanding the After-Tax Real Rate of Return
Over the course of a year, an investor might earn a nominal rate of return of 12% on his stock investment, but his real rate of return, the money he gets to put in his pocket at the end of the day, will be less than 12%. Inflation might have been 3% for the year, knocking his real rate of return down to 9%. And since he sold his stock at a profit, he will have to pay taxes on those profits, taking another, say 2%, off his return.
The commission he paid to buy and sell the stock also diminishes his return. Thus, in order to truly grow their nest eggs over time, investors must focus on the after-tax real rate of return, not the nominal return.
The after-tax real rate of return is a more accurate measure of investment earnings and usually differs significantly from an investment's nominal (gross) rate of return, or its return before fees, inflation, and taxes. However, investments in tax-advantaged securities, such as municipal bonds and inflation-protected securities, such as Treasury Inflation Protected Securities (TIPS), as well as investments held in tax-advantaged accounts, such as Roth IRAs, will show less discrepancy between nominal returns and after-tax real rates of return.
Example of the After-Tax Real Rate of Return
Let’s be more specific on how the after-tax real rate of return is determined. The return is calculated by, first of all, determining the after-tax return before inflation, which is calculated as Nominal Return x (1 - tax rate). For example, consider an investor whose nominal return on his equity investment is 17% and his applicable tax rate is 15%. His after-tax return is, therefore:
0.17 \times (1 - 0.15) = 0.1445 = 14.45\%
0.17×(1−0.15)=0.1445=14.45%
Let's assume that the inflation rate during this period is 2.5%. To calculate the real rate of return after tax, divide 1 plus the after-tax return by 1 plus the inflation rate. Dividing by inflation reflects the fact a dollar in hand today is worth more than a dollar in hand tomorrow. In other words, future dollars have less purchasing power than today’s dollars.
Following our example, the after-tax real rate of return is:
\frac{(1 + 0.1445)}{(1 + 0.025)} - 1 = 1.1166 - 1 = 0.1166 = 11.66\%
(1+0.025)(1+0.1445)−1=1.1166−1=0.1166=11.66%
That figure is quite a bit lower than the 17% gross return received on the investment. As long as the real rate of return after taxes is positive, however, an investor will be ahead of inflation. If it’s negative, the return will not be sufficient to sustain an investor’s standard of living in the future.
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2Al
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In this activity you will be using a Venn diagram to organise all of that interesting information you’ve gathered about Earth and Mars.
Before you begin though, you will refer to the Reading: Earth vs Mars which summarises a lot of the information you heard in Mission video 18: Earth vs Mars.
2Al
Having completed your Earth vs Mars Venn diagram should make it a lot easier to visualise what Mars and Earth have in common and how they are different. Now you are ready to think about what that potentially means for human life on Mars.
2Al
2Al
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Your team is in charge of games at the CPM Amusement Park. One of the games involves a robotic arm that randomly grabs a stuffed animal out of a large bin. You need to set up the game so that the probability of a customer’s grabbing a teddy bear is exactly
\frac { 1 } { 2 }
How would you set up the bin? Explain.
What fraction of the stuffed animals in the bin should be teddy bears?
If the probability of grabbing a teddy bear is
0.5
0.5
of the stuffed animals in the bin should be teddy bears.
What if you returned to check on the bin and found that there were 4 teddy bears left and 12 other animals? What could you add to or remove from the bin to return the probability of selecting a teddy bear to
\frac { 1 } { 2 }
How should the number of teddy bears compare to the number of other stuffed animals?
There are two possible answers. You could add teddy bears or remove other stuffed animals.
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3.1 Is Earth just another rocky planet? - Big History School
To work through '3.1 Is Earth just another rocky planet?' you need to complete the Activities in order. So first complete ‘Learning Plan’ then move to Activity '3.1.1' followed by Activity '3.1.2' through to '3.1.4', and then finish with ‘Learning Summary’.
Is Earth just another rocky planet?
In Is Earth just another rocky planet? you will learn all about what young Earth was like, its structured layers and how plate tectonics are constantly reshaping the Earth’s crust.
To get started, read carefully through the Is Earth just another rocky planet? learning goals below. Make sure you tick each of the check boxes to show that you have read all your Is Earth just another rocky planet? learning goals.
As you read through the learning goals you may come across some words that you haven’t heard before. Please don’t worry. By the time you finish Is Earth just another rocky planet? you will have become very familiar with them!
You will come back to these learning goals at the end of Is Earth just another rocky planet? to see if you have confidently achieved them.
Young Earth and its layers
Begin to understand what young Earth was like
Identify Earth’s layers
Understand what Geologists study
Understand Continental Drift Theory
Identify the three types of tectonic plate boundaries
Explain the consequences of tectonic plate movement
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Welcome to Mission Phase 3 - Planet!
In this Mission Phase you will learn all about Earth and how it changed over time to become a unique planet full of an amazing variety of life.
Mission video 14: young Earth and its layers begins by describing what young Earth was like and explains how its structure settled into distinctive layers.
While you watch Mission video 14: Young Earth and its layers look out for the answers to the following questions:
1. What was young Earth like?
2. What are geologists? What do they do?
3. What are the 4 main layers of Earth?
4. Why is the atmosphere important to us?
So where does ‘Earth formed’ appear on our History of the Universe Timeline?
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Now that you have learned about the different layers in the Earth’s structure, you will create a colourful poster to demonstrate what you have learned.
Follow the instructions on the Poster: the Earth’s layers worksheet step-by-step to create your poster:
Step 1. Colour the ‘Inner Core’ yellow; the ‘Outer Core’ orange; the ‘Mantle’ red and the ‘Crust’ brown.
Step 2. Cut out the Earth and paste it onto blue or white paper/cardboard.
Step 3. Paste the correct labels on each layer.
Step 4. Draw around the ‘Crust’ all the things we see on the land around us, for example, mountains, forests, skyscrapers, oceans etc.
Step 5. Draw in the ‘Atmosphere’ all the things you see in the sky, for example, clouds, birds, planes and anything else that you know forms part of the atmosphere.
Step 6. Write the heading ‘The Earth’s Layers’ on the top of your poster.
If you are part of a class, your teacher will ask you and your classmates to share your completed Earth’s layers posters.
Notice what a small portion of the Earth is made up of the crust - this is the part of the Earth where we live our entire lives!
And what did you include in your atmosphere? Did you include things that you can’t see as well as those that you can? For example, oxygen, pollution and the ozone layer? Can you think of anything else?
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When you learned about Earth’s structure in the last couple of activities, you may have been left with the impression that Earth is an unmoving, unchanging planet. Nothing could be further from the truth - especially on the Earth’s crust!
Mission video 15: Earth's changing surface introduces you to continental drift theory, which explains how the Earth’s crust is constantly changing and what the consequences are for our planet and for humans.
While you watch Mission video 15: Earth's changing surface look out for the answers to the following questions:
1. What is continental drift theory?
2. What was the evidence for this theory?
3. What do we now know causes continental drift?
4. What are the three types of tectonic plate boundaries? What effect do they have?
5. What are two things that make Earth unique?
To help you visualise the relationship between tectonic plate boundaries and the number of incidences of earthquakes and locations of volcanoes, launch the PBS interactive in Helpful Resources. Take note of some of the continents/countries which appear to have the most earthquakes and volcanoes are they near tectonic plate boundaries? What impact do earthquakes and volcanic eruptions have on our planet, on plant and animal habitats and on human life?
https://www.pbslearningmedia.org/resource/ess05.sci.ess.earthsys.tectonic/tectonic-plates-earthquakes-and-volcanoes/#.W4SuxugzaUm
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Now that you have learned about the theory of plate tectonics and you have seen some examples of how the movement of tectonic plates changes the Earth’s crust, you will undertake a hands-on demonstration of the three different types of tectonic plates boundaries.
In this demonstration you will use an Oreo or similar sandwich-style biscuit to demonstrate the differences between the three types of tectonic boundaries.
Follow the steps on the Demo: tectonic plate boundaries instruction sheet:
Step 1. Always keep your biscuit above your paper plate so that any crumbs land on the plate.
Step 2. Hold the bottom of your biscuit while gently twisting the top of the biscuit so that it separates from the soft centre and the rest of the biscuit.
Step 3. Hold the top part of the biscuit and, using both your thumb nails, place gentle pressure on the middle of the biscuit so that it breaks in half. These two halves represent two tectonic plates.
Step 4. Place the two tectonic plates back together onto the soft centre of the bottom half of the biscuit. The soft centre represents the sludgy mantle.
Step 5. To simulate a divergent boundary place a thumb on each of the two tectonic plates and gently slide them apart. Imagine that this is what is currently happening in the Kenyan Rift Valley.
Step 6. Slide the tectonic plates back together again.
Step 7. To simulate a transform boundary, place pressure on both of the tectonic plates with your thumbs and slide one up while you try to slide the other one down. Imagine how this could create an earthquake as the huge plates grind past each other.
Step 9. To simulate a convergent boundary place pressure on both of the tectonic plates with your thumbs and push them towards each other, allowing one to slide beneath the other. Imagine how this could cause mountains to rise from the ocean and create islands.
For a final summary of the three different types of tectonic plate boundaries, launch the PBS interactive, ‘Mountain Maker Earth Shaker.’ You will find a link in Helpful Resources. Drag the arrows to see what happens with each different tectonic plate movement and to read what the consequences are.
https://www.pbslearningmedia.org/resource/ess05.sci.ess.earthsys.shake/mountain-maker-earth-shaker/?#.WtgNpohuaUk
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In Is Earth just another rocky planet you learned all about what young Earth was like, its structured layers and how plate tectonics are constantly reshaping the Earth’s crust.
Now it’s time to revisit your Is Earth just another rocky planet? learning goals and read through them again carefully.
Well done on completing your learning summary. Click here to go to 3.2 What are the goldilocks conditions for life?
Once you have checked the boxes to confirm you have achieved your learning goals for Sequence '3.1 Is Earth just another rocky planet?' click on the 'I have achieved my learning goals' button below.
Go to 3.2 What are the goldilocks conditions for life? »
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To Francis Darwin 15 August [1873]1
Down, | Beckenham, Kent. [Bassett, Southampton.]
I am very glad to hear, & to see, the nectar-holes in the Lathyrus.2 Now I know that they occur in 4 British species, which no doubt are crossed occasionally, & they do not occur in 2 foreign species which I believe are never crossed; & this really looks as if the Bees were bothered by there being no nectar holes.3 But I saw at Abinger a Bombus lapidarius going to the base of the standard of the sweet Pea (L. odoratus) on one side & sucking nectar, so that he knew how to get to the nectar, but he did not stand on the keel & could have done nothing for fertilisation.4 It is curious about the bees biting holes (I could not find them in the
\frac{2}{3}
withered flowers which you sent) on the right side of the keel, for from the position of the stigma bees ought to suck on the left side (standard being in front of the beholder); & if they had bitten holes on the left side I shd. have inferred that there was more honey on left side.— Perhaps the keel is more exposed on the right side. From what William has said I now remember that I observed L. maritimus at Shanklin & I have somewhere got notes on the bees sucking (not biting holes) on one side of flower;5 but W. is almost sure that the pistil in some flowers was bent to one side & in others to the opposite. It was bent so that stigma faced to the left in all the flowers which were sent by you.6
With respect to bees biting holes the sole rule which I could ever made out, has been that when many plants of the same species with a somewhat closed or irregular flowers grow close together & are visited by many bees, they then bite holes: I imagine that the honey is exhausted in many flowers, & the bees then bite holes so as to visit them more quickly.
T. H. Farrer has made out a curious case with another Legum. plant, Coronella, about which I will hereafter tell you.—7
I am very glad that the worms are flourishing: Amy seems to have the soul of a true naturalist.8
Many thanks about Drosera: the bread ought to be left embraced until the tentacles just begin to reexpand.— Do put a little bit of raw meat on a leaf, & as you are a great histologist, compare after several days its state with a bit on moss.9
It would now I think be too late in the season, but it wd. be a fine experiment to cover up under net about 4 young plants of about same size of Drosera; & leave 2 without any insects & give every leaf of the other 2 plants, (as often they expand & reexpand) insects, & then compare size of plants at end of autumn. They could then be weighed. But I fancy it would not be possible now to find young plants which had not already caught many insects.
I have been working at Mimosa here, and everything has turned out as [perversely] as possible10
Yours affecte Father | C. Darwin
When soil is loose as round banked up Celery plant, I find that the worms always make their castings within their burrows.11
The year is established by CD’s reference to his visit to Abinger; this visit took place between 5 and 9 August 1873 (‘Journal’ (Appendix II)).
See letter from Francis Darwin, 14 August [1873] and n. 2. CD refers to Lathyrus maritimus.
In a note dated 17 August 1873 (DAR 77: 33), CD listed Lathyrus praetensis, L. sylvestris, L. maritimus, and L. macrorrhizus (now L. linifolius) as species in which he had found nectar holes. He found no holes in L. odoratus (sweet pea) and L. grandiflorus (two-flowered everlasting pea). In the note ‘maritimus’ has been crossed out. In Cross and self fertilisation, p. 155, CD wrote that in three native species of Lathyrus, the staminal tube was perforated by nectar passages but that the passages were covered in L. odoratus and missing in L. grandiflorus. See also letter from Francis Darwin, 14 August [1873] and n. 3
Bombus lapidarius is the red-tailed bumble-bee. Lathyrus odoratus is pollinated in its native habitat by leafcutter bees (Megachile spp.). In Cross and self fertilisation, pp. 155–6, CD described the activity of B. lapidarius in sucking nectar without depressing the keel.
CD stayed at Freshwater on the Isle of Wight in July and August 1868; William Erasmus Darwin visited twice during CD’s stay (Emma Darwin’s diary (DAR 242)). A planned two-day visit to Shanklin on the Isle of Wight is mentioned in a letter from H. E. Darwin to G. H. Darwin, [18 July 1868] (DAR 245: 296). CD’s notes on bees and Lathyrus maritimus observed at Shanklin have not been found.
The specimens that Francis sent were later identified by him as Lathyrus sylvestris, not L. maritimus (see letter from Francis Darwin, [16 or 17 August 1873]). In L. sylvestris, one side has a more easily accessible nectar passage owing to the bending of the keel.
Thomas Henry Farrer published his observations on Coronilla in 1874. See letter to T. H. Farrer, 14 August 1873 and n. 3.
In his letter of 14 August [1873], Francis mentioned the worm garden he and Amy Ruck had set up. Amy had made observations on worm-castings for CD in 1872 and helped with an experiment investigating the effect of formic acid on the development of spawn (see Correspondence vol. 20, letter to Amy Ruck, 24 February [1872], and this volume, letter to Francis Darwin, [before 15 April 1873]).
See letter from Francis Darwin, 14 August [1873] and n. 4. Francis’s training in the natural sciences at Cambridge and in medicine at St George’s Hospital, London, would have familiarised him with histological techniques. In Insectivorous plants, p. 15, CD described differences after forty-eight hours in the state of raw meat left on a leaf of Drosera (sundew) compared with meat surrounded by wet moss.
In a note dated 15 August 1873, CD described the closing up of leaves of Mimosa when they were moved from shade to bright sunshine, comparing the action to that of Drosera leaves in hot water. He also described a further experiment to test the effect of a few drops of water on a leaf. Although CD reported that the leaf moved in this case, he was doubtful of the cause, suspecting that the wind might have played a part (DAR 209.2: 43). Notes on further experiments carried out in August 1873 are in DAR 209.2: 44–7.
In his letter of 14 August [1873], Francis had noted that the worms in his worm garden made castings in the burrow itself but not on the surface.
Observations on bees’ biting holes in Lathyrus.
Suggests an experiment FD could carry out with Drosera.
CD is working on Mimosa, and "everything has turned out as perversely as possible".
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Continuity equation - zxc.wiki
A continuity equation is a certain partial differential equation that belongs to a conserved quantity (see below). It links the temporal change in the density associated with this conservation quantity with the spatial change in its current density :
{\ displaystyle {\ frac {\ partial} {\ partial t}}}
{\ displaystyle \ rho}
{\ displaystyle {\ vec {j}}}
{\ displaystyle {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla}} \ cdot {\ vec {j}} = 0}
For the mathematical definition of see divergence of a vector field .
{\ displaystyle {\ vec {\ nabla}} \ cdot}
The continuity equation occurs in all field theories of physics . The sizes obtained can be:
the crowd ,
the electric charge ,
the energy ,
the probability and
some particle numbers ( lepton number , baryon number ).
The generalization of the continuity equation to physical quantities that are not conserved quantities is the balance equation . An additional source term appears in it on the right-hand side of the equation .
1 relationship with a conservation quantity
2 Special continuity equations
3 Further applications: general conserved quantities
Connection with a conservation quantity
The “charge” contained in a volume V (the volume integral over the density) can only change due to the continuity equation in that unbalanced currents flow out of the surface of the volume. Accordingly, the total charge does not change over time and is a conservation quantity if no (net) currents flow through the surface of the volume under consideration.
{\ displaystyle V \ to \ infty}
Because the change in charge over time , given by
{\ displaystyle Q_ {V}}
{\ displaystyle Q_ {V} = \ iiint _ {V} \ mathrm {d} ^ {3} x \, \ rho (t, {\ vec {x}})}
in a volume that does not change over time , is because of the continuity equation according to Gauss's integral theorem
{\ displaystyle V}
{\ displaystyle {\ frac {\ mathrm {d} Q_ {V}} {\ mathrm {d} t}} = \ iiint _ {V} \ mathrm {d} ^ {3} x \, {\ frac {\ partial \ rho} {\ partial t}} = - \ iiint _ {V} \! \ mathrm {d} ^ {3} x \, {\ vec {\ nabla}} \ cdot {\ vec {j}} = - \ oint _ {\ partial V} \, {\ vec {j}} \; \ cdot {\ vec {n}} \, \ mathrm {d} S \ ,,}
equal to the area integral over the edge area of the volume over the proportion of the current density that flows outward in the direction of the surface normal . The charge in the volume only changes if unbalanced currents flow through the edge surface in the specified manner.
{\ displaystyle \ partial V}
{\ displaystyle {\ vec {j}}}
{\ displaystyle {\ vec {n}}}
Special continuity equations
If the mass density changes in hydrodynamics because the liquid flows with the speed along the trajectories , then the corresponding current density is
{\ displaystyle \ rho (t, {\ vec {x}})}
{\ displaystyle {\ vec {u}} = {\ tfrac {\ mathrm {d} {\ vec {x}}} {\ mathrm {d} t}}}
{\ displaystyle {\ vec {x}} (t)}
{\ displaystyle {\ vec {j}} = \ rho \, {\ vec {u}}}
and the equation of continuity is
{\ displaystyle {\ begin {alignedat} {2} & {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla}} \ cdot (\ rho {\ vec {u}}) && = 0 \\\ Leftrightarrow & {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla}} \ rho \ cdot {\ vec {u}} + \ rho {\ vec { \ nabla}} \ cdot {\ vec {u}} && = 0 \ end {alignedat}}}
(Reason: product rule )
For the change in density over time for a particle passing through the orbit , this says:
{\ displaystyle {\ vec {x}} (t)}
{\ displaystyle {\ begin {alignedat} {2} & {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla}} \ rho \ cdot {\ frac {\ mathrm {d} {\ vec {x}}} {\ mathrm {d} t}} && = - \ rho \ cdot {\ vec {\ nabla}} \ cdot {\ vec {u}} \\\ Leftrightarrow & {\ frac { \ mathrm {d}} {\ mathrm {d} t}} \ rho (t, {\ vec {x}} (t)) && = - \ rho \ cdot {\ vec {\ nabla}} \ cdot \, {\ vec {u}} \ end {alignedat}}}
(Reason: total differential ).
Along a trajectory, the density changes with the divergence of the flow
{\ displaystyle {\ vec {u}}.}
The flow is incompressible if the density remains constant along a trajectory:
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ rho (t, {\ vec {x}} (t)) = 0}
It follows that in this case the divergence of the flow is zero:
{\ displaystyle \ Rightarrow {\ vec {\ nabla}} \ cdot {\ vec {u}} = {\ frac {\ partial u} {\ partial x}} + {\ frac {\ partial v} {\ partial y }} + {\ frac {\ partial w} {\ partial z}} = 0}
In electrodynamics , the continuity equation for the electrical charge density and the electrical current density results from the identity and the two inhomogeneous Maxwell equations
{\ displaystyle \ rho}
{\ displaystyle {\ vec {j}}}
{\ displaystyle {\ vec {\ nabla}} \ cdot {\ vec {\ nabla}} \ times \ dots = 0}
{\ displaystyle 0 \ \ {\ stackrel {\ operatorname {div} \ \ operatorname {red} \ = \ 0} {=}} \ \ {\ vec {\ nabla}} \ cdot \ left ({\ vec {\ nabla}} \ times {\ vec {H}} \ right) \ \ {\ stackrel {\ text {Maxwell}} {=}} \ \ {\ vec {\ nabla}} \ cdot \ left ({\ frac { \ partial} {\ partial t}} {\ vec {D}} + {\ vec {j}} \ right) = {\ frac {\ partial} {\ partial t}} {\ vec {\ nabla}} \ cdot {\ vec {D}} + {\ vec {\ nabla}} \ cdot {\ vec {j}} = {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla} } \ cdot {\ vec {j}} \,}
d. H. it follows with the other inhomogeneous Maxwell equation
{\ displaystyle {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla}} \ cdot {\ vec {j}} = 0 \.}
In semiconductors describes the violation of the continuity equation
{\ displaystyle {\ frac {\ partial \ rho} {\ partial t}} + {\ vec {\ nabla}} \ cdot {\ vec {j}} = - r + g}
the change in the space charge density due to the recombination rate per volume,, and the generation rate .
{\ displaystyle \ rho}
{\ displaystyle r}
{\ displaystyle g}
From the Maxwell equations of electrodynamics it follows (in CGS units) for the energy density
{\ displaystyle u = {\ frac {1} {8 \ pi}} \ left ({\ vec {E}} ^ {2} + {\ vec {B}} ^ {2} \ right)}
and the energy flux density (also Poynting vector )
{\ displaystyle {\ vec {S}} = {\ frac {c} {4 \ pi}} \ left ({\ vec {E}} \ times {\ vec {H}} \ right)}
almost a continuity equation:
{\ displaystyle {\ frac {\ partial u} {\ partial t}} + {\ vec {\ nabla}} \ cdot {\ vec {S}} = - {\ vec {j}} \ cdot {\ vec { E}} \ ,.}
The continuity equation for the energy in the electromagnetic field is fulfilled where the electrical current density disappears, for example in a vacuum. There energy density can only change through energy flows. Where the electrical current density does not disappear, the electrical field does work and exchanges energy with the charge carriers.
{\ displaystyle {\ vec {j}}}
{\ displaystyle {\ vec {j}}}
{\ displaystyle {\ vec {E}}}
The continuity equation for electromagnetic field energy is Poynting's theorem .
In the relativistic formulation of electrodynamics with Minkowski vectors , cρ and j are combined into a four-vector . As above, it follows from the Maxwell equations that its four-way divergence vanishes. This formulation is independent of the Minkowski signature chosen, equivalent to the continuity equation and can be generalized to relativistic field theories.
{\ displaystyle (j ^ {\ alpha}) = (c \ rho, j_ {x}, j_ {y}, j_ {z}) \ ,,}
{\ displaystyle \ partial _ {\ alpha} j ^ {\ alpha} = {\ frac {c \ partial \ rho} {c \ partial t}} + {\ frac {\ partial j_ {x}} {\ partial x }} + {\ frac {\ partial j_ {y}} {\ partial y}} + {\ frac {\ partial j_ {z}} {\ partial z}} = 0 \ ,.}
In non-relativistic quantum mechanics, the state of a particle, such as a single electron , is described by a wave function .
{\ displaystyle \ Psi ({\ vec {x}}, t)}
The square of the amount
{\ displaystyle \ rho ({\ vec {x}}, t) = | \ Psi ({\ vec {x}}, t) | ^ {2}}
is the probability density to ensure a particle at the time at the location to be found. With the associated probability current density
{\ displaystyle t}
{\ displaystyle {\ vec {x}}}
{\ displaystyle {\ vec {j}} = - {\ frac {i \ hbar} {2m}} (\ Psi ^ {*} {\ vec {\ nabla}} \ Psi - \ Psi {\ vec {\ nabla }} \ Psi ^ {*})}
Without an external magnetic field, the continuity equation applies as a consequence of the Schrödinger equation
{\ displaystyle {\ frac {\ partial} {\ partial t}} \ rho + {\ vec {\ nabla}} \ cdot {\ vec {j}} = 0 \,}
If there is an external magnetic field, the Pauli equation must be used and this results
{\ displaystyle {\ vec {j}} = - {\ frac {i \ hbar} {2m}} (\ Psi ^ {\ dagger} {\ vec {\ nabla}} \ Psi - ({\ vec {\ nabla }} \ Psi ^ {\ dagger}) \ Psi) - {\ frac {q} {m}} {\ vec {A}} \ Psi ^ {\ dagger} \ Psi + {\ frac {\ hbar} {2m }} {\ vec {\ nabla}} \ times (\ Psi ^ {\ dagger} {\ vec {\ sigma}} \ Psi)}
where stand for the Pauli matrices . The last term disappears when the divergence is formed and cannot be derived directly from the Pauli equation, but results from the non-relativistic limit case of the Dirac equation.
{\ displaystyle \ sigma}
In the context of relativistic quantum mechanics, particles obey the Klein-Gordon equation (for scalar bosons ) or the Dirac equation (for fermions ). Since the equations obey the special theory of relativity, the continuity equations for these cases can be manifestly covariant
{\ displaystyle \ partial _ {\ mu} j ^ {\ mu} = \ partial _ {t} \ rho + {\ vec {\ nabla}} \ cdot {\ vec {j}} = 0}
be written and it arises
{\ displaystyle j _ {\ text {KG}} ^ {\ mu} = \ mathrm {i} \ left (\ phi ^ {*} \ partial ^ {\ mu} \ phi - \ phi \ partial ^ {\ mu} \ phi ^ {*} \ right)}
{\ displaystyle j _ {\ text {Dirac}} ^ {\ mu} = \ psi ^ {\ dagger} \ gamma ^ {0} \ gamma ^ {\ mu} \ psi}
where and respectively stand for the scalar bosonic / vector valued fermionic wave function and are the Dirac matrices .
{\ displaystyle \ phi}
{\ displaystyle \ psi}
{\ displaystyle \ gamma}
In the context of the Klein-Gordon continuity equation - in contrast to the nonrelativistic or fermion case - the quantity cannot be interpreted as a probability density, since this quantity is not positive semidefinite.
{\ displaystyle j ^ {0} = {\ frac {1} {c}} \ mathrm {i} \ left (\ phi ^ {*} \ partial _ {t} \ phi - \ phi \ partial _ {t} \ phi ^ {*} \ right)}
Further applications: general conserved quantities
The analogy to the “electrical” case shows that continuity equations must always apply when a charge-like quantity and a current-like quantity are related as stated above. Another concrete example could be the heat flow , which is important in thermodynamics . When integrated over the entire space, the “charge density” must result in a conservation quantity, e.g. B. the total electrical charge, or - in the case of quantum mechanics - the total probability, 1 , or in the third case, the total heat supplied, in systems whose heat content can be viewed as "preserved" (e.g. heat diffusion ).
In fluid mechanics, the continuity law for (incompressible) fluids follows from the continuity equation .
Batchelor, GK : An introduction to fluid dynamics, Cambridge university press, 2000, ISBN 0-521-66396-2
Video: equation of continuity . Institute for Scientific Film (IWF) 2004, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-14818 .
↑ When deriving u. a. the divergence of the so-called Maxwell's complement is formed and the interchangeability of the partial derivative with the divergence operator is used.
{\ displaystyle {\ frac {\ partial {\ vec {D}}} {\ partial t}}}
{\ displaystyle {\ frac {\ partial} {\ partial t}}}
↑ Torsten Fließbach : Electrodynamics Spectrum Akademischer Verlag, 3rd edition, p. 159.
This page is based on the copyrighted Wikipedia article "Kontinuit%C3%A4tsgleichung" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Prime Sieves in Agda - Donnacha Oisín Kidney
Prime Sieves in Agda
Part 2 of a 2-part series on Prime Sieves
Prime numbers in Agda are slow. First, they’re Peano-based, so a huge chunk of optimizations we might make in other languages are out of the window. Second, we really often want to prove that they’re prime, so the generation code has to carry verification logic with it (I won’t do that today, though). And third, as always in Agda, you have to convince the compiler of termination. With all of that in mind, let’s try and write a (very slow, very basic) prime sieve in Agda.
First, we can make an “array” of numbers that we cross off as we go.
primes : ∀ n → List (Fin n)
primes (suc (suc (suc m))) = sieve (tabulate (just ∘ Fin.suc))
cross-off : Fin _ → List (Maybe (Fin _)) → List (Maybe (Fin _))
sieve : List (Maybe (Fin _)) → List (Fin _)
sieve (just x ∷ xs) = suc x ∷ sieve (cross-off x xs)
cross-off p fs = foldr f (const []) fs p
B = ∀ {i} → Fin i → List (Maybe (Fin (2 + m)))
f : Maybe (Fin (2 + m)) → B → B
f _ xs zero = nothing ∷ xs p
f x xs (suc y) = x ∷ xs y
Very simple so far: we run through the list, filtering out the multiples of each prime as we see it. Unfortunately, this won’t pass the termination checker. This recursive call to sieve is the problem:
sieve (just x ∷ xs) = suc x ∷ sieve (cross-off x xs)
Agda finds if a function is terminating by checking that at least one argument gets (structurally) smaller on every recursive call. sieve only takes one argument (the input list), so that’s the one that needs to get smaller. In the line above, if we replaced it with the following:
sieve (just x ∷ xs) = suc x ∷ sieve xs
We’d be good to go: xs is definitely smaller than (just x ∷ xs). cross-off x xs, though? The thing is, cross-off returns a list of the same length that it’s given. But the function call is opaque: Agda can’t automatically see the fact that the length stays the same. Reaching for a proof here is the wrong move, though: you can get all of the same benefit by switching out the list for a length-indexed vector.
cross-off : ∀ {n} → Fin _ → Vec (Maybe _) n → Vec (Maybe _) n
sieve : ∀ {n} → Vec (Maybe (Fin (2 + m))) n → List (Fin (3 + m))
cross-off p fs = foldr B f (const []) fs p
f : ∀ {n} → Maybe (Fin (2 + m)) → B n → B (suc n)
Actually, my explanation above is a little bit of a lie. Often, the way I think about dependently-typed programs has a lot to do with my intuition for “proofs” and so on. But this leads you down the wrong path (and it’s why writing a proof that cross-off returns a list of the same length is the wrong move).
The actual termination checking algorithm is very simple, albeit strict: the argument passed recursively must be structurally smaller. That’s it. Basically, the recursive argument has to be contained in one of the arguments passed. It has nothing to do with Agda “seeing” inside the function cross-off or anything like that. What we’ve done above (to make it terminate) is add another argument to the function: the length of the vector. The argument is implicit, but if we were to make it explicit in the recursive call:
sieve {suc n} (just x ∷ xs) = suc x ∷ sieve {n} (cross-off x xs)
We can see that it does indeed get structurally smaller.
Adding the Squaring Optimization
A simple improvement we should be able to make is stopping once we hit the square root of the limit. Since we don’t want to be squaring as we go, we’ll use the following identity:
(n + 1)^2 = n^2 + 2n + 1
to figure out the square of the next number from the previous. In fact, we’ll just pass in the limit, and reduce it by
2n + 1
each time, until it reaches zero:
cross-off : ∀ {n} → ℕ → Vec (Maybe _) n → Vec (Maybe _) n
sieve : ∀ {n} → ℕ → ℕ → Vec (Maybe (Fin (3 + m))) n → List (Fin (3 + m))
sieve i (suc l) (just x ∷ xs) = x ∷ sieve (suc i) (l ∸ i ∸ i) (cross-off i xs)
cross-off p fs = Vec.foldr B f (const []) fs p
f : ∀ {i} → Maybe (Fin (3 + m)) → B i → B (suc i)
A slight variation on the code above (the first version) will give us the prime factors of a number:
primeFactors : ∀ n → List (Fin n)
primeFactors zero = []
primeFactors (suc zero) = []
primeFactors (suc (suc zero)) = []
primeFactors (suc (suc (suc m))) = sieve (Vec.tabulate (just ∘ Fin.suc))
sieve : ∀ {n} → Vec (Maybe (Fin (2 + m))) n → List (Fin (3 + m))
sieve (nothing ∷ xs) = sieve xs
sieve (just x ∷ xs) = Vec.foldr B remove b xs sieve x
B = λ n → ∀ {i}
→ (Vec (Maybe (Fin (2 + m))) n
→ List (Fin (3 + m)))
→ Fin i
→ List (Fin (3 + m))
b : B 0
b k zero = suc x ∷ k []
b k (suc _) = k []
remove : ∀ {n} → Maybe (Fin (2 + m)) → B n → B (suc n)
remove y ys k zero = ys (k ∘ (nothing ∷_)) x
remove y ys k (suc j) = ys (k ∘ (y ∷_)) j
Adding the squaring optimization complicates things significantly:
primeFactors (suc (suc (suc m))) = sqr (suc m) m suc sieve
_2F-_ : ∀ {n} → ℕ → Fin n → ℕ
x 2F- zero = x
zero 2F- suc y = zero
suc zero 2F- suc y = zero
suc (suc x) 2F- suc y = x 2F- y
sqr : ∀ n
→ ℕ
→ (Fin n → Fin (2 + m))
→ (∀ {i} → Vec (Maybe (Fin (2 + m))) i → ℕ → List (Fin (3 + m)))
sqr n zero f k = k [] n
sqr zero (suc l) f k = k [] zero
sqr (suc n) (suc l) f k =
let x = f zero
in sqr n (l 2F- x) (f ∘ suc) (k ∘ (just x ∷_))
sieve : ∀ {n} → Vec (Maybe (Fin (2 + m))) n → ℕ → List (Fin (3 + m))
sieve xs′ i = go xs′
go : ∀ {n} → Vec (Maybe (Fin (2 + m))) n → List (Fin (3 + m))
go (nothing ∷ xs) = go xs
go (just x ∷ xs) = Vec.foldr B remove (b i) xs x go
→ (Vec (Maybe (Fin (2 + m))) n → List (Fin (3 + m)))
b : ℕ → B 0
b zero zero k = suc x ∷ k []
b zero (suc y) k = k []
b (suc n) zero k = b n x k
b (suc n) (suc y) k = b n y k
remove y ys zero k = ys x (k ∘ (nothing ∷_))
remove y ys (suc j) k = ys j (k ∘ (y ∷_))
The above sieve aren’t “true” in that each remove is linear, so the performance is
\mathcal{O}(n^2)
overall. This is the same problem we ran into with the naive infinite sieve in Haskell.
Since it bears such a similarity to the infinite sieve, we have to ask: can this sieve be infinite? Agda supports a notion of infinite data, so it would seem like it:
infixr 5 _◂_
constructor _◂_
primes : Stream ℕ
primes = sieve 1 nats
nats : Stream ℕ
head nats = 0
tail nats = nats
sieve : ℕ → Stream ℕ → Stream ℕ
head (sieve i xs) = suc i
tail (sieve i xs) = remove i (head xs) (tail xs) (sieve ∘ suc ∘ (_+ i))
remove : ℕ → ℕ → Stream ℕ → (ℕ → Stream ℕ → Stream ℕ) → Stream ℕ
remove zero zero zs k = remove i (head zs) (tail zs) (k ∘ suc)
remove zero (suc z) zs k = remove i z zs (k ∘ suc)
remove (suc y) zero zs k = k zero (remove y (head zs) (tail zs) _◂_)
remove (suc y) (suc z) zs k = remove y z zs (k ∘ suc)
But this won’t pass the termination checker. What we actually need to prove to do so is that there are infinitely many primes: a nontrivial task in Agda.
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4.1 What makes humans uniquely successful? - Big History School
To work through '4.1 What makes humans uniquely successful?' you need to complete the Activities in order. So first complete ‘Learning Plan’ then move to Activity '4.1.1' followed by Activity '4.1.2' through to '4.1.5', and then finish with ‘Learning Summary’.
What makes humans uniquely successful?
In What makes humans uniquely successful? you will learn about the differences between humans and other species and how collective learning led to the technological innovations which shape our connected world.
To get started, read carefully through the What makes humans uniquely successful? learning goals below. Make sure you tick each of the check boxes to show that you have read all your What makes humans uniquely successful? learning goals.
As you read through the learning goals you may come across some words that you haven’t heard before. Please don’t worry. By the time you finish What makes humans uniquely successful? you will have become very familiar with them!
You will come back to these learning goals at the end of What makes humans uniquely successful? to see if you have confidently achieved them.
Junior Pre 4.1
Differences between humans and other species
Use claim testers to evaluate claims about humans and other species
Identify physical and cognitive differences between humans and other species
Define collective learning
Understand what an anthropologist studies
Collective learning and technological innovation
Identify key developments in the history of technological innovation
Begin to understand the link between collective learning, technological innovation and population growth
2Al
You’ve probably heard a lot of claims about the differences between humans and other species. Well before you explore ‘humans’ any further, you’re going to play a game of Snap Judgement. In this game you will have to decide which claims about ‘humans’ you trust and which ones you don’t trust. Once you watch Mission Video 20: Differences between humans and other species in the next activity, you will have an opportunity to come back to your ‘snap judgements’ and review your responses.
Do you remember what the four claim testers are? If you need a reminder, take a look at the Infographic: the four claim testers in Helpful Resources.
If you’re playing this game with other students your teacher may have already placed some ‘claims’ on display for you. Your teacher will ask you to go to each one of the three claims on display and write on a post-it note if you trust or don’t trust the claim with a short sentence explaining why. You then need to decide which claim tester you are using, e.g. intuition, logic, authority or evidence, and place your post-it note under that claim tester heading before moving on to the next claim.
If your teacher has instructed you to complete this activity on your own or in a pair, you will use the Claims: snap judgement human worksheet. Read each one of the three claims on the worksheet carefully and write down whether you trust or don’t trust each of the claims. Write at least one sentence explaining why and circle which claim testers you used to come to your decision.
You will learn about the differences between humans and other species in the next activity. You will then have an opportunity to come back to the 'snap judgements' that you made in this activity, think about your responses and see if you’ve changed your mind.
2Al
In Mission Phase 4 you will learn all about humans and how they came to be the most successful species on the planet. You’ll look at the positives and negatives of the connected world we live in today and begin to imagine what the future may hold for us all.
Let’s start by watching Mission video 20: Differences between humans and other species which highlights some of the differences between humans and other animal species and begins to explain what it means to be human.
While you watch Mission video 20: Differences between humans and other species look out for the answers to the following questions:
1. What do anthropologists study?
2. What are three physical differences between humans and other species?
3. What are three cognitive differences between humans and other species?
4. How has collective learning helped humans become so successful?
So where does ‘first humans’ appear on our History of the Universe Timeline?
Either your teacher will help you identify where it appears on the classroom display or you will refer back to the Timeline: history of the Universe worksheet you completed when you did the “3 big mission phase questions” activity You will find a copy of Timeline: history of the Universe example in Helpful Resources.
So now that you have a clearer idea about the differences between humans and other animal species, go back to the claims you responded to in the Snap Judgement game in the last activity.
If you are part of a class, your teacher will lead a class discussion where you and your classmates will share your responses to the claims, analyze which claim testers you used and decide whether, after watching Mission video 20: Differences between humans and other species any of you have changed your minds about any of the claims and why.
2Al
Mission video 20: Differences between humans and other species highlighted three physical differences which give humans an advantage over other animal species. An additional physical difference is the opposable thumb.
An opposable thumb is simply a thumb which can stretch across and touch all the other fingers on your hand. Compare this to, for example, a cat’s front paws. Cats have five ‘digits’ on their front paws but none of them can stretch across and touch each other.
While there are other animals which have opposable thumbs, especially other primates, the human thumb is more flexible than most.
To help you understand how an opposable thumb helps to make humans one of the most successful animal species, you will undertake a hands-on opposable thumb demonstration.
In this demonstration you will choose at least six tasks that you normally find quite simple to complete and then try to carry out that task without the help of your opposable thumb. Your teacher will instruct you if you will be working with a partner.
On the Demo: opposable thumbs worksheet you will find some suggested tasks:
Fold a piece of paper into quarters
Use a pen to write a sentence
Tie a knot on a string or shoelaces
Tap out a rhythm on a tabletop
Twist a lid off a bottle
Cut a piece of paper with scissors.
Follow the step-by-step instructions on the Demo: opposable thumbs worksheet:
Step 1: Locate the materials you need to carry out your chosen tasks.
Step 2: Have your classmate gently wrap tape around your thumb and forefinger. If you are right-handed, tape your right hand. If you are left-handed, tape your left hand.
Step 3: Try each of your chosen tasks with your taped hand.
Step 4: Refer to the Results Table. Under the heading ‘Task’ describe the task you performed. Under the heading ‘Could it be done without an opposable thumb?’ tick the relevant box.
If you are part of a class, your teacher will lead a class discussion where you and your classmates will share your responses to the reflection questions on the Demo: opposable thumbs worksheet:
Which types of tasks were you still able to complete without thumbs? Why?
Which types of tasks were you unable to complete without thumbs? Why?
How do you think having opposable thumbs has helped to make humans so successful?
2Al
In the previous activities you learned about some of the ways in which humans are different to other animal species. One of the most important differences is our ability to learn collectively by storing, sharing and building on information from generation to generation.
In Mission video 21: Collective learning and technological innovation you will see how collective learning, over time, has led to waves of technological innovation and how that has led to the world that we live in today...
While you watch Mission Video 21: Collective learning and technological innovation look out for the answers to the following questions:
1. How did humans live 300,000 (3 lakh) years ago?
2. Which wave of innovation began 12,000 years ago?
3. Which wave of innovation began 200 years ago?
5. Which wave of innovation began 20 years ago?
6. Which wave of innovation is happening right now?
7. What effect has each wave of innovation had on the human population?
Mission video 21: Collective learning and technological innovation explained the connection between collective learning, technological innovation and population growth. In Helpful Resources you will find a link to a video which traces human population growth from about 2000 years ago to the year 2050 on a map. Watch the video carefully, paying particular attention to the moments when the biggest jumps in population growth occur. Do they coincide with waves of innovation, for example, industrial innovation which began in the 1800s? Can you find any other examples of big jumps in the population?
http://worldpopulationhistory.org/map/101/mercator/1/0/25/
2Al
Mission video 21: Collective learning and technological innovation explored how, over hundreds of thousands of years, collective learning led to waves of technological innovation which have made us the uniquely successful species we are today.
In this activity you will place the most important waves of technological innovation in chronological order on a timeline.
Your teacher will provide you with a copy of the Timeline: technological innovation worksheet. Take a careful look and see what you notice about the timeline:
How long ago does it begin?
What is happening to the size of the gaps between the waves of innovation as you get closer to today?
Cut out each of the five ‘Innovations’ on the strip of paper provided by your teacher. Paste them into the correct boxes on the Timeline: technological innovation worksheet. Finally, answer the question, ‘What is the link between collective learning, technological innovation and population growth?’
Take a look at your completed Timeline: technological innovation worksheet. What do you notice about how often the waves of innovation occur? Can you see how there was a huge gap before the first wave of innovation (almost 200,000 years!) and there is a gap of only 20 years between the most recent waves of innovation? This shows us how collective learning and human innovation are accelerating. And we know that this leads to the population increasing as well.
As a fun final reflection on the differences between our lives and those of humans who lived in the stone ages 12,000 years ago, watch the BBC video clip in Helpful Resources and notice the differences between the life of a 10 year old then compared to your life today. Which technologies do you use and enjoy today that kids didn’t have 12,000 years ago? And who do you think has it better - the Stone-Agers or you?
https://www.youtube.com/watch?v=cE6OeRZB_Wc
2Al
In What makes humans uniquely successful? you learned about the differences between humans and other species and how collective learning led to the technological innovations which shape our connected world.
Now it’s time to revisit your What makes humans uniquely successful? learning goals and read through them again carefully.
Junior Post 4.1
Well done on completing your learning summary. Click here to go to 4.2 What are the challenges of human success?
Once you have checked the boxes to confirm you have achieved your learning goals for Sequence '4.1 What makes humans uniquely successful?' click on the 'I have achieved my learning goals' button below.
Go to 4.2 What are the challenges of human success? »
2Al
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Carborane - Wikipedia
Ball-and-stick model of o-carborane
Carboranes are electron-delocalized (non-classically bonded) clusters composed of boron, carbon and hydrogen atoms.[1] Like many of the related boron hydrides, these clusters are polyhedra or fragments of polyhedra. Carboranes are one class of heteroboranes.[2]
In terms of scope, carboranes can have as few as 5 and as many as 14 atoms in the cage framework. The majority have two cage carbon atoms. The corresponding C-alkyl and B-alkyl analogues are also known in a few cases.
2.1 Monocarba derivatives
2.2 Dicarba clusters
3 Classification by cage size
3.1 Small, open carboranes
3.2 Small, closed carboranes
3.3 Intermediate-sized carboranes
3.4 Icosahedral carboranes
Carboranes and boranes adopt 3-dimensional cage (cluster) geometries in sharp contrast to typical organic compounds. Cages are compatible with sigma—delocalized bonding, whereas hydrocarbons are typically chains or rings
Like for other electron-delocalized polyhedral clusters, the electronic structure of these cluster compounds can be described by the Wade–Mingos rules. Like the related boron hydrides, these clusters are polyhedra or fragments of polyhedra, and are similarly classified as closo-, nido-, arachno-, hypho-, etc., based on whether they represent a complete (closo-) polyhedron or a polyhedron that is missing one (nido-), two (arachno-), three (hypho-), or more vertices. Carboranes are a notable example of heteroboranes.[2][3] The essence, these rules emphasize delocalized, multi-centered bonding for B-B, C-C, and B-C interactions.
Geometrical isomers of carboranes can exist on the basis of the various locations of carbon within the cage. Isomers necessitate the use of the numerical prefixes in a compound's name. The closo dicarbadecaborane can exist in three isomers: 1,2-, 1,7-, and 1,12-C2B10H12.
Carboranes have been prepared by many routes, the most common being addition of alkynyl reagents to boron hydride clusters to form dicarbon carboranes. For example, the high-temperature reaction of pentaborane(9) with acetylene affords several closo-carboranes as well as other products:
nido-B5H9 + C2H2
{\displaystyle {\overset {500{-}600\ ^{\circ }{\text{C}}}{\longrightarrow }}}
closo-1,5-C2B3H5, closo-1,6-C2B4H6, 2,4-C2B5H7
When the reaction is conducted at lower temperatures, an open-cage carborane is obtained:
{\displaystyle {\overset {200\ ^{\circ }{\text{C}}}{\longrightarrow }}}
nido-2,4-C2B4H8
Other procedures generate carboranes containing three or four cage carbon atoms (refs. [1],[4],[5][clarify])
Monocarba derivatives[edit]
Monocarboranes are clusters with BnC cages. The 12-vertex derivative is best studied, but several are known.
Typically they are prepared by the addition of one-carbon reagents to boron hydride clusters. One-carbon reagents include cyanide, isocyanides, and formaldehyde. For example, monocarbadodecaborate ([CB11H12]−) is produced from decaborane and formaldehyde, followed by addition of borane dimethylsulfide.[4][5] Monocarboranes are precursors to weakly coordinating anions.[6]
Dicarba clusters[edit]
Dicarbaboranes can be prepared from boron hydrides using alkynes as the source of the two carbon centers. In addition to the closo-C2BnHn+2 series mentioned above, several open-cage dicarbon species are known including nido-C2B3H7 (isostructural and isoelectronic with B5H9) and arachno-C2B7H13.
Structure of nido-C2B4H8, highlighting some trends: carbon at the low connectivity sites, bridging hydrogen between B centers on open face[7]
Syntheses of icosahedral closo-dicarbadodecaborane derivatives (R2C2B10H10) employ alkynes as the R2C2 source and decaborane (B10H14) to supply the B10 unit.
Classification by cage size[edit]
The following classification is adapted from Grimes's book on carboranes.[1]
Small, open carboranes[edit]
This family of clusters includes the nido cages CB5H9, C2B4H8, C3B3H7, C4B2H6, and C2B3H7. Relatively little work has been devoted to these compounds. Pentaborane[9] reacts with acetylene to give nido-1,2-C2B4H8. Upon treatment with sodium hydride, latter forms the salt Na[1,2-C2B4H7].
Small, closed carboranes[edit]
This family of clusters includes the closo cages C2B3H5, C2B4H6, C2B5H7, and CB5H7. This family of clusters are also lightly studied owing to synthetic difficulties. Also reflecting synthetic challenges, many of these compounds are best known as their alkyl derivatives. 1,5-C2B3H5 is the only known isomer of this five-vertex cage.
Intermediate-sized carboranes[edit]
Structure of 1,3-C2B7H13 (all unlabeled vertices are BH).
This family of clusters includes the closo cages C2B6H8, C2B7H9, C2B8H10, and C2B9H11.
Isomerism is well established in this family: 1,2- and 1,6-C2B6H8, and 2,3- and 2,4-C2B5H7, as well as in open-cage carboranes such as 2,3- and 2,4-C2B4H8 and 1,2 and 1,3-C2B9H13. In general, isomers having non-adjacent cage carbon atoms are more thermally stable than those with adjacent carbons, so that heating tends to induce mutual separation of the carbon atoms in the framework.
Carboranes of intermediate nuclearity are most efficiently generated by degradations starting with carborane as summarized in these equations, starting from C2B9H12−:
C2B9H12− + H+ → C2B9H13
C2B9H13 → C2B9H11 + H2
In contrast, smaller carboranes are usually prepared by building-up routes, e.g. from pentaborane + alkyne.
Chromate oxidation of this 11-vertex cluster results in deboronation, giving C2B7H13. From that species, other clusters result by pyrolysis, sometimes in the presence of diborane: C2B6H8, C2B8H10, and C2B7H9.[1]
Icosahedral carboranes[edit]
The icosahedral charge-neutral closo-carboranes, 1,2-, 1,7-, and 1,12-C2B10H12 (respectively ortho-, meta-, and para-carborane in the informal nomenclature) are particularly stable and are commercially available.[8][9] The 1,2-isomer forms first upon the reaction of decarborane and acetylene. It converts quantitatively to the 1,7-isomer upon heating in an inert atmosphere. The formation of the 1,2- to 1,12-isomer requires 700 °C, proceeding in ca. 25% yield.[1]
[CB11H12]− is also well established.
Base-induced degradation of carboranes give anionic nido derivatives, which can be employed as ligands for transition metals, generating metallacarboranes, which are carboranes containing one or more transition metal or main group metal atoms in the cage framework. Most famous are the dicarbollide, complexes with the formula M[C2B9H11]2-.[10]
Dicarbollide complexes have been evaluated for many applications for many years, but commercial applications are rare. The bis(dicarbollide) [Co(C2B9H11)2]− has been used as a precipitant for removal of 137Cs+ from radiowastes.[11]
The medical applications of carboranes have been explored.[12][13] C-functionalized carboranes represent a source of boron for boron neutron capture therapy.[14]
The compound H(CHB11Cl11) is a superacid, forming an isolable salt with protonated benzene, C6H+
7.[15] It protonates fullerene, C60.[16]
Azaborane
Heteroborane
Dicarbollide
^ a b c d Grimes, R. N., Carboranes 3rd Ed., Elsevier, Amsterdam and New York (2016), ISBN 9780128018941.
^ a b Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 181–189. ISBN 978-0-08-037941-8.
^ The Wade–Mingos rules were first stated by Kenneth Wade in 1971 and expanded by Michael Mingos in 1972:
Wade, Kenneth (1971). "The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds". J. Chem. Soc. D. 1971 (15): 792–793. doi:10.1039/C29710000792.
Mingos, D. M. P. (1972). "A general theory for cluster and ring compounds of the main group and transition elements". Nature Physical Science. 236 (68): 99–102. Bibcode:1972NPhS..236...99M. doi:10.1038/physci236099a0.
They are sometimes known as simply "Wade's rules".
Welch, Alan J. (2013). "The significance and impact of Wade's rules". Chem. Commun. 49 (35): 3615–3616. doi:10.1039/C3CC00069A. PMID 23535980.
^ W. H. Knoth (1967). "1-B9H9CH− and B11H11CH−". J. Am. Chem. Soc. 89 (5): 1274–1275. doi:10.1021/ja00981a048.
^ Tanaka, N.; Shoji, Y.; Fukushima, T. (2016). "Convenient Route to Monocarba-closo-dodecaborate Anions". Organometallics. 35 (11): 2022–2025. doi:10.1021/acs.organomet.6b00309. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ Reed, Christopher A. (2010). "H+, CH3+, and R3Si+Carborane Reagents: When Triflates Fail". Accounts of Chemical Research. 43 (1): 121–128. doi:10.1021/ar900159e. PMC 2808449. PMID 19736934.
^ G. S. Pawley (1966). "Further Refinements of Some Rigid Boron Compounds". Acta Crystallogr. 20 (5): 631–638. doi:10.1107/S0365110X66001531.
^ Jemmis, E. D. (1982). "Overlap Control and Stability of Polyhedral Molecules. Closo-Carboranes". Journal of the American Chemical Society. 104 (25): 7017–7020. doi:10.1021/ja00389a021.
^ Spokoyny, A. M. (2013). "New ligand platforms featuring boron-rich clusters as organomimetic substituents". Pure and Applied Chemistry. 85 (5): 903–919. doi:10.1351/PAC-CON-13-01-13. PMC 3845684. PMID 24311823.
^ Sivaev, I. B.; Bregadze, V. I. (2000). "Chemistry of Nickel and Iron Bis(dicarbollides). A Review". Journal of Organometallic Chemistry. 614–615: 27–36. doi:10.1016/S0022-328X(00)00610-0. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ Dash, B. P.; Satapathy, R.; Swain, B. R.; Mahanta, C. S.; Jena, B. B.; Hosmane, N. S. (2017). "Cobalt bis(dicarbollide) anion and its derivatives". J. Organomet. Chem. 849–850: 170–194. doi:10.1016/j.jorganchem.2017.04.006. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ Issa, F.; Kassiou, M.; Rendina, L. M. (2011). "Boron in drug discovery: carboranes as unique pharmacophores in biologically active compounds". Chem. Rev. 111 (9): 5701–5722. doi:10.1021/cr2000866. PMID 21718011. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ Stockmann, Philipp; Gozzi, Marta; Kuhnert, Robert; Sárosi, Menyhárt B.; Hey-Hawkins, Evamarie (2019). "New keys for old locks: carborane-containing drugs as platforms for mechanism-based therapies". Chemical Society Reviews. 48 (13): 3497–3512. doi:10.1039/C9CS00197B. PMID 31214680.
^ Soloway, A. H.; Tjarks, W.; Barnum, B. A.; Rong, F.-G.; Barth, R. F.; Codogni, I. M.; Wilson, J. G. (1998). "The Chemistry of Neutron Capture Therapy". Chemical Reviews. 98 (4): 1515–1562. doi:10.1021/cr941195u. PMID 11848941.
^ Olah, G. A.; Prakash, G. K. S.; Sommer, J.; Molnar, A. (2009). Superacid Chemistry (2nd ed.). Wiley. p. 41. ISBN 978-0-471-59668-4.
^ Reed Christopher A (2013). "Myths about the Proton. The Nature of H+ in Condensed Media". Acc. Chem. Res. 46 (11): 2567–2575. doi:10.1021/ar400064q. PMC 3833890. PMID 23875729.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Carborane&oldid=1085698079"
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empty script base - Maple Help
Home : Support : Online Help : System : Error Message Guide : empty script base
Error, empty script base
Error, invalid base
These are 2-D math parser errors. They occur when you attempt to execute an expression that has a superscript or subscript with no base.
Tip: If an expression is executed accidentally, it can lead to error messages (if it is not valid Maple syntax). If this happens, toggling the expression to nonexecutable math removes the error message and changes the math to nonexecutable. To change an expression to nonexecutable math use the shortcut key Shift + F5. For more information, see Executable and Nonexecutable Math or example 4a below.
{}^{b}+{x}^{2}
This error occurred because the
b
exponent is not attached to a base. To create this error, you must actively go back after creating the exponent and erase the base. Using standard keystroke or palette entry, the base is included before a superscript or subscript can be added.
To fix this error, ensure that any superscripts or subscripts are attached to a base.
{x}^{b}+{x}^{2}
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}
{}_{b}-{x}^{5}
In this example, a subscript is missing the required base.
To fix the error, ensure the subscript is attached to a base.
{x}_{b}-{x}^{5}
\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{b}}
{x}^{2}-{ }^{b}
This is the same type of error, but the missing base for the superscript
b
is found inside the expression. The 2-D math parser returns a slightly different error, because it is interpreting the superscript as a base, which is invalid.
Just like the "empty script base" error, you must ensure that any superscripts or subscripts include a valid base.
{x}^{2}-{x}^{b}
{\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}{b}}
Example 4a - Expression in text
To write an expression but not execute it, you can use nonexecutable math. That can help avoid parsing errors for expressions. For example, suppose you want to write Radium-229 in isotope notation in text, either in a paragraph (or in a plot—see Example 4b).
As a first attempt, write a^229 Ra and then delete a. If executed, this gives a parsing error.
Radioactive Decay of
{}^{229}\mathrm{Ra}
Change the expression to nonexecutable math: click the expression, and use the shortcut key Shift + F5. The error message is removed. For more information, see Executable and Nonexecutable Math.
{}^{229}\mathrm{Ra}
Use the Layout palette element
{}^{\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{b}}\textcolor[rgb]{0,0.627450980392157,0.313725490196078}{A}
or use the shortcut key for pre-superscript.
{}^{229}\mathrm{Ra}
Example 4b - Expression in text in a plot
Now suppose you want to write use Radium-229 in text in a plot, perhaps as a title or in a legend.
(This example is based on one from the MaplePortal/ChemicalIsotope help page.)
\mathrm{\lambda } ≔ \frac{0.693}{\mathrm{evalf}\left(\mathrm{ScientificConstants}:-\mathrm{Element}\left({\mathrm{Ra}}_{229}, \mathrm{halflife}\right)\right)};
\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.002887500000}
\mathrm{plot}\left({ⅇ}^{-\mathrm{\lambda }\cdot t},t=0..2 {10}^{3},\mathrm{labels}=["Time \left(s\right)","Activity"],\mathrm{title}=\mathrm{typeset}\left("Radioactive Decay of"{,}^{229}\mathrm{Ra}\right)\right)
Use the Layout palette or use the shortcut key for pre-superscript to get an expression which doesn't have parsing issues.
\mathrm{plot}\left({ⅇ}^{-\mathrm{\lambda }\cdot t},t=0..2 {10}^{3},\mathrm{labels}=["Time \left(s\right)","Activity"],\mathrm{title}=\mathrm{typeset}\left("Radioactive Decay of",{}^{229}\mathrm{Ra}\right)\right)
For more tips on fixing parsing errors in text in a plot, including using atomic identifiers, see Typesetting in Plots.
2-D Math Shortcut Keys
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EUDML | Links associated with generic immersions of graphs. EuDML | Links associated with generic immersions of graphs.
Links associated with generic immersions of graphs.
Kawamura, Tomomi. "Links associated with generic immersions of graphs.." Algebraic & Geometric Topology 4 (2004): 571-594. <http://eudml.org/doc/124191>.
author = {Kawamura, Tomomi},
keywords = {divide; graph divide; quasipositive link; slice Euler characteristic; four-dimensional clasp number},
title = {Links associated with generic immersions of graphs.},
AU - Kawamura, Tomomi
TI - Links associated with generic immersions of graphs.
KW - divide; graph divide; quasipositive link; slice Euler characteristic; four-dimensional clasp number
divide, graph divide, quasipositive link, slice Euler characteristic, four-dimensional clasp number
{S}^{3}
Articles by Kawamura
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Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow | EMS Press
Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow
CMLA-ENS, Cachan, France
In this paper we compute the explicit evolution of the Minimizing Total Variation flow when the initial condition is the characteristic function of a convex set in
\mathbb{R}^2
, or a finite number of them which are sufficiently separated. We shall also obtain some explicit solutions of the Total Variation formulation of the denoising problem in image processing. We illustrate these results with some experiments.
Antonin Chambolle, Vicent Caselles, François Alter, Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. Interfaces Free Bound. 7 (2005), no. 1, pp. 29–53
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Warning, if d is meant to be the differential symbol (and not just a variable d), use command completion or palettes to enter this expression, or use the diff command - Maple Help
Home : Support : Online Help : Warning, if d is meant to be the differential symbol (and not just a variable d), use command completion or palettes to enter this expression, or use the diff command
ⅇ,
ⅆ
ⅈ
d
ⅆ
d
\mathrm{dx}
\frac{d}{\mathrm{dx}}{x}^{2}
\frac{\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{\mathrm{dx}}}
\frac{ⅆ}{ⅆx}{x}^{2}
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
ⅆ
\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{x}
{x}^{2}
x
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}
\textcolor[rgb]{0,0,1}{x}
\frac{ⅆ}{ⅆ x}\left({x}^{2}\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
{x}^{2}-40 x-6
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{40}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}
\frac{ⅆ}{ⅆx}\left({x}^{2}-40 x-6\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{40}
\mathrm{diff}\left({x}^{2},x\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
\mathrm{dt}≔0.01
\textcolor[rgb]{0,0,1}{\mathrm{dt}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.01}
\frac{d}{\mathrm{dt}}
\textcolor[rgb]{0,0,1}{100.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}
\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{parserwarnings}=\mathrm{false}\right):
\frac{d}{\mathrm{dt}}
\textcolor[rgb]{0,0,1}{100.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}
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cos(x)^{2}/x
x^{-1}(1 + 0 - x^{2} ....)
1/x - x + ...
x^{s_1} f_1(x)
x^{s_2} f_2 (x)
x^{s_1+s_2}f_1(x)f_2(x)
You are right. I will be working on them. :)
Should I add HolonomicFunction in srepr for a valid python output or change the default one to use f = Function('f')?
According to issue the str should be valid python.
I think the better way to do this can be:
str will use f = Function('f') in printing so that one can use .subs(). So it should return HolonomicFunction(f(x) + Derivative(f(x), x, x), x), f(0) = 0, f'(0) = 1 for our example.
And calling srepr on the example should return HolonomicFunction(1 + Dx**2, x, 0, [0, 1]) which is valid python.
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Heath-Jarrow-Morton Model (HJM) Definition
What Is the Heath-Jarrow-Morton (HJM) Model?
The Heath-Jarrow-Morton Model (HJM Model) is used to model forward interest rates. These rates are then modeled to an existing term structure of interest rates to determine appropriate prices for interest-rate-sensitive securities.
The Heath-Jarrow-Morton Model (HJM Model) is used to model forward interest rates using a differential equation that allows for randomness.
These rates are then modeled to an existing term structure of interest rates to determine appropriate prices for interest-rate sensitive securities such as bonds or swaps.
Today, it is used mainly by arbitrageurs seeking arbitrage opportunities, as well as analysts pricing derivatives.
Formula for the HJM Model
In general, the HJM model and those that are built on its framework follow the formula:
\begin{aligned} &\text{d}f(t,T) = \alpha (t,T)\text{d}t + \sigma (t,T)\text{d}W(t)\\ &\textbf{where:}\\ &\text{d}f(t,T) = \text{The instantaneous forward interest rate of}\\&\text{zero-coupon bond with maturity T, is assumed to satisfy}\\&\text{the stochastic differential equation shown above.}\\ &\alpha, \sigma = \text{Adapted}\\ &W = \text{A Brownian motion (random-walk) under the}\\&\text{risk-neutral assumption}\\ \end{aligned}
df(t,T)=α(t,T)dt+σ(t,T)dW(t)where:df(t,T)=The instantaneous forward interest rate ofzero-coupon bond with maturity T, is assumed to satisfythe stochastic differential equation shown above.α,σ=AdaptedW=A Brownian motion (random-walk) under therisk-neutral assumption
What Does the HJM Model Tell You?
A Heath-Jarrow-Morton Model is very theoretical and is used at the most advanced levels of financial analysis. It is used mainly by arbitrageurs seeking arbitrage opportunities, as well as analysts pricing derivatives. The HJM Model predicts forward interest rates, with the starting point being the sum of what’s known as drift terms and diffusion terms. The forward rate drift is driven by volatility, which is known as the HJM drift condition. In the basic sense, an HJM Model is any interest rate model driven by a finite number of Brownian motions.
The HJM Model is based on the work of economists David Heath, Robert Jarrow, and Andrew Morton in the 1980s. The trio wrote a series of notable papers in the late 1980s and early 1990sthat laid the groundwork for the framework, among them "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation", " Contingent Claims Valuation with a Random Evolution of Interest Rates", and “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation".
There are various additional models built on the HJM Framework. They all generally look to predict the entire forward rate curve, not just the short rate or another point on the curve. The biggest issue with HJM Models is that they tend to have infinite dimensions, making it almost impossible to compute. There are various models that look to express the HJM Model as a finite state.
HJM Model and Option Pricing
The HJM Model is also used in option pricing, which refers to finding the fair value of a derivative contract. Trading institutions may use models to price options as a strategy for finding under- or overvalued options.
Option pricing models are mathematical models that use known inputs and predicted values, such as implied volatility, to find the theoretical value of options. Traders will use certain models to figure out the price at a certain point in time, updating the value calculation based on changing risk.
For an HJM Model, to calculate the value of an interest rate swap, the first step is to form a discount curve based on current option prices. From that discount curve, forward rates can be obtained. From there, the volatility of forwarding interest rates must be input, and if the volatility is known the drift can be determined.
David Heath, Robert Jarrow and Andrew Morton. "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation". Journal of Financial and Quantitative Analysis. Volume 25, No. 4, 1990, Pages 419-440.
The Cox-Ingersoll-Ross model is a mathematical formula used to model interest rate movements and is driven by a sole source of market risk.
The Vasicek interest rate model predicts interest rate movement based on market risk, time and long-term equilibrium interest rate values.
The Hull-White model is used to price derivatives under the assumption that short rates have a normal distribution and revert to the mean.
The Brace Gatarek Musiela (BGM) Model is a nonlinear financial model that uses LIBOR rates to price interest rate derivatives.
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Genus–differentia definition - Wikipedia
Type of intensional definition
Find sources: "Genus–differentia definition" – news · newspapers · books · scholar · JSTOR (December 2021) (Learn how and when to remove this template message)
A genus–differentia definition is a type of intensional definition, and it is composed of two parts:
the differentia: The portion of the definition that is not provided by the genus.
For example, consider these two definitions:
a triangle: A plane figure that has 3 straight bounding sides.
a quadrilateral: A plane figure that has 4 straight bounding sides.
Those definitions can be expressed as one genus and two differentiae:
one genus:
the genus for both a triangle and a quadrilateral: "A plane figure"
two differentiae:
the differentia for a triangle: "that has 3 straight bounding sides."
the differentia for a quadrilateral: "that has 4 straight bounding sides."
The use of genus and differentia in constructing definitions goes back at least as far as Aristotle (384–322 BCE).[1]
1 Differentiation and Abstraction
Differentiation and Abstraction[edit]
The process of producing new definitions by extending existing definitions is commonly known as differentiation (and also as derivation). The reverse process, by which just part of an existing definition is used itself as a new definition, is called abstraction; the new definition is called an abstraction and it is said to have been abstracted away from the existing definition.
A part of that definition may be singled out (using parentheses here):
a square: (a quadrilateral that has interior angles which are all right angles), and that has bounding sides which all have the same length.
and with that part, an abstraction may be formed:
Then, the definition of a square may be recast with that abstraction as its genus:
Similarly, the definition of a square may be rearranged and another portion singled out:
a square: (a quadrilateral that has bounding sides which all have the same length), and that has interior angles which are all right angles.
leading to the following abstraction:
a rhombus: a quadrilateral that has bounding sides which all have the same length.
a square: a rhombus that has interior angles which are all right angles.
More generally, a collection of
{\displaystyle n>1}
equivalent definitions (each of which is expressed with one unique genus) can be recast as one definition that is expressed with
{\displaystyle n}
genera.[citation needed] Thus, the following:
a Definition: a Genus1 that is a Genus2 and that is a Genus3 and that is a… and that is a Genusn-1 and that is a Genusn, which has some non-genus Differentia.
a Definition: a Genusn-1 that is a Genus1 and that is a Genus2 and that is a Genus3 and that is a… and that is a Genusn, which has some non-genus Differentia.
a Definition: a Genusn that is a Genus1 and that is a Genus2 and that is a Genus3 and that is a… and that is a Genusn-1, which has some non-genus Differentia.
a Definition: a Genus1 and a Genus2 and a Genus3 and a… and a Genusn-1 and a Genusn, which has some non-genus Differentia.
A genus of a definition provides a means by which to specify an is-a relationship:
A square is a rectangle, which is a quadrilateral, which is a plane figure, which is a…
A square is a rhombus, which is a quadrilateral, which is a plane figure, which is a…
A square is a quadrilateral, which is a plane figure, which is a…
A square is a plane figure, which is a…
A square is a…
The non-genus portion of the differentia of a definition provides a means by which to specify a has-a relationship:
A square has an interior angle that is a right angle.
A square has a straight bounding side.
A square has a…
When a system of definitions is constructed with genera and differentiae, the definitions can be thought of as nodes forming a hierarchy or—more generally—a directed acyclic graph; a node that has no predecessor is a most general definition; each node along a directed path is more differentiated (or more derived) than any one of its predecessors, and a node with no successor is a most differentiated (or a most derived) definition.
When a definition, S, is the tail of each of its successors (that is, S has at least one successor and each direct successor of S is a most differentiated definition), then S is often called the species of each of its successors, and each direct successor of S is often called an individual (or an entity) of the species S; that is, the genus of an individual is synonymously called the species of that individual. Furthermore, the differentia of an individual is synonymously called the identity of that individual. For instance, consider the following definition:
[the] John Smith: a human that has the name 'John Smith'.
The whole definition is an individual; that is, [the] John Smith is an individual.
The genus of [the] John Smith (which is "a human") may be called synonymously the species of [the] John Smith; that is, [the] John Smith is an individual of the species [a] human.
The differentia of [the] John Smith (which is "that has the name 'John Smith'") may be called synonymously the identity of [the] John Smith; that is, [the] John Smith is identified among other individuals of the same species by the fact that [the] John Smith is the one "that has the name 'John Smith'".
As in that example, the identity itself (or some part of it) is often used to refer to the entire individual, a phenomenon that is known in linguistics as a pars pro toto synecdoche.
^ Parry, William Thomas; Hacker, Edward A. (1991). Aristotelian Logic. G - Reference,Information and Interdisciplinary Subjects Series. Albany: State University of New York Press. p. 86. ISBN 9780791406892. Retrieved 8 Feb 2019. Aristotle recognized only one method of real definition, namely, the method of genus and differentia, applied to defining real things, not words.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Genus–differentia_definition&oldid=1082893767"
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Uninstall - Maple Help
Home : Support : Online Help : Programming : Package Tools : Uninstall
uninstall a workbook-based package
Uninstall( package )
The Uninstall command removes an installed package given the name or unique integer identifier of the package.
The Uninstall command irreversibly removes the files that were installed by the PackageTools:-Install command.
By default packages are installed in "maple/toolbox/package" relative to your home directory. They can also optionally be installed in "$MAPLE/toolbox/package", where $MAPLE is your Maple installation directory.
\mathrm{with}\left(\mathrm{PackageTools}\right):
\mathrm{Uninstall}\left("mypack.maple"\right)
\mathrm{Uninstall}\left(5095707910340608,\mathrm{version}=1\right)
The PackageTools[Uninstall] command was introduced in Maple 2017.
module,package
worksheet,cloud,overview
worksheet,expressions,VariableManager
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Asymptotic stability of harmonic maps under the Schrödinger flow
1 December 2008 Asymptotic stability of harmonic maps under the Schrödinger flow
Stephen Gustafson, Kyungkeun Kang, Tai-Peng Tsai
Stephen Gustafson,1 Kyungkeun Kang,2 Tai-Peng Tsai1
2Department of Mathematics, Sungkyunkwan University and Institute of Basic Science
For Schrödinger maps from
{\mathbb{R}}^{2}×{\mathbb{R}}^{+}
2
{\mathbb{S}}^{2}
, it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space--dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map
Stephen Gustafson. Kyungkeun Kang. Tai-Peng Tsai. "Asymptotic stability of harmonic maps under the Schrödinger flow." Duke Math. J. 145 (3) 537 - 583, 1 December 2008. https://doi.org/10.1215/00127094-2008-058
Stephen Gustafson, Kyungkeun Kang, Tai-Peng Tsai "Asymptotic stability of harmonic maps under the Schrödinger flow," Duke Mathematical Journal, Duke Math. J. 145(3), 537-583, (1 December 2008)
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Probabilistic method - Wikipedia
Nonconstructive method for mathematical proofs
The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero. Although the proof uses probability, the final conclusion is determined for certain, without any possible error.
This method has now been applied to other areas of mathematics such as number theory, linear algebra, and real analysis, as well as in computer science (e.g. randomized rounding), and information theory.
2 Two examples due to Erdős
If every object in a collection of objects fails to have a certain property, then the probability that a random object chosen from the collection has that property is zero.
Similarly, showing that the probability is (strictly) less than 1 can be used to prove the existence of an object that does not satisfy the prescribed properties.
Another way to use the probabilistic method is by calculating the expected value of some random variable. If it can be shown that the random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value.
Alternatively, the probabilistic method can also be used to guarantee the existence of a desired element in a sample space with a value that is greater than or equal to the calculated expected value, since the non-existence of such element would imply every element in the sample space is less than the expected value, a contradiction.
Common tools used in the probabilistic method include Markov's inequality, the Chernoff bound, and the Lovász local lemma.
Two examples due to Erdős[edit]
Although others before him proved theorems via the probabilistic method (for example, Szele's 1943 result that there exist tournaments containing a large number of Hamiltonian cycles), many of the most well known proofs using this method are due to Erdős. The first example below describes one such result from 1947 that gives a proof of a lower bound for the Ramsey number R(r, r).
First example[edit]
Suppose we have a complete graph on n vertices. We wish to show (for small enough values of n) that it is possible to color the edges of the graph in two colors (say red and blue) so that there is no complete subgraph on r vertices which is monochromatic (every edge colored the same color).
To do so, we color the graph randomly. Color each edge independently with probability 1/2 of being red and 1/2 of being blue. We calculate the expected number of monochromatic subgraphs on r vertices as follows:
{\displaystyle S_{r}}
{\displaystyle r}
vertices from our graph, define the variable
{\displaystyle X(S_{r})}
to be 1 if every edge amongst the
{\displaystyle r}
vertices is the same color, and 0 otherwise. Note that the number of monochromatic
{\displaystyle r}
-subgraphs is the sum of
{\displaystyle X(S_{r})}
over all possible subsets
{\displaystyle S_{r}}
. For any individual set
{\displaystyle S_{r}^{i}}
{\displaystyle X(S_{r}^{i})}
is simply the probability that all of the
{\displaystyle C(r,2)}
edges in
{\displaystyle S_{r}^{i}}
are the same color:
{\displaystyle E[X(S_{r}^{i})]=2\cdot 2^{-{r \choose 2}}}
(the factor of 2 comes because there are two possible colors).
This holds true for any of the
{\displaystyle C(n,r)}
possible subsets we could have chosen, i.e.
{\displaystyle i}
ranges from 1 to
{\displaystyle C(n,r)}
. So we have that the sum of
{\displaystyle E[X(S_{r}^{i})]}
{\displaystyle S_{r}^{i}}
{\displaystyle \sum _{i=1}^{C(n,r)}E[X(S_{r}^{i})]={n \choose r}2^{1-{r \choose 2}}.}
The sum of expectations is the expectation of the sum (regardless of whether the variables are independent), so the expectation of the sum (the expected number of all monochromatic
{\displaystyle r}
-subgraphs) is
{\displaystyle E[X(S_{r})]={n \choose r}2^{1-{r \choose 2}}.}
Consider what happens if this value is less than 1. Since the expected number of monochromatic r-subgraphs is strictly less than 1, there exists a coloring satisfying the condition that the number of monochromatic r-subgraphs is strictly less than 1. The number of monochromatic r-subgraphs in this random coloring is a non-negative integer, hence it must be 0 (0 is the only non-negative integer less than 1). It follows that if
{\displaystyle E[X(S_{r})]={n \choose r}2^{1-{r \choose 2}}<1}
(which holds, for example, for n=5 and r=4), there must exist a coloring in which there are no monochromatic r-subgraphs. [a]
By definition of the Ramsey number, this implies that R(r, r) must be bigger than n. In particular, R(r, r) must grow at least exponentially with r.
A weakness of this argument is that it is entirely nonconstructive. Even though it proves (for example) that almost every coloring of the complete graph on (1.1)r vertices contains no monochromatic r-subgraph, it gives no explicit example of such a coloring. The problem of finding such a coloring has been open for more than 50 years.
^ The same fact can be proved without probability, using a simple counting argument:
The total number of r-subgraphs is
{\displaystyle {n \choose r}}
Each r-subgraphs has
{\displaystyle {r \choose 2}}
edges and thus can be colored in
{\displaystyle 2^{r \choose 2}}
different ways.
Of these colorings, only 2 colorings are 'bad' for that subgraph (the colorings in which all vertices are red or all vertices are blue).
Hence, the total number of colorings that are bad for some (at least one) subgraph is at most
{\displaystyle 2{n \choose r}}
{\displaystyle 2^{r \choose 2}>2{n \choose r}}
, there must be at least one coloring which is not 'bad' for any subgraph.
Second example[edit]
A 1959 paper of Erdős (see reference cited below) addressed the following problem in graph theory: given positive integers g and k, does there exist a graph G containing only cycles of length at least g, such that the chromatic number of G is at least k?
It can be shown that such a graph exists for any g and k, and the proof is reasonably simple. Let n be very large and consider a random graph G on n vertices, where every edge in G exists with probability p = n1/g−1. We show that with positive probability, G satisfies the following two properties:
Property 1. G contains at most n/2 cycles of length less than g.
Proof. Let X be the number cycles of length less than g. Number of cycles of length i in the complete graph on n vertices is
{\displaystyle {\frac {n!}{2\cdot i\cdot (n-i)!}}\leq {\frac {n^{i}}{2}}}
and each of them is present in G with probability pi. Hence by Markov's inequality we have
{\displaystyle \Pr \left(X>{\tfrac {n}{2}}\right)\leq {\frac {2}{n}}E[X]\leq {\frac {1}{n}}\sum _{i=3}^{g-1}p^{i}n^{i}={\frac {1}{n}}\sum _{i=3}^{g-1}n^{\frac {i}{g}}\leq {\frac {g}{n}}n^{\frac {g-1}{g}}=gn^{-{\frac {1}{g}}}=o(1).}
Thus for sufficiently large n, property 1 holds with a probability of more than 1/2.
Property 2. G contains no independent set of size
{\displaystyle \lceil {\tfrac {n}{2k}}\rceil }
Proof. Let Y be the size of the largest independent set in G. Clearly, we have
{\displaystyle \Pr(Y\geq y)\leq {n \choose y}(1-p)^{\frac {y(y-1)}{2}}\leq n^{y}e^{-{\frac {py(y-1)}{2}}}=e^{-{\frac {y}{2}}\cdot (py-2\ln n-p)}=o(1),}
{\displaystyle y=\left\lceil {\frac {n}{2k}}\right\rceil .}
Thus, for sufficiently large n, property 2 holds with a probability of more than 1/2.
For sufficiently large n, the probability that a graph from the distribution has both properties is positive, as the events for these properties cannot be disjoint (if they were, their probabilities would sum up to more than 1).
Here comes the trick: since G has these two properties, we can remove at most n/2 vertices from G to obtain a new graph G′ on
{\displaystyle n'\geq n/2}
vertices that contains only cycles of length at least g. We can see that this new graph has no independent set of size
{\displaystyle \left\lceil {\frac {n'}{k}}\right\rceil }
. G′ can only be partitioned into at least k independent sets, and, hence, has chromatic number at least k.
This result gives a hint as to why the computation of the chromatic number of a graph is so difficult: even when there are no local reasons (such as small cycles) for a graph to require many colors the chromatic number can still be arbitrarily large.
Alon, Noga; Spencer, Joel H. (2000). The probabilistic method (2ed). New York: Wiley-Interscience. ISBN 0-471-37046-0.
Erdős, P. (1959). "Graph theory and probability". Can. J. Math. 11: 34–38. doi:10.4153/CJM-1959-003-9. MR 0102081.
Erdős, P. (1961). "Graph theory and probability, II". Can. J. Math. 13: 346–352. CiteSeerX 10.1.1.210.6669. doi:10.4153/CJM-1961-029-9. MR 0120168.
J. Matoušek, J. Vondrak. The Probabilistic Method. Lecture notes.
Alon, N and Krivelevich, M (2006). Extremal and Probabilistic Combinatorics
Retrieved from "https://en.wikipedia.org/w/index.php?title=Probabilistic_method&oldid=1066963110"
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Introducing RcppArrayFire | R-bloggers
Introducing RcppArrayFire
The RcppArrayFire package provides an interface from R to and from the ArrayFire library, an open source library that can make use of GPUs and other hardware accelerators via CUDA or OpenCL.
The official R bindings expose ArrayFire data structures as S4 objects in R, which would require a large amount of code to support all the methods defined in ArrayFire’s C/C++ API. RcppArrayFire instead, which is derived from RcppFire by Kazuki Fukui, follows the lead of packages like RcppArmadillo or RcppEigen to provide seamless communication between R and ArrayFire at the C++ level.
Please note that RcppArrayFire is developed and tested on Linux systems. There is preliminary support for Mac OS X.
In order to use RcppArrayFire you will need the ArrayFire library and header files. While ArrayFire has been packaged for Debian, I currently prefer using upstream’s binary installer or building from source.
RcppArrayFire is not on CRAN, but you can install the current version via drat:
#install.packages("drat")
install.packages("RcppArrayFire")
If you have installed ArrayFire in a non-standard directory, you have to use the configure argument --with-arrayfire, e.g.:
install.packages("RcppArrayFire", configure.args = "--with-arrayfire=/opt/arrayfire-3")
Let’s look at the classical example of calculating
\pi
via simulation. The basic idea is to generate a large number of random points within the unit square. An approximation for
\pi
can then be calculated from the ratio of points within the unit circle to the total number of points. A vectorized implementation in R might look like this:
piR <- function(N) {
system.time(cat("pi ~= ", piR(10^6), "\n"))
pi ~= 3.13999
A simple way to use C++ code in R is to use the inline package or cppFunction() from Rcpp, which are both possible with RcppArrayFire. An implementation in C++ using ArrayFire might look like this:
double piAF (const int N) {
array x = randu(N, f32);
array y = randu(N, f32);
return 4.0 * sum<float>(sqrt(x*x + y*y) < 1.0) / N;
Rcpp::cppFunction(code = src, depends = "RcppArrayFire", includes = "using namespace af;")
RcppArrayFire::arrayfire_set_seed(42)
cat("pi ~= ", piAF(10^6), "\n") # also used for warm-up
pi ~= 3.1451
system.time(piAF(10^6))
The syntax is almost identical. Besides the need for using types and a different function name when generating random numbers, the argument f32 to randu as well as the float type catches the eye. These instruct ArrayFire to use single precision floats, since not all devices support double precision floating point numbers. If you want and can to use double precision, you have to specify f64 and double.
The results are not the same since ArrayFire uses a different random number generator.
The speed-up can be quite impressive. However, the first invocation of a function is often not as fast as expected due to the just-in-time compilation used by ArrayFire. This can be circumvented by using a warm-up call with (normally) fewer computations.
Up to now we have only considered simple types like double or int as function parameters and return values. However, we can also use arrays. Consider the case of an European put option that was recently handled with R, Rcpp and RcppArmadillo. The Armadillo based function from this post reads:
This function can be applied to a range of spot prices:
Porting this code to RcppArrayFire is straight forward:
#include <RcppArrayFire.h>
// [[Rcpp::depends(RcppArrayFire)]]
using af::array;
using af::log;
using af::erfc;
array normcdf(array x) {
return erfc(-x / sqrt(2.0)) / 2.0;
array put_option_pricer_af(RcppArrayFire::typed_array<f32> s, double k, double r,
double y, double t, double sigma) {
array d1 = (log(s / k) + (r - y + sigma * sigma / 2.0) * t) / (sigma * sqrt(t));
array d2 = d1 - sigma * sqrt(t);
return normcdf(-d2) * k * exp(-r * t) - s * exp(-y * t) * normcdf(-d1);
Compared with the implementations in R, Rcpp and RcppArmadillo the syntax is again almost the same. One exception is that ArrayFire does not contain a function for the cumulative normal distribution function. However, the closely related error function is available.
Since an object of type af::array can contain different data types, the templated wrapper class RcppArrayFire::typed_array<> is used to indicate the desired data type when converting from R to C++. Again single precision floats are used with ArrayFire, which leads to differences of the order
10^{-6}
compared to the results from R, Rcpp and RcppArmadillo:
put_option_pricer_af(s = 55:60, 60, .01, .02, 1, .05)
The reason to use hardware accelerators is of course the quest for increased performance. How does ArrayFire fare in this respect? Using the same benchmark as in the R, Rcpp and RcppArmadillo comparison:
rbenchmark::benchmark(Arma = put_option_pricer_arma(s, 60, .01, .02, 1, .05),
AF = put_option_pricer_af(s, 60, .01, .02, 1, .05),
2 AF 100 0.489 1.000
Here a Nvidia GeForce GT 1030 is used together with ArrayFire’s CUDA backend. With a build-in Intel HD Graphics 520 using the OpenCL backend the ArrayFire solution is about 6 times faster. Even without a high performance GPU the performance boost from using ArrayFire can be quite impressive. However, the results change dramatically, if fewer options are evaluated:
s <- matrix(seq(0, 100, by = 1), ncol = 1)
# use more replications to get run times of more than 10 ms
replications = 1000)[,1:4]
1 Arma 1000 0.007 1.000
2 AF 1000 0.088 12.571
But is it realistic to process
10^6
options at once? Probably not in the way used in the benchmark where only the spot price is allowed to vary. However, one can alter the function to process not only arrays of spot prices but also arrays of strikes, risk free rates etc.:
using af::sqrt; // arrayfire function instead of standard function
using af::exp; // arrayfire function instead of standard function
array put_option_pricer_af(RcppArrayFire::typed_array<f32> s,
RcppArrayFire::typed_array<f32> k,
RcppArrayFire::typed_array<f32> r,
RcppArrayFire::typed_array<f32> y,
RcppArrayFire::typed_array<f32> t,
RcppArrayFire::typed_array<f32> sigma) {
Note that ArrayFire does not recycle elements if arrays with non-matching dimensions are combined. In this particular case this means that all arrays must have the same length. One can ensure that by using a data frame for the values:
# 1000 * 21 * 3 * 3 * 3 * 3 = 1,701,000 different options
options <- expand.grid(
s = rnorm(1000, mean = 60, sd = 20),
k = 50:70,
r = c(0.01, 0.005, 0.02),
t = c(1, 0.5, 2),
sigma = c(0.05, 0.025, 0.1)
head(within(options,
p <- put_option_pricer_af(s, k, r, y, t, sigma)))
s k r y t sigma p
1 87.4192 50 0.01 0.02 1 0.05 7.45937e-29
2 48.7060 50 0.01 0.02 1 0.05 2.09402e+00
The ArrayFire library provides a convenient way to use hardware accelerators without the need to write low-level OpenCL or CUDA code. The C++ syntax is actually quite similar to properly vectorized R code. The RcppArrayFire package makes this available to useRs. However, one still has to be careful: using hardware accelerators is not a “silver bullet” due to the inherent memory transfer overhead.
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Abelian group - Citizendium
In the mathematical field of abstract algebra, an abelian group is a type of group in which the group operation is commutative – abelian groups are also known as commutative groups. Abelian groups are named for the mathematician Niels Henrik Abel; even so, most modern authors leave the term "abelian" uncapitalized.
Many common number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers are abelian groups with the group operation being addition. In contrast, symmetry groups and permutation groups, which describe the symmetry of a figure and the ways to rearrange the elements in a list respectively, are often non-commutative. Symmetry groups and permutation groups consist of maps; a group consisting of maps is commutative if and only if the equality
{\displaystyle \scriptstyle f\circ g=g\circ f}
(it means
{\displaystyle f(g(x))=g(f(x))}
for all x) holds for all maps f, g in the group.
Computations with an abelian group operation are usually easier than computations with non-abelian operations because one is allowed to rearrange the group elements in the computation and collect "like terms". The structure of abelian groups is generally easier both to obtain and to describe than that of non-abelian groups. When an abelian group is also finitely generated, the possible group structure is exceptionally simple, being described by the fundamental theorem of finitely generated abelian groups.
Retrieved from "https://citizendium.org/wiki/index.php?title=Abelian_group&oldid=466717"
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Craig is practicing his baseball pitching. He kept track of the speed of each of his throws yesterday, and made the histogram at right.
Can you tell the speed of Craig’s fastest pitch? Explain.
When looking at the intervals of the fastest pitch speeds, can you tell the exact mph of his pitches or do the bars just tell you how many pitches were thrown in a certain mph range?
Between what speeds does Craig usually pitch?
Examine the graph around the tallest bars to see the range of speed where most of Craig's throws lie.
Which bar frequencies add up to more than half of his total throws?
Which speed intervals do these bars represent?
50\text{-}55
50\text{-}65
Based on this data, what is the probability that Craig will pitch the ball between
70
75
miles per hour?
How many of Craig's pitches were between
70
75
mph? How many pitches did Craig throw in total?
\frac{2}{16}=\frac{1}{8}
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Dobson, Matthew ; Luskin, Mitchell
The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O(
h
) in the discrete
{\ell }^{\infty }
{w}^{1,1}
norms where
h
is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O(
h
) at distance O(
h|logh|
) in the atomistic region and distance O(
h
) in the continuum region. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete
{\ell }^{\infty }
{w}^{1,p}
norms.
Classification : 65Z05, 70C20
Mots clés : quasicontinuum, atomistic to continuum, ghost force
author = {Dobson, Matthew and Luskin, Mitchell},
title = {An analysis of the effect of ghost force oscillation on quasicontinuum error},
AU - Dobson, Matthew
AU - Luskin, Mitchell
TI - An analysis of the effect of ghost force oscillation on quasicontinuum error
Dobson, Matthew; Luskin, Mitchell. An analysis of the effect of ghost force oscillation on quasicontinuum error. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) no. 3, pp. 591-604. doi : 10.1051/m2an/2009007. http://www.numdam.org/articles/10.1051/m2an/2009007/
[1] M. Arndt and M. Luskin, Goal-oriented atomistic-continuum adaptivity for the quasicontinuum approximation. Int. J. Mult. Comp. Eng. 5 (2007) 407-415.
[2] M. Arndt and M. Luskin, Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model. Multiscale Model. Simul. 7 (2008) 147-170. | MR 2399541 | Zbl 1160.82313
[3] M. Arndt and M. Luskin, Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model. Comp. Meth. App. Mech. Eng. 197 (2008) 4298-4306. | MR 2463663
[4] S. Badia, M.L. Parks, P.B. Bochev, M. Gunzburger and R.B. Lehoucq, On atomistic-to-continuum (AtC) coupling by blending. Multiscale Model. Simul. 7 (2008) 381-406. | MR 2399551 | Zbl 1160.65338
[5] X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN 39 (2005) 797-826. | Numdam | MR 2165680
[6] W. Curtin and R. Miller, Atomistic/continuum coupling in computational materials science. Model. Simul. Mater. Sc. 11 (2003) R33-R68.
[7] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum method. ESAIM: M2AN 42 (2008) 113-139. | Numdam | MR 2387424 | Zbl 1140.74006
[8] W. E and P. Ming. Analysis of the local quasicontinuum method, in Frontiers and Prospects of Contemporary Applied Mathematics, T. Li and P. Zhang Eds., Higher Education Press, World Scientific (2005) 18-32. | MR 2249291
[9] W. E, J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115.
[11] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675 (electronic). | MR 1954960 | Zbl 1010.74003
[12] P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal. 45 (2007) 313-332. | MR 2285857
[13] R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des. 9 (2002) 203-239.
[14] R. Miller, L. Shilkrot and W. Curtin. A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films. Acta Mater. 52 (2003) 271-284.
[15] P. Ming and J.Z. Yang, Analysis of a one-dimensional nonlocal quasicontinuum method. Preprint. | Zbl 1177.74169
[16] J.T. Oden, S. Prudhomme, A. Romkes and P. Bauman, Multi-scale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comput. 28 (2006) 2359-2389. | MR 2272265 | Zbl 1126.74006
[17] C. Ortner and E. Süli, A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Research Report NA-06/13, Oxford University Computing Laboratory (2006).
[18] C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension. ESAIM: M2AN 42 (2008) 57-91. | Numdam | MR 2387422 | Zbl 1139.74004
[20] S. Prudhomme, P.T. Bauman and J.T. Oden, Error control for molecular statics problems. Int. J. Mult. Comp. Eng. 4 (2006) 647-662.
[21] D. Rodney and R. Phillips, Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett. 82 (1999) 1704-1707.
[22] V. Shenoy, R. Miller, E. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mech. Phys. Solids 47 (1999) 611-642. | MR 1675219 | Zbl 0982.74071
[23] T. Shimokawa, J. Mortensen, J. Schiotz and K. Jacobsen, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic regions. Phys. Rev. B 69 (2004) 214104.
[24] G. Strang and G. Fix, Analysis of the Finite Elements Method. Prentice Hall (1973). | MR 443377 | Zbl 0356.65096
[25] E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A 73 (1996) 1529-1563.
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Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces
15 January 2009 Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces
Frédéric Bourgeois, Alexandru Oancea
Frédéric Bourgeois,1 Alexandru Oancea2
2Université Louis Pasteur
Duke Math. J. 146(1): 71-174 (15 January 2009). DOI: 10.1215/00127094-2008-062
We define Floer homology for a time-independent or autonomous Hamiltonian on a symplectic manifold with contact-type boundary under the assumption that its
1
-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between the linearized contact homology of a fillable contact manifold and the symplectic homology of its filling
Frédéric Bourgeois. Alexandru Oancea. "Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces." Duke Math. J. 146 (1) 71 - 174, 15 January 2009. https://doi.org/10.1215/00127094-2008-062
Frédéric Bourgeois, Alexandru Oancea "Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces," Duke Mathematical Journal, Duke Math. J. 146(1), 71-174, (15 January 2009)
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EUDML | A lambda-graph system for the Dyck shift and its -groups. EuDML | A lambda-graph system for the Dyck shift and its -groups.
A lambda-graph system for the Dyck shift and its
K
Krieger, Wolfgang; Matsumoto, Kengo
Krieger, Wolfgang, and Matsumoto, Kengo. "A lambda-graph system for the Dyck shift and its -groups.." Documenta Mathematica 8 (2003): 79-96. <http://eudml.org/doc/50513>.
@article{Krieger2003,
author = {Krieger, Wolfgang, Matsumoto, Kengo},
keywords = {Cantor horizon subshift; Shannon graph; -graph system; Dyck shift; -groups; Bowen-Franks groups; -graph system; -groups},
title = {A lambda-graph system for the Dyck shift and its -groups.},
AU - Krieger, Wolfgang
TI - A lambda-graph system for the Dyck shift and its -groups.
KW - Cantor horizon subshift; Shannon graph; -graph system; Dyck shift; -groups; Bowen-Franks groups; -graph system; -groups
Cantor horizon subshift, Shannon graph,
\lambda
-graph system, Dyck shift,
K
-groups, Bowen-Franks groups,
\lambda
-graph system,
K
Articles by Krieger
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Logic gate - formulasearchengine
{{#invoke:main|main}} To build a functionally complete logic system, relays, valves (vacuum tubes), or transistors can be used. The simplest family of logic gates using bipolar transistors is called resistor-transistor logic (RTL). Unlike simple diode logic gates (which do not have a gain element), RTL gates can be cascaded indefinitely to produce more complex logic functions. RTL gates were used in early integrated circuits. For higher speed and better density, the resistors used in RTL were replaced by diodes resulting in diode-transistor logic (DTL). Transistor-transistor logic (TTL) then supplanted DTL. As integrated circuits became more complex, bipolar transistors were replaced with smaller field-effect transistors (MOSFETs); see PMOS and NMOS. To reduce power consumption still further, most contemporary chip implementations of digital systems now use CMOS logic. CMOS uses complementary (both n-channel and p-channel) MOSFET devices to achieve a high speed with low power dissipation.
There are two sets of symbols for elementary logic gates in common use, both defined in ANSI/IEEE Std 91-1984 and its supplement ANSI/IEEE Std 91a-1991. The "distinctive shape" set, based on traditional schematics, is used for simple drawings, and derives from MIL-STD-806 of the 1950s and 1960s. It is sometimes unofficially described as "military", reflecting its origin. The "rectangular shape" set, based on ANSI Y32.14 and other early industry standards, as later refined by IEEE and IEC, has rectangular outlines for all types of gate and allows representation of a much wider range of devices than is possible with the traditional symbols.[3] The IEC standard, IEC 60617-12, has been adopted by other standards, such as EN 60617-12:1999 in Europe and BS EN 60617-12:1999 in the United Kingdom.
A third style of symbols was in use in Europe and is still preferred by some, see the column "DIN 40700" in the table in the German Wikipedia.
{\displaystyle A\cdot B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle A+B}
{\displaystyle {\overline {A}}}
or ~
{\displaystyle A}
{\displaystyle {\overline {A\cdot B}}}
{\displaystyle A|B}
{\displaystyle {\overline {A+B}}}
{\displaystyle A-B}
{\displaystyle A\oplus B}
{\displaystyle {\overline {A\oplus B}}}
{\displaystyle {A\odot B}}
Two more gates are (1) the exclusive-OR or XOR function and (2) its complement, the exclusive-NOR or XNOR or EQV (equivalent) function. The two input exclusive-OR is true only when the two input values are different, false if they are equal, regardless of the value. If there are more than two inputs, the gate generates a true at its output if the number of trues at its input is odd ([1]). In practice, these gates are built from combinations of simpler logic gates.
This leads to an alternative set of symbols for basic gates that use the opposite core symbol (AND or OR) but with the inputs and outputs negated. Use of these alternative symbols can make logic circuit diagrams much clearer and help to show accidental connection of an active high output to an active low input or vice-versa. Any connection that has logic negations at both ends can be replaced by a negationless connection and a suitable change of gate or vice-versa. Any connection that has a negation at one end and no negation at the other can be made easier to interpret by instead using the De Morgan equivalent symbol at either of the two ends. When negation or polarity indicators on both ends of a connection match, there is no logic negation in that path (effectively, bubbles "cancel"), making it easier to follow logic states from one symbol to the next. This is commonly seen in real logic diagrams - thus the reader must not get into the habit of associating the shapes exclusively as OR or AND shapes, but also take into account the bubbles at both inputs and outputs in order to determine the "true" logic function indicated.
Logic gates can also be used to store data. A storage element can be constructed by connecting several gates in a "latch" circuit. More complicated designs that use clock signals and that change only on a rising or falling edge of the clock are called edge-triggered "flip-flops". The combination of multiple flip-flops in parallel, to store a multiple-bit value, is known as a register. When using any of these gate setups the overall system has memory; it is then called a sequential logic system since its output can be influenced by its previous state(s).
The binary number system was refined by Gottfried Wilhelm Leibniz (published in 1705) and he also established that by using the binary system, the principles of arithmetic and logic could be combined. In an 1886 letter, Charles Sanders Peirce described how logical operations could be carried out by electrical switching circuits.[7] Eventually, vacuum tubes replaced relays for logic operations. Lee De Forest's modification, in 1907, of the Fleming valve can be used as AND logic gate. Ludwig Wittgenstein introduced a version of the 16-row truth table as proposition 5.101 of Tractatus Logico-Philosophicus (1921). Walther Bothe, inventor of the coincidence circuit, got part of the 1954 Nobel Prize in physics, for the first modern electronic AND gate in 1924. Konrad Zuse designed and built electromechanical logic gates for his computer Z1 (from 1935–38). Claude E. Shannon introduced the use of Boolean algebra in the analysis and design of switching circuits in 1937. Active research is taking place in molecular logic gates.
In principle any method that leads to a gate that is functionally complete (for example, either a NOR or a NAND gate) can be used to make any kind of digital logic circuit. Note that the use of 3-state logic for bus systems is not needed, and can be replaced by digital multiplexers.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Logic_gate&oldid=220070"
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Earnings growth of Mexican immigrants: new versus traditional destinations | IZA Journal of Development and Migration | Full Text
\begin{array}{l}\mathit{Ln}{\left(\mathit{\text{Wage}}\right)}_{\mathit{ijt}}={X}_{\mathit{it}}\beta +{Z}_{\mathit{pt}}\gamma +\sum _{m=1}^{M}{\alpha }_{m}*\mathit{YS}{I}_{\mathit{im}}+\sum _{m=1}^{M}{\alpha }_{\mathit{mh}}*\left(\mathit{YS}{I}_{\mathit{im}}*\mathit{NH}\right)\\ \phantom{\rule{6.6em}{0ex}}+\sum _{m=1}^{M}{\alpha }_{\mathit{ml}}*\left(\mathit{YS}{I}_{\mathit{im}}*\mathit{NL}\right)+\\ {\eta }_{t}+{\lambda }_{k}+{\sigma }_{j-\left(t-k\right)}+{u}_{\mathit{ijkt}}\\ \mathrm{k}=1980‒1989,1990‒2000,2001‒2009\phantom{\rule{1em}{0ex}}\left(\mathit{\text{period}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{arrival}}\right)\\ \mathrm{t}=2001,....,2009\phantom{\rule{1.75em}{0ex}}\left(\mathit{\text{year}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{survey}}\right)\\ {\mathrm{YSI}}_{\mathrm{m}}=0‒3,3‒7,\phantom{\rule{0.5em}{0ex}}7‒11,11‒15,15‒20,\phantom{\rule{0.5em}{0ex}}20‒29\phantom{\rule{0.25em}{0ex}}\mathit{\text{years}}\phantom{\rule{0.25em}{0ex}}\left(\mathit{\text{years}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{since}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{immigration}}\right)\end{array}
\begin{array}{l}\mathit{Ln}{\left(\mathit{\text{Wage}}\right)}_{\mathit{ijt}}={\pi }_{i}+{X}_{\mathit{it}}\stackrel{˜}{\beta }+{Z}_{\mathit{pt}}\stackrel{˜}{\gamma }+{\stackrel{˜}{\delta }}_{j}+{\stackrel{˜}{\eta }}_{t}\\ \phantom{\rule{7em}{0ex}}+\sum _{m=1}^{M}\stackrel{˜}{\alpha }{2}_{\mathit{mt}}\left(\mathit{YS}{I}_{i\left(t-1\right)m}*\mathit{\text{YEAR}}_T*\mathit{Trad}\right)+\\ \phantom{\rule{7em}{0ex}}\sum _{m=1}^{M}\stackrel{˜}{\alpha }{2}_{\mathit{mh}}\left(\mathit{YS}{I}_{i\left(t-1\right)\mathit{mh}}*\mathit{\text{YEAR}}_T*\mathit{NH}\right)\\ \phantom{\rule{7em}{0ex}}+\sum _{m=1}^{M}\stackrel{˜}{\alpha }{2}_{\mathit{ml}}\left(\mathit{YS}{I}_{i\left(t-1\right)m}*\mathit{\text{YEAR}}_T*\mathit{NL}\right)+{u}_{\mathit{ijkt}}\end{array}
There are three things to note about Equation (2). First, the equation includes person-specific fixed effects (π i ). Second, each person is in the sample for two periods: t-1 and t, and the value of years since immigration in the US (YSI) is fixed at year t-1. Third, we allow the effect of YSI to differ by whether the observation is from year t-1 or t. In Equation (2) this choice is reflected by the interaction term (YSI i(t-1)m * YEAR_T ). The parameters of interest are:
\stackrel{˜}{\alpha }{2}_{\mathit{mt}}
\stackrel{˜}{\alpha }{2}_{\mathit{mh}}
\stackrel{˜}{\alpha }{2}_{\mathit{ml}}
, which measure changes in earnings of Mexican immigrants, between t-1 and t, at the traditional, new high-growth and low-growth destinations, respectively. Note that the main effect of years-since-immigration drops out of the model because in the longitudinal analysis this variable is time invariant for a specific immigrant.
{L}_{\mathit{it}}={D}_{\mathit{it}}+I{M}_{\mathit{it}}+{E}_{\mathit{it}}+N{M}_{\mathit{it}}
N{M}_{\mathit{st}}={L}_{\mathit{st}}-{D}_{\mathit{st}}-I{M}_{\mathit{st}}
{\stackrel{^}{E}}_{\mathit{it}}={L}_{\mathit{it}}-{\stackrel{^}{D}}_{\mathit{it}}-I{\stackrel{^}{M}}_{\mathit{it}}-N{\stackrel{^}{M}}_{\mathit{it}}
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Homotopy classification of twisted complex projective spaces of dimension 4
April, 2005 Homotopy classification of twisted complex projective spaces of dimension 4
We study the existence problem of a
2n
dimensional Poincaré complex whose homology is isomorphic to that of
n
dimensional complex projective space when
n=4
Juno MUKAI. Kohhei YAMAGUCHI. "Homotopy classification of twisted complex projective spaces of dimension 4." J. Math. Soc. Japan 57 (2) 461 - 489, April, 2005. https://doi.org/10.2969/jmsj/1158242066
Keywords: homotopy type , Poincaré complex , Whitehead product
Juno MUKAI, Kohhei YAMAGUCHI "Homotopy classification of twisted complex projective spaces of dimension 4," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 57(2), 461-489, (April, 2005)
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Reorder eigenvalues in QZ factorization - MATLAB ordqz - MathWorks Italia
ordqz
Reorder QZ Factorization
AASBBSQSZS
Quasitriangular
Reorder eigenvalues in QZ factorization
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,select)
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,keyword)
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,clusters)
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,select) reorders the QZ factorization Q*A*Z = AA and Q*B*Z = BB produced by [AA,BB,Q,Z] = qz(A,B) and returns the reordered matrix pair (AAS,BBS) along with orthogonal matrices (QS,ZS), such that QS*A*ZS = AAS and QS*B*ZS = BBS.
In this reordering, the selected cluster of eigenvalues appears in the leading (upper left) diagonal blocks of the quasitriangular pair (AAS,BBS). The leading columns of ZS span the corresponding invariant subspace. The logical vector select specifies the selected cluster as e(select), where e = ordeig(AA,BB).
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,keyword) sets the selected cluster to include all eigenvalues in the region specified by keyword.
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,clusters) reorders multiple clusters simultaneously. ordqz sorts the specified clusters in descending order along the diagonal of (AAS,BBS), with the cluster of highest index appearing in the upper left corner.
Compute the QZ factorization of a pair of matrices, and then reorder the factors according to a specified ordering of the eigenvalues.
Find the QZ factorization, or generalized Schur decomposition, of a pair of matrices A and B. This decomposition results in the factors
\mathrm{AA}=\mathrm{QAZ}
\mathrm{BB}=\mathrm{QBZ}
14.5272 -2.3517 8.5757 -0.2350 -1.4432
0 -19.7471 2.1824 4.5417 7.2059
0 0 -17.9538 8.9292 -9.6961
0 0 0.0210 0.1006 -0.1341
0 0 0 0.0623 -1.1380
Since AA and BB are triangular, use ordeig to extract the eigenvalues from the diagonal blocks of AA and BB.
Separate the eigenvalues into clusters, with the real positive eigenvalues (
\mathit{e}>0
) forming the leading cluster. Reorder the matrices AA, BB, Q, and Z according to this ordering of the eigenvalues.
[AAS,BBS,QS,ZS] = ordqz(AA,BB,Q,Z,'rhp')
AAS = 5×5
0 21.7128 -19.1784 -1.8380 9.1187
0 0 60.3083 8.4452 -6.4304
0 0 0 -18.2081 3.3783
BBS = 5×5
QS = 5×5
ZS = 5×5
Examine the new eigenvalue order.
E2 = ordeig(AAS,BBS)
AA, BB — Matrix factors
Matrix factors, specified as the matrices returned by [AA,BB,Q,Z] = qz(A,B). These matrices satisfy Q*A*Z = AA and Q*B*Z = BB. For complex matrices, AA and BB are triangular.
If AA and BB do not form a valid QZ decomposition, then ordqz does not produce an error and returns incorrect results.
Q, Z — Unitary matrices
Unitary matrices, specified as the matrices returned by [AA,BB,Q,Z] = qz(A,B). These matrices satisfy Q*A*Z = AA and Q*B*Z = BB.
select — Cluster selector
Cluster selector, specified as a logical vector with length equal to the number of generalized eigenvalues. The generalized eigenvalues appear along the diagonal of AA-λ*BB.
keyword — Eigenvalue region keyword
'lhp' | 'rhp' | 'udi' | 'udo'
Eigenvalue region keyword, specified as one of the options in this table.
(e = ordeig(AA,BB))
'lhp'
Left-half plane (real(e) < 0)
'rhp'
Right-half plane (real(e) >= 0)
'udi'
Interior of unit disk (abs(e) < 1)
Exterior of unit disk (abs(e) >= 1)
clusters — Cluster indices
Cluster indices, specified as a vector of positive integers with length equal to the number of eigenvalues. clusters assigns each eigenvalue returned by e = ordeig(AA,BB) to a different cluster. All eigenvalues with the same index value in clusters form one cluster.
Example: ordqz(AA,BB,Q,Z,[1 1 2 3 3]) groups five eigenvalues into three clusters.
AAS, BBS, QS, ZS — Reordered matrices
Reordered matrices, returned as matrices that satisfy QS*A*ZS = AAS and QS*B*ZS = BBS.
QS and ZS are unitary, while AAS is quasitriangular and BBS is triangular.
An upper quasitriangular matrix can result from the Schur decomposition or generalized Schur (QZ) decomposition of real matrices. These matrices are block upper triangular, with 1-by-1 and 2-by-2 blocks along the diagonal.
The eigenvalues of these diagonal blocks are also the eigenvalues of the matrix. The 1-by-1 blocks correspond to real eigenvalues, and the 2-by-2 blocks correspond to complex conjugate eigenvalue pairs.
If AA has complex conjugate pairs (nonzero elements on the subdiagonal), then you should move the pair to the same cluster. Otherwise, ordqz acts to keep the pair together:
If select is not the same for two eigenvalues in a conjugate pair, then ordqz treats both as selected.
If clusters is not the same for two eigenvalues in a conjugate pair, then ordqz treats both as part of the cluster with larger index.
[1] Kressner, Daniel. “Block Algorithms for Reordering Standard and Generalized Schur Forms.” ACM Transactions on Mathematical Software 32, no. 4 (December 2006): 521–532. https://doi.org/10.1145/1186785.1186787.
ordeig | ordschur | qz
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Please answer this : 21 If the area of a circle of radius 7 cm is - Maths - Practical Geometry - 11947001 | Meritnation.com
21. If the area of a circle of radius 7 cm is 154 sq cm, what is the area of the shaded portion of the figure shown below ?
A. 9 sq cm
B. 10.5 sq cm
C. 12.25 sq cm
D. 16.5 sq cm
From the figure we can see that , OA=OC=7 cm \left[ radius of the circle\right]\phantom{\rule{0ex}{0ex}}Also, AB=OC=OA=BC=7cm\phantom{\rule{0ex}{0ex}}Now area of circle={\mathrm{\pi r}}^{2} , \mathrm{where} \mathrm{r} \mathrm{is} \mathrm{the} \mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{circle}.\phantom{\rule{0ex}{0ex}}=\frac{22}{7}×{7}^{2}\phantom{\rule{0ex}{0ex}}=154 {\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{Now}, \mathrm{area} \mathrm{of} \mathrm{square} \mathrm{OABC} \left(\mathrm{being} \mathrm{all} \mathrm{sides} =7 \mathrm{cm}\right)\phantom{\rule{0ex}{0ex}}={\left(\mathrm{side}\right)}^{2}\phantom{\rule{0ex}{0ex}}={7}^{2}\phantom{\rule{0ex}{0ex}}=49 {\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{Now} \mathrm{area} \mathrm{of} \mathrm{sector} \mathrm{OCEA}=\frac{\mathrm{\theta }}{360°}×{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}=\frac{90°}{360°}×154 \left(\mathrm{\theta }=90° \mathrm{as} \mathrm{assuming} \mathrm{OCEA} \mathrm{is} \mathrm{a} \mathrm{quadrant} \mathrm{made} \mathrm{by} \mathrm{two} \mathrm{radiuss} \mathrm{at} \mathrm{right} \mathrm{angles} \mathrm{connecting} \mathrm{arc}\right)\phantom{\rule{0ex}{0ex}}=\frac{77}{2} {\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{So}, \mathrm{area} \mathrm{of} \mathrm{shaded} \mathrm{region}=\mathrm{area} \mathrm{of} \mathrm{square} \mathrm{OABC}- \mathrm{area} \mathrm{of} \mathrm{sector} \mathrm{OCEA}\phantom{\rule{0ex}{0ex}}=\left[49-\frac{77}{2}\right] {\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=10.5 {\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}
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Exergy Analysis of Condensation of a Binary Mixture With One Noncondensable Component in a Shell and Tube Condenser | J. Heat Transfer | ASME Digital Collection
e-mail: yousef.haseli@mycampus.uoit.ca
I. Dincer,
e-mail: greg.naterer@uoit.ca
Haseli, Y., Dincer, I., and Naterer, G. F. (June 2, 2008). "Exergy Analysis of Condensation of a Binary Mixture With One Noncondensable Component in a Shell and Tube Condenser." ASME. J. Heat Transfer. August 2008; 130(8): 084504. https://doi.org/10.1115/1.2909610
The exergy (second-law) efficiency is formulated for a condensation process in a shell and one-path tube exchanger for a fixed control volume. The exergy efficiency
ηex
is expressed as a function of the inlet and outlet temperatures and mass flow rates of the streams. This analysis is utilized to assess the trend of local exergy efficiency along the condensation path and evaluate its value for the entire condenser, i.e., overall exergy efficiency. The numerical results for an industrial condenser, with a steam-air mixture and cooling water as working fluids, indicate that
ηex
is significantly affected by the inlet cooling water and environment temperatures. Further investigation shows that other performance parameters, such as the upstream mixture temperature, air mass flow rate, and ratio of cooling water mass flow rate to upstream steam mass flow rate, do not have considerable effects on
ηex
. The investigations involve a dimensionless ratio of the temperature difference of the cooling water and environment to the temperature difference of condensation and the environment. Numerical results for various operational conditions enable us to accurately correlate both the local and overall exergy efficiency as linear functions of dimensionless temperature.
condensation, cooling, exergy, mass transfer, pipe flow, exergy efficiency, condensation, steam-air mixture, shell and tube condenser
Condensation, Condensers (steam plant), Cooling, Exergy, Shells, Steam, Temperature, Flow (Dynamics), Water
The Role of Exergy in Energy Policy Making
Evaluation of Heat Transfer and Exergy Loss in a Concentric Double Pipe Exchanger Equipped With Helical Wires
Second Law Analysis on the Heat Transfer of the Horizontal Concentric Tube Heat Exchanger
Second-Law Analysis of a Wet Crossflow Heat Exchanger
Selbaş
Kızılkan
Thermoeconomic Optimization of Subcooled and Superheated Vapor Compression Refrigeration Cycle
Determining the Optimal Configuration of a Heat Exchanger (With a Two-Phase Refrigerant) Using Exergoeconomics
Fundamental of Engineering Thermodynamics
A Calculation Method for Analysis Condensation of a Pure Vapor in the Presence of a Non-Condensable Gas on a Shell and Tube Condenser
, Charlotte, NC, Vol.
Simultaneous Modeling of Heat and Mass Transfer of Steam-Air Mixture on a Shell and Tube Condenser Based on Film Theory
Comparative Study of Using R-410A, R-407C, R-22 and R-134a as Cooling Medium in the Condenser of a Steam Power Plant
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Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping | EMS Press
Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping
We consider the nonlinear hyperbolic equation \begin{align*} u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p} \end{align*} posed in
Q:=(0,\infty )\times \mathbb{R}^{N},
D_{+}^{\alpha }u
% 0<\alpha <1
is a time fractional derivative, with given initial position and velocity
u(0,x)=u_{0}(x)
u_{t}(0,x)=u_{1}(x).
We find the Fujita's exponent which separates in terms of
p,\alpha
N,
the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on
u_{1}(x)
h(x,t)
assuring non-existence of local solutions.
Nasser-edine Tatar, Mokhtar Kirane, Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping. Z. Anal. Anwend. 25 (2006), no. 2, pp. 131–142
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Difference between revisions of "User:Jtoal" - SFU_Public
Difference between revisions of "User:Jtoal"
{| style="background:lightblue; color:black" border="0" cellspacing="10" cellpadding="10" align="right" width="50%"
{| cellspacing="10" cellpadding="10" border="0" width="50%" align="right" style="background: none repeat scroll 0% 0% lightblue; color: black;"
|'''Hello. My name is...
| '''Hello. My name is...'''
Please feel free to contact me via [mailto:jtoal@sfu.ca email] or phone: '''778.268.7243'''
<br> '''Links'''
<br> [[Image:Blue rss.gif]]<rss>http://feeds.feedburner.com/educationbuilding/</rss>
[[Image:Blue rss.gif]]<rss>http://feeds.feedburner.com/educationbuilding/</rss>
[[Image:Jason.gif]]
I'm a designer in the [http://www.lidc.sfu.ca LIDC], with a special interest and focus on the (web)user experience. That means I talk to 'users' (people) to find out what they like and dislike about certain websites. I'll do this formally, (user testing, interviews) and informally, (blog reading/writing, the water cooler!) but always try to generate some useful artifacts or documentation to inform the project process (See ''Service Scenarios'' below).
Training, seminars/workshops on a various computer technologies, (everything from email to 2ndlife)
Online resources for above. Possible topics: (How to podcast, blogging, etc)
Templates for blogs, webCT courses, whatever really
Presentation support. Perhaps something to augment the Vocalization workshop. Includes powerpoint tips and such
Digital Organization - How do you keep all your digital artifacts together and manage them over your academic career?
=== My Sites ===
Education Building (my blog at work) - http://jasontoal.blogs.elinc.ca/ - an effort to blog for sfu educators interested in the use of social software for teaching & learning, and the general goings on around SFU webscape.
=== Service Scenarios ===
**Student Services Screenshots** - A high level view of the various sub sections within the Student services website. Each of these sections represents a different working group and different web development process and strategies. (not to mention timelines and people.) Yet, for the user, these distinctions are inconsequential and it is important that navigation and information be organized across these sites as a whole.
**Student Services Sitemap** - An actual sitemap of all the pages within the first level of the Student Services website. This representation also attempts to show which pages are displayed in which menus.
<br> Co-operative Education Sitemap - a more generic site map style showing strictly a representation of the existing information of the website and how each page is interrelated. You can tell from this diagram that some of the information is buried seven clicks beneath the front page, typically seen as a user experience no no.
'''Recommended Readings'''
<br> '''Recommended Readings'''
http://www.google.com/reader/public/atom/user/03514583726113949065/state/com.google/starred
<rss>http://www.google.com/reader/public/atom/user/03514583726113949065/state/com.google/starred</rss>
SFU Second Life events (also not working)
http://www.google.com/calendar/feeds/gg4ctnork30q2fgdo9nl92vle8%40group.calendar.google.com/public/basic
<rss>http://www.google.com/calendar/feeds/gg4ctnork30q2fgdo9nl92vle8%40group.calendar.google.com/public/basic</rss>
{\displaystyle {\sqrt {2}}}
|
fit(deprecated)/leastmed - Maple Help
Home : Support : Online Help : fit(deprecated)/leastmed
stats[fit, leastmediansquare]
Fit a Curve to data using the least median of squares method
stats[fit, leastmediansquare[vars]](data)
fit[leastmediansquare[vars]](data)
The function leastmediansquare of the subpackage stats[fit, ...] fits a curve to the given data using the method of least median of squares.
The equation to fit will be linear (affine) in the unknown parameters. The equation itself need not be linear. For example, specifying vars to be
[x,y]
implies fitting to the equation
y=a{x}^{2}+bx+c
The well-known least square regression method suffer from leverage points. By adding a single point to the data, one can change the result to an arbitrary extent. The least median of squares method allows one to add up to half the number of points without changing the result. The price to pay for this robustness is a substantial increase in computation cost.
The command with(stats[fit],leastmediansquare) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{stats}\right):
\mathrm{data}≔\mathrm{convert}\left(\mathrm{linalg}[\mathrm{transpose}]\left([[1,3],[2,4],[3,5],[1,2]]\right),\mathrm{listlist}\right):
\mathrm{fit}[\mathrm{leastmediansquare}[[x,y]]]\left(\mathrm{data}\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
This is calculated as follows. For example, pick two points, say [1,3] and [2,4]. Pass a straight line through them. In this case it is y=2+x. For each point compute the square of the distance of the line to the point This gives [0,0,0,sqrt(2)/2]. Find the median through these distances: this gives 0. Now minimize over all possible lines. This is the result.
\mathrm{data}≔\mathrm{convert}\left(\mathrm{linalg}[\mathrm{transpose}]\left([[1,3],[2,4],[3,5],[1,\mathrm{Weight}\left(2,4\right)]]\right),\mathrm{listlist}\right):
\mathrm{fit}[\mathrm{leastmediansquare}[[x,y]]]\left(\mathrm{data}\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}
Rousseeuw, P. J. "Least Median of Squares Regression." Journal of the American Statistical Association, (December 1984): 871-880.
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qr - Maple Help
Home : Support : Online Help : Mathematics : Numerical Computations : Matlab : qr
compute the QR orthogonal-triangular decomposition of a MapleMatrix or MatlabMatrix in MATLAB(R), where X*P = Q*R
qr(X, output=R)
qr(X, output=QR)
qr(X, output=QRP)
return the upper triangular matrix R
return unitary matrix Q and upper triangular R matrix
return Q, R, and permutation matrix P
The qr command computes the QR orthogonal-triangular decomposition of a matrix (either a Maple matrix or a MatlabMatrix) in MATLAB®. When output=QRP, the result is computed where
XP=QR
. When output=QR, the result is computed where
X=QR
The matrix X can be either square or rectangular.
The matrix X is expressed as product of an upper triangular matrix and either a real orthonormal matrix or a complex unitary matrix.
The default if no output option is specified is to return the matrices Q and R.
\mathrm{with}\left(\mathrm{Matlab}\right):
\mathrm{maplematrix_a}≔\mathrm{Matrix}\left([[3,1,3],[1,6,4],[6,7,8],[3,3,7]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{maplematrix_a}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{7}\end{array}]
The QR decomposition of this MapleMatrix is computed and returns Q and R, as follows:
Q,R≔\mathrm{Matlab}[\mathrm{qr}]\left(\mathrm{maplematrix_a}\right)
Q, R :=
[-0.404519917477945468 , 0.418121005003545431 , -0.120768607347027060 , -0.804334137667873206]
[-0.134839972492648424 , -0.903141370807658106 , 0.0315048540905287777 , -0.406400406400609482]
[-0.809039834955890602 , -0.0836242010007090253 , -0.399061485146697980 , 0.423333756667301608]
[-0.404519917477945301 , 0.0501745206004254873 , 0.908389959610246600 , 0.0931334264668064182]
[-7.41619848709566209 -8.09039834955890668 -11.0568777443971715]
[ 0. -5.43557306504609006 -2.67597443202269013]
[ 0. 0. 2.92995143041917760 ]
The QR decomposition returning only the R matrix is as follows:
M≔\mathrm{Matlab}[\mathrm{qr}]\left(\mathrm{maplematrix_a},\mathrm{output}='R'\right)
[-7.41619848709566209 , -8.09039834955890668 , -11.0568777443971715]
[0.0960043149368622339 , -5.43557306504609006 , -2.67597443202269013]
To force the lower triangle entries to zero, use
\mathrm{Matrix}\left(M,\mathrm{shape}=\mathrm{triangular}[\mathrm{upper}]\right)
Note that the R in
\mathrm{output}='R'
is surrounded by quotation marks, since the variable R was assigned previously. QR decomposition returning Q, R, and P matrices is as follows:
Q,R,P≔\mathrm{Matlab}[\mathrm{qr}]\left(\mathrm{maplematrix_a},\mathrm{output}=\mathrm{QRP}\right)
Matlab[chol]
|
(-)-endo-fenchol synthase Wikipedia
(-)-endo-fenchol synthase
(−)-endo-fenchol synthase
In enzymology, a (−)-endo-fenchol synthase (EC 4.2.3.10) is an enzyme that catalyzes the chemical reaction
{\displaystyle \rightleftharpoons }
(−)-endo-fenchol + diphosphate
Thus, the two substrates of this enzyme are geranyl diphosphate and H2O, whereas its two products are (−)-endo-fenchol and diphosphate.
This enzyme belongs to the family of lyases, specifically those carbon-oxygen lyases acting on phosphates. The systematic name of this enzyme class is geranyl-diphosphate diphosphate-lyase [cyclizing, (−)-endo-fenchol-forming]. Other names in common use include (−)-endo-fenchol cyclase, and geranyl pyrophosphate:(−)-endo-fenchol cyclase. This enzyme participates in monoterpenoid biosynthesis.
Croteau R, Miyazaki JH, Wheeler CJ (1989). "Monoterpene biosynthesis: mechanistic evaluation of the geranyl pyrophosphate:(−)-endo-fenchol cyclase from fennel (Foeniculum vulgare)". Arch. Biochem. Biophys. 269 (2): 507–16. doi:10.1016/0003-9861(89)90134-3. PMID 2919880.
Croteau R, Satterwhite DM, Wheeler CJ, Felton NM (1988). "Biosynthesis of monoterpenes. Stereochemistry of the enzymatic cyclization of geranyl pyrophosphate to (−)-endo-fenchol". J. Biol. Chem. 263 (30): 15449–53. doi:10.1016/S0021-9258(19)37609-4. PMID 3170591.
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Some existence results for the Toda system on closed surfaces | EMS Press
Given a compact closed surface
\Sig
, we consider the {\em generalized Toda} system of equations on
\Sig
- \D u_i = \sum_{j=1}^2 \rho_j a_{ij} \left( \frac{h_j e^{u_j}}{\int_\Sig h_j e^{u_j} dV_g} - 1 \right)
i = 1, 2
\rho_1, \rho_2
are real parameters and
h_1, h_2
are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of
\rho_1
\rho_2 < 4 \pi
Andrea Malchiodi, Cheikh Birahim Ndiaye, Some existence results for the Toda system on closed surfaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 4, pp. 391–412
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EUDML | -star products on dual of Lie algebras. EuDML | -star products on dual of Lie algebras.
K
-star products on dual of Lie algebras.
Ben Amar, Nabiha
Ben Amar, Nabiha. "-star products on dual of Lie algebras.." Journal of Lie Theory 13.2 (2003): 329-357. <http://eudml.org/doc/123410>.
@article{BenAmar2003,
author = {Ben Amar, Nabiha},
keywords = {Kontsevich star products},
title = {-star products on dual of Lie algebras.},
AU - Ben Amar, Nabiha
TI - -star products on dual of Lie algebras.
KW - Kontsevich star products
Nabiha Ben Amar, Mouna Chaabouni, Mabrouka Hfaiedh, Kontsevich Deformation Quantization on Lie Algebras
Kontsevich star products
Poisson manifolds; Poisson groupoids and algebroids
Articles by Ben Amar
|
Mogensen–Scott encoding - Wikipedia
Mogensen–Scott encoding
In computer science, Scott encoding is a way to represent (recursive) data types in the lambda calculus. Church encoding performs a similar function. The data and operators form a mathematical structure which is embedded in the lambda calculus.
Whereas Church encoding starts with representations of the basic data types, and builds up from it, Scott encoding starts from the simplest method to compose algebraic data types.
Mogensen–Scott encoding extends and slightly modifies Scott encoding by applying the encoding to Metaprogramming. This encoding allows the representation of lambda calculus terms, as data, to be operated on by a meta program.
3.1 Scott encoding
3.2 Mogensen–Scott encoding
4 Comparison to the Church encoding
Scott encoding appears first in a set of unpublished lecture notes by Dana Scott[1] whose first citation occurs in the book Combinatorial Logic, Volume II.[2] Michel Parigot gave a logical interpretation of and strongly normalizing recursor for Scott-encoded numerals,[3] referring to them as the "Stack type" representation of numbers. Torben Mogensen later extended Scott encoding for the encoding of Lambda terms as data.[4]
Lambda calculus allows data to be stored as parameters to a function that does not yet have all the parameters required for application. For example,
{\displaystyle ((\lambda x_{1}\ldots x_{n}.\lambda c.c\ x_{1}\ldots x_{n})\ v_{1}\ldots v_{n})\ f}
May be thought of as a record or struct where the fields
{\displaystyle x_{1}\ldots x_{n}}
have been initialized with the values
{\displaystyle v_{1}\ldots v_{n}}
. These values may then be accessed by applying the term to a function f. This reduces to,
{\displaystyle f\ v_{1}\ldots v_{n}}
c may represent a constructor for an algebraic data type in functional languages such as Haskell. Now suppose there are N constructors, each with
{\displaystyle A_{i}}
arguments;
{\displaystyle {\begin{array}{c|c|c}{\text{Constructor}}&{\text{Given arguments}}&{\text{Result}}\\\hline ((\lambda x_{1}\ldots x_{A_{1}}.\lambda c_{1}\ldots c_{N}.c_{1}\ x_{1}\ldots x_{A_{1}})\ v_{1}\ldots v_{A_{1}})&f_{1}\ldots f_{N}&f_{1}\ v_{1}\ldots v_{A_{1}}\\((\lambda x_{1}\ldots x_{A_{2}}.\lambda c_{1}\ldots c_{N}.c_{2}\ x_{1}\ldots x_{A_{2}})\ v_{1}\ldots v_{A_{2}})&f_{1}\ldots f_{N}&f_{2}\ v_{1}\ldots v_{A_{2}}\\\vdots &\vdots &\vdots \\((\lambda x_{1}\ldots x_{A_{N}}.\lambda c_{1}\ldots c_{N}.c_{N}\ x_{1}\ldots x_{A_{N}})\ v_{1}\ldots v_{A_{N}})&f_{1}\ldots f_{N}&f_{N}\ v_{1}\ldots v_{A_{N}}\end{array}}}
Each constructor selects a different function from the function parameters
{\displaystyle f_{1}\ldots f_{N}}
. This provides branching in the process flow, based on the constructor. Each constructor may have a different arity (number of parameters). If the constructors have no parameters then the set of constructors acts like an enum; a type with a fixed number of values. If the constructors have parameters, recursive data structures may be constructed.
Let D be a datatype with N constructors,
{\displaystyle \{c_{i}\}_{i=1}^{N}}
, such that constructor
{\displaystyle c_{i}}
has arity
{\displaystyle A_{i}}
Scott encoding[edit]
The Scott encoding of constructor
{\displaystyle c_{i}}
of the data type D is
{\displaystyle \lambda x_{1}\ldots x_{A_{i}}.\lambda c_{1}\ldots c_{N}.c_{i}\ x_{1}\ldots x_{A_{i}}}
Mogensen–Scott encoding[edit]
Mogensen extends Scott encoding to encode any untyped lambda term as data. This allows a lambda term to be represented as data, within a Lambda calculus meta program. The meta function mse converts a lambda term into the corresponding data representation of the lambda term;
{\displaystyle {\begin{aligned}\operatorname {mse} [x]&=\lambda a,b,c.a\ x\\\operatorname {mse} [M\ N]&=\lambda a,b,c.b\ \operatorname {mse} [M]\ \operatorname {mse} [N]\\\operatorname {mse} [\lambda x.M]&=\lambda a,b,c.c\ (\lambda x.\operatorname {mse} [M])\\\end{aligned}}}
The "lambda term" is represented as a tagged union with three cases:
Constructor a - a variable (arity 1, not recursive)
Constructor b - function application (arity 2, recursive in both arguments),
Constructor c - lambda-abstraction (arity 1, recursive).
{\displaystyle {\begin{array}{l}\operatorname {mse} [\lambda x.f\ (x\ x)]\\\lambda a,b,c.c\ (\lambda x.\operatorname {mse} [f\ (x\ x)])\\\lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ \operatorname {mse} [f]\ \operatorname {mse} [x\ x])\\\lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ (\lambda a,b,c.a\ f)\ \operatorname {mse} [x\ x])\\\lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ (\lambda a,b,c.a\ f)\ (\lambda a,b,c.b\ \operatorname {mse} [x]\ \operatorname {mse} [x]))\\\lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ (\lambda a,b,c.a\ f)\ (\lambda a,b,c.b\ (\lambda a,b,c.a\ x)\ (\lambda a,b,c.a\ x)))\end{array}}}
Comparison to the Church encoding[edit]
The Scott encoding coincides with the Church encoding for booleans. Church encoding of pairs may be generalized to arbitrary data types by encoding
{\displaystyle c_{i}}
of D above as[citation needed]
{\displaystyle \lambda x_{1}\ldots x_{A_{i}}.\lambda c_{1}\ldots c_{N}.c_{i}(x_{1}c_{1}\ldots c_{N})\ldots (x_{A_{i}}c_{1}\ldots c_{N})}
compare this to the Mogensen Scott encoding,
{\displaystyle \lambda x_{1}\ldots x_{A_{i}}.\lambda c_{1}\ldots c_{N}.c_{i}x_{1}\ldots x_{A_{i}}}
With this generalization, the Scott and Church encodings coincide on all enumerated datatypes (such as the boolean datatype) because each constructor is a constant (no parameters).
Concerning the practicality of using either the Church or Scott encoding for programming, there is a symmetric trade-off:[5] Church-encoded numerals support a constant-time addition operation and have no better than a linear-time predecessor operation; Scott-encoded numerals support a constant-time predecessor operation and have no better than a linear-time addition operation.
Type definitions[edit]
Church-encoded data and operations on them are typable in system F, as are Scott-encoded data and operations. However, the encoding is significantly more complicated.[6]
The type of the Scott encoding of the natural numbers is the positive recursive type:
{\displaystyle \mu X.\forall R.R\to (X\to R)\to R}
Full recursive types are not part of System F, but positive recursive types can are expressible in System F via the encoding:
{\displaystyle \mu X:G[X]=\forall X:((G[X]\to X)\to X)}
Combining these twos fact yields the System F type of the Scott encoding:
{\displaystyle \forall X.(((\forall R.R\to (X\to R)\to R)\to X)\to X)}
This can be contrasted with the type of the Church encoding:
{\displaystyle \forall X.X\to (X\to X)\to X}
The Church encoding is a second-order type, but the Scott encoding is fourth-order!
^ Scott, Dana, A system of functional abstraction (1968). Lectures delivered at University of California, Berkeley, (1962)
^ Curry, Haskell (1972). Combinatorial Logic, Volume II. North-Holland Publishing Company. ISBN 0-7204-2208-6.
^ Parigot, Michel (1988). "Programming with proofs: a second order type theory". European Symposium on Programming. Lecture Notes in Computer Science. 300: 145–159. doi:10.1007/3-540-19027-9_10. ISBN 978-3-540-19027-1.
^ Mogensen, Torben (1994). "Efficient Self-Interpretation in Lambda Calculus". Journal of Functional Programming. 2 (3): 345–364. doi:10.1017/S0956796800000423.
^ Parigot, Michel (1990). "On the representation of data in lambda calculus". International Workshop on Computer Science Logic. Lecture Notes in Computer Science. 440: 209–321. doi:10.1007/3-540-52753-2_47. ISBN 978-3-540-52753-4.
^ See the note "Types for the Scott numerals" by Martín Abadi, Luca Cardelli and Gordon Plotkin (February 18, 1993).
Stump, A. (2009). Directly reflective meta-programming. Higher-Order and Symbolic Computation, 22, 115-144.
Mogensen, T.Æ. (1992). Efficient Self-Interpretations in lambda Calculus. J. Funct. Program., 2, 345-363.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mogensen–Scott_encoding&oldid=1071212526"
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Compute sequence components (positive, negative, and zero) of three-phase phasor signal - Simulink - MathWorks Nordic
Sequence Analyzer (Phasor)
Compute sequence components (positive, negative, and zero) of three-phase phasor signal
The Sequence Analyzer (Phasor) block computes the three-sequence phasor components (positive sequence u1, negative sequence u2, and zero sequence u0) of a three-phase phasor signal as follows:
\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\\ {u}_{0}\end{array}\right]=\frac{1}{3}\left[\begin{array}{ccc}1& a& {a}^{2}\\ 1& {a}^{2}& a\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]
\alpha
is the complex operator performing a 2
\pi /3
a={e}^{\frac{j2\pi }{3}}
Select the sequence components: Positive (default), Negative, Zero, or all three sequences Positive Negative Zero.
The three-phase phasor signal [ua ub uc].
Returns the magnitude of the selected sequence phasor component.
Returns the angle, in degrees, of the selected sequence phasor component.
The power_PhasorMeasurements example shows the use of the Sequence Analyzer (Phasor) block.
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1-D Wavelet Packet Analysis - MATLAB & Simulink - MathWorks 日本
Starting the Wavelet Packet 1-D Tool
Importing a Signal
Analyzing a Signal
Computing the Best Tree
Compressing a Signal Using Wavelet Packets
Selecting a Threshold for Compression
Compressing a Signal
De-Noising a Signal Using Wavelet Packets
Starting the Wavelet Packet 1–D Tool
We now turn to the Wavelet Packet 1-D tool to analyze a synthetic signal that is the sum of two linear chirps.
From the MATLAB® prompt, type waveletAnalyzer. The Wavelet Analyzer appears.
Click the Wavelet Packet 1-D menu item.
load sumlichr;
In the Wavelet Packets 1-D tool, select File > Import from Workspace > Import Signal. When the Import from Workspace dialog box appears, select the sumlichr variable. Click OK to import the data.
The sumlichr signal is loaded into the Wavelet Packet 1-D tool.
Make the appropriate settings for the analysis. Select the db2 wavelet, level 4, entropy threshold, and for the threshold parameter type 1. Click the Analyze button.
The available entropy types are listed below.
Nonnormalized entropy involving the logarithm of the squared value of each signal sample — or, more formally,
ââ{s}_{i}^{2}\mathrm{log}\left({s}_{i}^{2}\right).
The number of samples for which the absolute value of the signal exceeds a threshold ε.
The concentration in lp norm with 1 ≤ p.
The logarithm of “energy,†defined as the sum over all samples:
â\mathrm{log}\left({s}_{i}^{2}\right).
SURE (Stein's Unbiased Risk Estimate)
A threshold-based method in which the threshold equals
\sqrt{2{\mathrm{log}}_{e}\left(n{\mathrm{log}}_{2}\left(n\right)\right)}
where n is the number of samples in the signal.
An entropy type criterion you define in a file.
For more information about the available entropy types, user-defined entropy, and threshold parameters, see the wentropy reference page and Choosing the Optimal Decomposition.
Many capabilities are available using the command area on the right of the Wavelet Packet 1-D window.
Because there are so many ways to reconstruct the original signal from the wavelet packet decomposition tree, we select the best tree before attempting to compress the signal.
Click the Best Tree button.
After a pause for computation, the Wavelet Packet 1-D tool displays the best tree. Use the top and bottom sliders to spread nodes apart and pan over to particular areas of the tree, respectively.
Observe that, for this analysis, the best tree and the initial tree are almost the same. One branch at the far right of the tree was eliminated.
Click the Compress button.
The Wavelet Packet 1-D Compression window appears with an approximate threshold value automatically selected.
The leftmost graph shows how the threshold (vertical black dotted line) has been chosen automatically (1.482) to balance the number of zeros in the compressed signal (blue curve that increases as the threshold increases) with the amount of energy retained in the compressed signal (purple curve that decreases as the threshold increases).
This threshold means that any signal element whose value is less than 1.482 will be set to zero when we perform the compression.
Threshold controls are located to the right (see the red box in the figure above). Note that the automatic threshold of 1.482 results in a retained energy of only 81.49%. This may cause unacceptable amounts of distortion, especially in the peak values of the oscillating signal. Depending on your design criteria, you may want to choose a threshold that retains more of the original signal's energy.
Adjust the threshold by typing 0.8938 in the text field opposite the threshold slider, and then press the Enter key.
The value 0.8938 is a number that we have discovered through trial and error yields more satisfactory results for this analysis.
After a pause, the Wavelet Packet 1-D Compression window displays new information.
Note that, as we have reduced the threshold from 1.482 to 0.8938,
The vertical black dotted line has shifted to the left.
The retained energy has increased from 81.49% to 90.96%.
The number of zeros (equivalent to the amount of compression) has decreased from 81.55% to 75.28%.
The Wavelet Packet 1-D tool compresses the signal using the thresholding criterion we selected.
The original (red) and compressed () signals are displayed superimposed. Visual inspection suggests the compression quality is quite good.
Looking more closely at the compressed signal, we can see that the number of zeros in the wavelet packets representation of the compressed signal is about 75.3%, and the retained energy about 91%.
If you try to compress the same signal using wavelets with exactly the same parameters, only 89% of the signal energy is retained, and only 59% of the wavelet coefficients set to zero. This illustrates the superiority of wavelet packets for performing compression, at least on certain signals.
You can demonstrate this to yourself by returning to the main Wavelet Packet 1-D window, computing the wavelet tree, and then repeating the compression.
We now use the Wavelet Packet 1-D tool to analyze a noisy chirp signal. This analysis illustrates the use of Stein's Unbiased Estimate of Risk (SURE) as a principle for selecting a threshold to be used for de-noising.
This technique calls for setting the threshold T to
T=\sqrt{2{\mathrm{log}}_{e}\left(n{\mathrm{log}}_{2}\left(n\right)\right)}
A more thorough discussion of the SURE criterion appears in Choosing the Optimal Decomposition. For now, suffice it to say that this method works well if your signal is normalized in such a way that the data fit the model x(t) = f(t) + e(t), where e(t) is a Gaussian white noise with zero mean and unit variance.
If you've already started the Wavelet Packet 1-D tool and it is active on your computer's desktop, skip ahead to step 3.
The tool appears on the desktop.
load noischir;
In the Wavelet Packet 1-D tool, select File > Import from Workspace > Import Signal. When the Import from Workspace dialog box appears, select the sumlichr variable. Click OK to import the data
You can use File > Load > Signal to load a signal by navigating to its location.
The signal's length is 1024. This means we should set the SURE criterion threshold equal to sqrt(2.*log(1024.*log2(1024))), or 4.2975.
Make the appropriate settings for the analysis. Select the db2 wavelet, level 4, entropy type sure, and threshold parameter 4.2975. Click the Analyze button.
There is a pause while the wavelet packet analysis is computed.
Many capabilities are available using the command area on the right of the Wavelet Packet 1-D window. Some of them are used in the sequel. For a more complete description, see Wavelet Packet Tool Features (1-D and 2-D).
Computing the Best Tree and Performing De-Noising
Computing the best tree makes the de-noising calculations more efficient.
Click the De-noise button. This brings up the Wavelet Packet 1-D De-Noising window.
Click the De-noise button located at the center right side of the Wavelet Packet 1-D De-Noising window.
The results of the de-noising operation are quite good, as can be seen by looking at the thresholded coefficients. The frequency of the chirp signal increases quadratically over time, and the thresholded coefficients essentially capture the quadratic curve in the time-frequency plane.
You can also use the wpdencmp function to perform wavelet packet de-noising or compression from the command line.
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When the theories meet: Khovanov homology as Hochschild homology of links | EMS Press
When the theories meet: Khovanov homology as Hochschild homology of links
We show that Khovanov homology and Hochschild homology theories share a common structure. In fact they overlap: Khovanov homology of the (2,n) torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov–Rozansky sl(n) homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to the
\mathbb{M}
-reduced case, and in the case of a polygon extended to noncommutative algebras. In this framework we prove that for any unital algebra
\mathcal{A}
the Hochschild homology of
\mathcal{A}
is isomorphic to graph cohomology over
\mathcal{A}
of a polygon. We expect that this paper will encourage a flow of ideas in both directions between Hochschild/cyclic homology and Khovanov homology theories.
Józef H. Przytycki, When the theories meet: Khovanov homology as Hochschild homology of links. Quantum Topol. 1 (2010), no. 2, pp. 93–109
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Filter outliers using Hampel identifier - Simulink - MathWorks í•œêµ
Specify threshold from input port
Threshold for outlier detection (standard deviations)
Output outlier status
The Hampel Filter block detects and removes the outliers of the input signal by using the Hampel identifier. The Hampel identifier is a variation of the three-sigma rule of statistics, which is robust against outliers. For each sample of the input signal, the block computes the median of a window composed of the current sample and
\frac{Lenâ1}{2}
adjacent samples on each side of the current sample. Len is the window length you specify through the Window length parameter. The block also estimates the standard deviation of each sample about its window median by using the median absolute deviation. If a sample differs from the median by more than the threshold multiplied by the standard deviation, the filter replaces the sample with the median. For more information, see Algorithms.
The block accepts multichannel inputs, that is, m-by-n size inputs, where m ≥ 1, and n ≥ 1. m is the number of samples in each frame (channel), and n is the number of channels. The block also accepts variable-size inputs. That is, you can change the size of each input channel during simulation. However, the number of channels cannot change.
This port is unnamed until you select the Specify threshold from input port parameter.
real scalar greater than or equal to 0
Threshold for outlier detection, specified as a real scalar greater than or equal to 0. For information on how this parameter is used to detect the outlier, see Algorithms.
This port appears when you select the Specify threshold from input port parameter.
The size and data type of this output matches the size and data type of the input.
This port is unnamed until you select the Output outlier status check box.
outlier — Output the outlier status
A value of 1 in this output indicates that the corresponding element in the input is an outlier. This output has the same size as the input.
To enable this port, select the Output outlier status check box.
Length of the sliding window, specified as a positive odd scalar integer. The window of finite length slides over the data, and the block computes the median and median absolute deviation of the data in the window.
Specify threshold from input port — Flag to specify threshold
When you select this check box, the threshold is input through the T port. When you clear this check box, the threshold is specified on the block dialog through the Threshold for outlier detection (standard deviations) parameter.
Threshold for outlier detection (standard deviations) — Threshold
3 (default) | real scalar greater than or equal to 0
This parameter appears when you clear the Specify threshold from input port check box.
Output outlier status — Flag to output the outlier status
Select this parameter to output a matrix of boolean values that has the same size as the input. Each element in this matrix indicates whether the corresponding element in the input is an outlier. A value of 1 indicates an outlier.
{m}_{i}=\mathrm{median}\left({x}_{iâk},{x}_{iâk+1},{x}_{iâk+2},â¦,{x}_{i},â¦,{x}_{i+kâ2},{x}_{i+kâ1},{x}_{i+k}\right)
{\mathrm{Ï}}_{i}=\mathrm{κ}\mathrm{median}\left(|{x}_{iâk}â{m}_{i}|,â¦,|{x}_{i+k}â{m}_{i}|\right),
\mathrm{κ}=\frac{1}{\sqrt{2}{\mathrm{erfc}}^{â1}\left(1/2\right)}â1.4826
|{x}_{i}â{m}_{i}|>{n}_{\mathrm{Ï}}{\mathrm{Ï}}_{i}
|{x}_{s}â{m}_{i}|>{n}_{\mathrm{Ï}}Ã{\mathrm{Ï}}_{i}
{\left[\frac{Lenâ1}{2}+1\right]}^{\text{th}}
ma{d}_{i}=\mathrm{median}\left(|{x}_{iâk}â{m}_{i}|,â¦,|{x}_{i+k}â{m}_{i}|\right)
mad=\mathrm{median}\left(|0â0|,â¦,|1â0|\right)=0
\mathrm{κ}=\frac{1}{\sqrt{2}{\mathrm{erfc}}^{â1}\left(1/2\right)}â1.4826
\left[|{x}_{s}â{m}_{i}|=0\right]>\left[\left({n}_{\mathrm{Ï}}Ã{\mathrm{Ï}}_{i}\right)=0\right]
{\left[Endâ\frac{Lenâ1}{2}\right]}^{\text{th}}
\frac{Lenâ1}{2}
dsp.HampelFilter | dsp.MedianFilter | dsp.MovingAverage
Median Filter | Moving Average
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No, I have denoted by D the scale invariant differential operator x d/dx. It is particularly suitable for differential equations of functions defined in the positive real axis.
It seems the hypergeometric series itself diverges when
p > q+1
\mathbf{a_{p}}>0
i.e. all the top parameters are positive. Also I think there doesn't exist a closed relation between hypergeometric functions with argument z and 1/z in this case.
The more important case is:
p > q+1
and one or more of the a- parameters is nonpositive. In this case the series terminates to become a polynomial. One can deal with this case using this equation.
I think it diverges always when
p > q+1
, even if some of the
a_i
are negative (non-integers, of course).
p = q+1
, the radius of convergence is 1.
This should follow from the simple ratio test.
Yes you are right. The series only terminates if there is one or more nonpositive integer in a- parameters.
I actually didn't consider complex roots in the final solution yet. I will add them too.
There should be no essential difference in the treatment. (Probably no difference at all.)
Hi Kalevi. I probably won't be able to contribute for the next couple of days. Will be travelling back to college.
Hi. There is no need to be coding all the time, though you have been doing that. Besides, you have already completed about all that I would have expected.
cos(x)^{2}/x
x^{-1}(1 + 0 - x^{2} ....)
1/x - x + ...
x^{s_1} f_1(x)
x^{s_2} f_2 (x)
x^{s_1+s_2}f_1(x)f_2(x)
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EUDML | Common periodic points for a class of continuous commuting mappings on an interval. EuDML | Common periodic points for a class of continuous commuting mappings on an interval.
Common periodic points for a class of continuous commuting mappings on an interval.
Kang, Shin Min; Wang, Weili
Kang, Shin Min, and Wang, Weili. "Common periodic points for a class of continuous commuting mappings on an interval.." International Journal of Mathematics and Mathematical Sciences 2003.16 (2003): 1043-1046. <http://eudml.org/doc/50030>.
author = {Kang, Shin Min, Wang, Weili},
keywords = {-class; -class; common periodic points; self-mappings; interval; -class; -class},
title = {Common periodic points for a class of continuous commuting mappings on an interval.},
TI - Common periodic points for a class of continuous commuting mappings on an interval.
KW - -class; -class; common periodic points; self-mappings; interval; -class; -class
H
-class,
C
-class, common periodic points, self-mappings, interval,
H
C
-class
Articles by Kang
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Planetary gear set of carrier, inner planet, and outer planet wheels with adjustable gear ratio and friction losses - MATLAB - MathWorks í•œêµ
Inner planet-carrier power threshold
Inner planet-carrier viscous friction coefficient
Planetary gear set of carrier, inner planet, and outer planet wheels with adjustable gear ratio and friction losses
The Planet-Planet gear block represents a carrier and two inner-outer planet gear couples. Both planet gears are connected to and rotate with respect to the carrier. The planet gears corotate with a fixed gear ratio that you specify. For model details, see Equations.
The Planet-Planet block imposes one kinematic and one geometric constraint on the three connected axes:
{r}_{\text{C}}{\mathrm{Ï}}_{\text{C}}={r}_{\text{Po}}{\mathrm{Ï}}_{\text{Po}}+{r}_{\text{Pi}}{\mathrm{Ï}}_{\text{Pi}}
{r}_{\text{C}}={r}_{\text{Po}}+{r}_{\text{Pi}}
The outer planet-to-inner planet gear ratio is
{g}_{\text{oi}}={r}_{\text{Po}}/{r}_{\text{Pi}}={N}_{\text{Po}}/{N}_{\text{Pi}},
where N is the number of teeth on each gear. In terms of this ratio, the key kinematic constraint is
\left(\text{1 }+{g}_{\text{oi}}\right){\mathrm{Ï}}_{\text{C}}={\mathrm{Ï}}_{\text{Pi}}+{g}_{\text{oi}}{\mathrm{Ï}}_{\text{Po}}.
The three degrees of freedom reduce to two independent degrees of freedom. The gear pair is (1, 2) = (Pi,Po).
{g}_{\text{oi}}{\mathrm{Ï}}_{\text{Pi}}+{\mathrm{Ï}}_{\text{Po}}–{\mathrm{Ï}}_{\text{loss}}=\text{ }0.
Po — Outer planet gear
Rotational mechanical conserving port associated with the outer planet gear.
Pi — Inner planet gear
Rotational mechanical conserving port associated with the inner planet gear.
Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi) — Outer to inner planet gear rotation ratio
Ratio, goi, of the outer planet gear to inner planet gear rotations as defined by the number of outer planet gear teeth divided by the number of inner planet gear teeth. This gear ratio must be strictly positive.
Torque transfer efficiency, ηPP, for the outer and inner planet gear wheel pair meshing. This value must be in the range (0,1].
Vector of output-to-input power ratios that describe the power flow from the outer planet gear to the inner planet gear, ηPP. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Inner planet-carrier power threshold — Minimum efficiency power threshold for the inner planet-carrier gear coupling
Inner planet-carrier viscous friction coefficient — Gear viscous friction
Viscous friction coefficient μPi for the inner planet-carrier gear motion.
Planetary Gear | Ravigneaux Gear | Ring-Planet | Sun-Planet | Sun-Planet Bevel
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Proof theory - formulasearchengine
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.[1]
Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.
3 Kinds of proof calculi
7 Tableau systems
9 Logics from proof analysis
Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system; then his incompleteness theorems showed that this is unattainable. All of this work was carried out with the proof calculi called the Hilbert systems.
In parallel, the foundations of structural proof theory were being founded. Jan Łukasiewicz suggested in 1926 that one could improve on Hilbert systems as a basis for the axiomatic presentation of logic if one allowed the drawing of conclusions from assumptions in the inference rules of the logic. In response to this Stanisław Jaśkowski (1929) and Gerhard Gentzen (1934) independently provided such systems, called calculi of natural deduction, with Gentzen's approach introducing the idea of symmetry between the grounds for asserting propositions, expressed in introduction rules, and the consequences of accepting propositions in the elimination rules, an idea that has proved very important in proof theory.[2] Gentzen (1934) further introduced the idea of the sequent calculus, a calculus advanced in a similar spirit that better expressed the duality of the logical connectives,[3] and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the first combinatorial proof of the consistency of Peano arithmetic. Together, the presentation of natural deduction and the sequent calculus introduced the fundamental idea of analytic proof to proof theory.
Formal proofs are constructed with the help of computers in interactive theorem proving. Significantly, these proofs can be checked automatically, also by computer. (Checking formal proofs is usually simple, whereas finding proofs (automated theorem proving) is generally hard.) An informal proof in the mathematics literature, by contrast, requires weeks of peer review to be checked, and may still contain errors.
Kinds of proof calculi
The three most well-known styles of proof calculi are:
Each of these can give a complete and axiomatic formalization of propositional or predicate logic of either the classical or intuitionistic flavour, almost any modal logic, and many substructural logics, such as relevance logic or linear logic. Indeed it is unusual to find a logic that resists being represented in one of these calculi.
As previously mentioned, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable
{\displaystyle \Pi _{1}^{0}}
{\displaystyle \Pi _{1}^{0}}
sentence.
Much investigation has been carried out on this topic since, which has in particular led to:
The recent discovery of self-verifying theories, systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument.
Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz. The definition is slightly more complex: we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Girard's proof nets also support a notion of analytic proof.
Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. This provides the foundation for the intuitionistic type theory developed by Per Martin-Löf, and is often extended to a three way correspondence, the third leg of which are the cartesian closed categories.
Analytic tableaux apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics.
Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis.
Logics from proof analysis
Several important logics have come from insights into logical structure arising in structural proof theory.
↑ Template:Harvtxt.
G. Gentzen (1935/1969). Investigations into logical deduction. In M. E. Szabo, editor, Collected Papers of Gerhard Gentzen. North-Holland. Translated by Szabo from "Untersuchungen über das logische Schliessen", Mathematisches Zeitschrift 39: 176-210, 405-431.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Proof_theory&oldid=225469"
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FAQ - HunnyDAO
PCV is the Protocol Controlled Value. As HunnyDAO controls the funds in its treasuries, LOVE can only be minted or burned by the protocol. As HunnyDAO accumulates more PCV, more runway is guaranteed for the stakers. This means the stakers can be confident that the current staking APY can be sustained for a longer-term because more funds are available in the treasuries.
This is one of the main reasons why having a secondary treasury like in HunnyDAO is crucial for the sustainability of the protocol.
HunnyDAO owns most of its liquidity thanks to its bond mechanism. This has several benefits:
HunnyDAO does not have to pay out high farming rewards to incentivize liquidity providers a.k.a renting liquidity.
HunnyDAO guarantees the market that liquidity is always there to facilitate sell or buy transactions.
Rebase is a mechanism by which your staked LOVE balance increases automatically. When new LOVE is minted by HunnyDAO, a large portion of it goes to the stakers. Because stakers only see staked LOVE balance instead of LOVE tokens, the protocol utilizes the rebase mechanism to increase the staked LOVE balance so that 1 staked LOVE is always redeemable for 1 LOVE token.
Reward yield is the percentage by which your staked LOVE balance increases on the next interlude. It is also known as rebase rate.
APY stands for Annual Percentage Yield. It measures the real rate of return on your principal by taking into account the effect of compounding interest. In the case of HunnyDAO, your staked LOVE represents your principal, and the compound interest is added periodically on every interlude (around 8 hours) thanks to the rebase mechanism.
APY = ( 1 + rewardYield)^ {1095}
rewardYield = LOVE_{distributed} / LOVE_{totalStaked}
The number of LOVE distributed to the staking contract is calculated from LOVE total supply using the following equation:
LOVE_{distributed} = LOVE_{totalSupply} * rewardRate
Note that the reward rate is subject to change by the DAO.
Let’s say HunnyDAO targets an APY range of 1,000% to 10,000%, this would translate to a minimum reward yield of about 0.2105%, or a daily growth of about 0.6328%. You can refer to the equation above to learn how APY is calculated from the reward yield.
If there are 100,000 LOVE tokens staked right now, HunnyDAO would need to mint an additional 632.8 LOVE to achieve this daily growth. This is achievable if HunnyDAO can bring in at least $632.80 of daily revenue from bond sales.
Even in the worst-case scenario whereby HunnyDAO doesn't bring in that much revenue, it can still sustain 1,000% APY for a considerable amount of time due to the excess reserve in the treasuries. This is the reason why HunnyDAO secondary treasury will play an important role here.
Do I have to unstake and stake LOVE on every interlude to get my rebase rewards?
No. Once you have staked LOVE with HunnyDAO, your staked LOVE balance will auto-compound on every interlude. That increase in balance represents your rebase rewards.
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Length of the ascending node - zxc.wiki
Length of the ascending node
Orbit elements of the elliptical orbit of a celestial body around a central body (sun / earth)
Six orbit elements
a : Length of the major semi-axis
e : Numerical eccentricity
i : Orbital inclination, inclination
Ω: Length / right ascension of the ascending node
ω: Argument of the periapsis, periapsis distance
t : Time of the Periapsis passage, periapsis time, epoch of the periapsis passage
Further designations
M: Ellipse center. B: focus, central body, sun / earth. P: periapsis. A: Apoapsis. AP: apse line. HK: Celestial body, planet / satellite. ☋: descending node. ☊: ascending node. ☋☊: knot line. ♈: spring equinox. ν: true anomaly. r: distance of the celestial body HK from the central body B
In celestial mechanics, the length of the ascending node ( node length for short ; symbol Ω ) of an orbit around the sun is the heliocentric angle to be measured in the ecliptic ( reference plane or reference plane) between the ascending node ☊ and the spring point ♈.
The length of the ascending node is one of the six orbit elements (see graphic) that suffice for a sufficient description of an - ideal - Kepler orbit. Together with the inclination i and the argument of the periapsis ω , it belongs to that subgroup of orbit elements that defines the position of the orbit plane in space .
Other central bodies or reference planes
In the case of central bodies other than the sun and / or other reference planes than the ecliptic, “length” generally means the first polar coordinate of a spherical coordinate system .
Depending on the type of object whose path is specified, they, the following reference are flat usual:
for solar orbital objects of the solar system , d. H. for planets , asteroids , comets : the ecliptic
where the length of the ascending node is its ecliptical length ( longitude of the ascending node, LOAN ), measured from the vernal equinox .
for objects that do not orbit the sun: the equatorial plane of the central body that the object orbits instead; For example, for earth satellites with a uniform orbit semi-axis : the plane of the earth or celestial equator (see satellite orbit elements ).
where the length of the ascending node is its right ascension (i.e. the equatorial length, English right ascension of the ascending node, RAAN ), again measured from the vernal equinox, but this time along the equator.
for the earth's moon : the ecliptic
where the length of the ascending node is its ecliptical length , measured geocentrically from the vernal equinox .
In the case of Kepler orbits (only two bodies in a vacuum ) the length of the knot is constant and the orbital plane remains stable in its alignment under the fixed stars .
In the case of gravitational disturbances from third bodies , the length of the node suffers small, sometimes periodic changes. The orbit element is therefore given as a series of oscillating terms with respect to an epoch , i.e. as an approximate solution that is valid at a certain point in time .
As a first approximation, the value for the length of the lunar knot is given as
{\ displaystyle \ Omega = 125 {,} 0445 ^ {\ circ} -1934 {,} 1363 ^ {\ circ} \ cdot T}
with T as time argument in Julian centuries since the epoch J2000.0 ( Lit .: Vollmann, 3.5 p. 26).
The approximately 19.34 ° in a Julian year (365.25 days) correspond to one complete rotation of the knot line in 18.61 years, the nutation period .
Andreas Guthman: Introduction to celestial mechanics and ephemeris calculation , theory, algorithms, numerics , 2nd edition. Spectrum Academic Publishing House, 2000
Jean Meeus: Astronomical Algorithms . Willmann-Bell Inc., 2009
Wolfgang Vollmann: Changing star locations . In: Hermann Mucke (Hrsg.): Modern astronomical phenomenology. 20th Sternfreunde Seminar, 1992/93. Planetarium of the City of Vienna - Zeiss Planetarium of the City of Vienna - Zeiss Planetarium and Austrian Astronomical Association , 1992, pp. 55–102 ( online )
This page is based on the copyrighted Wikipedia article "L%C3%A4nge_des_aufsteigenden_Knotens" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Compactification (physics) - Wikipedia
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic.
Compactification plays an important part in thermal field theory where one compactifies time, in string theory where one compactifies the extra dimensions of the theory, and in two- or one-dimensional solid state physics, where one considers a system which is limited in one of the three usual spatial dimensions.
At the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is dimensionally reduced.
{\displaystyle M\times C}
is compactified over the compact
{\displaystyle C}
and after Kaluza–Klein decomposition, we have an effective field theory over M.
1 Compactification in string theory
2 Flux compactification
Compactification in string theory[edit]
In string theory, compactification is a generalization of Kaluza–Klein theory.[1] It tries to reconcile the gap between the conception of our universe based on its four observable dimensions with the ten, eleven, or twenty-six dimensions which theoretical equations lead us to suppose the universe is made with.
For this purpose it is assumed the extra dimensions are "wrapped" up on themselves, or "curled" up on Calabi–Yau spaces, or on orbifolds. Models in which the compact directions support fluxes are known as flux compactifications. The coupling constant of string theory, which determines the probability of strings splitting and reconnecting, can be described by a field called a dilaton. This in turn can be described as the size of an extra (eleventh) dimension which is compact. In this way, the ten-dimensional type IIA string theory can be described as the compactification of M-theory in eleven dimensions. Furthermore, different versions of string theory are related by different compactifications in a procedure known as T-duality.
The formulation of more precise versions of the meaning of compactification in this context has been promoted by discoveries such as the mysterious duality.
Flux compactification[edit]
A flux compactification is a particular way to deal with additional dimensions required by string theory.
It assumes that the shape of the internal manifold is a Calabi–Yau manifold or generalized Calabi–Yau manifold which is equipped with non-zero values of fluxes, i.e. differential forms, that generalize the concept of an electromagnetic field (see p-form electrodynamics).
The hypothetical concept of the anthropic landscape in string theory follows from a large number of possibilities in which the integers that characterize the fluxes can be chosen without violating rules of string theory. The flux compactifications can be described as F-theory vacua or type IIB string theory vacua with or without D-branes.
^ Dean Rickles (2014). A Brief History of String Theory: From Dual Models to M-Theory. Springer, p. 89 n. 44.
Chapter 16 of Michael Green, John H. Schwarz and Edward Witten (1987). Superstring theory. Cambridge University Press. Vol. 2: Loop amplitudes, anomalies and phenomenology. ISBN 0-521-35753-5.
Brian R. Greene, "String Theory on Calabi–Yau Manifolds". arXiv:hep-th/9702155.
Mariana Graña, "Flux compactifications in string theory: A comprehensive review", Physics Reports 423, 91–158 (2006). arXiv:hep-th/0509003.
Michael R. Douglas and Shamit Kachru "Flux compactification", Rev. Mod. Phys. 79, 733 (2007). arXiv:hep-th/0610102.
Ralph Blumenhagen, Boris Körs, Dieter Lüst, Stephan Stieberger, "Four-dimensional string compactifications with D-branes, orientifolds and fluxes", Physics Reports 445, 1–193 (2007). arXiv:hep-th/0610327.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Compactification_(physics)&oldid=1080531833"
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LOVE Supply - HunnyDAO
LOVE_{supplyGrowth} = LOVE_{stakers} + LOVE_{bonders} + LOVE_{HunnyDAO}
LOVE token supply does not have a hard cap. Its supply increases when:
LOVE is minted and distributed to the stakers.
LOVE is minted for the bonder. This happens whenever someone purchases a bond.
LOVE is minted for the DAO. This happens whenever someone purchases a bond. The DAO gets the same number of LOVE as the bonder.
LOVE_{stakers} = LOVE_{totalSupply} * rewardRate
At the end of each interlude, the treasury mints LOVE at a set reward rate. These LOVE will be distributed to all the stakers in HunnyDAO. You can track the latest reward rate on HunnyDAO dashboard.
LOVE_{bonders} = bondPayout
Whenever someone purchases a bond, a pre-determined amount of LOVE will be minted. These LOVE will not be released to the bonder all at once - they are vested to the bonder linearly over time. The bond payout uses a different formula for different types of bonds. Check the bonding section above to see how it is calculated.
LOVE_{HunnyDAO} = LOVE_{bonders}
HunnyDAO received the same amount of LOVE as the bonder. This represents HunnyDAO Profit.
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ΔABC
ΔDEF
Complete a flowchart to justify the relationship between the two triangles. Assume the triangles at right are not drawn to scale.
Use the Triangle Angle Sum Theorem to find the unknown angle in each triangle.
This will help you determine the relationship between the two triangles.
{DF}
The corresponding sides of similar triangles will all have equal ratios to one another.
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EUDML | On spaces whose nowhere dense subsets are scattered. EuDML | On spaces whose nowhere dense subsets are scattered.
On spaces whose nowhere dense subsets are scattered.
Dontchev, Julian; Rose, David
Dontchev, Julian, and Rose, David. "On spaces whose nowhere dense subsets are scattered.." International Journal of Mathematics and Mathematical Sciences 21.4 (1998): 735-740. <http://eudml.org/doc/48184>.
author = {Dontchev, Julian, Rose, David},
keywords = {scattered; nowhere dense; -topology; -topology},
title = {On spaces whose nowhere dense subsets are scattered.},
AU - Dontchev, Julian
TI - On spaces whose nowhere dense subsets are scattered.
KW - scattered; nowhere dense; -topology; -topology
scattered, nowhere dense,
\alpha
\alpha
F
Pathological spaces
|
Vector - Simple English Wikipedia, the free encyclopedia
geometric object that has magnitude (or length) and direction
This article is about a mathematical concept. For other uses, see Vector (biology).
A vector is a mathematical object that has a size, called the magnitude, and a direction. It is often represented by boldface letters (such as
{\displaystyle \mathbf {u} }
{\displaystyle \mathbf {v} }
{\displaystyle \mathbf {w} }
), or as a line segment from one point to another (as in
{\displaystyle {\overrightarrow {AB}}}
For example, a vector would be used to show the distance and direction something moved in. When asking for directions, if one says "Walk one kilometer towards the North", that would be a vector, but if they say "Walk one kilometer", without showing a direction, then that would be a scalar.
We usually draw vectors as arrows. The length of the arrow is proportional to the vector's magnitude. The direction in which the arrow points to is the vector's direction.[3]
1 Examples of vectors
2 Examples of scalars
3 More examples of vectors
4 How to add vectors
4.1 Adding vectors on paper using the head to tail method
4.2 Using component form
5 How to multiply vectors
5.1 Using the dot product
5.2 Using the cross product
Examples of vectorsEdit
John walks north 20 meters. The direction "north" together with the distance "20 meters" is a vector.
An apple falls down at 10 meters per second. The direction "down" combined with the speed "10 meters per second" is a vector. This kind of vector is also called velocity.
Examples of scalarsEdit
The distance between two places is 10 kilometers. This distance is not a vector because it does not contain a direction.
The number of fruit in a box is not a vector.
A person pointing is not a vector because there is only a direction. There is no magnitude (the distance from the person's finger to a building, for example).
The length of an object.
A car drives at 100 kilometers per hour. This does not describe a vector, as there is only a magnitude, but no direction.
More examples of vectorsEdit
Displacement is a vector. Displacement is the distance that something moves in a certain direction. A measure of distance alone is a scalar.
Force that includes direction is a vector.[3]
Velocity is a vector, because it is a speed in a certain direction.[3][4]
Acceleration is the rate of change of velocity. An object is accelerating if it is changing speed or changing direction.
How to add vectorsEdit
Adding vectors on paper using the head to tail methodEdit
Head-to-tail Addition
The Head to Tail method of adding vectors is useful for doing an estimate on paper of the result of adding two vectors. To do it:
Each vector is drawn as an arrow with an amount of length behind it, where each unit of length on the paper represents a certain magnitude of the vector.
Draw the next vector, with the tail(end) of the second vector at the head(front) of the first vector.
Repeat for all further vectors: Draw the tail of the next vector at head of the previous one.
Draw a line from the tail of the first vector to the head of the last vector - that's the resultant(sum) of all the vectors.
It's called the "Head to Tail" method, because each head from the previous vector leads in to the tail of the next one.
Using component formEdit
[needs to be explained]
Using the component form to add two vectors literally means adding the components of the vectors to create a new vector.[5] For example, let a and b be two two-dimensional vectors. These vectors can be written in terms of their components.
{\displaystyle \mathbf {a} =(a_{x},a_{y})}
{\displaystyle \mathbf {b} =(b_{x},b_{y})}
Suppose c is the sum of these two vectors, so that c = a + b. This means that
{\displaystyle \mathbf {c} =(a_{x}+b_{x},a_{y}+b_{y})}
Here is an example of addition of two vectors, using their component forms:
{\displaystyle \mathbf {a} =(3,-1)}
{\displaystyle \mathbf {b} =(2,2)}
{\displaystyle {\begin{aligned}\mathbf {c} &=\mathbf {a} +\mathbf {b} \\&=(a_{x}+b_{x},a_{y}+b_{y})\\&=(3+2,-1+2)\\&=(5,1)\end{aligned}}}
This method works for all vectors, not just two dimensional ones.
How to multiply vectorsEdit
Using the dot productEdit
The dot product is one method to multiply vectors. It produces a scalar. It uses component form:
{\displaystyle {\begin{aligned}\mathbf {a} &=(2,3)\\\mathbf {b} &=(1,4)\\\mathbf {a} \cdot \mathbf {b} &=(2,3)\cdot (1,4)\\&=(2\cdot 1)+(3\cdot 4)\\&=2+12\\&=14\end{aligned}}}
Using the cross productEdit
The cross product is another method to multiply vectors. Unlike dot product, it produces a vector. Using component form:
{\displaystyle \mathbf {a} \times \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\sin(\theta )\mathbf {n} }
{\displaystyle |\mathbf {a} |}
means the length of
{\displaystyle \mathbf {a} }
{\displaystyle \mathbf {n} }
is the unit vector at right angles to both
{\displaystyle \mathbf {a} }
{\displaystyle \mathbf {b} }
Multiplying by a scalarEdit
To multiply a vector by a scalar (a normal number), you multiply the number by each component of the vector:
{\displaystyle c\,\mathbf {x} =(c\,x_{1},c\,x_{2},...,c\,x_{n})}
{\displaystyle {\begin{aligned}c&=5\\\mathbf {x} &=(3,4)\\c\,\mathbf {x} &=(5\cdot 3,5\cdot 4)\\&=(15,20)\end{aligned}}}
↑ Weisstein, Eric W. "Vector". mathworld.wolfram.com. Retrieved 2020-08-19.
↑ 3.0 3.1 3.2 "Vectors". www.mathsisfun.com. Retrieved 2020-08-19.
↑ "vector | Definition & Facts". Encyclopedia Britannica. Retrieved 2020-08-19.
↑ "1.1: Vectors". Mathematics LibreTexts. 2013-11-07. Retrieved 2020-08-19.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Vector&oldid=7900686"
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Symmetric difference - Wikipedia
{\displaystyle \{1,2,3\}}
{\displaystyle \{3,4\}}
{\displaystyle \{1,2,4\}}
{\displaystyle A\triangle B}
{\displaystyle ~\setminus ~}
{\displaystyle ~=~}
{\displaystyle A\ominus B,}
{\displaystyle A\operatorname {\triangle } B.}
2 n-ary symmetric difference
3 Symmetric difference on measure spaces
4 Hausdorff distance vs. symmetric difference
{\displaystyle ~(A\triangle B)\triangle C}
{\displaystyle ~\triangle ~}
{\displaystyle ~=~}
{\displaystyle A\,\triangle \,B=\left(A\setminus B\right)\cup \left(B\setminus A\right),}
{\displaystyle A\mathbin {\triangle } B=\{x:(x\in A)\oplus (x\in B)\}.}
{\displaystyle \chi }
{\displaystyle \chi _{(A\,\triangle \,B)}=\chi _{A}\oplus \chi _{B}}
{\displaystyle [x\in A\,\triangle \,B]=[x\in A]\oplus [x\in B]}
{\displaystyle A\,\triangle \,B=(A\cup B)\setminus (A\cap B),}
{\displaystyle A\mathbin {\triangle } B\subseteq A\cup B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle D=A\mathbin {\triangle } B}
{\displaystyle I=A\cap B}
{\displaystyle D}
{\displaystyle I}
{\displaystyle D}
{\displaystyle I}
{\displaystyle A\cup B}
{\displaystyle A\,\cup \,B=(A\,\triangle \,B)\,\triangle \,(A\cap B)}
{\displaystyle {\begin{aligned}A\,\triangle \,B&=B\,\triangle \,A,\\(A\,\triangle \,B)\,\triangle \,C&=A\,\triangle \,(B\,\triangle \,C).\end{aligned}}}
{\displaystyle {\begin{aligned}A\,\triangle \,\varnothing &=A,\\A\,\triangle \,A&=\varnothing .\end{aligned}}}
{\displaystyle (A\,\triangle \,B)\,\triangle \,(B\,\triangle \,C)=A\,\triangle \,C.}
{\displaystyle A\cap (B\,\triangle \,C)=(A\cap B)\,\triangle \,(A\cap C),}
{\displaystyle A\mathbin {\triangle } B=\emptyset }
{\displaystyle A=B}
{\displaystyle A\mathbin {\triangle } B=A^{c}\mathbin {\triangle } B^{c}}
{\displaystyle A^{c}}
{\displaystyle B^{c}}
{\displaystyle A}
{\displaystyle B}
{\displaystyle \left(\bigcup _{\alpha \in {\mathcal {I}}}A_{\alpha }\right)\triangle \left(\bigcup _{\alpha \in {\mathcal {I}}}B_{\alpha }\right)\subseteq \bigcup _{\alpha \in {\mathcal {I}}}\left(A_{\alpha }\mathbin {\triangle } B_{\alpha }\right)}
{\displaystyle {\mathcal {I}}}
{\displaystyle f:S\rightarrow T}
{\displaystyle A,B\subseteq T}
{\displaystyle f}
{\displaystyle f^{-1}\left(A\mathbin {\triangle } B\right)=f^{-1}\left(A\right)\mathbin {\triangle } f^{-1}\left(B\right).}
{\displaystyle x\,\triangle \,y=(x\lor y)\land \lnot (x\land y)=(x\land \lnot y)\lor (y\land \lnot x)=x\oplus y.}
{\displaystyle \triangle M=\left\{a\in \bigcup M:\left|\{A\in M:a\in A\}\right|{\text{ is odd}}\right\}.}
{\textstyle \bigcup M}
{\displaystyle M}
{\displaystyle M=\left\{M_{1},M_{2},\ldots ,M_{n}\right\}}
{\displaystyle n\geq 2}
{\displaystyle |\triangle M|}
{\displaystyle \triangle M}
{\displaystyle M}
{\displaystyle |\triangle M|=\sum _{l=1}^{n}(-2)^{l-1}\sum _{1\leq i_{1}<i_{2}<\ldots <i_{l}\leq n}\left|M_{i_{1}}\cap M_{i_{2}}\cap \ldots \cap M_{i_{l}}\right|.}
{\displaystyle d_{\mu }(X,Y)=\mu (X\,\triangle \,Y)}
{\displaystyle \mu (X\,\triangle \,Y)=0}
{\displaystyle \mu (X),\mu (Y)<\infty }
{\displaystyle |\mu (X)-\mu (Y)|\leq \mu (X\,\triangle \,Y)}
{\displaystyle {\begin{aligned}|\mu (X)-\mu (Y)|&=\left|\left(\mu \left(X\setminus Y\right)+\mu \left(X\cap Y\right)\right)-\left(\mu \left(X\cap Y\right)+\mu \left(Y\setminus X\right)\right)\right|\\&=\left|\mu \left(X\setminus Y\right)-\mu \left(Y\setminus X\right)\right|\\&\leq \left|\mu \left(X\setminus Y\right)\right|+\left|\mu \left(Y\setminus X\right)\right|\\&=\mu \left(X\setminus Y\right)+\mu \left(Y\setminus X\right)\\&=\mu \left(\left(X\setminus Y\right)\cup \left(Y\setminus X\right)\right)\\&=\mu \left(X\,\triangle \,Y\right)\end{aligned}}}
{\displaystyle S=\left(\Omega ,{\mathcal {A}},\mu \right)}
{\displaystyle F,G\in {\mathcal {A}}}
{\displaystyle F\triangle G\in {\mathcal {A}}}
{\displaystyle F}
{\displaystyle G}
{\displaystyle \mu \left(F\triangle G\right)=0}
{\displaystyle F=G\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {D}},{\mathcal {E}}\subseteq {\mathcal {A}}}
{\displaystyle {\mathcal {D}}\subseteq {\mathcal {E}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle D\in {\mathcal {D}}}
{\displaystyle E\in {\mathcal {E}}}
{\displaystyle D=E\left[{\mathcal {A}},\mu \right]}
{\displaystyle \subseteq \left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {D}}={\mathcal {E}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {D}}\subseteq {\mathcal {E}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {E}}\subseteq {\mathcal {D}}\left[{\mathcal {A}},\mu \right]}
{\displaystyle =\left[{\mathcal {A}},\mu \right]}
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {D}}}
{\displaystyle {\mathcal {A}}}
{\displaystyle =\left[{\mathcal {A}},\mu \right]}
{\displaystyle D\in {\mathcal {D}}}
{\displaystyle {\mathcal {D}}}
{\displaystyle {\mathcal {D}}}
{\displaystyle {\mathcal {D}}}
{\displaystyle \sigma }
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {D}}}
{\displaystyle F=G\left[{\mathcal {A}},\mu \right]}
{\displaystyle \left|\mathbf {1} _{F}-\mathbf {1} _{G}\right|=0}
{\displaystyle \left[{\mathcal {A}},\mu \right]}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Symmetric_difference&oldid=1087427574"
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Wiener Algebras of Operators, and Applications to Pseudodifferential Operators | EMS Press
Wiener Algebras of Operators, and Applications to Pseudodifferential Operators
\newcommand{\sR}{\mathbb R} \newcommand{\sZ}{\mathbb Z} We introduce a Wiener algebra of operators on
L^2(\sR^N)
which contains, for example, all pseudodifferential operators in the H\"ormander class
OPS^0_{0,0}
. A discretization based on the action of the discrete Heisenberg group associates to each operator in this algebra a band-dominated operator in a Wiener algebra of operators on
l^2(\sZ^{2N}, \, L^2(\sR^N))
. The (generalized) Fredholmness of these discretized operators can be expressed by the invertibility of their limit operators. This implies a criterion for the Fredholmness on
L^2(\sR^N)
of pseudodifferential operators in
OPS^0_{0,0}
in terms of their limit operators. Applications to Schr\"odinger operators with continuous potential and other partial differential operators are given.
Vladimir S. Rabinovich, Steffen Roch, Wiener Algebras of Operators, and Applications to Pseudodifferential Operators. Z. Anal. Anwend. 23 (2004), no. 3, pp. 437–482
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2015 Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators
Dongyuan Liu, Zigen Ouyang, Huilan Wang
We consider the following state dependent boundary-value problem
{D}_{0+}^{\alpha }y\left(t\right)-p{D}_{0+}^{\beta }g\left(t,y\left(\sigma \left(t\right)\right)\right)+f\left(t,y\left(\tau \left(t\right)\right)\right)=0
0<t<1
y\left(0\right)=0
\eta y\left(\sigma \left(1\right)\right)=y\left(1\right),
{D}^{\alpha }
is the standard Riemann-Liouville fractional derivative of order
1<\alpha <2
0<\eta <1
p\le 0
0<\beta <1
\beta +1-\alpha \ge 0
g
g\left(t,u\right):\left[0,1\right]×\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)
g\left(0,0\right)=0
f
f\left(t,u\right):\left[0,1\right]×\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)\sigma \left(t\right)
\tau \left(t\right)
t
0\le \sigma \left(t\right)
\tau \left(t\right)\le t
. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.
Dongyuan Liu. Zigen Ouyang. Huilan Wang. "Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators." Abstr. Appl. Anal. 2015 (SI08) 1 - 11, 2015. https://doi.org/10.1155/2015/263748
Dongyuan Liu, Zigen Ouyang, Huilan Wang "Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators," Abstract and Applied Analysis, Abstr. Appl. Anal. 2015(SI08), 1-11, (2015)
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